UNIVERSITÀ DEGLI STUDI DI BARI
Dottorato di Ricerca in Matematica
XX Ciclo – A.A. 2007/2008
Settore Scientifico-Disciplinare:
MAT/03 – Geometria
Tesi di Dottorato
Lightlike hypersurfaces of
semi-Riemannian manifolds
with remarkable structures
Candidato:
Letizia BRUNETTI
Supervisori della tesi:
Proff. A. M. PASTORE, S. IANUŞ
Coordinatore del Dottorato di Ricerca:
Prof. L. LOPEZ
Contents
Introduction
1 Preliminaries
1.1 Algebraic Preliminaries
1.2 Distributions . . . . . .
1.3 f -Structures . . . . . . .
1.4 Compatible metrics . . .
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2 Indefinite S-manifolds
2.1 Indefinite almost S-manifolds . . . . . . . . . .
2.2 Indefinite S-manifolds . . . . . . . . . . . . . .
2.3 Sectional Curvature and ϕ̄-Sectional Curvature
2.4 Sectional Curvature in the case ε = 0 . . . . . .
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50
3 Lightlike hypersurfaces
55
3.1 Screen distribution . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Gauss and Weingarten equations . . . . . . . . . . . . . . . . . 57
3.3 Choice of screen distribution . . . . . . . . . . . . . . . . . . . 63
3.4 Lightlike hypersurface and the D0 distribution for indefinite Smanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Totally umbilical lightlike hypersurface and totally umbilical
screen distribution . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Examples of indefinite S-manifolds
4.1 A first indefinite S-manifold M1 on R62 . .
4.2 A second indefinite S-manifold M2 on R62
4.3 A special indefinite S-manifold M3 on R41
4.4 Lightlike hypersufaces of M1 and M2 . . .
4.5 Induced geometrical object . . . . . . . .
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93
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iv
Index
4.6
4.7
4.8
Distribution D0 of the lightlike hypersurface of M2 . . . . . . . 114
Lightlike hypersufaces of M3 . . . . . . . . . . . . . . . . . . . 114
The curvature of M3 . . . . . . . . . . . . . . . . . . . . . . . . 116
Introduction
In the wide field of geometry of semi-Riemannian manifolds, the study of
lightlike, or degenerate, hypersurfaces and submanifolds comes now to fill a
gap in the general theory of submanifolds. In fact, while the geometry of
hypersurfaces and submanifolds of Riemannian manifolds, since ([20]), has
received powerful impulse, polarizing a lot of attention, with studies conducted
in great generality and developing a great variety of techniques, on the contrary
the study of degenerate geometry is a relatively new field of research. It
rises within the semi-Riemannian context, due to the existence of the socalled ”causal character” of geometrical objects: their spacelike, timelike or
lightlike nature, in fact, implies the existence of three types of hypersurfaces
and submanifolds. Among them, the spacelike and timelike cases have received
a systematic exposition in the fundamental book ([39]). We have to be looking
forward 1990 to finding the first studies about the lightlike case, when Bejancu
and Duggal introduced the lightlike geometry ([6, 7, 8, 9, 24]).
The primary difference between the lightlike hypersurfaces and non-degenerate hypersurfaces arises due to the fact that in the first case the normal vector
bundle intersects with the tangent vector bundle and this intersection is not
trivial, and moreover in a lightlike hypersurface the normal vector bundle is
contained in the tangent vector bundle. Thus, one fails to use the classical
theory of non-degenerate submanifolds to define the induced geometrical objects (as linear connection, second fundamental form, Gauss and Weingarten
equations) on the lightlike hypersurface.
Even if the degenerate case is more difficult, it is one of the most important
topics of differential geometry, the growing importance of lightlike geometry
is motivated by the extensive use in mathematical physics, in particular in
relativity. In fact, semi-Riemannian manifolds (M̄ , ḡ) with dimM̄ = n >
4 are natural generalizations of spacetime of general relativity and lightlike
hypersurfaces are models of different types of horizons separating domains
of (M̄ , ḡ) with different physical properties. More precisely, we recall the
2
Introduction
following definition of Killing horizon:
Definition 0.0.1 ([24]). Let (M, g) be a lightlike hypersurface of a semiRiemannian manifold (M̄ , ḡ), M is said to be a local isometry horizon
(abbreviated to LIH) with respect to a group of isometry if
a) M is invariant under the group.
b) Each null geodesic generator is a trajectory of the group.
A LIH lightlike hypersurface, respect to a 1-parameter group, is said to be
a Killing horizon. More simply a Killing horizon is a lightlike hypersurface
whose generating lightlike vector coincides with a Killing vector field. For
example, in the case of the black hole, there are metrics which allow the
existence of Killing horizons and these Killing horizons coincide with their
event horizons. However it should be emphasised that these two notions of
horizon are independent. Now, we find an example of Killing horizon of the
black hole, endowed by the Kerr-Newman metric. To this end, we recall the
expression of the Kerr-Newman metric of the black hole of mass M rotating
with angular momentum J and charge Q:
Λ2
sin2 θ 2
2
2
(dt
−
αsin
θdφ)
+
[(r + α2 )dφ − αdt]2
ρ2
ρ2
ρ2
+ 2 dr 2 + ρ2 dθ 2 .
Λ
ds2 = −
where we put
α2 =
J
,
M
ρ2 = r 2 + α2 cos2 θ,
Λ2 = r 2 − 2M r + α2 + Q2 .
For this spacetime, the Killing horizon is located at
r = r+ := M + M − Q2 − J 2 /M 2 ,
and we note that, in the usual coordinates, outside the Killing horizon, the
Killing vector field is timelike, while inside it is spacelike.
There exist a lot of reasons that motivate the study of the lightlike hypersurfaces of indefinite g.f.f -manifolds and in particular of the lightlike hypersurfaces of indefinite S-manifolds, the most important reasons are the following:
• in ([24]), Duggal and Bejancu proved the following theorem:
Introduction
3
Theorem 0.0.2. A lightlike framed hypersurface of a Lorentz C-manifold,
with an induced metric connection, is a Killing horizon,
• in a recent work ([28]), Duggal and Sahin begin to work on lightlike submanifolds of indefinite Sasakian manifolds because the contact geometry
has a significant use in differential equations, optics and phase spaces of
dynamical systems,
• Duggal, in ([21]), shows up that a globally hyperbolic spacetime and de
Sitter spacetime can carry a framed structure and these are two important example of spacetime.
The first chapter is a chapter of algebraic preliminaries, therefore we give
quickly the definition of the radical of a vector space, some properties of the
semi-Euclidean spaces and of its lightlike vector subspaces; we define the semiRiemannian manifolds, and we introduce some definitions about distributions.
Finally, we introduce the indefinite metric g.f.f -structure on semi-Riemannian
manifolds and we give some theorems for these manifolds.
In Chapter 2, we carry out an in-depth study of the indefinite almost Smanifolds and the indefinite S-manifolds. For these manifolds we study the
sectional curvature and we define the ϕ̄-sectional curvature, hence we obtain
that the sectional curvature is related to the ϕ̄-sectional curvature. Finally, we
find an expression of Riemannian curvature tensor field R̄ which characterizes
the S-space form, that is an indefinite S-manifold with constant ϕ̄-sectional
curvature. Moreover, in this chapter, we analyse the sectional curvature of the
special indefinite S-manifold in which the number of the characteristic vector
fields is even and the number of the spacelike characteristic vector fields is
equal to the number of the timelike characteristic vector fields.
In Chapter 3 we briefly give basic information needed for lightlike hypersurfaces and we write the Gauss and Weingarten equations of the lightlike
hypersurface. Afterwards, introduced a particular screen distribution, using
the properties of the indefinite S-manifolds, we find other decompositions of
S(T M ) and T M and we get two distributions D0 and D on M . We study
the existence of g.f.f -structure on a lightlike hypersurface and, under suitable
hypotheses, we obtain an indefinite S-structure on the leaves of D0 , if D0 is an
integrable distribution. Finally, we deal with the existence of totally umbilical
lightlike hypersurface of an indefinite S-space form.
In the last chapter we give three examples of indefinite S-manifolds and,
found a lightlike hypersurface, we apply the previous results. The third ex-
4
Introduction
ample is a special indefinite S-manifold of dimension 4, which turns out to be
an indefinite Lorentzian S-space form with ϕ̄-sectional curvature c = 0.
Acknowledgments. I would like to express my sincere thanks to Professor Anna Maria Pastore for the patience she demonstrated during our endless
discussions which have provided ideas for so many research activities, and especially for helping me understand how to follow a good method of research
but also to feel the passion for and the beauty of carrying out research in this
field.
I would also like to express my gratitude to Professor Stere Janus for the
advices and the hospitality given in the long afternoons spent at the University
in Bucharest. I am also grateful to all the research team in Geometry at the
University in Bari for the help received.
For the continuous moral support and the constant encouragements I would
like to thank all the PhD students and all the post Docs, also for helping and
guiding me through the labyrinth of bureaucracy.
Finally, I would like to thank my family, Leonardo and all my friends,
who have always been by my side giving me the encouragement that has been
essential for the realization of this project.
Special thanks go to my mother who has always believed in me.
Chapter 1
Preliminaries
1.1
Algebraic Preliminaries
We begin with some algebraic preliminaries ([24]).
Let V be a real n-dimensional vector space and g : V ×V → R a symmetric
bilinear mapping. We say that g is degenerate on V if there exists a vector
ξ = 0 such that
g(ξ, v) = 0 for all v ∈ V,
otherwise g is called non-degenerate.
The radical or the null space of V , with respect to g, is the subspace
Rad V of V defined by
Rad V = {x ∈ V | ∀v ∈ V
g(x, v) = 0}.
The dimension of Rad V is called the nullity degree of g, denoted by null V ;
moreover g is degenerate or non-degenerate on V if and only if null V > 0 or
null V = 0, respectively.
We call a non degenerate symmetric bilinear form g on a real vector space
V a scalar product on V ; it is known that it is always possible to find an
ordered basis (ei )1≤i≤n of V such that
g(ei , ei ) = −1
g(ei , ei ) = +1
g(ei , ej ) = 0
for 1 ≤ i ≤ ν
for ν + 1 ≤ i ≤ n
for i = j
where the integer ν is uniquely determined, and is called the index of g; it is
the largest dimension of a subspace W ⊂ V , on which g is negative definite.
6
Chapter 1. Preliminaries
Definition 1.1.1 ([24]). Let V be a real n-dimensional vector space and g
a scalar product with index ν, 0 < ν ≤ n; the pair (V, g) is said to be a
semi-Euclidean space.
We observe that if W ⊂ V is a subspace of a semi-Euclidean space (V, g),
then one of the following two cases occurs: either g|W is degenerate, or non
degenerate; in the first case, we call W a lightlike subspace of V ; otherwise,
W is said to be a non-degenerate subspace.
Proposition 1.1.2 ([24]). Let (V, g) be an n-dimensional semi-Euclidean
space and let (W, g = g|W ) be an m-dimensional lightlike subspace, such that
null W = r < m. Then any complementary subspace to Rad W in W is nondegenerate subspace. A complementary subspace to Rad W in W is called a
screen subspace of W .
Proposition 1.1.3 ([24]). Let (V, g) be an n-dimensional semi-Euclidean
space and W a subspace. Then we have:
dim W + dim W ⊥ = n,
(W ⊥ )⊥ = W,
and
Rad W = Rad W ⊥ = W ∩ W ⊥ .
Proposition 1.1.4 ([24]). Let (V, g) be an n-dimensional semi-Euclidean
space and W a subspace. Then the following assertions are equivalent:
1) W is non-degenerate subspace;
2) W ⊥ is non-degenerate subspace;
3) W and W ⊥ are complementary orthogonal subspace of V with respect to
g;
4) V is the orthogonal direct sum of W and W ⊥ , i.e. V = W ⊥W ⊥.
Definition 1.1.5 ([24]). Let (V, g) be an n-dimensional semi-Euclidean space
and B = {ei , ei , uα } a basis of V , where i = 1, . . . , r and α = 1, . . . , t with
t+2r = n. B is called a quasi-orthonormal basis if the following conditions
are satisfied:
g(ei , ej ) = g(ei , ej ) = 0;
g(uα , ei ) = g(uα , ei ) = 0;
g(ei , ej ) = δij
g(uα , uβ ) = εα δαβ ,
for any i, j ∈ {1, . . . , r} and α, β ∈ {1, . . . , t} and where εα = ±1.
7
Definition 1.1.6 ([24]). Let (V, g) be a semi-Euclidean space, W an m-dimensional lightlike subspace of V and B = {ei , ei , uα } a quasi-orthonormal basis
of V , where i = 1, . . . , r and α = 1, . . . , t. B is called a quasi-orthonormal
basis of V along W , if W = Span{e1 , . . . , er , u1 , . . . , us }, with m = s + r
and 1 ≤ s ≤ t, or W = Span{e1 , . . . , em }, if m ≤ r.
1.2
Distributions
We introduce some fundamental definition that we will use in the following
chapters.
Definition 1.2.1 ([39]). A semi-Riemannian manifold is a smooth manifold M̄ endowed with a semi-Riemannian metric ḡ.
Definition 1.2.2. Let (M̄ , ḡ) be a semi-Riemannian manifold and
S : p ∈ M̄ → Sp ⊂ Tp M̄
a distribution on M̄ . Then, S is said to be non-degenerate (degenerate) if
for any p ∈ M̄ the vector subspace (Sp , g|Sp ) is non-degenerate (degenerate).
We will name non-degenerate (sub-)bundle each subbundle of T M̄ , whose
associate distribution is non-degenerate.
Remark 1.2.3. We note that the previous definition works well both if
(M̄ , ḡ) is a semi-Riemannian manifold endowed with a non-degenerate semiRiemannian metric and if (M̄ , ḡ) is a semi-Riemannian manifold endowed with
a degenerate semi-Riemannian metric because we have that Tp M̄ , for any
p ∈ M̄ , is non-degenerate and degenerate respectively and a degenerate vector
space has non-degenerate vector subspace and degenerate vector subspace.
As in ([1]), we introduce some fundamental tensors associated to a distribution. Let (M̄ , ḡ) be a semi-Riemannian manifold, and V a q-dimensional
distribution on M̄ , not necessarily integrable. Denoting its orthogonal distribution V ⊥ by H, we can consider a decomposition into orthogonal direct sum
of subbundles:
T M̄ = V ⊥ H,
we can interpret the previous equality in term of distributions, we call V
the vertical distribution and H the associated horizontal distribution and we
denote the orthogonal projections by pH : T M̄ → H and pV : T M̄ → V.
8
Chapter 1. Preliminaries
By the unsymmetrized second fundamental form of V we mean the (1,2)-type
tensor AV defined by
¯ p X pV Y ),
AVX Y = pH (∇
V
¯ being the Levi-Civita connection on (M̄ , ḡ). This
for any X, Y ∈ Γ(T M̄ ), ∇
tensor is not symmetric but, starting from the unsymmetrized second fundamental form of V, we can define the symmetric second fundamental form of
V, B V , given by
1
B V (X, Y ) = (AVX Y + AVY X)
2
1
¯ VX VY ) + pH (∇
¯ VY VX)},
= {pH (∇
2
for any X, Y ∈ Γ(T M̄ ). The integrability tensor of V is the (1,2)-type tensor
I V given by
1
1
I V (X, Y ) = {AVX Y − AVY X} = pH ([VX, VY ]),
2
2
for any X, Y ∈ Γ(T M̄ ). We note that I V is an antisymmetric tensor and it
vanishes if and only if V is integrable. By the above definitions we have the
following decomposition of AV into its symmetric and antisymmetric parts:
AV = B V + I V .
When V is integrable, we have that AV = B V , which is the second fundamental
form of the leaves, regarded as submanifolds of M̄ . We can always define the
vector field, called the mean curvature of V, by
µVp
q
1
¯ eα eα )p = 1 trB V ,
=
pH (∇
p
q α=1
q
where {e1 , . . . , eq } is a local frame of V, for any p ∈ M̄ .
Definition 1.2.4 ([1]). A distribution V on M̄ is said to be:
1) minimal if, for any p ∈ M̄ , the mean curvature vanishes;
2) totally geodesic if, for any p ∈ M̄ , BpV vanishes.
9
1.3
f -Structures
Definition 1.3.1. Let M̄ be a smooth manifold; an f -structure on M̄ is a
non vanishing smooth (1, 1)-tensor field ϕ̄ on T M̄ of constant rank such that
ϕ̄ 3 + ϕ̄ = 0.
A smooth manifold M̄ , provided with an f -structure, is said to be an f manifold, and will be denoted with (M̄ , ϕ̄). Moreover, we can consider the
distributions Im ϕ̄ and ker ϕ̄.
Proposition 1.3.2. Let (M̄ , ϕ̄) be an f -manifold. Then we have the following
decomposition:
T M̄ = Im ϕ̄ ⊕ ker ϕ̄,
(1.3.1)
and, putting J := ϕ̄|Im ϕ̄ , we get an almost complex structure on Im ϕ̄, and
so the rank of ϕ̄ is even. As a consequence of the previous decomposition, we
have dim(M̄ ) = 2n + r, where r = dim(ker(ϕ̄)).
Definition 1.3.3. Let M̄ be a smooth manifold; an f -structure ϕ̄ on M̄
is called a globally framed f -structure (g.f.f -structure) if there exist r
globally defined vector fields ξ̄1 , ξ¯2 . . . , ξ¯r on M̄ and r globally defined 1-forms
η̄ 1 , η̄ 2 . . . , η̄ r on M̄ , such that:
ϕ̄2 = −I + η̄ α ⊗ ξ¯α ,
η̄ α (ξ̄β ) = δβα
for all α, β ∈ {1, . . . , r}.
Moreover, a smooth manifold M̄ provided with a g.f.f -structure is called
a g.f.f -manifold, and it is denoted with (M̄ , ϕ̄, ξ¯α , η̄ α ); the vector fields ξ¯α ,
(α = 1, ...., r), are called characteristic vector fields.
Remark 1.3.4. From the above definition it is easy to deduce the following
properties:
for any α ∈ {1, . . . , r}
ϕ̄(ξ̄α ) = 0
η̄ α ◦ ϕ̄ = 0
1.4
for any α ∈ {1, . . . , r}
Compatible metrics
Let M̄ be a smooth manifold and ḡ a symmetric non-degenerate (0, 2)-tensor
field on M̄ . The tensor field ḡ is a metric tensor on M̄ if and only if, by
10
Chapter 1. Preliminaries
definition, ḡ has a constant index, that is the index of the scalar product ḡp ,
induced by ḡ, on the tangent space Tp M̄ is the same for any p ∈ M̄ .
If M is a submanifold of (M̄ , ḡ), then we consider the induced tensor field g
on M , i.e. g is formally the pullback j ∗ (ḡ), where j : M → M̄ is the inclusion
map. When the metric tensor ḡ on M̄ is indefinite, then j ∗ (ḡ) need not be a
metric on M because in general the index of g is not constant.
We note that if M̄ is connected then the index of ḡ is constant.
Definition 1.4.1. Let (M̄ , ϕ̄) be a (2n + r)-dimensional f -manifold and ḡ
a semi-Riemannian metric on M̄ with index ν, 0 < ν ≤ 2n + r. Then, the
pair (ϕ̄, ḡ) is said to be an indefinite metric f -structure, and the triple
(M̄ , ϕ̄, ḡ) is called an indefinite metric f -manifold, if ϕ̄ is skew-symmetric
with rispect to ḡ, that is, for any X, Y ∈ Γ(T M̄ ):
ḡ(ϕ̄X, Y ) + ḡ(X, ϕ̄Y ) = 0.
(1.4.1)
Definition 1.4.2. Let (M̄ 2n+r , ϕ̄, ξ¯α , η̄ α ), with α = 1, . . . , r, be a g.f.f manifold, and ḡ a semi-Riemannian metric on M̄ with index ν, 0 < ν ≤ 2n+r.
Then, we say that the two structures are compatible if for any X, Y ∈ Γ(T M̄ )
ḡ(ϕ̄X, ϕ̄Y ) = ḡ(X, Y ) −
r
εα η̄ α (X)η̄ α (Y ),
(1.4.2)
α=1
and for any α ∈ {1, . . . , r}:
ḡ(X, ξ¯α ) = εα η̄ α (X),
(1.4.3)
where εα = ±1 according to whether ξ̄α is spacelike or timelike. The system
(M̄ 2n+r , ϕ̄, ξ¯α , η̄ α , ḡ) is then called an indefinite metric g.f.f -manifold.
Remark 1.4.3. Observe that, if ḡ is a semi-Riemannian metric on a g.f.f manifold (M̄ , ϕ̄, ξ¯α , η̄ α ) compatible with the f -structure ϕ̄, then the pair (ϕ̄, ḡ)
is necessarily an indefinite metric f -structure.
Remark 1.4.4 ([24]). Let (M̄ , ϕ̄, ξ¯α , η̄ α ), with α = 1, . . . , r, be a g.f.f manifold, and ḡ a compatible semi-Riemannian metric on M̄ . We know that
the decomposition (1.3.1) holds, and that the induced structure J on Im ϕ̄ is an
almost complex structure; then (Im ϕ̄, g = ḡ|Im ϕ̄ , J) is a indefinite Hermitian
distribution and the only possible signatures of g are (2p, 2q) with p + q = n;
therefore g cannot be a Lorentz metric, for n > 1. Moreover, Im ϕ̄ and ker ϕ̄
are orthogonal with respect to ḡ. Furthermore, Im ϕ̄ and ker ϕ̄ will be denoted
with D and D⊥ respectively. Later on, to denote a section of D ( D⊥ ) we will
write X ∈ D or equivalently X ∈ Γ(D) ( X ∈ D⊥ or X ∈ Γ(D⊥ )).
11
Theorem 1.4.5 ([24]). Let (M̄ , ϕ̄, ξ¯α , η̄ α ), with α = 1, . . . , r, be a g.f.f manifold and h0 a semi-Riemannian metric on M̄ ; we suppose that {ξ̄α }1≤α≤r
are orthonormal with respect to h0 and that h0 (ξ̄α , ξ¯α ) = −εα , for any α in
{1, . . . , r}. Then there exists a symmetric tensor field ḡ of type (0, 2) on M̄
satisfying the conditions (1.4.2) and (1.4.3).
Definition 1.4.6. Let (M̄ , ϕ̄, ξ̄α , η̄ α ), with α = 1, . . . , r, be a g.f.f -manifold,
and ḡ a compatible semi-Riemannian metric on M̄ , we can define a 2-form,
denoted with Φ, which we call fundamental 2-form, by putting, for any
X, Y ∈ Γ(T M̄ ), Φ(X, Y ) = ḡ(X, ϕ̄Y ). Obviously, (1.4.1) ensures that Φ is a
2-form.
Proposition 1.4.7 ([12]). Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite metric g.f.f ¯ satisfies the following equality:
manifold; then the Levi-Civita connection ∇
¯ X ϕ̄)Y, Z) = 3dΦ(X, ϕ̄Y, ϕ̄Z) − 3dΦ(X, Y, Z)
2ḡ((∇
r
+ ḡ(N (Y, Z), ϕ̄X) +
εα Nα(2) (Y, Z)η̄ α (X)
+2
r
α=1
εα dη̄ (ϕ̄Y, X)η̄ (Z) − 2
α
α=1
α
r
(1.4.4)
εα dη̄ α (ϕ̄Z, X)η̄ α (Y ),
α=1
for any X, Y, Z ∈ Γ(T M̄ ), where
N = Nϕ̄ + 2
r
dη̄ α ⊗ ξ̄α ,
α=1
Nϕ̄ (X, Y ) = [ϕ̄X, ϕ̄X] + ϕ̄2 [X, Y ] − ϕ̄[ϕ̄X, Y ] − ϕ̄[X, ϕ̄Y ],
Nα(2) (X, Y ) = (Lϕ̄X η̄ α )(Y ) − (Lϕ̄Y η̄ α )(X) = 2dη̄ α (ϕ̄X, Y ) − 2dη̄ α (ϕ̄Y, X).
Proof. The Levi-Civita connection satisfies the Koszul identity:
¯ X Y, Z) = X(ḡ(Y, Z)) + Y (ḡ(Z, X)) − Z(ḡ(X, Y ))
2ḡ(∇
− ḡ([Y, Z], X) + ḡ([Z, X], Y ) + ḡ([X, Y ], Z),
and we know that the exterior differential of a 2-form satisfies the following
relation:
3dΦ(X, Y, Z) = SX,Y,Z {X(Φ(Y, Z)) − Φ([X, Y ], Z)}.
12
Chapter 1. Preliminaries
Therefore we obtain
¯ X (ϕ̄Y ), Z) − 2ḡ(ϕ̄(∇
¯ X Y ), Z)
¯ X ϕ̄)Y, Z) = 2ḡ(∇
2ḡ((∇
¯ X Y, ϕ̄Z)
¯ X (ϕ̄Y ), Z) + 2ḡ(∇
= 2ḡ(∇
= X(ḡ(ϕ̄Y, Z)) + (ϕ̄Y )(ḡ(Z, X)) − Z(ḡ(X, ϕ̄Y ))
− ḡ([ϕ̄Y, Z], X) + ḡ([Z, X], ϕ̄Y ) + ḡ([X, ϕ̄Y ], Z)
+ X(ḡ(Y, ϕ̄Z)) + Y (ḡ(ϕ̄Z, X)) − (ϕ̄Z)(ḡ(X, Y ))
− ḡ([Y, ϕ̄Z], X) + ḡ([ϕ̄Z, X], Y ) + ḡ([X, Y ], ϕ̄Z)
r
εα η̄ α (Z)η̄ α (X))
= −X(Φ(Y, Z)) + (ϕ̄Y )(ḡ(ϕ̄Z, ϕ̄X) +
α=1
− Z(Φ(X, Y )) + Φ([Z, X], Y ) + ḡ(ϕ̄[X, ϕ̄Y ], ϕ̄Z)
r
εα η̄ α (Z)η̄ α ([X, ϕ̄Y ]) − ḡ(ϕ̄[ϕ̄Y, Z], ϕ̄X)
+
−
α=1
r
εα η̄ α (X)η̄ α ([ϕ̄Y, Z]) + X(Φ(Y, Z)) + Y (Φ(X, Z))
α=1
− (ϕ̄Z)(ḡ(ϕ̄X, ϕ̄Y ) +
r
εα η̄ α (X)η̄ α (Y )) + Φ([X, Y ], Z)
α=1
+ ḡ(ϕ̄[ϕ̄Z, X], ϕ̄Y ) +
− ḡ(ϕ̄[Y, ϕ̄Z], ϕ̄X) −
r
α=1
r
εα η̄ α (Y )η̄ α ([ϕ̄Z, X])
εα η̄ α (X)η̄ α ([Y, ϕ̄Z]).
α=1
It is easy to check that Φ(X, Y ) = Φ(ϕ̄X, ϕ̄Y ) and then the previous expression
becomes:
¯ X ϕ̄)Y, Z) = −X(Φ(Y, Z)) + (ϕ̄Y )(Φ(ϕ̄Z, X))
2ḡ((∇
r
r
α
α
εα η̄ (X)(ϕ̄Y )(η̄ (Z)) +
εα η̄ α (Z)(ϕ̄Y )(η̄ α (X))
+
α=1
α=1
− Z(Φ(X, Y )) − Φ([X, ϕ̄Y ], ϕ̄Z) −
r
εα η̄ α (Z)η̄ α ([ϕ̄Y, X])
α=1
r
+ Φ([Z, X], Y ) − ḡ(ϕ̄[ϕ̄Y, Z], ϕ̄X) −
α=1
εα η̄ α (X)η̄ α ([ϕ̄Y, Z])
13
+ X(Φ(ϕ̄Y, ϕ̄Z)) − Y (Φ(Z, X)) + (ϕ̄Z)(Φ(X, ϕ̄Y ))
r
r
α
α
εα η̄ (X)(ϕ̄Z)(η̄ (Y )) −
εα η̄ α (Y )(ϕ̄Z)(η̄ α (X))
−
α=1
α=1
+ Φ([X, Y ], Z) − Φ([ϕ̄Z, X], ϕ̄Y ) +
− ḡ(ϕ̄[Y, ϕ̄Z], ϕ̄X) +
r
r
εα η̄ α (Y )η̄ α ([ϕ̄Z, X])
α=1
εα η̄ α (X)η̄ α ([ϕ̄Z, Y ]) + Φ([Y, Z], X)
α=1
− ḡ([Y, Z], ϕ̄X) − Φ([ϕ̄Y, ϕ̄Z], X) + ḡ([ϕ̄Y, ϕ̄Z], ϕ̄X)
+ ḡ(2dη̄ α (Y, Z)ξ̄α , ϕ̄X) + ḡ(η̄ α [Y, Z]ξ̄α , ϕ̄X)
= 3dΦ(X, ϕ̄Y, ϕ̄Z) − 3dΦ(X, Y, Z) + ḡ(N (Y, Z), ϕ̄X)
r
r
(2)
α
εα Nα (Y, Z)η̄ (X) + 2
εα dη̄ α (ϕ̄Y, X)η̄ α (Z)
+
α=1
r
−2
α=1
εα dη̄ α (ϕ̄Z, X)η̄ α (Y ),
α=1
and this ends the proof.
Proposition 1.4.8. Let (M̄ , ϕ̄, ξ¯α , η̄α , ḡ) be an indefinite metric g.f.f -manifold.
(2)
The vanishing of N implies the vanishing of Nα , for any α ∈ {1, . . . , r}.
Proof. For β ∈ {1, . . . , r} putting Y = ξ̄β , we have
0 = N (X, ξ¯β ) = −[X, ξ¯β ] − ϕ̄[ϕ̄X, ξ¯β ] −
r
ξ̄β (η̄ α (X))ξ̄α .
(1.4.5)
α=1
Applying, for same γ ∈ {1, . . . , r}, η̄ γ to (1.4.5), we find
η̄ γ ([X, ξ¯β ]) + ξ̄β (η̄ γ (X)) = 0,
now, applying ϕ̄ to (1.4.5) and using (1.4.6), we get
0 = ϕ̄[X, ξ¯β ] + ϕ̄2 [ϕ̄X, ξ¯β ]
= ϕ̄[X, ξ¯β ] − [ϕ̄X, ξ¯β ] +
r
α=1
= ϕ̄[X, ξ¯β ] − [ϕ̄X, ξ¯β ],
η̄ α [ϕ̄X, ξ¯β ]ξ̄α
(1.4.6)
14
Chapter 1. Preliminaries
that implies
ϕ̄[X, ξ¯β ] = [ϕ̄X, ξ¯β ].
(1.4.7)
So, for any X, Y ∈ Γ(T M̄ ), computing N (ϕ̄X, Y ), we obtain
− [X, ϕ̄Y ] +
r
η̄ α (X)[ξ̄α , ϕ̄Y ] −
α=1
− ϕ̄[ϕ̄X, ϕ̄Y ] + ϕ̄[X, Y ] − ϕ̄(
r
r
(ϕ̄Y )(η̄ α (X))ξ̄α − [ϕ̄X, Y ]
α=1
η̄ α (X)[ξ̄α , Y ]) +
α=1
r
(1.4.8)
(ϕ̄X)(η̄ α (Y ))ξ̄α = 0,
α=1
and, for γ ∈ {1, . . . , r} applying η̄ γ to (1.4.8), we have
0 = −η̄ γ [X, ϕ̄Y ] − (ϕ̄Y )(η̄ γ (X)) − η̄ γ [ϕ̄X, Y ] + (ϕ̄X)(η̄ γ (Y ))
= 2dη̄ γ (ϕ̄X, Y ) + 2dη̄ γ (X, ϕ̄Y ).
Therefore, for any α ∈ {1, . . . , r} and for any X, Y ∈ Γ(T M̄ ), we obtain
Nα(2) (X, Y ) = 0.
Proposition 1.4.9. Let (M̄ , ϕ̄, ξ¯α , η̄α , ḡ) be an indefinite metric g.f.f -manifold.
Then the following statements hold:
a) (Lξ̄α Φ)(X, Y ) = (Lξ̄α ḡ)(X, ϕ̄Y )+ḡ(X, (Lξ̄α ϕ̄)Y ), for any α ∈ {1, . . . , r}.
¯ X ϕ̄)Z), for any X, Y, Z ∈ Γ(T M̄ ).
¯ X Φ)(Y, Z) = ḡ(Y, (∇
b) (∇
c) If Lξ̄α ϕ̄ = 0, then η̄ β [ϕ̄Z, ξ¯α ] = 0, for any β ∈ {1, . . . , r}.
Proof. Let α ∈ {1, . . . , r} and X, Y ∈ Γ(T M̄ )
(Lξ̄α Φ)(X, Y ) = ξ¯α (ḡ(X, ϕ̄Y )) − ḡ([ξ̄α , X], ϕ̄Y ) − ḡ(X, ϕ̄[ξ̄α , Y ])
¯ X, ϕ̄Y ) + ḡ(∇
¯ X ξ̄α , ϕ̄Y )
= ξ¯α (ḡ(X, ϕ̄Y )) − ḡ(∇
ξ̄α
+ ḡ(X, (Lξ̄α ϕ̄)Y ) − ḡ(X, [ξ̄α , ϕ̄Y ])
= (Lξ̄α ḡ)(X, ϕ̄Y ) + ḡ(X, (Lξ̄α ϕ̄)Y ).
¯ X Φ)(Y, Z),
We compute (∇
¯ X Y, ϕ̄Z) − ḡ(Y, ϕ̄(∇
¯ X Z))
¯ X Φ)(Y, Z) = X(ḡ(Y, ϕ̄Z)) − ḡ(∇
(∇
¯ X ϕ̄)Z).
= ḡ(Y, (∇
15
Finally, we have
(Lξ̄α ϕ̄)Z = 0,
for any Z ∈ Γ(T M̄ ) that implies
[ϕ̄Z, ξ¯α ] = ϕ̄[Z, ξ¯α ]
then η̄ β [ϕ̄Z, ξ¯α ] = 0.
Definition 1.4.10. Let (M̄ 2n+r , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite metric g.f.f manifold; M̄ is called indefinite K-manifold if it is normal and dΦ = 0.
Definition 1.4.11. Let (M̄ 2n+r , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite K-manifold; M̄
is called indefinite C-manifold if for any α ∈ {1, . . . , r}
dη̄ α = 0.
Remark 1.4.12. From (1.4.4), using Proposition 1.4.8, we deduce that the
¯ satisfies
Levi-Civita connection ∇
¯ X ϕ̄)Y, Z) =
ḡ((∇
r
εα (dη̄ α (ϕ̄Y, X)η̄ α (Z) − dη̄ α (ϕ̄Z, X)η̄ α (Y )),
α=1
in an indefinite K-manifold, and
¯ ϕ̄ = 0,
∇
in an indefinite C-manifold.
In the following chapter we will deal with the case of indefinite (almost)
S-manifold.
Chapter 2
Indefinite S-manifolds
The properties of (almost) S-manifold (with Riemannian metric) are studied
in ([26]) and in ([12]). Now, we discuss indefinite (almost) S-manifold and
their properties.
2.1
Indefinite almost S-manifolds
Definition 2.1.1. Let (M̄ 2n+r , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite metric g.f.f manifold; M̄ is called indefinite almost S-manifold if for any α ∈ {1, . . . , r}
dη̄ α = Φ.
From the above definition of indefinite almost S-manifold we deduce that
dΦ = 0.
Lemma 2.1.2. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite almost S-manifold; then
(2)
the tensor fields Nα vanish and for any X, Y ∈ Γ(D) and α ∈ {1, . . . , r}, we
have
η̄ α [ϕ̄X, Y ] = η̄ α [ϕ̄Y, X].
Proof. We know that dη̄ α = Φ, for any α ∈ {1, . . . , r}; therefore we have:
Nα(2) (X, Y ) = 2dη̄ α (ϕ̄X, Y ) − 2dη̄ α (ϕ̄Y, X)
= 2Φ(ϕ̄X, Y ) − 2Φ(ϕ̄Y, X)
= 2ḡ(ϕ̄X, ϕ̄Y ) − 2ḡ(ϕ̄X, ϕ̄Y ) = 0.
Chapter 2. Indefinite S-manifolds
18
Then, for any X, Y ∈ Γ(D)
0 = Nα(2) (X, Y ) = (Lϕ̄X η̄ α )Y − (Lϕ̄Y η̄ α )X
= (ϕ̄X)(η̄ α (Y )) − η̄ α [ϕ̄X, Y ]
− (ϕ̄Y )(η̄ α (X)) + η̄ α [ϕ̄Y, X]
= η̄ α [ϕ̄Y, X] − η̄ α [ϕ̄X, Y ],
and we have
η̄ α [ϕ̄X, Y ] = η̄ α [ϕ̄Y, X].
Proposition
2.1.3. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite almost S-manifold.
Putting η̄ := rα=1 εα η̄ α , the following statements hold:
¯ X ϕ̄)Y, Z) = ḡ(N (Y, Z), ϕ̄X) + 2ḡ(ϕ̄Y, ϕ̄X)η̄(Z)
2ḡ((∇
(2.1.1)
− 2ḡ(ϕ̄Z, ϕ̄X)η̄(Y ),
¯ ϕ̄ = 0
∇
ξ̄α
¯ ξ¯β = 0
∇
ξ̄α
for all α ∈ {1, . . . , r},
(2.1.2)
for all α, β ∈ {1, . . . , r}.
(2.1.3)
(2)
Proof. Let us prove (2.1.1). Since dΦ = 0 and Nα vanishes, (1.4.4) becomes:
¯ X ϕ̄)Y, Z) = ḡ(N (Y, Z), ϕ̄X) + 2
2ḡ((∇
−2
r
r
εα dη̄ α (ϕ̄Y, X)η̄ α (Z)
α=1
εα dη̄ α (ϕ̄Z, X)η̄ α (Y ).
α=1
Moreover, for any α ∈ {1, . . . , r}, dη̄ α = Φ, and replacing in the above equation, we obtain:
¯ X ϕ̄)Y, Z) = ḡ(N (Y, Z), ϕ̄X) + 2Φ(ϕ̄Y, X)
2ḡ((∇
− 2Φ(ϕ̄Z, X)
r
r
εα η̄ α (Z)
α=1
εα η̄ α (Y )
α=1
= ḡ(N (Y, Z), ϕ̄X) + 2ḡ(ϕ̄Y, ϕ̄X)η̄(Z) − 2ḡ(ϕ̄Z, ϕ̄X)η̄(Y ).
19
Now, putting X = ξ¯α in (2.1.1), we have:
¯ ϕ̄)Y, Z) = ḡ(N (Y, Z), ϕ̄ξ¯α ) + 2ḡ(ϕ̄Y, ϕ̄ξ̄α )η̄(Z)
2ḡ((∇
ξ̄α
− 2ḡ(ϕ̄Z, ϕ̄ξ¯α )η̄(Y ) = 0,
¯ ϕ̄ = 0, i.e. (2.1.2).
and since Y and Z are arbitrary, we deduce ∇
ξ̄α
Finally, we prove (2.1.3). Using (2.1.2) we have:
¯ ξ̄β ),
¯ ϕ̄)(ξ̄β ) = −ϕ̄(∇
0 = (∇
ξ̄α
ξ̄α
therefore
¯ ξ¯β ∈ D⊥ ,
∇
ξ̄α
(2.1.4)
which implies that [ξ̄α , ξ¯β ] ∈ D⊥ . On the other hand, for any γ ∈ {1, . . . , r}
0 = Φ(ξ̄α , ξ¯β ) = dη̄ γ (ξ̄α , ξ̄β )
1
1
= − η̄ γ [ξ̄α , ξ¯β ] = − εγ ḡ([ξ̄α , ξ¯β ], ξ¯γ ),
2
2
therefore [ξ̄α , ξ¯β ] ∈ D ∩ D⊥ , being D ∩ D⊥ = {0}, we obtain [ξ̄α , ξ¯β ] = 0 and
¯ ξ̄α . Now we check that ∇
¯ ξ̄β ∈ D, that is, we check that for any
¯ ξ¯β = ∇
∇
ξ̄α
ξ̄β
ξ̄α
γ ∈ {1, . . . , r}
¯ ξ̄β , ξ̄γ ) = 0.
ḡ(∇
ξ̄α
¯ = 0, using the covariant derivative with rispect
Being ḡ(ξ̄β , ξ¯γ ) = εβ δβγ and ∇ḡ
¯
to ξα , we obtain:
¯ ξ̄γ ) = 0,
¯ ξ¯β , ξ¯γ ) + ḡ(ξ̄β , ∇
(2.1.5)
ḡ(∇
ξ̄α
ξ̄α
and likewise, deriving ḡ(ξ̄α , ξ¯γ ) = εα δαγ with rispect to ξ¯β , we obtain:
¯ ξ¯γ ) = 0.
¯ ξ̄α , ξ¯γ ) + ḡ(ξ̄α , ∇
ḡ(∇
ξ̄β
ξ̄β
(2.1.6)
¯ ξ̄α , we have:
¯ ξ̄β = ∇
Subtracting (2.1.5) and (2.1.6), using ∇
ξ̄α
ξ̄β
¯ ξ̄γ ) = ḡ(ξ̄α , ∇
¯ ξ̄γ ).
ḡ(ξ̄β , ∇
ξ̄α
ξ̄β
Therefore, we can compute
¯ ξ̄β ) = ḡ(ξ̄α , ∇
¯ ξ¯γ )
¯ ξ¯β , ξ¯γ ) = ḡ(ξ̄α , ∇
ḡ(∇
ξ̄α
ξ̄γ
ξ̄β
¯
¯
¯
= −ḡ(∇ ξ̄α , ξγ ) = −ḡ(∇ ξ̄β , ξ̄γ ),
ξ̄β
ξ̄α
¯ ξ̄β ∈ D ∩ D⊥ ,
¯ ξ¯β , ξ¯γ ) = 0. This result and (2.1.4) imply ∇
from which ḡ(∇
ξ̄α
ξ̄α
¯ ξ̄β = 0.
that is ∇
ξ̄α
Chapter 2. Indefinite S-manifolds
20
Note that (2.1.2) and (2.1.3) are consequences of (2.1.1).
Proposition 2.1.4. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite almost S-manifold.
Then the following statements are true:
a) for any α ∈ {1, . . . , r} the operator hα = 12 Lξ̄α ϕ̄ is self-adjoint,
b) for any α, β ∈ {1, . . . , r}, hα (ξ̄β ) = 0.
Proof. As first step we prove that, for any X, Y ∈ Γ(T M̄ ) and for any α in
{1, . . . , r},
¯ ϕ̄X Y + ∇
¯ X (ϕ̄Y ))).
ḡ((Lξ̄α ϕ̄)X, Y ) = εα (−(ϕ̄X)(η̄ α (Y )) + η̄ α (∇
(2.1.7)
In fact:
ḡ((Lξ̄α ϕ̄)X, Y ) = ḡ([ξ̄α , ϕ̄X] − ϕ̄[ξ̄α , X], Y )
¯ ϕ̄X ξ¯α − ϕ̄(∇
¯ X) + ϕ̄(∇
¯ X ξ̄α ), Y )
¯ (ϕ̄X) − ∇
= ḡ(∇
ξ̄α
ξ̄α
¯ ϕ̄X ξ¯α + ϕ̄(∇
¯ X ξ̄α ), Y )
¯ ϕ̄)X − ∇
= ḡ((∇
ξ̄α
¯ X ξ̄α , ϕ̄Y )
¯ ϕ̄X ξ¯α , Y ) − ḡ(∇
= −ḡ(∇
¯ ϕ̄X Y ) − X(ḡ(ξ̄α , ϕ̄Y ))
= −(ϕ̄X)(ḡ(ξ̄α , Y )) + ḡ(ξ̄α , ∇
¯ X (ϕ̄Y ))
+ ḡ(ξ̄α , ∇
¯ ϕ̄X Y + ∇
¯ X (ϕ̄Y ))
= −(ϕ̄X)(εα η̄ α (Y )) + εα η̄ α (∇
¯ ϕ̄X Y + ∇
¯ X (ϕ̄Y ))).
= εα (−(ϕ̄X)(η̄ α (Y )) + η̄ α (∇
It follows, using Lemma 2.1.2 and the definition of hα , that
2ḡ(hα (X), Y ) − 2ḡ(X, hα (Y )) = ḡ((Lξ̄α ϕ̄)X, Y ) − ḡ((Lξ̄α ϕ̄)Y, X)
¯ ϕ̄X Y + ∇
¯ X (ϕ̄Y ))
= −εα (ϕ̄X)(η̄ α (Y )) + εα η̄ α (∇
α
α ¯
¯ Y (ϕ̄X))
+ εα (ϕ̄Y )(η̄ (X)) − εα η̄ (∇ϕ̄Y X + ∇
= −εα (ϕ̄X)(η̄ α (Y )) + εα η̄ α [ϕ̄X, Y ]
+ εα (ϕ̄Y )(η̄ α (X)) − εα η̄ α [ϕ̄Y, X]
= −εα (Lϕ̄X η̄ α )(Y ) + εα (Lϕ̄Y η̄ α )(X)
= −εα Nα(2) (X, Y ) = 0.
So we obtain:
ḡ(hα (X), Y ) = ḡ(X, hα (Y )),
and a) is proved.
21
Finally, for any α, β ∈ {1, . . . , r} we have:
1
1
(Lξ̄α ϕ̄)(ξ̄β ) = ([ξ̄α , ϕ̄ξ̄β ] − ϕ̄[ξ̄α , ξ¯β ]) = 0
2
2
¯
because ϕ̄ξ̄β = 0 and [ξ̄α , ξβ ] = 0, as already proved in Proposition 2.1.3.
hα (ξ̄β ) =
We use the Einstein summation convention and we omit the sum symbol
for repeated indices above and below.
Proposition 2.1.5. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite almost S-manifold,
for any X, Y ∈ Γ(T M̄ ), the following properties hold:
a) ϕ̄(N (X, Y )) + N (ϕ̄X, Y ) = 2η̄ α (X)hα (Y ),
b) N (ϕ̄X, Y ) ∈ D.
Proof. Using Lemma 2.1.2, we compute,
ϕ̄(N (X, Y )) + N (ϕ̄X, Y ) = ϕ̄([ϕ̄X, ϕ̄Y ] − [X, Y ] + η̄ α [X, Y ]ξ̄α − ϕ̄[ϕ̄X, Y ]
− ϕ̄[X, ϕ̄Y ] + 2dη̄ α (X, Y )ξ̄α ) + [ϕ̄2 X, ϕ̄Y ]
− [ϕ̄X, Y ] + η̄ α [ϕ̄X, Y ]ξ̄α − ϕ̄[ϕ̄2 X, Y ]
− ϕ̄[ϕ̄X, ϕ̄Y ] + 2dη̄ α (ϕ̄X, Y )ξ̄α
= −ϕ̄2 [ϕ̄X, Y ] − ϕ̄2 [X, ϕ̄Y ] − [X, ϕ̄Y ]
+ [η̄ α (X)ξ̄α , ϕ̄Y ] − [ϕ̄X, Y ] + η̄ α [ϕ̄X, Y ]ξ̄α
− ϕ̄[η̄ α (X)ξ̄α , Y ] + 2dη̄ α (ϕ̄X, Y )ξ̄α
= −η̄ α [ϕ̄X, Y ]ξ̄α − η̄ α [X, ϕ̄Y ]ξ̄α − (ϕ̄Y )(η̄ α (X))ξ̄α
+ η̄ α (X)[ξ̄α , ϕ̄Y ] + (ϕ̄X)(η̄ α (Y ))ξ̄α
− η̄ α (X)ϕ̄[ξ̄α , Y ] = −(Lϕ̄Y η̄ α )(X)ξ̄α
+ (Lϕ̄X η̄ α )(Y )ξ̄α + η̄ α (X)(Lξ̄α ϕ̄)(Y )
r
Nα(2) (X, Y )ξ̄α + 2η̄ α (X)hα (Y )
=
α=1
α
= 2η̄ (X)hα (Y ),
Finally, applying a), we have:
ḡ(N (ϕ̄X, Y ), ξ¯α ) = ḡ(2η̄ β (X)hβ (Y ), ξ¯α ) − ḡ(ϕ̄(N (X, Y )), ξ¯α )
= 2η̄ β (X)ḡ(hβ (Y ), ξ¯α )
= 2η̄ β (X)ḡ(Y, hβ (ξ̄α )) = 0.
Chapter 2. Indefinite S-manifolds
22
Corollary 2.1.6. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite almost S-manifold.
Then, for any X, Y ∈ Γ(T M̄ ), we deduce:
N (X, Y ) ∈ D.
Proof. We observe that for any α ∈ {1, . . . , r} we have [ξ̄α , D] ⊂ D, in fact, if
β ∈ {1, . . . , r} and X ∈ Γ(T M̄ ), we have
η̄ β [ξ̄α , ϕ̄X] = −2dη̄ β (ξ̄α , ϕ̄X) + ξ¯α (η̄ β (ϕ̄X)) − (ϕ̄X)(η̄ β (ξ̄α )) = 0
and in particular, if X ∈ D and α = β:
η̄ α [ξ̄α , X] = 0.
Therefore, if Z ∈ D then
N (ξ̄α , Z) = [ϕ̄ξ̄α , ϕ̄Z] − [ξ̄α , Z] + η̄ β [ξ̄α , Z]ξ̄β − ϕ̄[ϕ̄ξ¯α , Z]
− ϕ̄[ξ̄α , ϕ̄Z] + 2dη̄ β (ξ̄α , Z)ξ̄β
= −[ξ̄α , Z] − ϕ̄[ξ̄α , ϕ̄Z] ∈ D.
It is easy to check that N (ξ̄α , ξ¯β ) = 0 for any α, β ∈ {1, . . . , r}; moreover,
we have that N (ξ̄α , X) ∈ D for any X ∈ Γ(T M̄ ), in fact
N (ξ̄α , X) = N (ξ̄α , −ϕ̄2 X + η̄ β (X)ξ̄β )
= −N (ξ̄α , ϕ̄2 X) + η̄ β (X)N (ξ̄α , ξ¯β )
= −N (ξ̄α , ϕ̄2 X)
and N (ξ̄α , ϕ̄2 X) ∈ D, since ϕ̄2 X ∈ D.
Finally, if X, Y ∈ Γ(T M̄ ), we get
N (X, Y ) = −N (ϕ̄2 X, Y ) + η̄ α (X)N (ξ̄α , Y ),
and being N (ϕ̄2 X, Y ) ∈ D, by condition b) of the previous proposition, and
N (ξ̄α , Y ) ∈ D, as just seen, we can conclude that N (X, Y ) ∈ D.
Proposition 2.1.7. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite almost S-manifold.
Then, for any X ∈ Γ(T M̄ ) and for any α ∈ {1, . . . , r},
¯ X ξ¯α = −εα ϕ̄(X) − ϕ̄(hα X).
∇
23
Proof. Putting X = ξ¯α in ϕ̄(N (X, Y )) + N (ϕ̄X, Y ) = 2η̄ α (X)hα (Y ), we have
that for any Z, Y ∈ Γ(T M̄ )
ḡ(N (ξ̄α , Y ), ϕ̄Z) = −ḡ(ϕ̄(N (ξ̄α , Y )), Z)
= −2η̄ β (ξ̄α )ḡ(hβ (Y ), Z)
= −2ḡ(hα (Y ), Z).
Moreover, applying (2.1.1) of Proposition 2.1.3, for any α ∈ {1, . . . , r} we find:
¯ X ϕ̄)ξ̄α , Z)
¯ X ξ̄α ), Z) = ḡ((∇
ḡ(−ϕ̄(∇
1
= ḡ(N (ξ̄α , Z), ϕ̄X) − ḡ(ϕ̄Z, ϕ̄X)η̄(ξ̄α )
2
r
εβ η̄ β (X)η̄ β (Z))
= −ḡ(hα (Z), X) − εα (ḡ(Z, X) −
β=1
r
= −ḡ(hα (Z), X) − εα ḡ(Z, X) + εα
εβ η̄ β (X)η̄ β (Z)
β=1
= ḡ(−hα (X) − εα X + εα η̄ (X)ξ̄β , Z),
β
then, being Z arbitrary, we have
¯ X ξ¯α ) = hα (X) + εα X − εα η̄ β (X)ξ̄β ,
ϕ̄(∇
and, applying ϕ̄ again to the previous equation, we obtain:
¯ X ξ̄α = −ϕ̄(hα (X)) − εα ϕ̄(X).
∇
¯ X ξ̄α ∈ D.
Note that ∇
Proposition2.1.8. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite almost S-manifold.
Putting ξ¯ := rα=1 ξ̄α , we have
¯ ϕ̄X ϕ̄)ϕ̄Y = 2ḡ(ϕ̄X, ϕ̄Y )ξ̄ + η̄(Y )ϕ̄2 (X) − η̄ α (Y )hα (X),
¯ X ϕ̄)Y + (∇
(∇
for any X, Y ∈ Γ(T M̄ ).
Proof. Using (2.1.1), the previous proposition and Corollary 2.1.6, for any
Chapter 2. Indefinite S-manifolds
24
X, Y, Z ∈ Γ(T M̄ ) we have
¯ ϕ̄X ϕ̄)ϕ̄Y, Z) = ḡ(N (Y, Z), ϕ̄X)
¯ X ϕ̄)Y, Z) + 2ḡ((∇
2ḡ((∇
+ 2ḡ(ϕ̄Y, ϕ̄X)η̄(Z) − 2ḡ(ϕ̄Z, ϕ̄X)η̄(Y )
+ ḡ(N (ϕ̄Y, Z), ϕ̄2 X) + 2ḡ(ϕ̄2 Y, ϕ̄2 X)η̄(Z)
− 2ḡ(ϕ̄Z, ϕ̄2 X)η̄(ϕ̄Y )
= −ḡ(ϕ̄(N (Y, Z)), X) − ḡ(N (ϕ̄Y, Z), X)
+ η̄ α (X)ḡ(N (ϕ̄Y, Z), ξ¯α ) + 4ḡ(ϕ̄Y, ϕ̄X)η̄(Z)
− 2ḡ(ϕ̄Z, ϕ̄X)η̄(Y )
= −ḡ(ϕ̄(N (Y, Z)) + N (ϕ̄Y, Z), X)
+ 4ḡ(ϕ̄Y, ϕ̄X)η̄(Z) − 2ḡ(ϕ̄Z, ϕ̄X)η̄(Y )
= −ḡ(2η̄ α (Y )hα (Z), X)
r
ḡ(Z, ξ¯α ) + 2ḡ(Z, ϕ̄2 X)η̄(Y )
+ 4ḡ(ϕ̄Y, ϕ̄X)
α=1
¯
= −2ḡ(Z, η̄ (Y )hα (X)) + 4ḡ(ϕ̄Y, ϕ̄X)ḡ(Z, ξ)
α
+ 2ḡ(Z, η̄(Y )ϕ̄2 X)
= 2ḡ(Z, −η̄ α (Y )hα (X) + 2ḡ(ϕ̄Y, ϕ̄X)ξ̄ + η̄(Y )ϕ̄2 X).
Being Z arbitrarily chosen, we can deduce:
¯ ϕ̄X ϕ̄)ϕ̄Y = 2ḡ(ϕ̄X, ϕ̄Y )ξ̄ + η̄(Y )ϕ̄2 (X) − η̄ α (Y )hα (X).
¯ X ϕ̄)Y + (∇
(∇
Corollary 2.1.9. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite almost S-manifold.
Then, for any X, Y ∈ D:
¯ ϕ̄X ϕ̄)(ϕ̄Y ) = 2ḡ(X, Y )ξ̄,
¯ X ϕ̄)Y + (∇
a) (∇
¯ ϕ̄X ϕ̄)X.
¯ X ϕ̄)(ϕ̄X) = (∇
b) (∇
Proof. Observing that for any X, Y ∈ D we have η̄ α (X) = 0 and ḡ(ϕ̄X, ϕ̄Y ) =
ḡ(X, Y ), the proof of first statement follows from the above proposition.
Putting Y := ϕ̄X in a), we have
¯ ϕ̄X ϕ̄)(ϕ̄2 X) = 2ḡ(X, ϕ̄X)ξ̄ = 0,
¯ X ϕ̄)(ϕ̄X) + (∇
(∇
¯ X ϕ̄)(ϕ̄X) = (∇
¯ ϕ̄X ϕ̄)X.
therefore, being ϕ̄2 X = −X, we obtain (∇
25
Remark 2.1.10. We note that the statement b) can be written:
¯ ϕ̄X (ϕ̄X) = ϕ̄[ϕ̄X, X].
¯ XX + ∇
∇
(2.1.8)
In fact, taking X ∈ D into consideration, b) becomes:
¯ X ϕ̄X) = ∇
¯ ϕ̄X (ϕ̄X) − ϕ̄(∇
¯ ϕ̄X X),
¯ X (ϕ̄2 X) − ϕ̄(∇
∇
that is
¯ XX − ∇
¯ ϕ̄X (ϕ̄X) = −ϕ̄(∇
¯ ϕ̄X X − ∇
¯ X ϕ̄X),
−∇
and then
¯ ϕ̄X (ϕ̄X) = ϕ̄[ϕ̄X, X].
¯ XX + ∇
∇
2.2
Indefinite S-manifolds
Definition 2.2.1. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite metric g.f.f -manifold.
M̄ is said an indefinite S-manifold if it is a normal indefinite almost Smanifold.
Proposition 2.2.2. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite almost S-manifold,
then M̄ is an indefinite S-manifold if and only if the Levi-Civita connection
satisfies:
¯ X ϕ̄)Y = ḡ(X, Y )ξ̄ − η̄(Y )X
(∇
r
(εα η̄ α (X)η̄ α (Y ))ξ̄ + η̄(Y )η̄ α (X)ξ̄α ,
−
(2.2.1)
α=1
or equivalently, for any X, Y ∈ Γ(T M̄ )
¯ X ϕ̄)Y = ḡ(ϕ̄X, ϕ̄Y )ξ̄ + η̄(Y )ϕ̄2 (X).
(∇
(2.2.2)
Proof. Assuming that M̄ is an indefinite S-manifold, (2.1.1) becomes
¯ X ϕ̄)Y, Z) = ḡ(ϕ̄Y, ϕ̄X)η̄(Z) − ḡ(ϕ̄Z, ϕ̄X)η̄(Y )
ḡ((∇
¯ + ḡ(Z, ϕ̄2 X)η̄(Y )
= ḡ(ϕ̄Y, ϕ̄X)ḡ(Z, ξ)
= ḡ(Z, ḡ(ϕ̄Y, ϕ̄X)ξ̄ + η̄(Y )ϕ̄2 X),
from which
¯ X ϕ̄)Y = ḡ(ϕ̄X, ϕ̄Y )ξ̄ + η̄(Y )ϕ̄2 (X)
(∇
r
r
εα η̄ α (X)η̄ α (Y )ξ̄ − η̄(Y )X + η̄(Y )
η̄ α (X)ξ̄α .
= ḡ(X, Y )ξ̄ −
α=1
α=1
Chapter 2. Indefinite S-manifolds
26
¯ satisfies (2.2.1) or (2.2.2). From (2.2.2) we
Vice versa, we suppose that ∇
obtain
¯ X ϕ̄)Y, Z) = ḡ(ϕ̄Y, ϕ̄X)η̄(Z) − ḡ(ϕ̄Z, ϕ̄X)η̄(Y ),
ḡ((∇
and comparing the previous equation with (2.1.1), we deduce for any X, Y ∈
Γ(T M̄ )
ḡ(N (Y, Z), ϕ̄X) = 0.
From Corollary 2.1.6, we obtain that N (Y, Z) = 0 for any Y, Z ∈ Γ(T M̄ ), that
is M̄ is normal.
Remark 2.2.3. In an indefinite S-manifold (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) for any α in
{1, . . . , r}, the operators Lξ̄α ϕ̄, and then hα , vanish. In fact, we know that
an indefinite S-manifold is normal, then for any X ∈ Γ(T M̄ ) and for any
α ∈ {1, . . . , r} we find:
0 = N (ϕ̄X, ξ¯α ) = ϕ̄2 [ϕ̄X, ξ¯α ] − ϕ̄[ϕ̄2 X, ξ¯α ] + 2dη̄ β (ϕ̄X, ξ¯α )ξ̄β
= −[ϕ̄X, ξ¯α ] + η̄ β [ϕ̄X, ξ¯α ]ξ̄β + ϕ̄[X, ξ¯α ] − ϕ̄(η̄ γ (X)[ξ̄γ , ξ¯α ]
− ξ̄α (η̄ γ (X))ξ̄γ ) = −[ϕ̄X, ξ¯α ] + ϕ̄[X, ξ¯α ]
= (Lξ̄α ϕ̄)X = 2hα (X).
Using Proposition 2.1.7, we obtain, for any α ∈ {1, . . . , r},
¯ X ξ̄α = −εα ϕ̄X.
∇
Now, we want to give the condition of indefinite S-manifold in terms of
the fundamental 2-form:
Proposition 2.2.4. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite almost S-manifold.
Then M̄ is an indefinite S-manifold if and only if for any X, Y, Z ∈ Γ(T M̄ ):
¯ X Φ)(Y, Z) =
(∇
r
εα (ḡ(X, Z)η̄ α (Y ) − ḡ(X, Y )η̄ α (Z))
α=1
−
r
εα εβ η̄ α (X)(η̄ α (Z)η̄ β (Y ) − η̄ α (Y )η̄ β (Z))
β,α=1
= η̄(Y )ḡ(ϕ̄X, ϕ̄Z) − η̄(Z)ḡ(ϕ̄X, ϕ̄Y ).
(2.2.3)
27
Proof. In general, if M̄ is an indefinite almost S-manifold then the following
relation holds for any X, Y, Z ∈ Γ(T M̄ ):
¯ X Y, Z) − Φ(Y, ∇
¯ X Z)
¯ X Φ)(Y, Z) = X(Φ(Y, Z)) − Φ(∇
(∇
¯ X Z))
¯ X Y, ϕ̄Z) − ḡ(Y, ϕ̄(∇
= X(ḡ(Y, ϕ̄Z)) − ḡ(∇
¯ X (ϕ̄Z)) − ḡ(Y, ϕ̄(∇
¯ X Z))
= ḡ(Y, ∇
¯ X ϕ̄)Z).
= ḡ(Y, (∇
Then we know that M̄ is an indefinite S-manifold if and only if (2.2.1) holds,
and so, replacing this relation in the previous one, we get that M̄ is an indefinite S-manifold if and only if
¯ X Φ)(Y, Z) =
(∇
−
r
εα (ḡ(X, Z)η̄ α (Y ) − ḡ(X, Y )η̄ α (Z))
α=1
r
εα εβ η̄ α (X)(η̄ α (Z)η̄ β (Y ) − η̄ α (Y )η̄ β (Z)).
β,α=1
Proposition 2.2.5. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite metric g.f.f -manifold.
If ξ̄α are Killing vector fields, Lξ̄α η̄ β = 0 for any α, β ∈ {1, . . . , r} and if M̄
satisfies (2.2.2) or equivalently (2.2.3), then M̄ is an indefinite S-manifold.
¯ X Φ)(Y, Z), from (2.2.3) we get
Proof. Being 3dΦ(X, Y, Z) = SX,Y,Z (∇
dΦ = 0.
From Proposition 1.4.9 we know that
(Lξ̄α Φ)(X, Y ) = (Lξ̄α ḡ)(X, ϕ̄Y ) + ḡ(X, (Lξ̄α ϕ̄)Y ),
(2.2.4)
and since Lξ̄α Φ = iξ̄α dΦ + diξ̄α Φ
(Lξ̄α Φ)(X, Y ) = 0,
(2.2.5)
for any α ∈ {1, . . . , r} and X, Y ∈ Γ(T M̄ ). From (2.2.4) and (2.2.5), being ξ̄α
Killing vector fields, we find
Lξ̄α ϕ̄ = 0,
Chapter 2. Indefinite S-manifolds
28
from which, for any α ∈ {1, . . . , r} and X ∈ Γ(T M̄ ), we have
[ξ̄α , ϕ̄X] = ϕ̄[ξ̄α , X].
(2.2.6)
In these hypothesis, (1.4.4) becomes
¯ X ϕ̄)Y, Z) = ḡ(N (Y, Z), ϕ̄X) + 2
2ḡ((∇
r
εα [dη̄ α (ϕ̄Y, Z)η̄ α (X)
α=1
− dη̄ (ϕ̄Z, Y )η̄ (X) + dη̄ α (ϕ̄Y, X)η̄ α (Z)
α
α
− dη̄ α (ϕ̄Z, X)η̄ α (Y )],
and, using Proposition 1.4.9 and (2.2.3), we obtain
¯ X ϕ̄)Z) = η̄(Y )ḡ(ϕ̄X, ϕ̄Z) − η̄(Z)ḡ(ϕ̄X, ϕ̄Y )
ḡ(Y, (∇
therefore we deduce
ḡ(N (Y, Z), ϕ̄X) = −2
r
εα [(dη̄ α (ϕ̄Y, Z) − dη̄ α (ϕ̄Z, Y ))η̄ α (X)
(2.2.7)
α=1
α
+ (dη̄ (ϕ̄Y, X) − ḡ(ϕ̄X, ϕ̄Y ))η̄ α (Z)
− (dη̄ α (ϕ̄Z, X) − ḡ(ϕ̄X, ϕ̄Z))η̄ α (Y )].
Putting Y = ξ¯β in (2.2.7), we get
ḡ(N (ξ̄β , Z), ϕ̄X) = 2εβ (dη̄ β (ϕ̄Z, X) − ḡ(ϕ̄X, ϕ̄Z)).
(2.2.8)
On the other hand, we have
N (ξ̄β , Z) = −[ξ̄β , Z] − ϕ̄[ξ̄β , ϕ̄Z] + ξ̄β (η̄ α (Z))ξ̄α ,
therefore from (2.2.6)
ϕ̄N (ξ̄β , Z) = (Lξ̄α ϕ̄)Z − η̄ α [ξ̄β , ϕ̄Z]ξ̄α = 0.
(2.2.9)
Replacing (2.2.9) in (2.2.8), we find
dη̄ β (ϕ̄Z, X) = ḡ(ϕ̄X, ϕ̄Z) = Φ(ϕ̄Z, X).
Finally Lξ̄α η̄ β = 0 implies iξ̄α dη̄ β = 0 and for any Y ∈ Γ(T M̄ ) being Y =
−ϕ̄2 Y + η̄ α (Y )ξ̄α we obtain
dη̄ β (Y, X) = −dη̄ β (ϕ̄2 Y, X) + η̄ α (Y )dη̄ β (ξ̄α , X) = −Φ(ϕ̄2 Y, X)
= −g(ϕ̄2 Y, ϕ̄X) = g(Y, ϕ̄X) = Φ(Y, X).
Then M̄ satisfies the hypothesis of Proposition 2.2.2.
29
2.3
Sectional Curvature and ϕ̄-Sectional Curvature
Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite S-manifold. The curvature tensor R̄
of the Levi-Civita connection is defined by
¯Y∇
¯ [X,Y ] Z,
¯YZ − ∇
¯ XZ − ∇
¯ X∇
R̄(X, Y, Z) = ∇
for any X, Y, Z ∈ Γ(T M̄ ) and we can define another tensor field R̄ of type
(0,4) given by
R̄(X, Y, Z, W ) = ḡ(R̄(Z, W, Y ), X) = −ḡ(R̄(X, Y, Z), W ),
(2.3.1)
for any X, Y, Z, W ∈ Γ(T M̄ ).
A two-dimensional subspace π of the tangent space Tp M̄ is called nondegenerate if and only if we have ∆(π) = ḡp (X, X)ḡp (Y, Y )-ḡp (X, Y )2 = 0
for any basis {X, Y } of π. We know that if π is a non-degenerate 2-plane of
Tp M̄ then we can define the sectional curvature Kp (π) at p with respect to
2-plane π, putting
Kp (π) =
ḡp (R̄p (X, Y, Y ), X)
ḡp (R̄p (X, Y, X), Y )
R̄p (X, Y, X, Y )
=
=−
,
∆(π)
∆(π)
∆(π)
where π = span{X, Y }. In the following we denote Kp (π) = Kp (X, Y ) and
we know that it does not depend on the particular choice of the basis {X, Y }.
Lemma 2.3.1. In an indefinite S-manifold (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) one has:
a) the distribution ker ϕ̄ is integrable and flat;
b) the sectional curvatures K(X, Y ), with Y = ξ̄α , for any α ∈ {1, . . . , r},
and X ∈ Im ϕ̄ non lightlike, have values +1 or −1.
Proof. For X, Y ∈ ker ϕ̄ we have X = f α ξ¯α , Y = tβ ξ¯β then
[X, Y ] = [f α ξ̄α , tβ ξ̄β ] = f α ξ̄α (tβ )ξ̄β − tβ ξ̄β (f α )ξ̄α ∈ ker ϕ̄
¯ ξ̄β = 0 and [ξ̄α , ξ̄β ] = 0, we
and ker ϕ̄ is integrable. Furthermore, since ∇
ξ̄α
¯
¯
have R̄(ξ̄α , ξβ , ξγ ) = 0 and ker ϕ̄ is flat. Note that a) holds also for indefinite
almost S-manifold.
Chapter 2. Indefinite S-manifolds
30
¯ X ξ¯α = −εα ϕ̄X,
Now, being M̄ an indefinite S-manifold, we know that ∇
Lξ̄α ϕ̄ = 0 and we have
¯
¯ ∇
¯ ¯ ¯
¯
¯
R̄(ξ̄α , X, ξ¯β ) = ∇
ξ̄α X ξ̄β − ∇X ∇ξ̄α ξβ − ∇[ξ̄α ,X] ξβ
¯ (ϕ̄X) + εβ ϕ̄[ξ̄α , X]
= −εβ ∇
ξ̄α
¯ ϕ̄X ξ̄α )
= εβ (ϕ̄[ξ̄α , X] − [ξ̄α , ϕ̄X] − ∇
= εβ εα ϕ̄2 X.
Then from the definition of K for X ∈ Im ϕ̄ non lightlike and Y = ξ̄α ,we have
K(X, ξ¯α ) = −
ḡ(R̄(ξ̄α , X, ξ¯α ), X)
εα ḡ(ϕ̄2 X, X)
=−
= εα .
ḡ(X, X)εα
ḡ(X, X)
As usual, we say that a 2-plane π in Tp M̄ , p ∈ M̄ , is a ϕ̄-plane if π =
span{X, ϕ̄X} with X ∈ Dp , and the sectional curvature at p of such a plane,
with X a non lightlike vector field, is said the ϕ̄-sectional curvature at p
and is denoted by Hp (X). We want to show that on an indefinite S-manifold,
as in the Sasakian case, the ϕ̄-sectional curvatures determine the sectional
curvatures.
As in [13], we define a tensor field of type (0,4) given for any X, Y, Z, W
in Γ(T M̄ ) by
P (X, Y ; Z, W ) = Φ(X, Z)ḡ(Y, W ) − Φ(X, W )ḡ(Y, Z)
(2.3.2)
− Φ(Y, Z)ḡ(X, W ) + Φ(Y, W )ḡ(X, Z).
The following lemmas will help us to show the properties of the curvature
tensor field.
Lemma 2.3.2. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite S-manifold, then P satisfies the following properties:
1) P (X, Y ; Z, W ) = −P (Z, W ; X, Y ), for any X, Y, Z, W ∈ Γ(T M̄ ),
2) P (X, Y ; X, ϕ̄Y ) = ḡ(X, ϕ̄Y )2 + ḡ(X, Y )2 − εX εY , where X, Y are unit
vector fields of D and εX = ḡ(X, X) and εY = ḡ(Y, Y ).
31
Proof. From the definition of P we have:
P (Z, W ; X, Y ) = Φ(Z, X)ḡ(W, Y ) − Φ(Z, Y )ḡ(W, X)
− Φ(W, X)ḡ(Z, Y ) + Φ(W, Y )ḡ(Z, X)
= −Φ(X, Z)ḡ(Y, W ) + Φ(Y, Z)ḡ(X, W )
+ Φ(X, W )ḡ(Y, Z) − Φ(Y, W )ḡ(X, Z)
= −P (X, Y ; Z, W ).
Again, from (2.3.2)
P (X, Y ; X, ϕ̄Y ) = Φ(X, X)ḡ(Y, ϕ̄Y ) − Φ(X, ϕ̄Y )ḡ(Y, X)
− Φ(Y, X)ḡ(X, ϕ̄Y ) + Φ(Y, ϕ̄Y )ḡ(X, X)
= −Φ(Y, X)ḡ(X, ϕ̄Y ) + Φ(Y, ϕ̄Y )εX
− Φ(X, ϕ̄Y )ḡ(Y, X)
= ḡ(X, ϕ̄Y )2 + ḡ(X, Y )2 − εX εY .
Lemma 2.3.3. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold, then for any
X, Y, Z, W ∈ Γ(T M̄ )
ḡ(R̄(X, Y, ϕ̄Z), W ) + ḡ(R̄(X, Y, Z), ϕ̄W )
(2.3.3)
= −εP (X, Y ; Z, W ) − Q(X, Y ; Z, W )
where ε =
r
α=1 εα
and
Q(X, Y ; Z, W ) = ḡ(W, ϕ̄Y )(ε(ḡ(X, Z) − ḡ(ϕ̄X, ϕ̄Z)) − η̄(Z)η̄(X))
− ḡ(W, ϕ̄X)(ε(ḡ(Y, Z) − ḡ(ϕ̄Y, ϕ̄Z)) − η̄(Z)η̄(Y ))
− ḡ(Z, ϕ̄Y )(ε(ḡ(X, W ) − ḡ(ϕ̄X, ϕ̄W )) − η̄(X)η̄(W ))
+ ḡ(Z, ϕ̄X)(ε(ḡ(Y, W ) − ḡ(ϕ̄Y, ϕ̄W )) − η̄(Y )η̄(W )).
Moreover if X, Y, Z, W ∈ D then obviusly Q(X, Y ; Z, W ) = 0 and the following
statements hold:
1) ḡ(R̄(ϕ̄X, ϕ̄Y, ϕ̄Z), ϕ̄W ) = ḡ(R̄(X, Y, Z), W );
2) ḡ(R̄(X, ϕ̄X, Y ), ϕ̄Y ) = ḡ(R̄(X, Y, X), Y ) + ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y )
− 2εP (X, Y, X, ϕ̄Y );
3) ḡ(R̄(ϕ̄X, Y, ϕ̄X), Y ) = ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ).
Chapter 2. Indefinite S-manifolds
32
Proof. We remark that ε can vanish only if r is an even number and the number of timelike characteristic vector fields is equal to the number of spacelike
¯ = 0, i.e. ξ¯ is a
characteristic vector fields. Moreover, ε = 0 means that ḡ(ξ̄, ξ)
lightlike vector field.
For any X, Y, Z, W ∈ Γ(T M̄ ) we prove that
(R̄XY Φ)(Z, W ) = −ḡ(R̄(X, Y, ϕ̄Z), W ) − ḡ(R̄(X, Y, Z), ϕ̄W ).
(2.3.4)
In fact, we have:
(R̄XY Φ)(Z, W ) = R̄XY (Φ(Z, W )) − Φ(R̄XY Z, W ) − Φ(Z, R̄XY W )
= ḡ(R̄XY Z, ϕ̄W ) + ḡ(Z, R̄XY (ϕ̄W )) − ḡ(R̄XY Z, ϕ̄W )
− ḡ(Z, ϕ̄(R̄XY W ))
= −ḡ(R̄XY Z, ϕ̄W ) + ḡ(ϕ̄Z, R̄XY W )
= −ḡ(R̄(X, Y, ϕ̄Z), W ) − ḡ(R̄(X, Y, Z), ϕ̄W ).
On the other hand, we can compute (R̄XY Φ)(Z, W ), using (2.2.3). At first we
¯ Y Φ)(Z, W ). Avoiding too much details, we have:
¯ X∇
value the term (∇
¯ Y Φ)(Z, W )) − (∇
¯ Y Φ)(∇
¯ X Z, W )
¯ Y Φ)(Z, W ) = X((∇
¯ X∇
(∇
¯ XW )
¯ Y Φ)(Z, ∇
− (∇
r
εα (ḡ(Y, W )η̄ α (Z) − ḡ(Y, Z)η̄ α (W ))}
= X{
α=1
r
εα εβ η̄ α (Y )(η̄ α (Z)η̄ β (W ) − η̄ α (W )η̄ β (Z))}
− X{
α,β=1
r
¯ X Z) − ḡ(Y, ∇
¯ X Z)η̄ α (W ))
εα (ḡ(Y, W )η̄ α (∇
−
α=1
r
¯ X Z)η̄ β (W )
εα εβ η̄ α (Y )(η̄ α (∇
+
α,β=1
r
¯ X Z)) −
¯ X W )η̄ α (Z)
εα (ḡ(Y, ∇
− η̄ α (W )η̄ β (∇
α=1
¯ X W )) − η̄ α (∇
¯ X W )η̄ β (Z))
− ḡ(Y, Z)η̄ (∇
r
¯ XW )
εα εβ η̄ α (Y )(η̄ α (Z)η̄ β (∇
+
α
α,β=1
= ε(ḡ(W, ϕ̄X)ḡ(Y, Z) − ḡ(Z, ϕ̄X)ḡ(Y, W ))
r
¯ X Y, W ) − η̄ α (W )ḡ(∇
¯ X Y, Z)}
εα {η̄ α (Z)ḡ(∇
+
α=1
r
εβ η̄ β (Y )η̄ β (W )
+ ε(ḡ(Z, ϕ̄X)
β=1
r
εβ η̄ β (Y )η̄ β (Z))
− ḡ(W, ϕ̄X)
β=1
33
r
εα εβ η̄ β (Y )η̄ α (Z)
+ ḡ(W, ϕ̄X)
α,β=1
r
εα εβ η̄ β (Y )η̄ α (W )
− ḡ(Z, ϕ̄X)
α,β=1
r
¯ X Y )(η̄ β (W )η̄ α (Z) − η̄ β (Z)η̄ α (W ))
εα εβ η̄ β (∇
−
α,β=1
= ε(ḡ(W, ϕ̄X)ḡ(Y, Z) − ḡ(Z, ϕ̄X)ḡ(Y, W ))
¯ X Y, Z) + εḡ(Z, ϕ̄X)(ḡ(Y, W ) − ḡ(ϕ̄Y, ϕ̄W ))
− η̄(W )ḡ(∇
¯ X Y, W )
− εḡ(W, ϕ̄X)(ḡ(Y, Z) − ḡ(ϕ̄Y, ϕ̄Z)) + η̄(Z)ḡ(∇
+ ḡ(W, ϕ̄X)η̄(Y )η̄(Z) − ḡ(Z, ϕ̄X)η̄(Y )η̄(W )
¯ X Y, ϕ̄W )η̄(Z)
¯ X Y, W )η̄(Z) + ḡ(ϕ̄∇
− ḡ(∇
¯ X Y, Z)η̄(W ) − ḡ(ϕ̄∇
¯ X Y, ϕ̄Z)η̄(W ).
+ ḡ(∇
Now, exchanging X for Y in the previous relation, we get
¯ X Φ)(Z, W ) = ε(ḡ(W, ϕ̄Y )ḡ(X, Z) − ḡ(Z, ϕ̄Y )ḡ(X, W ))
¯Y∇
(∇
¯ Y X, Z)
¯ Y X, W ) − η̄(W )ḡ(∇
+ η̄(Z)ḡ(∇
+ εḡ(Z, ϕ̄Y )(ḡ(X, W ) − ḡ(ϕ̄X, ϕ̄W ))
− εḡ(W, ϕ̄Y )(ḡ(X, Z) − ḡ(ϕ̄X, ϕ̄Z))
+ ḡ(W, ϕ̄Y )η̄(X)η̄(Z) − ḡ(Z, ϕ̄Y )η̄(X)η̄(W )
¯ Y X))
¯ Y X) − ḡ(ϕ̄W, ϕ̄∇
− η̄(Z)(ḡ(W, ∇
¯ Y X)).
¯ Y X) − ḡ(ϕ̄Z, ϕ̄∇
+ η̄(W )(ḡ(Z, ∇
Then we obtain:
(R̄XY Φ)(Z, W ) = ε(ḡ(W, ϕ̄X)ḡ(Y, Z) − ḡ(Z, ϕ̄X)ḡ(Y, W )
(2.3.5)
− ḡ(W, ϕ̄Y )ḡ(X, Z) + ḡ(Z, ϕ̄Y )ḡ(X, W ))
+ εḡ(Z, ϕ̄X)(ḡ(Y, W ) − ḡ(ϕ̄Y, ϕ̄W )) − η̄(Y )η̄(W )ḡ(Z, ϕ̄X)
− εḡ(W, ϕ̄X)(ḡ(Y, Z) − ḡ(ϕ̄Y, ϕ̄Z)) + η̄(Y )η̄(Z)ḡ(W, ϕ̄X)
− εḡ(Z, ϕ̄Y )(ḡ(X, W ) − ḡ(ϕ̄X, ϕ̄W )) + ḡ(Z, ϕ̄Y )η̄(X)η̄(W )
+ εḡ(W, ϕ̄Y )(ḡ(X, Z) − ḡ(ϕ̄X, ϕ̄Z)) − ḡ(W, ϕ̄Y )η̄(X)η̄(Z)
= εP (X, Y ; Z, W ) + Q(X, Y ; Z, W ).
So, comparing (2.3.4) and (2.3.5), we obtain the first equation. From the
definition of Q is obvious that we have Q(X, Y ; Z, W ) = 0 for any X, Y, Z, W
in Γ(D), therefore (2.3.3), for any X, Y, Z, W in Γ(D), becomes
ḡ(R̄(X, Y, ϕ̄Z), W ) + ḡ(R̄(X, Y, Z), ϕ̄W ) = −εP (X, Y ; Z, W ).
(2.3.6)
Chapter 2. Indefinite S-manifolds
34
Thus, using this equation, we prove the statement 1), 2) and 3).
Writing (2.3.6) in term of ϕ̄X, ϕ̄Y , ϕ̄Z and W , we get:
−ḡ(R̄(ϕ̄X, ϕ̄Y, Z), W ) + ḡ(R̄(ϕ̄X, ϕ̄Y, ϕ̄Z), ϕ̄W ) = −εP (ϕ̄X, ϕ̄Y ; ϕ̄Z, W ),
and writing (2.3.6) in term of Z, W , ϕ̄X and Y , we have:
−ḡ(R̄(Z, W, X), Y ) + ḡ(R̄(Z, W, ϕ̄X), ϕ̄Y ) = −εP (Z, W ; ϕ̄X, Y ).
Now, summing the obtained equations, we get:
ḡ(R̄(ϕ̄X, ϕ̄Y, ϕ̄Z), ϕ̄W ) = ḡ(R̄(Z, W, X), Y ) = ḡ(R̄(X, Y, Z), W ),
because, using the definition of P , we find
P (ϕ̄X, ϕ̄Y ; ϕ̄Z, W ) = −P (Z, W ; ϕ̄X, Y ).
Namely,
P (ϕ̄X, ϕ̄Y ; ϕ̄Z, W ) = Φ(ϕ̄X, ϕ̄Z)ḡ(ϕ̄Y, W ) − Φ(ϕ̄X, W )ḡ(ϕ̄Y, ϕ̄Z)
− Φ(ϕ̄Y, ϕ̄Z)ḡ(ϕ̄X, W ) + Φ(ϕ̄Y, W )ḡ(ϕ̄X, ϕ̄Z)
= ḡ(ϕ̄X, Z)Φ(Y, W ) − Φ(ϕ̄X, W )ḡ(Y, Z)
− Φ(Y, Z)ḡ(ϕ̄X, W ) + ḡ(Y, W )Φ(ϕ̄X, ϕ̄Z)
= P (ϕ̄X, Y ; Z, W ) = −P (Z, W ; ϕ̄X, Y ).
To have 2), from the Bianchi identity, SY,Z,W R̄(X, Y, Z, W ) = 0, we have:
ḡ(R̄(X, ϕ̄X, Y ), ϕ̄Y ) + ḡ(R̄(X, Y, ϕ̄Y ), ϕ̄X) + ḡ(R̄(X, ϕ̄Y, ϕ̄X), Y ) = 0.
Applying (2.3.6) to ḡ(R̄(X, Y, ϕ̄Y ), ϕ̄X) and ḡ(R̄(X, ϕ̄Y, ϕ̄X), Y ), we have:
ḡ(R̄(X, Y, ϕ̄Y ), ϕ̄X) = −ḡ(R̄(X, Y, Y ), ϕ̄2 X) − εP (X, Y ; Y, ϕ̄X)
= −ḡ(R̄(X, Y, X), Y ) − εP (X, Y ; Y, ϕ̄X),
and
ḡ(R̄(X, ϕ̄Y, ϕ̄X), Y ) = −ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ) − εP (X, ϕ̄Y ; X, Y ).
Moreover we can prove that
P (X, Y ; Y, ϕ̄X) = P (X, ϕ̄Y ; X, Y ) = −P (X, Y ; X, ϕ̄Y )
35
using only the definition of P :
P (X, Y ; Y, ϕ̄X) = Φ(X, Y )ḡ(Y, ϕ̄X) − Φ(X, ϕ̄X)ḡ(Y, Y )
+ Φ(Y, ϕ̄X)ḡ(X, Y )
= −Φ(X, Y )ḡ(ϕ̄Y, X) + Φ(ϕ̄Y, Y )ḡ(X, X)
− Φ(ϕ̄Y, X)ḡ(X, Y )
= P (X, ϕ̄Y ; X, Y ) = −P (X, Y ; X, ϕ̄Y ).
Therefore, we deduce that
ḡ(R̄(X, ϕ̄X, Y ), ϕ̄Y ) = −ḡ(R̄(X, Y, ϕ̄Y ), ϕ̄X) − ḡ(R̄(X, ϕ̄Y, ϕ̄X), Y )
= ḡ(R̄(X, Y, X), Y ) − εP (X, Y ; X, ϕ̄Y )
+ ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ) − εP (X, Y ; X, ϕ̄Y )
= ḡ(R̄(X, Y, X), Y ) + ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y )
− 2εP (X, Y ; X, ϕ̄Y ).
Finally, to obtain 3), we use (2.3.6) twice, getting:
ḡ(R̄(ϕ̄X, Y, ϕ̄X), Y ) + ḡ(R̄(ϕ̄X, Y, X), ϕ̄Y ) = −εP (ϕ̄X, Y ; X, Y ),
ḡ(R̄(X, ϕ̄Y, ϕ̄X), Y ) + ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ) = −εP (X, ϕ̄Y ; X, Y ).
Then, noting that
ḡ(R̄(ϕ̄X, Y, X), ϕ̄Y ) = ḡ(R̄(X, ϕ̄Y, ϕ̄X), Y ),
and
P (ϕ̄X, Y ; X, Y ) = P (X, ϕ̄Y ; X, Y ),
and subtracting the previous equations, we have:
ḡ(R̄(ϕ̄X, Y, ϕ̄X), Y ) = ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ).
We put
B(X, Y ) = ḡ(R̄(X, Y, X), Y )
and
D(X) = B(X, ϕ̄X)
for any X ∈ Γ(D). The following Lemma gives the expression of B(X, Y ), for
any X, Y ∈ Γ(D).
Chapter 2. Indefinite S-manifolds
36
Lemma 2.3.4. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite S-manifold. Then, for
any X, Y ∈ Γ(D), we have
B(X, Y ) =
1
{3D(X + ϕ̄Y ) + 3D(X − ϕ̄Y ) − D(X + Y )
(2.3.7)
32
− D(X − Y ) − 4D(X) − 4D(Y ) + 24εP (X, Y ; X, ϕ̄Y )}.
Proof. The following links between D and R̄ are given by a lengthy but straight
computation, that we will not write for the sake of brevity,
3D(X + ϕ̄Y ) + 3D(X − ϕ̄Y ) = 3(B(X + ϕ̄Y, ϕ̄X − Y )
+ B(X − ϕ̄Y, ϕ̄X + Y ))
= 6ḡ(R̄(X, ϕ̄X, X), ϕ̄X) − 12ḡ(R̄(ϕ̄Y, Y, X), ϕ̄X)
+ 6ḡ(R̄(ϕ̄Y, ϕ̄X, ϕ̄Y ), ϕ̄X) + 12ḡ(R̄(X, Y, ϕ̄X), ϕ̄Y )
+ 6ḡ(R̄(X, Y, X), Y ) + 6ḡ(R̄(ϕ̄Y, Y, ϕ̄Y ), Y ),
and
−(D(X + Y ) + D(X − Y )) = −(B(X + Y, ϕ̄X + ϕ̄Y )
+ B(X − Y, ϕ̄X − ϕ̄Y ))
= −2ḡ(R̄(X, ϕ̄X, X), ϕ̄X) − 4ḡ(R̄(Y, ϕ̄Y, X), ϕ̄X)
− 2ḡ(R̄(X, ϕ̄Y, Y ), ϕ̄X) − 2ḡ(R̄(Y, ϕ̄X, X), ϕ̄Y )
− 2ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ) − 2ḡ(R̄(Y, ϕ̄Y, Y ), ϕ̄Y )
− 2ḡ(R̄(Y, ϕ̄X, Y ), ϕ̄X).
We begin to compute the right side of (2.3.7), using Lemma 2.3.3 and replacing
the previous links between D and R̄:
1
{3D(X + ϕ̄Y ) + 3D(X − ϕ̄Y ) − D(X + Y ) − D(X − Y ) − 4D(X)
32
− 4D(Y ) + 24εP (X, Y ; X, ϕ̄Y )}
1
= {6ḡ(R̄(X, ϕ̄X, X), ϕ̄X) − 12ḡ(R̄(ϕ̄Y, Y, X), ϕ̄X)
32
+ 6ḡ(R̄(ϕ̄Y, ϕ̄X, ϕ̄Y ), ϕ̄X) + 12ḡ(R̄(X, Y, ϕ̄X), ϕ̄Y )
+ 6ḡ(R̄(X, Y, X), Y ) + 6ḡ(R̄(ϕ̄Y, Y, ϕ̄Y ), Y )
− 2ḡ(R̄(X, ϕ̄X, X), ϕ̄X) − 4ḡ(R̄(Y, ϕ̄Y, X), ϕ̄X)
− 2ḡ(R̄(X, ϕ̄Y, Y ), ϕ̄X) − 2ḡ(R̄(Y, ϕ̄X, X), ϕ̄Y )
− 2ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ) − 2ḡ(R̄(Y, ϕ̄Y, Y ), ϕ̄Y )
− 2ḡ(R̄(Y, ϕ̄X, Y ), ϕ̄X) − 4ḡ(R̄(X, ϕ̄X, X), ϕ̄X)
37
− 4ḡ(R̄(ϕ̄Y, Y, ϕ̄Y ), Y ) + 24εP (X, Y ; X, ϕ̄Y )}
1
= {8[ḡ(R̄(X, Y, X), Y ) + ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y )
32
− 2εP (X, Y ; X, ϕ̄Y )] + 6ḡ(R̄(X, Y, X), Y )
+ 12[ḡ(R̄(X, Y, X), Y ) − εP (X, Y ; X, ϕ̄Y )]
+ 6ḡ(R̄(X, Y, X), Y ) − 4[ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y )
− εP (X, Y ; X, ϕ̄Y )] − 2ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y )
− 2ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ) + 24εP (X, Y ; X, ϕ̄Y )}
= ḡ(R̄(X, Y, X), Y ) = B(X, Y ).
Proposition 2.3.5. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and p in
M̄ . We consider a non degenerate 2-plane π = span{X, Y } of Dp , where X
and Y are unit vectors of Dp . Then the sectional curvature Kp (X, Y ) is given
by
Kp (X, Y ) =
1
{3(εX + εY + 2ḡ(X, ϕ̄Y ))2 Hp (X + ϕ̄Y )
32(εX εY − ḡ(X, Y )2 )
+ 3(εX + εY − 2ḡ(X, ϕ̄Y ))2 Hp (X − ϕ̄Y )
− (εX + εY + 2ḡ(X, Y ))2 Hp (X + Y )
− (εX + εY − 2ḡ(X, Y ))2 Hp (X − Y ) − 4Hp (X) − 4Hp (Y )
+ 24ε(ḡ(X, ϕ̄Y )2 + ḡ(X, Y )2 − εX εY )}.
Proof. We note that:
1. if X ∈ Dp we have:
Dp (X) = Bp (X, ϕ̄X) = ḡp (R̄p (X, ϕ̄X, X), ϕ̄X) = −ḡp (X, X)2 Hp (X).
2. if X and Y are unit vectors of Dp , using the notation of Lemma 2.3.2,
we find:
ḡ(X + ϕ̄Y, X + ϕ̄Y ) = εX + εY + 2ḡ(X, ϕ̄Y ),
ḡ(X − ϕ̄Y, X − ϕ̄Y ) = εX + εY − 2ḡ(X, ϕ̄Y ),
ḡ(X + Y, X + Y ) = εX + εY + 2ḡ(X, Y ),
ḡ(X − Y, X − Y ) = εX + εY − 2ḡ(X, Y ).
Chapter 2. Indefinite S-manifolds
38
Being ∆(π) = ḡp (X, X)ḡp (Y, Y ) − ḡp (X, Y )2 = εX εY − ḡp (X, Y )2 , from the
definition of the sectional curvature we have:
Kp (π) = −
ḡp (R̄p (X, Y, X), Y )
Bp (X, Y )
=−
.
∆(π)
∆(π)
Using the previous lemma, we get an expression of the sectional curvature in
term of ϕ̄-sectional curvature in the following way:
Bp (X, Y )
1
=−
{3Dp (X + ϕ̄Y ) + 3Dp (X − ϕ̄Y )
∆(π)
32∆(π)
− Dp (X + Y ) − Dp (X − Y ) − 4Dp (X) − 4Dp (Y )
Kp (π) = −
+ 24εPp (X, Y ; X, ϕ̄Y )}
1
{−3(εX + εY + 2ḡ(X, ϕ̄Y ))2 Hp (X + ϕ̄Y )
=−
32∆(π)
− 3(εX + εY − 2ḡ(X, ϕ̄Y ))2 Hp (X − ϕ̄Y )
+ (εX + εY + 2ḡ(X, Y ))2 Hp (X + Y )
+ (εX + εY − 2ḡ(X, Y ))2 Hp (X − Y ) + 4Hp (X) + 4Hp (Y )
+ 24ε(ḡ(X, ϕ̄Y )2 + ḡ(X, Y )2 − εX εY )}
1
{3(εX + εY + 2ḡ(X, ϕ̄Y ))2 Hp (X + ϕ̄Y )
=
32∆(π)
+ 3(εX + εY − 2ḡ(X, ϕ̄Y ))2 Hp (X − ϕ̄Y )
− (εX + εY + 2ḡ(X, Y ))2 Hp (X + Y )
− (εX + εY − 2ḡ(X, Y ))2 Hp (X − Y ) − 4Hp (X) − 4Hp (Y )
+ 24ε(ḡ(X, ϕ̄Y )2 + ḡ(X, Y )2 − εX εY )}.
Remark 2.3.6. We note that if X ∈ Γ(D) is a unit vector field we have
R̄(ξ̄α , X, ξ¯β ) = −εβ εα X,
¯
R̄(X, ξ¯α , X) = −εX εα ξ,
and these equalities are proved in the following way:
¯
¯ ∇
¯ ¯ ¯
¯
¯
R̄(ξ̄α , X, ξ¯β ) = ∇
ξ̄α X ξ̄β − ∇X ∇ξ̄α ξβ − ∇[ξ̄α ,X] ξβ
¯ (ϕ̄X) + εβ ϕ̄[ξ̄α , X]
= −εβ ∇
ξ̄α
¯ ϕ̄X ξ̄α )
= εβ (ϕ̄[ξ̄α , X] − [ξ̄α , ϕ̄X] − ∇
= εβ εα ϕ̄2 X = −εβ εα X,
39
and, if Y ∈ Γ(T M̄ ), being η̄ α (X) = 0 for any α ∈ {1, . . . , r}, then we can
compute
ḡ(R̄(X, ξ¯α , X), Y ) = −ḡ(R̄(X, Y, ξ¯α ), X)
¯ X (ϕ̄Y ) + εα ∇
¯ Y (ϕ̄X) + εα ϕ̄[X, Y ], X)
= −ḡ(−εα ∇
¯ X (ϕ̄Y ) − ∇
¯ Y (ϕ̄X) − ϕ̄[X, Y ], X)
= εα ḡ(∇
¯ X ϕ̄)Y − (∇
¯ Y ϕ̄)X, X)
= εα ḡ((∇
= εα ḡ(−η̄(Y )X + η̄(X)Y, X)
r
εα εβ η̄ β (Y )
= −εX εα η̄(Y ) = −εX
β=1
r
ξ̄β , Y ) = −εX εα ḡ(ξ̄, Y ),
= −εX εα ḡ(
β=1
then, being Y arbitrary, we have R̄(X, ξ¯α , X) = −εX εα ξ̄.
In general, if X, Y ∈ Γ(D) and Z ∈ Γ(T M̄ ) then we get
¯ Y (ϕ̄Z), X) − εα ḡ(∇
¯ Z (ϕ̄Y ), X)
ḡ(R̄(X, ξ¯α , Y ), Z) = −ḡ(R̄(Y, Z, ξ¯α ), X) = εα ḡ(∇
¯ Y ϕ̄)Z, X) − εα ḡ((∇
¯ Z ϕ̄)Y, X)
− εα ḡ(ϕ̄[Y, Z], X) = εα ḡ((∇
= −εα ḡ(Y, X)η̄(Z) = −εα ḡ(Y, X)ḡ(ξ̄, Z).
Theorem 2.3.7. The ϕ̄-sectional curvatures completely determine the sectional curvatures of an indefinite S-manifold.
Proof. We want to show that for any p ∈ M̄ and for any non degenerate 2plane π = span{X, Y } in Tp (M̄ ) the sectional curvature Kp (X, Y ) is uniquely
determined by the metric ḡ and the ϕ̄-sectional curvature. In the sequel of the
proof we suppose that p ∈ M̄ is fixed. If X, Y ∈ Dp then the thesis is given
by the previous Proposition; if X or Y is ξ̄α , for any α ∈ {1, . . . , r}, we have
already seen that Kp (X, Y ) = εα . If X, Y ∈ Tp M̄ , they can be written in the
following way:
X = X + X Y = Y + Y where X , Y ∈ Dp and X , Y ∈ D⊥
p . More precisely, we can suppose that
X = aZ,
and
X = η̄ α (X)ξ̄α ,
Y = bW
Y = η̄ β (Y )ξ̄β
where Z and W are unit vectors, that is ḡp (Z, Z) = εZ and ḡp (W, W ) = εW ,
and a and b must satisfy:
r
r
εα (η̄ α (X))2 , b2 εW = εY −
εα (η̄ α (Y ))2 .
a2 εZ = εX −
α=1
α=1
Chapter 2. Indefinite S-manifolds
40
Therefore, we compute
ḡp (R̄p (X, Y, X), Y ) = ḡp (R̄p (X , Y , X), Y ) + ḡp (R̄p (X , Y , X), Y )
(2.3.8)
+ ḡp (R̄p (X , Y , X), Y ) + ḡp (R̄p (X , Y , X), Y )
= ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
= ḡp (R̄p (X , Y , X ), Y ) + 2ḡp (R̄p (X , Y , X ), Y )
+ 2ḡp (R̄p (X , Y , X ), Y ) + 2ḡp (R̄p (X , Y , X ), Y )
+ ḡp (R̄p (X , Y , X ), Y ) + 2ḡp (R̄p (X , Y , X ), Y )
+ 2ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y )
+ 2ḡp (R̄p (X , Y , X ), Y ) + ḡp (R̄p (X , Y , X ), Y ).
Now, replacing X = aZ, Y = bW , X = η̄ α (X)ξ̄α and Y = η̄ β (Y )ξ̄β in
(2.3.8), we have
ḡp (R̄p (X, Y, X), Y ) = a2 b2 ḡp (R̄p (Z, W, Z), W )
(2.3.9)
+ 2a2 b η̄ β (Y )ḡp (R̄p (Z, W, Z), ξ¯β )
+ 2ab2 η̄ α (X)ḡp (R̄p (Z, W, ξ¯α ), W )
+ 2abη̄ α (X)η̄ β (Y )ḡp (R̄p (Z, W, ξ¯α ), ξ¯β )
+ a2 η̄ β (Y )η̄ δ (Y )ḡp (R̄p (Z, ξ¯β , Z), ξ¯δ )
+ 2abη̄ β (Y )η̄ α (X)ḡp (R̄p (Z, ξ¯β , ξ̄α ), W )
+ 2aη̄ β (Y )η̄ α (X)η̄ δ (Y )ḡp (R̄p (Z, ξ¯β , ξ̄α ), ξ¯δ )
+ b2 η̄ α (X)η̄ γ (X)ḡp (R̄p (ξ̄α , W, ξ¯γ ), W )
+ 2bη̄ α (X)η̄ β (Y )η̄ γ (X)ḡp (R̄p (ξ̄α , Z, ξ¯γ ), ξ¯β )
+ η̄ α (X)η̄ β (Y )η̄ γ (X)η̄ δ (Y )ḡp (R̄p (ξ̄α , ξ¯β , ξ¯γ ), ξ¯δ ).
Now, separately we take the terms of previous expression into account, using
41
Remark 2.3.6 and the Bianchi identity, as follows:
ḡp (R̄p (Z, W, Z), ξ¯β ) = ḡp (R̄p (Z, ξ¯β , Z), W ) = −εZ εβ ḡp (ξ̄, W ) = 0,
ḡp (R̄p (Z, W, ξ¯α ), W ) = ḡp (R̄p (ξ̄α , W, Z), W ) = ḡp (R̄p (W, ξ¯α , W ), Z)
= −εW εα ḡp (ξ̄, Z) = 0,
ḡp (R̄p (Z, W, ξ¯α ), ξ¯β ) = −ḡp (R̄p (Z, ξ¯α , ξ¯β ), W ) − ḡp (R̄p (Z, ξ¯β ), ξ¯α ), W )
= ḡp (R̄p (ξ̄α , Z, ξ¯β ), W ) + εβ ḡp (Z, W )ḡp (ξ̄), ξ¯α )
= −εβ εα ḡp (Z, W ) + εβ εα ḡp (Z, W ) = 0,
¯
ḡp (R̄p (Z, ξβ , ξ̄α ), W ) = −ḡp (R̄p (Z, ξ¯β , W )ξ̄α ) = εβ ḡp (Z, W )ḡp (ξ̄), ξ¯α )
= εβ εα ḡp (Z, W ),
ḡp (R̄p (Z, ξ¯β , ξ̄α ), ξ¯δ ) = −ḡp (R̄p (ξ̄β , Z, ξ¯α ), ξ¯δ ) = εβ εα ḡp (Z, ξ¯δ ) = 0,
ḡp (R̄p (ξ̄α , W, ξ¯γ ), ξ¯β ) = εγ εα ḡp (Z, ξ¯β ) = 0.
Therefore, replacing the previous expressions in (2.3.9), we have:
ḡp (R̄p (X, Y, X), Y ) = a2 b2 ḡp (R̄p (Z, W, Z), W )
2 β
(2.3.10)
− a η̄ (Y )η̄ (Y )εZ εβ ḡp (ξ̄, ξ¯δ )
δ
+ 2abη̄ β (Y )η̄ α (X)εβ εα ḡp (Z, W )
− b2 η̄ α (X)η̄ γ (X)εα εγ ḡp (W, W )
= a2 b2 ḡp (R̄p (Z, W, Z), W ) − a2 εZ η̄(Y )η̄(Y )
+ 2abη̄(Y )η̄(X)ḡp (Z, W ) − b2 εW η̄(X)η̄(X).
Hence, being Kp (X, Y ) = −εX εY ḡp (R̄p (X, Y, X), Y ), form (2.3.10) we deduce
Kp (X, Y ) = εX εY {a2 b2 ḡp (R̄p (Z, W, W ), Z) − 2abη̄(Y )η̄(X)ḡp (Z, W )
+ b2 εW η̄(X)2 + a2 εZ η̄(Y )2 }.
Now, we note that
1
ḡp (X − η̄ α (X)ξ̄α , Y − η̄ β (Y )ξ̄β )
ab
r
r
1
η̄ β (Y )ḡp (X, ξ¯β ) −
η̄ α (X)ḡp (ξ̄α , Y )
= {−
ab
α=1
ḡp (Z, W ) =
+
β=1
r
α
η̄ (X)η̄ β (Y )ḡp (ξ̄α , ξ¯β )} = −
α,β=1
1 εα η̄ α (X)η̄ α (Y ),
ab α=1
r
Chapter 2. Indefinite S-manifolds
42
and
ḡp (R̄p (Z, W, W ), Z) = [Z W − ḡp (Z, W )2 ]Kp (Z, W )
r
1
= 2 2 [a2 Z b2 W − (
εα η̄ α (X)η̄ α (Y ))2 ]Kp (Z, W )
a b
=
r
α=1
1
[(X −
εα η̄ α (X)2 )(Y −
εα η̄ α (Y )2 )
2
2
a b
α=1
α=1
−(
r
r
εα η̄ α (X)η̄ α (Y ))2 ]Kp (Z, W ).
α=1
Thus, using the previous expressions of ḡ(R̄(Z, W, W ), Z) and of ḡp (Z, W ), we
obtain
r
εα (η̄ α (X))2 )
Kp (X, Y ) = εX εY {[(εX −
α=1
r
εβ (η̄ β (Y ))2 )
(εY −
β=1
r
εα η̄ α (X)η̄ α (Y ))2 ]Kp (Z, W )
−(
α=1
r
+ 2η̄(Y )η̄(X)
εα η̄ α (X)η̄ α (Y )
α=1
r
εβ (η̄ β (Y ))2 )η̄(X)2
+ (εY −
β=1
r
+ (εX −
εα (η̄ α (X))2 )η̄(Y )2 },
α=1
and this completes the proof, since Kp (Z, W ) is given as in Proposition 2.3.5.
Lemma 2.3.8 ([39]). Let (V, g) be a semi-Euclidean vector space and R a
(0, 4)-type tensor on V such that for any X, Y, Z, W ∈ V the following conditions hold:
1) R(X, Y, Z, W ) = −R(Y, X, Z, W ),
2) R(X, Y, Z, W ) = −R(X, Y, W, Z),
3) R(X, Y, Z, W ) = R(Z, W, X, Y ),
4) SY,Z,W R(X, Y, Z, W ) = 0.
If R(X, Y, X, Y ) = 0 for any linearly indipendent and non lightlike vectors
X, Y ∈ V , then R = 0.
43
An obvious consequence of this lemma is the following result.
Lemma 2.3.9 ([39]). Let (V, g) be a semi-Euclidean vector space and R, S
(0, 4)-type tensors on V such that the conditions (1-4) of the previous Lemma
are satisfied. If R(X, Y, X, Y ) = S(X, Y, X, Y ) for any X, Y ∈ V linearly
indipendent and non lightlike vectors, then R = S.
Lemma 2.3.10. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and R, S
(0, 4)-type tensor fields on M̄ such that the following conditions hold:
1) for any X, Y, Z, W ∈ Γ(T M̄ )
R(X, Y, Z, W ) = −R(Y, X, Z, W ) and S(X, Y, Z, W ) = −S(Y, X, Z, W ),
2) for any X, Y, Z, W ∈ Γ(T M̄ )
R(X, Y, Z, W ) = −R(X, Y, W, Z) and S(X, Y, Z, W ) = −S(X, Y, W, Z),
3) for any X, Y, Z, W ∈ Γ(T M̄ )
R(X, Y, Z, W ) = R(Z, W, X, Y ) and S(X, Y, Z, W ) = S(Z, W, X, Y ),
4) for any X, Y, Z, W ∈ Γ(T M̄ )
SY,Z,W R(X, Y, Z, W ) = 0 and SY,Z,W S(X, Y, Z, W ) = 0,
5) for any X, Y, Z, W ∈ Γ(D)
R(X, Y, ϕ̄Z, W ) + R(X, Y, Z, ϕ̄W ) = εP (X, Y ; Z, W )
S(X, Y, ϕ̄Z, W ) + S(X, Y, Z, ϕ̄W ) = εP (X, Y ; Z, W )
6) for any X, Y ∈ Γ(D) and for any α, β, γ, δ ∈ {1, . . . , r}
(a) R(X, ξ¯α , X, Y ) = S(X, ξ¯α , X, Y ),
(b) R(ξ̄α , X, ξ¯β , Y ) = S(ξ̄α , X, ξ¯β , Y ),
(c) R(ξ̄α , X, ξ¯β , ξ¯γ ) = S(ξ̄α , X, ξ¯β , ξ¯γ ),
(d) R(ξ̄α , ξ̄β , ξ¯γ , ξ̄δ ) = S(ξ̄α , ξ¯β , ξ¯γ , ξ¯δ ).
Then, if R(X, ϕ̄X, X, ϕ̄X) = S(X, ϕ̄X, X, ϕ̄X) for any X ∈ Γ(D) non
lightlike vector field, R = S.
Chapter 2. Indefinite S-manifolds
44
Proof. We can prove, like in Lemma 2.3.3, that 5) implies that for any X , Y , Z , W in Γ(D)
R(ϕ̄X , ϕ̄Y , ϕ̄Z , ϕ̄W ) = R(X , Y , Z , W ),
and, using the above formula, we obtain
R(ϕ̄X , ϕ̄Y , Z , W ) = R(X , Y , ϕ̄Z , ϕ̄W ).
Analogously, for the tensor field S we have
S(ϕ̄X , ϕ̄Y , Z , W ) = S(X , Y , ϕ̄Z , ϕ̄W ).
Now, being ϕ̄p an almost complex structure on Dp for any p ∈ M̄ , we deduce,
from a well-known result analogue to the Lemma 2.3.9 ([2]), in the case of a
real vector space endowed with an almost complex structure, and finally using
the hypotheses, that for any X , Y , Z , W ∈ Γ(D)
R(X , Y , Z , W ) = S(X , Y , Z , W ).
Then, in particular, we have
R(X , Y , X , Y ) = S(X , Y , X , Y ).
Now, if X, Y ∈ Γ(T M̄ ) are linearly independent and non lightlike, then we
compute R(X, Y, X, Y ) and S(X, Y, X, Y ), writing X = X + X and Y =
Y + Y , where X = η̄ α (X)ξ̄α and Y = η̄ α (Y )ξ̄α , and likewise to (2.3.8) we
obtain:
R(X, Y, X, Y ) = R(X , Y , X , Y ) + 2R(X , Y , X , Y )
(2.3.11)
+ 2R(X , Y , X , Y ) + 2R(X , Y , X , Y )
+ R(X , Y , X , Y ) + 2R(X , Y , X , Y )
+ 2R(X , Y , X , Y ) + R(X , Y , X , Y )
+ 2R(X , Y , X , Y ) + R(X , Y , X , Y )
= R(X , Y , X , Y ) + 2η̄ β (Y )R(X , Y , X , ξ¯β )
+ 2η̄ α (X)R(X , Y , ξ¯α , Y )
+ 2η̄ α (X)η̄ β (Y )R(X , Y , ξ¯α , ξ¯β )
+ η̄ β (Y )η̄ δ (Y )R(X , ξ¯β , X , ξ¯δ )
+ 2η̄ α (X)η̄ β (Y )R(X , ξ̄β , ξ¯α , Y )
+ 2η̄ α (X)η̄ β (Y )η̄ δ (Y )R(X , ξ¯β , ξ¯α , ξ¯δ )
45
+ η̄ α (X)η̄ γ (X)R(ξ̄α , Y , ξ̄γ , Y )
+ 2η̄ α (X)η̄ β (Y )η̄ γ (X)R(ξ̄α , Y , ξ¯γ , ξ¯β )
+ η̄ α (X)η̄ β (Y )η̄ γ (X)η̄ δ (Y )R(ξ̄α , ξ¯β , ξ̄γ , ξ¯δ )
= R(X , Y , X , Y ) + 2η̄ β (Y )R(X , ξ¯β , X , Y )
+ 2η̄ α (X)R(Y , ξ¯α , Y , X )
+ 2η̄ α (X)η̄ β (Y )(R(ξ̄α , X , ξ¯β , Y ) − R(ξ̄α , Y , ξ̄β , X )
+ η̄ β (Y )η̄ δ (Y )R(ξ̄β , X , ξ¯δ , X )
− 2η̄ α (X)η̄ β (Y )R(ξ̄β , X , ξ̄α , Y )
− 2η̄ α (X)η̄ β (Y )η̄ δ (Y )R(ξ̄β , X , ξ¯α , ξ¯δ )
+ η̄ α (X)η̄ γ (X)R(ξ̄α , Y , ξ̄γ , Y )
+ 2η̄ α (X)η̄ β (Y )η̄ γ (X)R(ξ̄α , Y , ξ¯γ , ξ¯β )
+ η̄ α (X)η̄ β (Y )η̄ γ (X)η̄ δ (Y )R(ξ̄α , ξ¯β , ξ̄γ , ξ¯δ ).
Analogously, we obtain:
S(X, Y, X, Y ) = S(X , Y , X , Y ) + 2η̄ β (Y )S(X , ξ̄β , X , Y )
α
(2.3.12)
+ 2η̄ (X)S(Y , ξ̄α , Y , X )
+ 2η̄ α (X)η̄ β (Y )(S(ξ̄α , X , ξ¯β , Y ) − S(ξ̄α , Y , ξ¯β , X )
+ η̄ β (Y )η̄ δ (Y )S(ξ̄β , X , ξ¯δ , X )
− 2η̄ α (X)η̄ β (Y )S(ξ̄β , X , ξ¯α , Y )
− 2η̄ α (X)η̄ β (Y )η̄ δ (Y )S(ξ̄β , X , ξ̄α , ξ¯δ )
+ η̄ α (X)η̄ γ (X)S(ξ̄α , Y , ξ¯γ , Y )
+ 2η̄ α (X)η̄ β (Y )η̄ γ (X)S(ξ̄α , Y , ξ̄γ , ξ¯β )
+ η̄ α (X)η̄ β (Y )η̄ γ (X)η̄ δ (Y )S(ξ̄α , ξ¯β , ξ¯γ , ξ¯δ ),
therefore, using 6), (2.3.11) and (2.3.12), we give R(X, Y, X, Y ) = S(X, Y, X, Y ).
Remark 2.3.11. Using Remark 2.3.6 and Lemma 2.3.1, the Riemannian
(0,4)-type curvature tensor field R̄ satisfies the following properties:
1) for any X, Y, Z, W ∈ Γ(T M̄ )
R̄(X, Y, Z, W ) = −R̄(Y, X, Z, W ) = −R̄(X, Y, W, Z),
Chapter 2. Indefinite S-manifolds
46
2) for any X, Y, Z, W ∈ Γ(T M̄ )
R̄(X, Y, Z, W ) = R̄(Z, W, X, Y ),
3) for any X, Y, Z, W ∈ Γ(T M̄ )
SY,Z,W R̄(X, Y, Z, W ) = 0,
4) for any X, Y, Z, W ∈ Γ(D), by (2.3.6) we get
R̄(X, Y, ϕ̄Z, W ) + R̄(X, Y, Z, ϕ̄W ) = εP (X, Y ; Z, W ),
5) for any X, Y ∈ Γ(D) and for any α, β, γ, δ ∈ {1, . . . , r},
a) R̄(X, ξ¯α , X, Y ) = −ḡ(R̄(X, ξ¯α , X), Y ) = εα ḡ(X, X)ḡ(ξ̄, Y ) = 0,
b) R̄(ξ̄α , X, ξ¯β , Y ) = −ḡ(R̄(ξ̄α , X, ξ¯β ), Y ) = εα εβ ḡ(X, Y ),
c) R̄(ξ̄α , X, ξ¯β , ξ¯γ ) = −ḡ(R̄(ξ̄α , X, ξ¯β ), ξ¯γ ) = εα εβ ḡ(X, ξ¯γ ) = 0,
d) R̄(ξ̄α , ξ¯β , ξ¯γ , ξ¯δ ) = 0.
Moreover, by virtue of Lemma 2.3.10, we have that the unique (0,4)-type
tensor field on an indefinite S-manifold which satisfies the above properties
and has the same ϕ̄-sectional curvature of the Riemannian curvature tensor
field coincides the Riemannian (0,4)-type curvature tensor field R̄.
Theorem 2.3.12. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be a indefinite S-manifold. Then the
ϕ̄-sectional curvature c is pointwise constant, c ∈ F(M̄ ), if and only if the
Riemannian (0, 4)-type curvature tensor field R̄ is given by
c + 3ε
{ḡ(Y, Z)ḡ(X, W ) − ḡ(X, Z)ḡ(Y, W )
(2.3.13)
4
r
εβ η̄ β (Y )η̄ β (W )
+ ḡ(X, Z)
β=1
r
εβ η̄ β (X)η̄ β (Z)
+ ḡ(Y, W )
β=1
r
εβ η̄ β (Y )η̄ β (Z)
− ḡ(X, W )
β=1
r
εβ η̄ β (X)η̄ β (W )
− ḡ(Y, Z)
β=1
r
r
εβ η̄ β (X)η̄ β (W )
εγ η̄ γ (Y )η̄ γ (Z)
+
β=1
γ=1
r
r
εβ η̄ β (X)η̄ β (Z)
εγ η̄ γ (Y )η̄ γ (W )}
−
R̄(X, Y, Z, W ) = −
β=1
γ=1
47
c−ε
{Φ(W, X)Φ(Z, Y ) − Φ(Z, X)Φ(W, Y )
4
+ 2Φ(X, Y )Φ(W, Z)}
r
εβ εα {η̄ α (W )η̄ β (X)ḡ(ϕ̄Z, ϕ̄Y )
−
−
α,β=1
− η̄ α (W )η̄ β (Y )ḡ(ϕ̄Z, ϕ̄X) + η̄ α (Y )η̄ β (Z)ḡ(ϕ̄W, ϕ̄X)
− η̄ α (Z)η̄ β (X)ḡ(ϕ̄W, ϕ̄Y )},
or equivalently by
c + 3ε
{ḡ(ϕ̄Y, ϕ̄Z)ḡ(ϕ̄X, ϕ̄W )
(2.3.14)
4
c−ε
{Φ(W, X)Φ(Z, Y )
− ḡ(ϕ̄X, ϕ̄Z)ḡ(ϕ̄Y, ϕ̄W )} −
4
− Φ(Z, X)Φ(W, Y ) + 2Φ(X, Y )Φ(W, Z)}
R̄(X, Y, Z, W ) = −
− {η̄(W )η̄(X)ḡ(ϕ̄Z, ϕ̄Y ) − η̄(W )η̄(Y )ḡ(ϕ̄Z, ϕ̄X)
+ η̄(Y )η̄(Z)ḡ(ϕ̄W, ϕ̄X) − η̄(Z)η̄(X)ḡ(ϕ̄W, ϕ̄Y )}.
Proof. We suppose that the ϕ̄-sectional curvature c is pointwise constant and
in order to prove (2.3.14), denote by S(X, Y, Z, W ) the right-hand side of
(2.3.14). Obviously S is a tensor field of type (0,4) on M̄ , and we will
prove that S coincides with R̄. To this end it is easy to check that for any
X, Y, Z, W ∈ Γ(T M̄ ) we have the properties of skew-symmetry
−S(X, Y, W, Z) = S(X, Y, Z, W ) = −S(Y, X, Z, W )
and the Bianchi identity
SY,Z,W S(X, Y, Z, W ) = 0,
while the property 3) of Lemma 2.3.10, S(X, Y, Z, W ) = S(Z, W, X, Y ), follows
by the Bianchi identity and the skew-symmetries.
Now, we prove the following equality
S(X, Y, Z, ϕ̄W ) + S(X, Y, ϕ̄Z, W ) = εP (X, Y ; Z, W )
for any X, Y, Z, W ∈ Γ(D). We precise that in all the following calculations we
will use the orthogonality between D and ker ϕ̄ = span{ξ̄1 , . . . , ξ¯r }, without
pointing it out each time.
c + 3ε
{ḡ(ϕ̄Y, ϕ̄Z)ḡ(ϕ̄X, ϕ̄2 W )
4
− ḡ(ϕ̄X, ϕ̄Z)ḡ(ϕ̄Y, ϕ̄2 W ) + ḡ(ϕ̄Y, ϕ̄2 Z)ḡ(ϕ̄X, ϕ̄W )
S(X, Y, Z, ϕ̄W ) + S(X, Y, ϕ̄Z, W ) = −
Chapter 2. Indefinite S-manifolds
48
c−ε
{ḡ(ϕ̄W, ϕ̄X)Φ(Z, Y )
4
− Φ(Z, X)ḡ(ϕ̄W, Y ) + 2Φ(X, Y )ḡ(W, Z) + Φ(W, X)ḡ(Z, Y )
− ḡ(ϕ̄X, ϕ̄2 Z)ḡ(ϕ̄Y, ϕ̄W )} −
− ḡ(Z, X)Φ(W, Y ) − 2Φ(X, Y )ḡ(W, Z)}
c
= − {ḡ(Y, Z)Φ(X, W ) − ḡ(X, Z)Φ(Y, W ) + Φ(Y, Z)ḡ(X, W )
4
− Φ(X, Z)ḡ(Y, W ) + ḡ(W, X)Φ(Z, Y ) − Φ(Z, X)ḡ(W, Y )
+ Φ(W, X)ḡ(Z, Y ) − ḡ(Z, X)Φ(W, Y )}
ε
− {3Φ(X, W )ḡ(Z, Y ) − 3Φ(Y, W )ḡ(X, Z)
4
+ 3ḡ(X, W )Φ(Y, Z) − 3ḡ(Y, W )Φ(X, Z) + Φ(Y, Z)ḡ(W, X)
− Φ(X, Z)ḡ(W, Y ) + Φ(X, W )ḡ(Z, Y ) − Φ(Y, W )ḡ(Z, X)}
ε
= − {Φ(X, W )ḡ(Z, Y ) − Φ(X, Z)ḡ(Y, W ) − Φ(Y, W )ḡ(X, Z)
4
+ ḡ(X, W )Φ(Y, Z)} = εP (X, Y ; Z, W ).
We continue to verify (a), (b), (c) and (d) of 6) of Lemma 2.3.10:
c + 3ε
{ḡ(ϕ̄ξ¯α , ϕ̄X)ḡ(ϕ̄X, ϕ̄Y ) − ḡ(ϕ̄X, ϕ̄X)ḡ(ϕ̄ξ̄α , ϕ̄Y )}
4
c−ε
{Φ(Y, X)Φ(X, ξ¯α ) − Φ(X, X)Φ(Y, ξ¯α )
−
4
+ 2Φ(X, ξ¯α )Φ(Y, X)} − {η̄(Y )η̄(X)ḡ(ϕ̄X, ϕ̄ξ̄α )
S(X, ξ¯α , X, Y ) = −
− η̄(Y )η̄(Y )ḡ(ϕ̄X, ϕ̄X) + η̄(ξ̄α )η̄(X)ḡ(ϕ̄Y, ϕ̄X)
− η̄(X)η̄(X)ḡ(ϕ̄Y, ϕ̄ξ̄α )} = 0 = R̄(X, ξ¯α , X, Y ),
and
c + 3ε
{ḡ(ϕ̄X, ϕ̄ξ̄β )ḡ(ϕ̄ξ̄α , ϕ̄Y ) − ḡ(ϕ̄ξ̄α , ϕ̄ξ̄β )ḡ(ϕ̄X, ϕ̄Y )}
S(ξ̄α , X, ξ¯β , Y ) = −
4
c−ε
{Φ(Y, ξ¯α )Φ(ξ̄β , X) − Φ(ξ̄β , ξ¯α )Φ(Y, X)
−
4
+ 2Φ(ξ̄α , X)Φ(Y, ξ¯β )} − {η̄(Y )η̄(ξ̄α )ḡ(ϕ̄ξ̄β , ϕ̄X)
− η̄(Y )η̄(X)ḡ(ϕ̄ξ̄β , ϕ̄ξ¯α ) + η̄(X)η̄(ξ̄β )ḡ(ϕ̄Y, ϕ̄ξ̄α )
− η̄(ξ̄β )η̄(ξ̄α )ḡ(ϕ̄Y, ϕ̄X)} = εα εβ ḡ(X, Y ) = R̄(ξ̄α , X, ξ¯β , Y ),
and moreover
c + 3ε
{ḡ(ϕ̄X, ϕ̄ξ¯β )ḡ(ϕ̄ξ¯α , ϕ̄ξ¯γ ) − ḡ(ϕ̄ξ¯α , ϕ̄ξ¯β )ḡ(ϕ̄X, ϕ̄ξ¯γ )}
4
c−ε
{Φ(ξ̄γ , ξ̄α )Φ(ξ̄β , X) − Φ(ξ̄β , ξ¯α )Φ(ξ̄γ , X)
−
4
S(ξ̄α , X, ξ¯β , ξ¯γ ) = −
49
+ 2Φ(ξ̄α , X)Φ(ξ̄γ , ξ¯β )} − {η̄(ξ̄γ )η̄(ξ̄α )ḡ(ϕ̄ξ¯β , ϕ̄X)
− η̄(ξ̄γ )η̄(X)ḡ(ϕ̄ξ̄β , ϕ̄ξ¯α ) + η̄(X)η̄(ξ̄β )ḡ(ϕ̄ξ¯γ , ϕ̄ξ¯α )
− η̄(ξ̄β )η̄(ξ̄α )ḡ(ϕ̄ξ¯γ , ϕ̄X)} = 0 = R̄(ξ̄δ , X, ξ¯β , ξ¯γ ),
and finally
c + 3ε
{ḡ(ϕ̄ξ̄δ , ϕ̄ξ̄β )ḡ(ϕ̄ξ̄α , ϕ̄ξ̄γ ) − ḡ(ϕ̄ξ̄α , ϕ̄ξ¯β )ḡ(ϕ̄ξ̄δ , ϕ̄ξ¯γ )}
S(ξ̄α , ξ¯δ , ξ¯β , ξ̄γ ) = −
4
c−ε
{Φ(ξ̄γ , ξ¯α )Φ(ξ̄β , ξ¯δ ) − Φ(ξ̄β , ξ¯α )Φ(ξ̄γ , ξ̄δ )
−
4
+ 2Φ(ξ̄α , ξ¯δ )Φ(ξ̄γ , ξ¯β )} − {η̄(ξ̄γ )η̄(ξ̄α )ḡ(ϕ̄ξ̄β , ϕ̄ξ̄δ )
− η̄(ξ̄γ )η̄(ξ̄δ )ḡ(ϕ̄ξ̄β , ϕ̄ξ̄α ) + η̄(ξ̄δ )η̄(ξ̄β )ḡ(ϕ̄ξ̄γ , ϕ̄ξ̄α )
− η̄(ξ̄β )η̄(ξ̄α )ḡ(ϕ̄ξ¯γ , ϕ̄ξ̄δ )} = 0 = R̄(ξ̄δ , ξ̄δ , ξ̄β , ξ¯γ ),
We compute, for any X ∈ Γ(D) non lightlike vector field, S(X, ϕ̄X, X, ϕ̄X):
c + 3ε
{ḡ(ϕ̄2 X, ϕ̄X)ḡ(ϕ̄X, ϕ̄2 X)
(2.3.15)
4
c−ε
{Φ(ϕ̄X, X)Φ(X, ϕ̄X)
− ḡ(ϕ̄X, ϕ̄X)ḡ(ϕ̄2 X, ϕ̄2 X)} −
4
− Φ(X, X)Φ(ϕ̄X, ϕ̄X) + 2Φ(X, ϕ̄X)Φ(ϕ̄X, X)}
S(X, ϕ̄X, X, ϕ̄X) == −
− {η̄(ϕ̄X)η̄(X)ḡ(ϕ̄X, ϕ̄2 X) − η̄(ϕ̄X)η̄(ϕ̄X)ḡ(ϕ̄X, ϕ̄X)
+ η̄(ϕ̄X)η̄(X)ḡ(ϕ̄2 X, ϕ̄X) − η̄(X)η̄(X)ḡ(ϕ̄2 X, ϕ̄2 X)}
c−ε
c + 3ε
ḡ(X, X)2 −
{−ḡ(X, X)2 − 2ḡ(X, X)2 }
=
4
4
c−ε
c + 3ε
ḡ(X, X)2 + 3
ḡ(X, X)2 = cḡ(X, X)2 .
=
4
4
Moreover, we note that from the definition of ϕ̄-sectional curvature we have
c=
R̄(X, ϕ̄X, X, ϕ̄X)
,
ḡ(X, X)2
then we deduce
R̄(X, ϕ̄X, X, ϕ̄X) = cḡ(X, X)2 .
From (2.3.15) and (2.3.16) we have
R̄(X, ϕ̄X, X, ϕ̄X) = S(X, ϕ̄X, X, ϕ̄X),
(2.3.16)
Chapter 2. Indefinite S-manifolds
50
from which, using Lemma 2.3.10, the previous Remark and the properties of
tensor field S, we have
R̄(X, Y, Z, W ) = S(X, Y, Z, W ),
for any X, Y, Z, W ∈ Γ(T M̄ ), that is the formula (2.3.14).
Conversely, if we suppose that the formula (2.3.13) or equivalently (2.3.14)
holds, choosing a point p ∈ M̄ and a ϕ̄-plane π = span{X, ϕ̄X}, with X ∈ Dp
non lightlike vector, we have, for simplicity omitting the point p:
c + 3ε
R̄(X, ϕ̄X, X, ϕ̄X)
=−
{ḡ(ϕ̄2 X, ϕ̄X)ḡ(ϕ̄X, ϕ̄2 X)
2
ḡ(X, X)
4ḡ(X, X)2
c−ε
{Φ(ϕ̄X, X)Φ(X, ϕ̄X)
− ḡ(ϕ̄X, ϕ̄X)ḡ(ϕ̄2 X, ϕ̄2 X)} −
4ḡ(X, X)2
− Φ(X, X)Φ(ϕ̄X, ϕ̄X) + 2Φ(X, ϕ̄X)Φ(ϕ̄X, X)}
1
−
{η̄(ϕ̄X)η̄(X)ḡ(ϕ̄X, ϕ̄2 X) − η̄(ϕ̄X)η̄(ϕ̄X)ḡ(ϕ̄X, ϕ̄X)
ḡ(X, X)2
+ η̄(ϕ̄X)η̄(X)ḡ(ϕ̄2 X, ϕ̄X) − η̄(X)η̄(X)ḡ(ϕ̄2 X, ϕ̄2 X)}
c−ε
c + 3ε
ḡ(X, X)2 −
{−ḡ(X, X)2 − 2ḡ(X, X)2 }
=
2
4ḡ(X, X)
4ḡ(X, X)2
c−ε
c + 3ε
ḡ(X, X)2 + 3
ḡ(X, X)2 = c.
=
2
4ḡ(X, X)
4ḡ(X, X)2
H(X) =
2.4
Sectional Curvature in the case ε = 0
In this paragraph we consider the case ε = 0, as already pointed out, r = 2p
and ξ¯1 , . . . , ξ¯p are timelike vector field, ξ̄p+1 , . . . , ξ¯2p are spacelike vector field.
We call such a manifold a special indefinite S-manifold. The previous
results specialize as follows
Lemma 2.4.1. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be a special indefinite S-manifold, then
for any X, Y, Z, W ∈ Γ(T M̄ )
ḡ(R̄(X, Y, ϕ̄Z), W ) + ḡ(R̄(X, Y, Z), ϕ̄W ) = −Q(X, Y ; Z, W )
where
Q(X, Y ; Z, W ) = −ḡ(W, ϕ̄Y )η̄(Z)η̄(X) + ḡ(W, ϕ̄X)η̄(Z)η̄(Y )
+ ḡ(Z, ϕ̄Y )η̄(X)η̄(W ) − ḡ(Z, ϕ̄X)η̄(Y )η̄(W ).
(2.4.1)
51
Moreover, being Q(X, Y ; Z, W ) = 0 for any X, Y, Z, W ∈ D, we have
1) ḡ(R̄(ϕ̄X, ϕ̄Y, ϕ̄Z), ϕ̄W ) = ḡ(R̄(X, Y, Z), W );
2) ḡ(R̄(X, ϕ̄X, Y ), ϕ̄Y ) = ḡ(R̄(X, Y, X), Y ) + ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y );
3) ḡ(R̄(ϕ̄X, Y, ϕ̄X), Y ) = ḡ(R̄(X, ϕ̄Y, X), ϕ̄Y ).
Lemma 2.4.2. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be a special indefinite S-manifold. Then,
for any X, Y ∈ Γ(D) we have
B(X, Y ) =
1
{3D(X + ϕ̄Y ) + 3D(X − ϕ̄Y ) − D(X + Y )
32
− D(X − Y ) − 4D(X) − 4D(Y )}.
(2.4.2)
Proposition 2.4.3. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be a special indefinite S-manifold
and p in M̄ . We consider a non degenerate 2-plane π = span{X, Y } of Dp ,
where X and Y are unit vectors of Dp . Then the sectional curvature Kp (X, Y )
is given by
Kp (X, Y ) =
1
{3(εX + εY + 2ḡ(X, ϕ̄Y ))2 Hp (X + ϕ̄Y )
32(εX εY − ḡ(X, Y )2 )
+ 3(εX + εY − 2ḡ(X, ϕ̄Y ))2 Hp (X − ϕ̄Y )
− (εX + εY + 2ḡ(X, Y ))2 Hp (X + Y )
− (εX + εY − 2ḡ(X, Y ))2 Hp (X − Y ) − 4Hp (X) − 4Hp (Y )}.
Theorem 2.4.4. The ϕ̄-sectional curvatures completely determine the sectional curvatures of a special indefinite S-manifold.
Lemma 2.4.5. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be a special indefinite S-manifold and R,
S (0, 4)-type tensor fields on M̄ such that the following conditions hold:
1) for any X, Y, Z, W ∈ Γ(T M̄ )
R(X, Y, Z, W ) = −R(Y, X, Z, W ) and S(X, Y, Z, W ) = −S(Y, X, Z, W ),
2) for any X, Y, Z, W ∈ Γ(T M̄ )
R(X, Y, Z, W ) = −R(X, Y, W, Z) and S(X, Y, Z, W ) = −S(X, Y, W, Z),
3) for any X, Y, Z, W ∈ Γ(T M̄ )
R(X, Y, Z, W ) = R(Z, W, X, Y ) and S(X, Y, Z, W ) = S(Z, W, X, Y ),
Chapter 2. Indefinite S-manifolds
52
4) for any X, Y, Z, W ∈ Γ(T M̄ )
SY,Z,W R(X, Y, Z, W ) = 0 and SY,Z,W S(X, Y, Z, W ) = 0,
5) for any X, Y, Z, W ∈ Γ(D)
R(X, Y, ϕ̄Z, W ) + R(X, Y, Z, ϕ̄W ) = 0
S(X, Y, ϕ̄Z, W ) + S(X, Y, Z, ϕ̄W ) = 0
6) for any X, Y ∈ Γ(D) and for any α, β, γ ∈ {1, . . . , r}
(a) R(X, ξ¯α , X, Y ) = S(X, ξ¯α , X, Y ),
(b) R(ξ̄α , X, ξ¯β , Y ) = S(ξ̄α , X, ξ¯β , Y ),
(c) R(ξ̄α , X, ξ¯β , ξ¯γ ) = S(ξ̄α , X, ξ¯β , ξ¯γ ),
(d) R(ξ̄α , ξ¯β , ξ¯γ , ξ¯δ ) = S(ξ̄α , ξ¯β , ξ̄γ , ξ¯δ ).
Then, if R(X, ϕ̄X, X, ϕ̄X) = S(X, ϕ̄X, X, ϕ̄X) for any X ∈ Γ(D) non
lightlike vector, then R = S.
Remark 2.4.6. Using Remark 2.3.6 and Lemma 2.3.1, the Riemannian (0,4)type curvature tensor field R̄ satisfies the following properties:
1) for any X, Y, Z, W ∈ Γ(T M̄ )
R̄(X, Y, Z, W ) = −R̄(Y, X, Z, W ) = −R̄(X, Y, W, Z),
2) for any X, Y, Z, W ∈ Γ(T M̄ )
R̄(X, Y, Z, W ) = R̄(Z, W, X, Y ),
3) for any X, Y, Z, W ∈ Γ(T M̄ )
SY,Z,W R̄(X, Y, Z, W ) = 0,
4) for any X, Y, Z, W ∈ Γ(D), we find
R̄(X, Y, ϕ̄Z, W ) + R̄(X, Y, Z, ϕ̄W ) = 0,
5) for any X, Y ∈ Γ(D) and for any α, β, γ ∈ {1, . . . , r},
a) R̄(X, ξ¯α , X, Y ) = −ḡ(R̄(X, ξ¯α , X), Y ) = εα ḡ(X, X)ḡ(ξ̄, Y ) = 0,
53
b) R̄(ξ̄α , X, ξ¯β , Y ) = −ḡ(R̄(ξ̄α , X, ξ¯β ), Y ) = εα εβ ḡ(X, Y ),
c) R̄(ξ̄α , X, ξ¯β , ξ¯γ ) = −ḡ(R̄(ξ̄α , X, ξ¯β ), ξ¯γ ) = εα εβ ḡ(X, ξ¯γ ) = 0,
d) R̄(ξ̄α , ξ̄β , ξ¯γ , ξ̄δ ) = 0.
Moreover, by virtue of Lemma 2.4.5, we have that the unique (0,4)-type tensor field on an indefinite S-manifold which satisfies the above properties and
has the same ϕ̄-sectional curvature of the Riemannian curvature tensor field
coincides the Riemannian (0,4)-type curvature tensor field R̄.
Theorem 2.4.7. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be a special indefinite S-manifold. Then
the ϕ̄-sectional curvature c is pointwise constant, c ∈ F(M̄ ), if and only if the
Riemannian (0, 4)-type curvature tensor field R̄ is given by
c
(2.4.3)
R̄(X, Y, Z, W ) = − {ḡ(Y, Z)ḡ(X, W ) − ḡ(X, Z)ḡ(Y, W )
4
r
εβ η̄ β (Y )η̄ β (W )
+ ḡ(X, Z)
β=1
r
εβ η̄ β (X)η̄ β (Z)
+ ḡ(Y, W )
β=1
r
εβ η̄ β (Y )η̄ β (Z)
− ḡ(X, W )
β=1
r
εβ η̄ β (X)η̄ β (W )
− ḡ(Y, Z)
β=1
r
r
εβ η̄ β (X)η̄ β (W )
εγ η̄ γ (Y )η̄ γ (Z)
+
β=1
γ=1
r
r
εβ η̄ β (X)η̄ β (Z)
εγ η̄ γ (Y )η̄ γ (W )
−
β=1
γ=1
+ Φ(W, X)Φ(Z, Y ) − Φ(Z, X)Φ(W, Y )
+ 2Φ(X, Y )Φ(W, Z)}
r
εβ εα {η̄ α (W )η̄ β (X)ḡ(ϕ̄Z, ϕ̄Y )
−
α,β=1
β
− η̄ (W )η̄ (Y )ḡ(ϕ̄Z, ϕ̄X) + η̄ α (Y )η̄ β (Z)ḡ(ϕ̄W, ϕ̄X)
α
− η̄ α (Z)η̄ β (X)ḡ(ϕ̄W, ϕ̄Y )},
or equivalently
c
R̄(X, Y, Z, W ) = − {ḡ(ϕ̄Y, ϕ̄Z)ḡ(ϕ̄X, ϕ̄W )
4
− ḡ(ϕ̄X, ϕ̄Z)ḡ(ϕ̄Y, ϕ̄W ) + Φ(W, X)Φ(Z, Y )
(2.4.4)
− Φ(Z, X)Φ(W, Y ) + 2Φ(X, Y )Φ(W, Z)}
− {η̄(W )η̄(X)ḡ(ϕ̄Z, ϕ̄Y ) − η̄(W )η̄(Y )ḡ(ϕ̄Z, ϕ̄X)
+ η̄(Y )η̄(Z)ḡ(ϕ̄W, ϕ̄X) − η̄(Z)η̄(X)ḡ(ϕ̄W, ϕ̄Y )}.
Chapter 3
Lightlike hypersurfaces
In this chapter we study lightlike hypersurfaces of an indefinite g.f.f -manifold
(M̄ , ϕ̄, ξ¯α , η̄ α , ḡ). Let M be a lightlike hypersurface of M̄ (that is (M, g),
where g = ḡ|M , is a hypersurface of M̄ and the metric g is degenerate). We
can consider for any p ∈ M the following vector spaces:
Tp M ⊥ = {Ep ∈ Tp M̄ | ḡp (Ep , W ) = 0 for all W ∈ Tp M },
Rad Tp M = {V ∈ Tp M | gp (V, W ) = 0 for all W ∈ Tp M } = Tp M ⊥ ∩ Tp M.
Remark 3.0.8. Note that
1) Being M a lightlike hypersurface, for any p ∈ M we have RadTp M = {0}
and dim(Tp M ⊥ ) = 1, therefore we deduce that Rad Tp M = Tp M ⊥ ⊂
Tp M . We can consider the distribution
T M ⊥ : p ∈ M → Tp M ⊥ ,
that is a 1-dimensional degenerate distribution of M.
2) From (1.4.1), for any Z in T M ⊥ , one has ḡ(ϕ̄Z, Z) = 0, therefore ϕ̄Z in
T M , i.e. if we define
ϕ̄(T M ⊥ ) : p ∈ M → ϕ̄(Tp M ⊥ ),
we get a distribution on M ; being, obviously, ϕ̄(T M ⊥ ) ⊂ Im ϕ̄, on which
ϕ̄ acts like an almost complex structure, we have that
T M ⊥ = {0} ⇒ ϕ̄(T M ⊥ ) = {0};
and, ϕ̄(T M ⊥ ) is a 1-dimensional distribution on M .
56
Chapter 3. Lightlike hypersurfaces
3.1
Screen distribution
We recall some results, proved in the case of a semi-Riemannian manifold
(M̄ , ḡ) ([24]), and introduce a screen distribution:
Definition 3.1.1 ([24]). Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite g.f.f -manifold,
and M a lightlike hypersurface of M̄ . For any p ∈ M , the tangent space Tp M
decomposes in the following way
Tp M = Rad Tp M ⊥S(Tp M ),
(3.1.1)
so defining a distribution, denoted with S(T M ), such that
S(T M ) : p ∈ M → S(Tp M ) ⊂ Tp M,
which is called a screen distribution.
Remark 3.1.2. The decomposition (3.1.1) is not uniquely determined, therefore there exist several screen distributions.
1) Using Definition 1.2.2 and Proposition 1.1.2, we notice that a screen
distribution is a non-degenerate distribution on M .
2) Equation (3.1.1) can be written in the following way
Tp M = Tp M ⊥ ⊥S(Tp M ) for all
p ∈ M,
because M is a hypersurface. Moreover, we can write
T M = Rad T M ⊥S(T M ),
or rather
T M = T M ⊥ ⊥S(T M ).
(3.1.2)
From now on, we suppose that a screen distribution is chosen, and since
it is a non-degenerate distribution, we can consider the complementary vector
bundle to S(T M ) in T M̄|M with respect to ḡ, that is
T M̄ |M = S(T M )⊥S(T M )⊥ .
(3.1.3)
Theorem 3.1.3 ([24]). Let (M, g, S(T M )) be a lightlike hypersurface of an
indefinite g.f.f -manifold (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ). Then there exists a unique rank one
57
vector subbundle ltr(M ) of T M̄ , with base space M , such that for any nonzero section E of T M ⊥ on a coordinate neighbourhood U ⊂ M , there exists a
unique section N of ltr(M ) on U satisfying:
ḡ(N, E) = 1,
ḡ(N, N ) = 0,
(3.1.4)
ḡ(N, W ) = 0 for all W ∈ Γ(S(T M )|U ).
(3.1.5)
The vector bundle ltr(M ) is called lightlike transversal vector bundle of
M with respect to S(T M ).
From the above theorem we deduce the following decomposition
S(T M )⊥ = T M ⊥ ⊕ ltr(M ).
(3.1.6)
Using (3.1.2), (3.1.3) and (3.1.6), we obtain
T M̄ |M = T M ⊕ ltr(M ).
3.2
(3.1.7)
Gauss and Weingarten equations
Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite g.f.f -manifold, and M a lightlike hyper¯ the Levi-Civita connection on M̄ . Let us
surface of M̄ . We denote by ∇
suppose that S(T M ) and ltr(M ) are a screen distribution and the corresponding lightlike transversal vector bundle of M , respectively. Using (3.1.7), we
deduce
¯ X Y = ∇X Y + h(X, Y ),
(3.2.1)
∇
and
¯ X V = −AV X + ∇lt V,
∇
X
(3.2.2)
for any X, Y ∈ Γ(T M ) and V ∈ Γ(ltrM ), where
∇ : Γ(T M ) × Γ(T M ) → Γ(T M ) is a torsion-free linear connection on M ,
h : Γ(T M ) × Γ(T M ) → Γ(ltrM ) is a symmetric F(M )-bilinear form
on Γ(T M ),
AV : Γ(T M ) → Γ(T M ) is an F(M )-linear operator on Γ(T M ),
∇lt : Γ(T M ) × Γ(ltrM ) → Γ(ltrM ) is a linear connection on ltr(M ).
58
Chapter 3. Lightlike hypersurfaces
∇ and ∇lt are called the induced connections on M and ltr(M ) respectively,
and as in the classical theory of Riemannian hypersurfaces, h and AV are
called the second fundamental form and the shape operator respectively.
Further, we call (3.2.1) and (3.2.2) the Gauss and Weingarten equation,
respectively. Locally, let E, N and U be as in Theorem 3.1.3, then for any
X, Y ∈ Γ(T M |U ) we define:
B(X, Y ) = ḡ(h(X, Y ), E),
τ (X) = ḡ(∇lt
X N, E).
Then for any X, Y ∈ Γ(T M |U )
h(X, Y ) = B(X, Y )N,
(3.2.3)
∇lt
X N = τ (X)N.
(3.2.4)
Using (3.2.3) and (3.2.4), on U (3.2.1) and (3.2.2) become
¯ X Y = ∇X Y + B(X, Y )N,
∇
(3.2.5)
¯ X N = −AN X + τ (X)N,
∇
(3.2.6)
and
respectively. B is called the local second fundamental form of M , because
it determines h on U . Since the study of a lightlike hypersurface is related
to the choice of a screen distribution, in the following proposition we see the
relationships between the objects of two different screen distributions.
Proposition 3.2.1. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite g.f.f -manifold, and
M a lightlike hypersurface of M̄ ; S(T M ) and S(T M ) two screen distributions
on M and B and B the local second fundamental forms with respect to ltr(M )
and ltr(M ) , respectively. Then the local second fundamental form of M on U
is independent of the choice of screen distribution, that is B = B on U .
Proof. For any X, Y ∈ Γ(T M |U ), using the definition (3.2.3) of B and B , we
have:
¯ X Y, E) = ḡ(h (X, Y ), E) = B (X, Y ).
B(X, Y ) = ḡ(h(X, Y ), E) = ḡ(∇
Proposition 3.2.2. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite g.f.f -manifold, M
a lightlike hypersurface of M̄ and S(T M ) a fixed screen distributions on M .
Then the local second fundamental form of M is degenerate.
59
¯ is a metric connection, for any X ∈ Γ(T M |U ), considering a
Proof. Since ∇
non zero section E of T M ⊥ , we have:
¯ X ḡ)(E, E) = X(ḡ(E, E)) − ḡ(∇
¯ X E, E) − ḡ(E, ∇
¯ X E)
0 = (∇
¯ X E, E) = −2B(X, E).
= −2ḡ(∇
Hence, for any X ∈ Γ(T M |U ) we obtain
B(X, E) = 0.
The decomposition (3.1.2) allows to define a canonical projection
P : Γ(T M ) → Γ(S(T M )).
Then for any X, Y ∈ Γ(T M ) and U ∈ Γ(T M ⊥ ) we can write
∗
∗
∇X P Y = ∇X P Y + h(X, P Y ),
∗
(3.2.7)
∗
∇X U = −AU X + ∇tX U,
(3.2.8)
where
∗
∇ : Γ(T M ) × Γ(S(T M )) → Γ(S(T M )) is a linear connection on S(T M ),
∗
h : Γ(T M ) × Γ(S(T M )) → Γ(T M ⊥ ) is F(M )-bilinear,
∗
AU : Γ(T M ) → Γ(S(T M )) which, reduced to Γ(S(T M )), is an F(M )-linear
operator on Γ(S(T M )),
∗
∇t : Γ(T M ) × Γ(T M ⊥ ) → Γ(T M ⊥ ) is a linear connection on T M ⊥ .
∗
∗
We call h and AU the second fundamental form and the shape operator
of the screen distribution S(T M ), respectively, and (3.2.7) and (3.2.8) the
Gauss equation and Weingarten equation. Using (3.2.1), (3.2.2), (3.2.7)
and (3.2.8), we find a relation between geometrical objects of the lightlike
hypersurface and of the screen distribution:
1) for any X, Y ∈ Γ(T M ) and V ∈ Γ(ltr(M ))
∗
g(AV X, P Y ) = ḡ(V, h(X, P Y )),
(3.2.9)
60
Chapter 3. Lightlike hypersurfaces
2) for any X ∈ Γ(T M ) and V ∈ Γ(ltr(M ))
ḡ(AV X, V ) = 0,
(3.2.10)
3) for any X, Y ∈ Γ(T M ), U ∈ Γ(T M ⊥ )
∗
g(AU X, P Y ) = ḡ(U, h(X, P Y )),
(3.2.11)
4) for any X ∈ Γ(T M ), U ∈ Γ(T M ⊥ ) and V ∈ Γ(ltr(M ))
∗
ḡ(AU X, V ) = 0.
(3.2.12)
Let us check the above formulas.
1) For any X, Y ∈ Γ(T M ) and V ∈ Γ(ltr(M ))
∗
∗
ḡ(V, h(X, P Y )) = ḡ(V, ∇X P Y − ∇X P Y ) = ḡ(V, ∇X P Y )
¯ X P Y − h(X, P Y )) = ḡ(V, ∇
¯ XPY )
= ḡ(V, ∇
¯ X V, P Y )
¯ X V, P Y ) = −ḡ(∇
= X(ḡ(V, P Y )) − ḡ(∇
= −ḡ(−AV X + ∇lt
X V, P Y ) = ḡ(AV X, P Y )
= g(AV X, P Y ).
2) For any X ∈ Γ(T M ) and V ∈ Γ(ltr(M ))
¯
¯
ḡ(AV X, V ) = ḡ(∇lt
X V − ∇X V, V ) = −ḡ(∇X V, V ) = 0
¯ a metric connection, we have
because, being ∇
¯ X ḡ)(V, V )
0 = (∇
¯ XV )
¯ X V, V ) − ḡ(V, ∇
= X(ḡ(V, V )) − ḡ(∇
¯ X V, V ),
= −2ḡ(∇
3) For any X, Y ∈ Γ(T M ) and U ∈ Γ(T M ⊥ )
¯ XPY )
¯ X P Y − ∇X P Y ) = ḡ(U, ∇
ḡ(U, h(X, P Y )) = ḡ(U, ∇
¯ XU)
= X(ḡ(U, P Y )) − ḡ(P Y, ∇
¯ X U ) = −ḡ(P Y, ∇X U + h(X, U ))
= −ḡ(P Y, ∇
∗
∗
= −ḡ(P Y, ∇X U ) = −ḡ(P Y, −AU X + ∇tX U )
∗
∗
= ḡ(P Y, AU X) = g(P Y, AU X).
61
4) Equation (3.2.12) follows since for any X ∈ Γ(T M ), U ∈ Γ(T M ⊥ ) and
∗
V ∈ Γ(ltr(M )) AU X ∈ Γ(S(T M )).
Locally, let U be a coordinate neighbourhood of M , and E, N a pair of sections
on U , as in Theorem 3.1.3; then for any X, Y ∈ Γ(T M |U ) we define:
∗
C(X, P Y ) = ḡ(h(X, P Y ), N ),
∗
e(X) = ḡ(∇tX E, N ).
(3.2.13)
We deduce for any X, Y ∈ Γ(T M |U )
∗
h(X, P Y ) = C(X, P Y )E,
(3.2.14)
∗
∇tX E = e(X)E.
(3.2.15)
Using (3.2.13), (3.2.8), (3.1.5), (3.2.5), (3.1.4) and (3.2.6), we find
∗
∗
e(X) = ḡ(∇tX E, N ) = ḡ(∇X E + AE X, N ) = ḡ(∇X E, N )
¯ X E, N )
¯ X E − B(X, E)N, N ) = ḡ(∇
= ḡ(∇
(3.2.16)
¯ XN)
¯ X N ) = −ḡ(E, ∇
= X(ḡ(E, N )) − ḡ(E, ∇
= −ḡ(E, −AN X + τ (X)N ) = −τ (X)
Equations (3.2.7) and (3.2.8), using (3.2.14), (3.2.15) and (3.2.16), locally on
U become
∗
∇X P Y = ∇X P Y + C(X, P Y )E,
and
∗
∇X E = −AE X − τ (X)E,
(3.2.17)
respectively. Finally, (3.2.9), (3.2.10), (3.2.11) and (3.2.12) can be rewritten
as:
1) for any X, Y ∈ Γ(T M )
g(AN X, P Y ) = C(X, P Y ),
2) for any X ∈ Γ(T M )
ḡ(AN X, N ) = 0,
(3.2.18)
62
Chapter 3. Lightlike hypersurfaces
3) for any X, Y ∈ Γ(T M )
∗
g(AE X, P Y ) = B(X, P Y ),
4) for any X ∈ Γ(T M )
(3.2.19)
∗
ḡ(AE X, N ) = 0,
respectively.
Remark 3.2.3. We note that:
∗
1) AE E = 0.
Using (3.2.19) and Proposition 3.2.2, we have
∗
g(AE E, P X) = B(E, P X) = 0
∗
for any X ∈ T M . Being AE E and P X element of S(T M ), S(T M )
non-degenerate and P X arbitrary in S(T M ), we deduce
∗
AE E = 0.
¯ E E = ∇E E = −τ (E)E.
2) ∇
Using (3.2.5) and Proposition 3.2.2, we have
¯ E E = ∇E E + B(E, E)N = ∇E E.
∇
Using (3.2.17), we go on with calculation:
∗
¯ E E = ∇E E = −AE E − τ (E)E = −τ (E)E.
∇
3) The induced connection ∇ on M is independent of the choice of S(T M )
if and only if the second fundamental form h of M vanishes identically.
4) The induced connection ∇ on M is not a metric connection, in fact it
satisfies
(∇X g)(Y, Z) = B(X, Y )θ(Z) + B(X, Z)θ(Y ),
for any X, Y, Z ∈ Γ(T M |U ), where θ(X) = ḡ(X, N ).
5) From the previous formula, if we choose Y, Z ∈ Γ(S(T M )), we get
(∇X g)(Y, Z) = 0, and using (3.2.7) we easily obtain that the linear
∗
connection ∇ on S(T M ) is a metric connection.
63
3.3
Choice of screen distribution
Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite g.f.f -manifold and M a ligthlike hypersurface. Resuming all the previous decompositions and notations and supposing that all ξ¯α are tangent to M , we fix a particular screen distribution as
follows. We would like to point out that this hypothesis is justified by the
example 2 in the Chapter 4.
Proposition 3.3.1. Let (M, g) be a lightlike hypersurface of an indefinite
g.f.f -manifold (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) such that the ξ̄α are tangent to M and E a
non zero section of T M ⊥ . Then there exists a screen distribution S(T M )
such that, for any α ∈ {1, . . . , r}, ξ̄α and ϕ̄(E) belong to Γ(S(T M )).
Proof. Now, we prove that E, ϕ̄E, ξ¯1 , . . . , ξ¯r are linearly indipendent. First
step, ϕ̄E and E are linearly indipendent. In fact, if ϕ̄E = λE, then ϕ̄2 E =
λ2 E. On the other hand, ϕ̄2 E = −E + η̄ α (E)ξ̄α = −E, therefore we have
λ2 = −1 which is a contradiction. Second step, we check that E, ϕ̄E, ξ¯1 , . . . , ξ¯r
are linearly indipendent. To this end, we consider
λE + λ ϕ̄E + λα ξ̄α = 0,
(3.3.1)
where λα ∈ R for any α ∈ {1, . . . , r} and λ, λ ∈ R, and for any β ∈ {1, . . . , r}
we get
0 = ḡ(ξ̄β , λE + λ ϕ̄E + λα ξ¯α ) = λα ḡ(ξ̄β , ξ̄α ) = λα εβ δβα = εβ λβ ,
then, using (3.3.1) and the first step, λ = λ = 0.
Third step, we consider E, ϕ̄E, ξ¯1 , . . . , ξ¯r and choose v1 , . . . , v2n−3 such
that (E, ϕ̄E, ξ¯1 , . . . , ξ¯r , v1 , . . . , v2n−3 ) is a local basis of T M . Now, we define
the screen distribution S(T M ), as spanned by ϕ̄E, ξ¯1 , . . . , ξ¯r , v1 , . . . , v2n−3 .
S(T M ) is not degenerate. Namely, if v ∈ S(Tp M ), for any p ∈ M , such that
ḡ(v, u) = 0,
for any u ∈ S(Tp M ), then, v being orthogonal to all vector of Tp M , we have
that v ∈ Rad(Tp M ), from which we obtain that v = 0.
Remark 3.3.2. Now, we want to fix all elements of a lightlike hypersurface
(M, g). To this aim, let E be a fixed non zero section of T M ⊥ , then we
consider the screen distribution S(T M ) such that ξ¯α , ϕ̄E ∈ Γ(S(T M )) and,
using Theorem 3.1.3, we fix ltr(M ) and N . In the sequel, we will denote a
lightlike hypersurface by (M, g, S(T M )) and S(T M ), N and ltr(M ) will be
fixed by the choice of E and the above proposition.
64
Chapter 3. Lightlike hypersurfaces
1) Since ϕ̄E ∈ Γ(S(T M )) and N ∈ Γ(ltr(M )), we have
Remark 3.3.3.
ḡ(N, ϕ̄E) = 0.
2) If we consider the vector field ϕ̄N ∈ Γ(T M̄ |M ), then ϕ̄N ∈ Γ(S(T M )).
Using (3.1.6), we prove that ϕ̄N is orthogonal to S(T M )⊥ :
ḡ(ϕ̄N, E) = −ḡ(N, ϕ̄E) = 0,
and obviously
ḡ(ϕ̄N, N ) = 0.
3) ḡ(ϕ̄N, ϕ̄E) = 1.
Using Theorem 3.1.3 and (1.4.1), we find
ḡ(ϕ̄N, ϕ̄E) = ḡ(N, E) −
r
εα η̄ α (N )η̄ α (E) = ḡ(N, E) = 1,
α=1
because η̄ α (N ) = ḡ(N, ξ¯α ) = 0.
Proposition 3.3.4. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite g.f.f -manifold and
(M, g, S(T M )) a lightlike hypersurface of M̄ . We consider the vector bundles
T M ⊥ and ltrM and we can define the rank 2 vector subbundle ϕ̄(T M ⊥ ) ⊕
ϕ̄(ltrM ) of S(T M ). Then this vector subbundle is non-degenerate.
Proof. We prove that ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM ) is non-degenerate, showing that
for any p ∈ M , the vector space ϕ̄(Tp M ⊥ ) ⊕ ϕ̄(ltrp M ) is non-degenerate; let
p ∈ M and v ∈ ϕ̄(Tp M ⊥ ) ⊕ ϕ̄(ltrp M ) such that
∀u ∈ ϕ̄(Tp M ⊥ ) ⊕ ϕ̄(ltrp M )
gp (v, u) = 0.
(3.3.2)
We consider ϕ̄Ep that is the generator of ϕ̄(Tp M ⊥ ) and, resuming the notation
of Theorem 3.1.3, let U be a coordinate neighbourhood on M and N the unique
section of ltr(M ) on U ; then ϕ̄Np is the generator of ϕ̄(ltrp M ). Therefore,
v ∈ ϕ̄(Tp M ⊥ ) ⊕ ϕ̄(ltrp M ) can be written in the following way:
v = α ϕ̄Ep + β ϕ̄Np
where α, β ∈ R. We use (3.3.2), putting u = ϕ̄Np ,
gp (v, ϕ̄Np ) = αgp (ϕ̄Ep , ϕ̄Np ) = 0,
(3.3.3)
65
that is
α = 0,
and, putting u = ϕ̄Ep ,
gp (v, ϕ̄Ep ) = βgp (ϕ̄Np , ϕ̄Ep ) = 0,
that is
β = 0,
obtaining v = 0.
Let (M, g, S(T M )) be a lightlike hypersurface of an indefinite g.f.f -manifold
(M̄ , ϕ̄, ξ¯α , η̄ α , ḡ). Since the vector subbundle ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM ) and the distribution S(T M ) are non-degenerate, from Proposition 1.1.4 and Definition
1.2.2, we can define the unique non-degenerate distribution D0 such that
S(T M ) = (ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM ))⊥D0 .
(3.3.4)
Proposition 3.3.5. The distribution D0 satifies:
1) for any α ∈ {1, . . . , r} ξ̄α ∈ D0 .
2) D0 is ϕ̄-invariant, that is ϕ̄(D0 ) ⊂ D0 .
Proof. For any α ∈ {1, . . . , r} we get:
g(ξ̄α , ϕ̄E) = ḡ(ξ̄α , ϕ̄E) = −ḡ(ϕ̄ξ̄α , E) = 0,
g(ξ̄α , ϕ̄N ) = ḡ(ξ̄α , ϕ̄N ) = −ḡ(ϕ̄ξ¯α , N ) = 0.
Then, each ξ̄α is orthogonal to arbitrary vector fields of (ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM )),
therefore ξ̄α ∈ D0 . Now, let X ∈ D0 , we prove that ϕ̄X ∈ D0 , proving that
g(ϕ̄X, ϕ̄N ) = 0,
and
g(ϕ̄X, ϕ̄E) = 0,
being obviously ḡ(ϕ̄X, E) = −ḡ(X, ϕ̄E) = 0 and ḡ(ϕ̄X, N ) = −ḡ(X, ϕ̄N ) = 0.
In fact:
g(ϕ̄X, ϕ̄N ) = ḡ(X, N ) −
r
α=1
εα η̄ α (X)η̄ α (N ) = ḡ(X, N ) = 0,
66
Chapter 3. Lightlike hypersurfaces
where the last equality holds because X belongs to S(T M ) and N belongs to
ltr(M ). Furthermore,
g(ϕ̄X, ϕ̄E) = ḡ(X, E) −
r
εα η̄ α (X)η̄ α (E) = ḡ(X, E) = 0,
α=1
where the last equality holds because X belongs to S(T M ) and E belongs to
T M ⊥.
Using (3.1.2), (3.1.7) and (3.3.4), we can write:
T M = D0 ⊥(ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM ))⊥T M ⊥ ,
T M̄ |M = D0 ⊥(ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM ))⊥(T M ⊥ ⊕ ltr(M )).
(3.3.5)
(3.3.6)
Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite g.f.f -manifold and (M, g, S(T M )) be
a lightlike hypersurface of M̄ , we consider the distributions on M :
D := D0 ⊥ϕ̄(T M ⊥ )⊥T M ⊥ , D := ϕ̄(ltrM ).
(3.3.7)
Proposition 3.3.6. The distribution D is ϕ̄-invariant and
T M = D ⊕ D .
Proof. From (3.3.5) and the definition of D and D , we immediately deduce
that
(3.3.8)
T M = D ⊕ D .
Let us show that D is a ϕ̄-invariant distribution, i.e. for any p ∈ M and for
any X ∈ Dp we have
ϕ̄X ∈ Dp .
Since D = D0 ⊥ϕ̄(T M ⊥ )⊥T M ⊥ , there exist α, β, γ ∈ R such that
X = α X0 + β ϕ̄Ep + γ Ep
where X0 ∈ D0p . Applying ϕ̄ we obtain
ϕ̄X = α ϕ̄X0 + β ϕ̄2 Ep + γ ϕ̄Ep = α ϕ̄X0 − β Ep + γ ϕ̄Ep ∈ Dp
since D0 is ϕ̄-invariant and ϕ̄2 Ep = −Ep + εα η̄ α (Ep )ξ̄α = −Ep .
67
We consider the local lightlike vector fields
U := −ϕ̄N and V := −ϕ̄E,
we note that U ∈ D and V ∈ D. From (3.3.8) any X ∈ Γ(T M ) can be written
as
X = SX + QX and QX = u(X)U,
(3.3.9)
where S : T M → D and Q : T M → D are the canonical projection maps,
and u is a local 1-form on M defined by
u(X) := g(X, V ).
We note, for later use, that the following relations hold:
u(U ) = 1,
(3.3.10)
∀Y ∈ Γ(D) u(Y ) = 0,
(3.3.11)
ϕ̄2 N = −N.
(3.3.12)
In fact, using the point 3) of Remark 3.3.3, Proposition 3.3.6 and the property
of a g.f.f -manifold we compute:
u(U ) = g(−ϕ̄N, −ϕ̄E) = g(ϕ̄N, ϕ̄E) = 1,
u(Y ) = g(Y, −ϕ̄E) = g(ϕ̄Y, E) = 0,
ϕ̄2 N = −N + η̄ α (N )ξ̄α = −N.
Applying ϕ̄ to (3.3.9) and using (3.3.12), we obtain
ϕ̄X = ϕ̄(SX) + u(X)ϕ̄U = ϕ̄(SX) − u(X)ϕ̄2 N = ϕ̄(SX) + u(X)N.
For any X ∈ Γ(T M ) we put
ϕX := ϕ̄(SX),
obtaining a tensor field ϕ of type (1, 1) on M ; from the above equality we get
then
ϕ̄X = ϕX + u(X)N
(3.3.13)
and applying again ϕ̄ to (3.3.13), we have
ϕ̄2 X = ϕ̄ϕX + u(X)ϕ̄N
(3.3.14)
68
Chapter 3. Lightlike hypersurfaces
that is
−X + η̄ α (X)ξ̄α = ϕ̄ϕX − u(X)U.
We note that if X ∈ Γ(T M ), applying S : T M → D, SX ∈ D and, using
again Proposition 3.3.6, we obtain
ϕX = ϕ̄(SX) ∈ D,
therefore
S(ϕX) = ϕX.
We observe
ϕ̄(ϕX) = ϕ̄(SϕX) = ϕ̄S(ϕX) = ϕ2 X.
(3.3.15)
Using the equations (3.3.14) and (3.3.15), we can write
−X + η̄ α (X)ξ̄α = ϕ2 X − u(X)U
that is
ϕ2 X = −X + η̄ α (X)ξ̄α + u(X)U.
(3.3.16)
Finally, we have
1) ϕU = ϕ̄(SU ) = 0, since U ∈ D ,
2) η̄ α ◦ ϕ = η̄ α ◦ ϕ̄S = 0,
3) for any X ∈ Γ(T M )
u(ϕX) = g(ϕX, V ) = g(ϕX, −ϕ̄E) = −g(ϕ̄SX, ϕ̄E)
= −(g(SX, E) − εα η̄ α (SX)η̄ α (E)) = 0.
Using the (1, 1)-type tensor field ϕ on M , we want to introduce an f structure on M . From (3.3.8) we can locally write any X ∈ Γ(T M ) as
X = SX + u(X)U,
where u(X) = g(X, V ) which is a local 1-form on M and U := −ϕ̄N and
V := −ϕ̄E which are local lightlike vector fields on M . From (3.3.13) we have
ϕ̄X = ϕX + u(X)N,
that is for any X ∈ T M ϕX is the tangent component of ϕ̄X along M .
69
3.4
Lightlike hypersurface and the D0 distribution
for indefinite S-manifolds
Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold. As already seen in Proposition 2.2.2, for any X̄, Ȳ ∈ Γ(T M̄ )
¯ ϕ̄)Ȳ = ḡ(X̄, Ȳ )ξ̄ − η̄(Ȳ )X̄
(∇
X̄
r
εα η̄ α (X̄)η̄ α (Ȳ )ξ̄ + η̄(Ȳ )η̄ α (X̄)ξ̄α ,
−
α=1
where ξ̄ = rβ=1 ξ̄β and η̄ = rβ=1 εβ η̄ β , and let (M, g, S(T M )) be a lightlike
hypersurface of M̄ , with ker ϕ̄ ⊂ S(T M ) and ϕ̄E ∈ Γ(S(T M )).
Using the following Theorem we define a g.f.f -structure on M .
Theorem 3.4.1. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and consider (M, g, S(T M )) a lightlike hypersurface of M̄ such that E and N are
globally defined on M and ker(ϕ̄) ⊂ S(T M ) and ϕ̄E ∈ Γ(S(T M )). Then
(M, ϕ, ξ¯α , U, η α , u) is a g.f.f -manifold.
Proof. We put η α = η̄ α |M , applying ϕ̄ to ϕ̄X = ϕX + u(X)N , from (3.3.16)
we obtain
ϕ2 X = −X + η α (X)ξ̄α + u(X)U.
Moreover, we find
u(U ) = g(U, V ) = ḡ(ϕ̄N, ϕ̄E) = 1,
for any α ∈ {1, . . . , r}
u(ξ̄α ) = g(ξ̄α , V ) = −ḡ(ξ̄α , ϕ̄E) = 0,
η α (U ) = η̄ α (U ) = 0,
η α (ξ̄β ) = η̄ α (ξ̄β ) = δβα .
(M 2(n−1)+r+1 , ϕ, ξ¯α , U, η α , u) is a g.f.f -manifold. Note that ker ϕ = span{ξ̄α, U }.
70
Chapter 3. Lightlike hypersurfaces
We compute, for any X, Y ∈ Γ(T M ), the field (∇X ϕ)Y , using (3.3.13),
(3.2.5) and (3.2.6):
¯ X (ϕ̄Y ) − ϕ̄(∇
¯ XY )
¯ X ϕ̄)Y = ∇
(∇
¯ X (ϕY + u(Y )N ) − ϕ̄(∇X Y + B(X, Y )N )
=∇
¯ X N − ϕ̄(∇X Y ) + B(X, Y )U
¯ X (ϕY ) + X(u(Y ))N + u(Y )∇
=∇
= ∇X (ϕY ) + B(X, ϕY )N + X(u(Y ))N − u(Y )AN X
+ u(Y )τ (X)N − ϕ(∇X Y ) − u(∇X Y )N + B(X, Y )U
= (∇X ϕ)Y − u(Y )AN X + B(X, Y )U
+ (B(X, ϕY ) + (∇X u)Y + u(Y )τ (X))N.
Then we have:
(∇X ϕ)Y − u(Y )AN X + B(X, Y )U + (B(X, ϕY ) + (∇X u)Y + u(Y )τ (X))N
r
εα η̄ α (X)η̄ α (Y )ξ̄
= ḡ(X, Y )ξ̄ − η̄(Y )X −
α=1
+ η̄(Y )η̄ α (X)ξ̄α .
Now, comparing the components along T M and ltr(M ), according to the
decomposition (3.1.7), we have:
(∇X ϕ)Y − u(Y )AN X + B(X, Y )U
r
εα η̄ α (X)η̄ α (Y )ξ̄ + η̄(Y )η̄ α (X)ξ̄α ,
= ḡ(X, Y )ξ̄ − η̄(Y )X −
α=1
and
B(X, ϕY ) + (∇X u)Y + u(Y )τ (X) = 0.
So we deduce
(∇X ϕ)Y = u(Y )AN X − B(X, Y )U + ḡ(ϕ̄X, ϕ̄Y )ξ̄ + η̄(Y )ϕ̄2 X,
(3.4.1)
and
(∇X u)Y = −B(X, ϕY ) − u(Y )τ (X).
Definition 3.4.2. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite g.f.f -manifold and
(M, g, S(T M )) a lightlike hypersurface of M̄ . Then M is called totally
geodesic lightlike hypersurface if any geodesic of M with respect to the
¯
induced connection ∇ is a geodesic of M̄ with respect to ∇.
71
In the book of Bejancu and Duggal ([24]) it is proved that the previous
definition does not depend the choice of a screen distribution.
Proposition 3.4.3. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and let
(M, g, S(T M )) be a lightlike hypersurface of M̄ . Then M is totally geodesic if
and only if for any X ∈ Γ(T M ) and for any Y ∈ Γ(D)
(∇X ϕ)Y = −η̄(Y )X+g(X, Y )ξ̄−
r
εα η̄ α (X)η̄ α (Y )ξ̄+η̄(Y )η̄ α (X)ξ̄α , (3.4.2)
α=1
and
AN X = −ϕ(∇X U ) − g(X, U )ξ̄.
(3.4.3)
Proof. We suppose that M is totally geodesic, that B(X, Y ) = 0 for any
X, Y ∈ Γ(T M ); therefore, the condition (3.4.1) becomes:
(∇X ϕ)Y = u(Y )AN X + ḡ(X, Y )ξ̄ − η̄(Y )X
r
−
εα η̄ α (X)η̄ α (Y )ξ̄ + η̄(Y )η̄ α (X)ξ̄α .
(3.4.4)
α=1
From (3.4.4), using the hypothesis that Y ∈ Γ(D) and (3.3.11), we have:
(∇X ϕ)Y = ḡ(X, Y )ξ̄ − η̄(Y )X −
r
εα η̄ α (X)η̄ α (Y )ξ̄ + η̄(Y )η̄ α (X)ξ̄α .
α=1
From (3.4.4), putting Y = U , we have:
(∇X ϕ)U = u(U )AN X + ḡ(X, U )ξ̄ − η̄(U )X
r
εα η̄ α (X)η̄ α (U )ξ̄ + η̄(U )η̄ α (X)ξ̄α ,
−
α=1
and using (3.3.10), the above equation becomes
(∇X ϕ)U = AN X + ḡ(X, U )ξ̄ − η̄(U )X
r
εα η̄ α (X)η̄ α (U )ξ̄ + η̄(U )η̄ α (X)ξ̄α .
−
(3.4.5)
α=1
We note that for any α ∈ {1, . . . , r}
η̄ α (U ) = η̄ α (−ϕ̄N ) = −η̄ α ϕ̄N = 0,
(3.4.6)
72
Chapter 3. Lightlike hypersurfaces
therefore
η̄(U ) =
r
εα η̄ α (U ) = 0;
(3.4.7)
α=1
at the same time, using ϕU = 0, we have
(∇X ϕ)U = ∇X (ϕU ) − ϕ(∇X U ) = −ϕ(∇X U ).
(3.4.8)
Replacing (3.4.6), (3.4.7) and (3.4.8) in (3.4.5), we obtain
−ϕ(∇X U ) = AN X + ḡ(X, U )ξ̄,
from which we have
AN X = −ϕ(∇X U ) − ḡ(X, U )ξ̄.
Conversely, we suppose that the conditions (3.4.2) and (3.4.3) hold. We have
to prove that the local second fundamental form B vanishes. If Y ∈ Γ(T M ),
using the decomposition (3.3.8), we have that there exists α ∈ F(U ) such that
Y = Yd + αU,
therefore we obtain, for any X ∈ Γ(T M )
B(X, Y ) = B(X, Yd ) + αB(X, U ).
Using (3.4.1), (3.4.2) with Y = Yd , we find
−η̄(Yd )X + g(X, Yd )ξ̄ −
r
εα η̄ α (X)η̄ α (Yd )ξ̄ + η̄(Yd )η̄ α (X)ξ̄α
α=1
= u(Yd )AN X − B(X, Yd )U + ḡ(X, Yd )ξ̄ − η̄(Yd )X
r
εα η̄ α (X)η̄ α (Yd )ξ̄ + η̄(Yd )η̄ α (X)ξ̄α ,
−
α=1
therefore,
B(X, Yd )U = u(Yd )AN X = 0,
which implies
B(X, Yd ) = 0.
73
From (3.4.1), putting Y = U , we find
(∇X ϕ)U = u(U )AN X − B(X, U )U + ḡ(X, U )ξ̄ − η̄(U )X
r
εα η̄ α (X)η̄ α (U )ξ̄ + η̄(U )η̄ α (X)ξ̄α .
−
α=1
Then, (3.4.1), (3.4.6), (3.4.7) and (3.4.8) and replacing in the previous equation, we have
−ϕ(∇X U ) = AN X − B(X, U )U + ḡ(X, U )ξ̄.
Again by (3.4.3) we obtain
−ϕ(∇X U ) = −ϕ(∇X U ) − g(X, U )ξ̄ − B(X, U )U + ḡ(X, U )ξ̄
which implies
B(X, U ) = 0.
Now, we consider the distribution D0 , defined in (3.3.4), using (3.3.5) and
putting F := (ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM ))⊥T M ⊥ , for any X in Γ(T M ), for any Y
in Γ(D0 ) and for any W in Γ(F) we have:
◦
◦
∇X Y = ∇X Y + h(X, Y ),
(3.4.9)
◦
∇X W = −AW X + ∇F
W X.
where
◦
∇ : Γ(T M ) × Γ(D0 ) → Γ(D0 ) is a linear connection on D0 ,
◦
h : Γ(T M ) × Γ(D0 ) → Γ(F) is F(M )-bilinear,
◦
A : Γ(T M ) → Γ(D0 ) which, reduced to Γ(D0 ), is an F(M )-linear
operator on Γ(D0 ),
F
∇ : Γ(T M ) × Γ(F) → Γ(F) is a linear connection on F.
Let U ⊂ M be a coordinate neighbourhood as fixed in Theorem 3.1.3 and let
X, Y ∈ Γ(D0 |U ). Then, using (3.3.5), we put
◦
H1 (X, Y ) = g(h(X, Y ), ϕ̄N ),
74
Chapter 3. Lightlike hypersurfaces
◦
H2 (X, Y ) = g(h(X, Y ), ϕ̄E),
◦
H3 (X, Y ) = g(h(X, Y ), N ),
so (3.4.9) can be written (locally) in the following way:
◦
∇X Y = ∇X Y + H1 (X, Y )ϕ̄E + H2 (X, Y )ϕ̄N + H3 (X, Y )E.
(3.4.10)
Now, we express the Hi , i ∈ {1, 2, 3}, in terms of B and C, proving previously
the following Lemma.
Lemma 3.4.4. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and (M, g,
S(T M )) a lightlike hypersurface of M̄ . Let U ⊂ M be a coordinate neighbourhood as fixed in Theorem 3.1.3, then for any X, Y ∈ Γ(D0 )
¯ X ϕ̄)Y, E) = 0,
ḡ((∇
(3.4.11)
¯ X ϕ̄)Y, N ) = 0.
ḡ((∇
(3.4.12)
and
Proof.
Being ḡ(ξ̄α , N ) = 0 and ḡ(ξ̄α , E) = 0 for any α ∈ {1, . . . , r} and putting
ξ̄ = rβ=1 ξ̄β , we have:
ḡ(ξ̄, N ) = 0,
ḡ(ξ̄, E) = 0.
Now, we prove (3.4.11) and (3.4.12). In fact for any X, Y ∈ Γ(D0 ) we have
¯ X ϕ̄)Y, E) = ḡ(X, Y )ḡ(ξ̄, E) − η̄(Y )g(X, E)
ḡ((∇
r
εα η̄ α (X)η̄ α (Y )ḡ(ξ̄, E) + η̄(Y )η̄ α (X)ḡ(ξ̄α , E) = 0,
−
α=1
and
¯ X ϕ̄)Y, N ) = ḡ(X, Y )ḡ(ξ̄, N ) − η̄(Y )g(X, N )
ḡ((∇
r
εα η̄ α (X)η̄ α (Y )ḡ(ξ̄, N ) + η̄(Y )η̄ α (X)ḡ(ξ̄α , N ) = 0.
−
α=1
75
As a first step, let us compute H3 .
For any X, Y ∈ Γ(D0 |U ), being X, Y ∈ Γ(S(T M )|U ), we know that
∗
∇X Y = ∇X Y + C(X, Y )E,
and so from (3.4.10) we have that
◦
g(∇X Y, N ) = g(∇X Y + H1 (X, Y )ϕ̄E + H2 (X, Y )ϕ̄N
+ H3 (X, Y )E, N ) = H3 (X, Y ),
on the other hand
∗
g(∇X Y + C(X, Y )E, N ) = C(X, Y )
then
H3 (X, Y ) = C(X, Y ).
For H2 we have:
◦
g(∇X Y, ϕ̄E) = g(∇X Y + H1 (X, Y )ϕ̄E + H2 (X, Y )ϕ̄N
(3.4.13)
+ H3 (X, Y )E, ϕ̄E) = H2 (X, Y )
¯ a metric
and, using (3.4.11), (3.2.5), (3.2.17) and being D0 ϕ̄-invariant and ∇
connection, we have, on the other hand:
g(∇X Y, ϕ̄E) = ḡ(∇X Y, ϕ̄E) = −ḡ(ϕ̄(∇X Y ), E)
¯ X Y ) − B(X, Y )ϕ̄N, E)
= −ḡ(ϕ̄(∇
(3.4.14)
¯ X Y ), E) + B(X, Y )ḡ(ϕ̄N, E)
= −ḡ(ϕ̄(∇
¯ X ϕ̄)Y, E) − ḡ(∇
¯ X (ϕ̄Y ), E)
¯ X Y ), E) = ḡ((∇
= −ḡ(ϕ̄(∇
¯ X E) − X(ḡ(ϕ̄Y, E))
¯ X (ϕ̄Y ), E) = ḡ(ϕ̄Y, ∇
= −ḡ(∇
= ḡ(ϕ̄Y, ∇X E + B(X, E)N ) = ḡ(ϕ̄Y, ∇X E)
∗
∗
= ḡ(ϕ̄Y, −AE X − τ (X)E) = −ḡ(AE X, ϕ̄Y )
= −B(X, ϕ̄Y ),
where the last equality holds, using (3.2.19). Then, comparing (3.4.13) and
(3.4.14), we obtain:
H2 (X, Y ) = −B(X, ϕ̄Y ).
76
Chapter 3. Lightlike hypersurfaces
Again, for H1 we compute:
◦
g(∇X Y, ϕ̄N ) = g(∇X Y + H1 (X, Y )ϕ̄E + H2 (X, Y )ϕ̄N
(3.4.15)
+ H3 (X, Y )E, ϕ̄N ) = H1 (X, Y ),
¯ a metric conand, using (3.4.12), (3.2.5) (3.2.6), being D0 ϕ̄-invariant and ∇
nection, we get
g(∇X Y, ϕ̄N ) = ḡ(∇X Y, ϕ̄N ) = −ḡ(ϕ̄(∇X Y ), N )
¯ X Y − B(X, Y )N ), N )
= −ḡ(ϕ̄(∇
(3.4.16)
¯ X Y ), N ) + B(X, Y )ḡ(ϕ̄N, N )
= −ḡ(ϕ̄(∇
¯ X ϕ̄)Y, N ) − ḡ(∇
¯ X (ϕ̄Y ), N )
¯ X Y ), N ) = ḡ((∇
= −ḡ(ϕ̄(∇
¯ X N ) − X(ḡ(ϕ̄Y, N ))
¯ X (ϕ̄Y ), N ) = ḡ(ϕ̄Y, ∇
= −ḡ(∇
¯ X N ) = ḡ(ϕ̄Y, −AN X + τ (X)N )
= ḡ(ϕ̄Y, ∇
= −ḡ(ϕ̄Y, AN X) = −g(AN X, ϕ̄Y )
= −C(X, ϕ̄Y ).
where the last equality holds by (3.2.18). Then, comparing (3.4.15) and
(3.4.16), we obtain:
H1 (X, Y ) = −C(X, ϕ̄Y ).
Then (3.4.10) becomes
◦
∇X Y = ∇X Y − C(X, ϕ̄Y )ϕ̄E − B(X, ϕ̄Y )ϕ̄N + C(X, Y )E,
(3.4.17)
◦
and the local expression of h is
◦
h(X, Y ) = −C(X, ϕ̄Y )ϕ̄E − B(X, ϕ̄Y )ϕ̄N + C(X, Y )E.
(3.4.18)
Theorem 3.4.5. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and let
(M, g, S(T M )) be a lightlike hypersurface of M̄ , with the induced g.f.f -structure.
The distribution D0 on M is integrable if and only if for any X, Y ∈ Γ(D0 ):
C(X, Y ) = C(Y, X),
C(X, ϕ̄Y ) = C(ϕ̄X, Y ),
B(X, ϕ̄Y ) = B(ϕ̄X, Y ).
77
Proof. First of all, being ∇ a torsion-free connection, for any X, Y ∈ Γ(T M )
we find
[X, Y ] = ∇X Y − ∇Y X,
therefore, using (3.4.17), for any X, Y ∈ Γ(D0 ) we get
[X, Y ] = ∇X Y − ∇Y X
(3.4.19)
◦
= ∇X Y − C(X, ϕ̄Y )ϕ̄E − B(X, ϕ̄Y )ϕ̄N + C(X, Y )E
◦
− ∇Y X + C(Y, ϕ̄X)ϕ̄E + B(Y, ϕ̄X)ϕ̄N − C(Y, X)E
◦
◦
= ∇X Y − ∇Y X + (C(Y, ϕ̄X) − C(X, ϕ̄Y ))ϕ̄E
+ (B(Y, ϕ̄X) − B(X, ϕ̄Y ))ϕ̄N − (C(Y, X) − C(X, Y ))E.
So, let us suppose that D0 is integrable, that is, for any X, Y ∈ Γ(D0 ) we have
[X, Y ] ∈ Γ(D0 ). Then the components of [X, Y ] with respect to ϕ̄E, ϕ̄N, E
must vanish, therefore from (3.4.19) we have:
C(X, Y ) = C(Y, X),
C(X, ϕ̄Y ) = C(Y, ϕ̄X),
B(X, ϕ̄Y ) = B(Y, ϕ̄X).
Vice versa, if the three above equalities are satisfied then, using (3.4.19), for
any X, Y ∈ Γ(D0 ) we have
◦
◦
[X, Y ] = ∇X Y − ∇Y X
therefore,
[X, Y ] ∈ Γ(D0 ).
◦
Remark 3.4.6. We deduce that h is symmetric on D0 if and only if the
D0 is integrable. If D0 is integrable, looking at (3.4.18) and using the above
theorem, we have
◦
h(Y, X) = −C(Y, ϕ̄X)ϕ̄E − B(Y, ϕ̄X)ϕ̄N + C(Y, X)E
= −C(ϕ̄Y, X)ϕ̄E − B(ϕ̄Y, X)ϕ̄N + C(Y, X)E
= −C(X, ϕ̄Y )ϕ̄E − B(X, ϕ̄Y )ϕ̄N + C(X, Y )E
◦
= h(X, Y ),
78
Chapter 3. Lightlike hypersurfaces
◦
i.e. h is symmetric.
◦
Vice vera, if h is symmetric, we have
−C(Y, ϕ̄X)ϕ̄E − B(Y, ϕ̄X)ϕ̄N + C(Y, X)E = −C(X, ϕ̄Y )ϕ̄E − B(X, ϕ̄Y )ϕ̄N
+ C(X, Y )E,
and, being ϕ̄E, ϕ̄N and E linearly indipendent, we get
C(X, Y ) = C(Y, X),
C(X, ϕ̄Y ) = C(Y, ϕ̄X),
B(X, ϕ̄Y ) = B(Y, ϕ̄X).
◦
Moreover, the integrability of D0 implies that ∇ is a linear symmetric connection on the integral manifolds.
Lemma 3.4.7. If X, Y ∈ Γ(D0 ) and if we suppose that D0 is an integrable
distribution, then we have
B(ϕ̄X, ϕ̄Y ) = −B(X, Y ),
(3.4.20)
C(ϕ̄X, ϕ̄Y ) = −C(X, Y ).
(3.4.21)
Proof. Being D0 integrable, we have
B(ϕ̄X, ϕ̄Y ) = B(ϕ̄2 X, Y ),
C(ϕ̄X, ϕ̄Y ) = C(ϕ̄2 X, Y ).
Computing B(ξ̄α , Y ) and C(ξ̄α , Y ), we find
¯ Y, E) = ḡ(∇
¯ Y ξ̄α , E) = −εα ḡ(ϕ̄Y, E) = 0,
B(ξ̄α , Y ) = ḡ(∇
ξ̄α
and
¯ Y, N ) = ḡ(∇
¯ Y ξ¯α , N ) = −εα ḡ(ϕ̄Y, N ) = 0.
C(ξ̄α , Y ) = ḡ(∇ξ̄α Y, N ) = ḡ(∇
ξ̄α
Therefore, using the above relations, we obtain
B(ϕ̄X, ϕ̄Y ) = B(ϕ̄2 X, Y ) = −B(X, Y ),
and
C(ϕ̄X, ϕ̄Y ) = C(ϕ̄2 X, Y ) = −C(X, Y ).
79
Corollary 3.4.8. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and (M, g,
S(T M )) a lightlike hypersurface of M̄ , we consider the distribution D0 on M
and we suppose that D0 is integrable. Then, we have
◦
trace(h) = 0,
that is all the integral manifolds of D0 are minimal submanifolds of M with
respect to the symmetric connection ∇ on M .
Proof. We note that
rank (D0 ) = 2n + r − 4 = 2(n − 2) + r,
and we can consider the adapted frame, defined by
{Xa , ϕ̄Xa , ξ¯α | a = 1, . . . , n − 2, α = 1, . . . , r};
then we have
◦
trace(h) =
n−2
◦
◦
εa (h(Xa , Xa ) + h(ϕ̄Xa , ϕ̄Xa )) +
a=1
r
◦
εα h(ξ̄α , ξ¯α ),
(3.4.22)
α=1
where the values of εa derive from the causale type of vector fields Xa .
We compute separately the two terms:
◦
h(ξ̄α , ξ¯α ) = −C(ξ̄α , ϕ̄ξ̄α )ϕ̄E − B(ξ̄α , ϕ̄ξ̄α )ϕ̄N + C(ξ̄α , ξ̄α )E = C(ξ̄α , ξ¯α )E,
and, using (3.2.18) and (3.2.6) and (2.1.3), we obtain:
¯ N, ξ¯α )
C(ξ̄α , ξ¯α ) = g(AN ξ̄α , ξ¯α ) = ḡ(AN ξ̄α , ξ¯α ) = −ḡ(∇
ξ̄α
¯
= ḡ(N, ∇ ξ̄α ) = 0.
ξ̄α
then, we conclude
◦
h(ξ̄α , ξ̄α ) = 0.
(3.4.23)
We know that ϕ̄2 Xa = −Xa + η̄ α (Xa )ξ̄α = −Xa + εα ḡ(Xa , ξ¯α )ξ̄α = −Xa ,
80
Chapter 3. Lightlike hypersurfaces
therefore we deduce
◦
◦
h(Xa , Xa ) + h(ϕ̄Xa , ϕ̄Xa ) = −C(Xa , ϕ̄Xa )ϕ̄E − B(Xa , ϕ̄Xa )ϕ̄N
+ C(Xa , Xa )E − C(ϕ̄Xa , ϕ̄2 Xa )ϕ̄E
− B(ϕ̄Xa , ϕ̄2 Xa )ϕ̄N + C(ϕ̄Xa , ϕ̄Xa )E
= −C(Xa , ϕ̄Xa )ϕ̄E − B(Xa , ϕ̄Xa )ϕ̄N
+ C(Xa , Xa )E + C(ϕ̄Xa , Xa )ϕ̄E
+ B(ϕ̄Xa , Xa )ϕ̄N + C(ϕ̄Xa , ϕ̄Xa )E
= (C(ϕ̄Xa , Xa ) − C(Xa , ϕ̄Xa ))ϕ̄E
+ (C(Xa , Xa ) + C(ϕ̄Xa , ϕ̄Xa ))E
= (C(Xa , Xa ) + C(ϕ̄Xa , ϕ̄Xa ))E.
Furthermore, using (3.4.21), we get
C(Xa , Xa ) + C(ϕ̄Xa , ϕ̄Xa ) = 0.
So, we have
◦
◦
h(Xa , Xa ) + h(ϕ̄Xa , ϕ̄Xa ) = 0,
(3.4.24)
◦
and, replacing (3.4.23) and (3.4.24) in (3.4.22), we get trace(h) = 0.
Referring to the arguments at page 7 and Definition 1.2.4, D0 can be called
minimal distribution with respect to the connection ∇ on M , supposing D0
integrable, even if ∇ is not the Levi-Civita connection. We have the following
decomposition T M = D0 ⊥ F and the symmetric connection ∇, then we can
define, as in page 7, the unsymmetrized second fundamental form of D0 as the
(1,2)-type tensor AD0 given by
0
AD
X Y = pF (∇X0 Y0 ),
for any X, Y ∈ Γ(T M ), where X0 and Y0 are the projection of X and Y onto
D0 and pF : T M → F is the canonical projection on F. Its symmetric part
B D0 is given by
1
0
0
Y + AD
B D0 (X, Y ) = (AD
Y X)
2 X
1
= {pF (∇X0 Y0 ) + pF (∇Y0 X0 )}
2
◦
◦
◦
◦
1
= {pF (∇X0 Y0 + h(X0 , Y0 )) + pF (∇Y0 X0 + h(Y0 , X0 ))}
2
◦
= h(X0 , Y0 ),
81
for any X, Y ∈ Γ(T M ). Then, for any X, Y ∈ Γ(D0 ), we have the mean
curvature µD0 of D0
µ D0 =
=
1
2(n − 2) + r
1
2(n − 2) + r
2(n−2)+r
B D0 (Ea , Ea ) =
a=1
2(n−2)+r
◦
h(Ea , Ea )
a=1
◦
1
tr h = 0,
=
2(n − 2) + r
where {Ea | a = 1, . . . , 2(n − 2) + r} is a local frame of D0 ; so we have that
the mean curvature of D0 vanishes, in this sense D0 is said to be minimal.
We consider the distribution D0 and, using (3.3.6) in which we put
E := (ϕ̄(T M ⊥ ) ⊕ ϕ̄(ltrM )}⊥{T M ⊥ ⊕ ltr(M )),
for any X ∈ Γ(T M ), Y ∈ Γ(D0 ) and W ∈ Γ(E), we have:
XY + ¯ XY = ∇
h(X, Y ),
∇
(3.4.25)
W X + ∇EW X.
¯ X W = −A
∇
where
: Γ(T M ) × Γ(D0 ) → Γ(D0 ) is a linear connection on D0 ,
∇
h : Γ(T M ) × Γ(D0 ) → Γ(E) is F(M )-bilinear,
: Γ(T M ) → Γ(D0 ) which, reduced to Γ(D0 ), is an F(M )-linear
A
operator on Γ(D0 ),
∇E : Γ(T M ) × Γ(E) → Γ(E) is a linear connection on E.
Let U ⊂ M be a coordinate neighbourhood as fixed in Theorem 3.1.3 and
X, Y ∈ Γ(D0 |U ). Then, using (3.3.6), we put
h(X, Y ), ϕ̄N ),
F1 (X, Y ) = ḡ(
h(X, Y ), ϕ̄E),
F2 (X, Y ) = ḡ(
h(X, Y ), N ),
F3 (X, Y ) = ḡ(
82
Chapter 3. Lightlike hypersurfaces
F4 (X, Y ) = ḡ(
h(X, Y ), E),
so (3.4.25) can be written locally in the following way:
X Y + F1 (X, Y )ϕ̄E + F2 (X, Y )ϕ̄N + F3 (X, Y )E
¯ XY = ∇
∇
(3.4.26)
+ F4 (X, Y )N.
Now, we express the Fi , i ∈ {1, 2, 3, 4}, in terms of B and C. We begin to
compute F3 and F4 . For any X, Y ∈ Γ(D0 |U ), from (3.4.26), using (3.2.6),
¯ a metric connection, we have
(3.2.18) and being ∇
¯ X Y, N ) = −ḡ(Y, ∇
¯ X N ) = −ḡ(Y, −AN X + τ (X)N )
F3 (X, Y ) = ḡ(∇
= −ḡ(Y, −AN X) = g(AN X, Y ) = C(X, Y ),
and, again from (3.4.26), using (3.2.5), (3.2.17), (3.2.19), we have
¯ X Y, E) = −ḡ(Y, ∇
¯ X E) = −ḡ(Y, ∇X E + B(X, E)N )
F4 (X, Y ) = ḡ(∇
= −ḡ(Y, ∇X E) = −ḡ(Y, −AE X − τ (X)E) = g(AE X, Y )
= B(X, Y ).
For F2 we have, using (3.4.11), (3.2.5), (3.2.17), (3.2.19), and being D0 a
ϕ̄-invariant distribution:
¯ X Y, ϕ̄E) = −ḡ(ϕ̄(∇
¯ X Y ), E)
F2 (X, Y ) = ḡ(∇
¯ X (ϕ̄Y ), E)
¯ X ϕ̄)Y, E) − ḡ(∇
= ḡ((∇
¯ X E)
¯ X (ϕ̄Y ), E) = ḡ(ϕ̄Y, ∇
= −ḡ(∇
= ḡ(ϕ̄Y, ∇X E + B(X, E)N )
∗
= ḡ(ϕ̄Y, ∇X E) = ḡ(ϕ̄Y, −AE X − τ (X)E)
∗
= −g(AE X, ϕ̄Y ) = −B(X, ϕ̄Y ).
Again, for F1 we compute, using (3.4.12), (3.2.6) and (3.2.18), also by the
ϕ̄-invariance of distribution D0 :
¯ X Y, ϕ̄N ) = −ḡ(ϕ̄(∇
¯ X Y ), N )
F1 (X, Y ) = ḡ(∇
¯ X (ϕ̄Y ), N )
¯ X ϕ̄)Y, N ) − ḡ(∇
= ḡ((∇
¯ XN)
¯ X (ϕ̄Y ), N ) = ḡ(ϕ̄Y, ∇
= −ḡ(∇
= ḡ(ϕ̄Y, −AN X + τ (X)N )
= −g(AN X, ϕ̄Y ) = −C(X, ϕ̄Y ).
83
Then, (3.4.26) becomes
X Y − C(X, ϕ̄Y )ϕ̄E − B(X, ϕ̄Y )ϕ̄N
¯ XY = ∇
∇
+ C(X, Y )E + B(X, Y )N,
and the local expression of h is
h(X, Y ) = −C(X, ϕ̄Y )ϕ̄E − B(X, ϕ̄Y )ϕ̄N + C(X, Y )E + B(X, Y )N
◦
= h(X, Y ) + B(X, Y )N.
Corollary 3.4.9. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and let
(M, g, S(T M )) be a lightlike hypersurface of M̄ . Supposing D0 integrable, we
have
trace(
h) = 0,
that is all the integral manifolds of D0 are minimal submanifolds of M̄ and
from Definition 1.2.4 D0 is minimal.
Proof. We write the following expression for the trace of h:
trace(
h) =
n−2
εa (
h(Xa , Xa ) + h(ϕ̄Xa , ϕ̄Xa )) +
a=1
r
εα hεα (ξ̄α , ξ¯α ), (3.4.27)
α=1
where the values of εa derive from the causale type of vector fields Xa .
We compute separately the two terms:
◦
h(ξ̄α , ξ¯α ) = h(ξ̄α , ξ¯α ) + B(ξ̄α , ξ̄α )N.
Using (3.2.18), (3.2.17), (3.2.5) and (2.1.3), we have
∗
∗
B(ξ̄α , ξ¯α ) = g(AE ξ̄α , ξ¯α ) = ḡ(AE ξ¯α , ξ̄α )
= −ḡ(∇ E + τ (ξ̄α )E, ξ¯α )
ξ̄α
= −ḡ(∇ξ̄α E, ξ¯α )
¯ E − B(ξ̄α , E)N, ξ¯α )
= −ḡ(∇
ξ̄α
¯ ξ̄α ) = 0,
¯ E, ξ¯α ) = ḡ(E, ∇
= −ḡ(∇
ξ̄α
ξ̄α
and therefore, using (3.4.23), we find
h(ξ̄α , ξ̄α ) = 0.
(3.4.28)
84
Chapter 3. Lightlike hypersurfaces
Using (3.4.20) and (3.4.24), we get
◦
h(ϕ̄Xa , ϕ̄Xa ) = h(Xa , Xa ) + B(Xa , Xa )N
h(Xa , Xa ) + (3.4.29)
◦
+ h(ϕ̄Xa , ϕ̄Xa ) + B(ϕ̄Xa , ϕ̄Xa )N = 0.
Replacing (3.4.29) and (3.4.28) in (3.4.27), we obtain trace(
h) = 0.
Proposition 3.4.10. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite S-manifold and
let (M, g, S(T M )) be a lightlike hypersurface of M̄ . If D0 is an integrable
distribution, then the leaves of D0 have an indefinite S-structure.
Proof. Let M0 be a leaf of D0 , then for any p ∈ M0 we have Tp M0 = (D0 )p .
If X0 ∈ T M0 , we have
ϕX0 = ϕ̄SX0 = ϕ̄X0 ,
since, being S : Γ(T M ) → Γ(D) and D = D0 ⊥ϕ̄(T M ⊥ )⊥T M ⊥ , we have that
SX0 = X0 . We put
◦
ϕ := ϕ|D0 ,
and for any α ∈ {1, . . . , r}
◦α
η := η̄ α |D0
◦
ϕ defines an (1, 1)-type tensor field on M0 because D0 is ϕ̄-invariant. Now
◦
we consider (M0 , ϕ, ξ¯α , η̄ α , g) and we will check that this is an indefinite Sstructure. We know that ϕ2 X = −X + η̄ α (X)ξ̄α +u(X)U , for any X ∈ Γ(T M ),
and that u(Y ) = 0 for any Y ∈ Γ(D), so we deduce that
◦2
◦α
ϕ X0 = −X0 + η (X0 )ξ̄α ,
for any X0 ∈ Γ(T M0 ). For any α, β ∈ {1, . . . , r}
◦α
η (ξ̄β ) = δβα .
◦
◦
Then (M0 , ϕ, ξ¯α , η ) is a g.f.f -manifold. Now we want to prove the compatibility between the g.f.f -structure and the metric g on M0 . Using the definition
ϕ̄X = ϕX + u(X)N , for any X0 , Y0 ∈ Γ(T M0 ) we have
α
◦
◦
g(ϕX0 , ϕY0 ) = g(ϕX0 , ϕY0 ) = ḡ(ϕ̄X0 − u(X0 )N, ϕ̄Y0 − u(Y0 )N )
= ḡ(ϕ̄X0 , ϕ̄Y0 ) − u(Y0 )ḡ(ϕ̄X0 , N )
− u(X0 )ḡ(N, ϕ̄Y0 ) + u(X0 )u(Y0 )ḡ(N, N )
r
◦α
◦α
εα η (X0 )η (Y0 ).
= g(X0 , Y0 ) −
α=1
85
It is easy to check that, for any X0 , Y0 ∈ Γ(T M0 ) and α ∈ {1, . . . , r}, we have
◦α
dη̄ α (X0 , Y0 ) = dη (X0 , Y0 ). For any X0 , Y0 ∈ Γ(T M0 ) and α ∈ {1, . . . , r}
dη̄ α (X0 , Y0 ) = X0 (η̄ α (Y0 )) − Y0 (η̄ α (X0 )) − η̄ α [X0 , Y0 ]
◦α
◦α
(3.4.30)
◦α
= X0 (η (Y0 )) − Y0 (η (X0 )) − η [X0 , Y0 ]
◦α
= dη (X0 , Y0 ).
◦
◦
Now, using (3.4.30), we prove that (M0 , ϕ, ξ¯α , η , g) is an indefinite almost
S-manifold. In fact, for any X0 , Y0 ∈ Γ(T M0 ) and α ∈ {1, . . . , r}, we have
α
◦
◦
Φ(X0 , Y0 ) = g(X0 , ϕY0 ) = ḡ(X0 , ϕ̄Y0 )
◦α
= dη̄ α (X0 , Y0 ) = dη (X0 , Y0 ).
Furthermore, using again (3.4.30), we get
◦
◦
◦
◦α
◦ ◦
N(X0 , Y0 ) = [ϕX0 , ϕY0 ] − [X0 , Y0 ] + η [X0 , Y0 ]ξ̄α − ϕ[ϕX0 , Y0 ]
◦
◦
◦α
− ϕ[X0 , ϕY0 ] + 2dη (X0 , Y0 )ξ̄α
= [ϕ̄X0 , ϕ̄Y0 ] − [X0 , Y0 ] + η̄ α [X0 , Y0 ]ξ̄α − ϕ̄[ϕ̄X0 , Y0 ]
− ϕ̄[X0 , ϕ̄Y0 ] + 2dη̄ α (X0 , Y0 )ξ̄α = 0,
◦
◦
this, and together the above statement, means that (M0 , ϕ, ξ¯α , η , g) is an
α
◦
indefinite S-manifold. Moreover, ∇ is the Levi-Civita connection on M0 . In
fact, being D0 ⊂ S(T M ), for any X0 , Y0 , Z0 ∈ ΓT M0 we have
◦
◦
◦
(∇X0 g)(Y0 , Z0 ) = X0 (g(Y0 , Z0 )) − g(∇X0 Y0 , Z0 ) − g(Y0 , ∇X0 Z0 )
◦
= X0 (g(Y0 , Z0 )) − g(∇X0 Y0 , Z0 ) + g(h(X0 , Y0 ), Z0 )
◦
− g(Y0 , ∇X0 Z0 ) + g(Y0 , h(X0 , Z0 )) = (∇X0 g)(Y0 , Z0 )
= B(X0 , Y0 )g(Z0 , N ) + B(X0 , Z0 )g(Y0 , N ) = 0.
◦
Hence ∇ is a metric connection. By Remark 3.4.6 it is also symmetric, thus
it is the Levi-Civita connection and, from (2.2.2), we have
◦
◦
◦
◦
◦2
(∇X0 ϕ)Y0 = g(ϕX0 , ϕY0 )ξ̄ + η̄(Y0 )ϕ (X0 ).
86
Chapter 3. Lightlike hypersurfaces
Lemma 3.4.11. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and consider
¯ X ϕ̄)Y along
(M, g, S(T M )) a lightlike hypersurface. Then the component of (∇
ltr(T M ) vanishes, for any X ∈ Γ(T M ) and Y ∈ Γ(T M̄ ).
Proof. Let X ∈ Γ(T M ) and Y ∈ Γ(T M̄ ). We compute the component of
¯ X ϕ̄)Y along ltr(T M ):
(∇
¯ X ϕ̄)Y, E) = ḡ(X, Y )ḡ(ξ̄, E) − η̄(Y )ḡ(X, E)
ḡ((∇
r
εα η̄ α (X)η̄ α (Y )ḡ(ξ̄, E) + η̄(Y )η̄ α (X)ḡ(ξ̄α , E) = 0.
−
α=1
Proposition 3.4.12. Let (M̄ , ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite S-manifold and
(M, g, S(T M )) a lightlike hypersurface. Then D = D0 ⊥ϕ̄(T M ⊥ )⊥T M ⊥ is
integrable if and only if B satisfies the following conditions:
a) B(X, ϕ̄Y ) = B(ϕ̄X, Y ), for any X, Y ∈ Γ(D0 )
b) B(X, V ) = 0, for any X ∈ Γ(D0 )
c) B(V, V ) = 0,
where V = −ϕ̄E.
Proof. At first, for any X, Y ∈ Γ(D) we compute the component of [X, Y ]
along ϕ̄(ltr(T M ))
¯ Y X, E) = −ḡ(∇
¯ X (ϕ̄Y ), E)
¯ X Y, E) + ḡ(ϕ̄∇
ḡ([X, Y ], ϕ̄E) = −ḡ(ϕ̄∇
¯ X E) − ḡ(ϕ̄X, ∇
¯ Y E)
¯ Y (ϕ̄X), E) = ḡ(ϕ̄Y, ∇
+ ḡ(∇
∗
∗
= −g(ϕ̄Y, AE X) + g(ϕ̄X, AE Y ).
From the definition of D we get
X = αE + β ϕ̄E + X0 ,
Y = δE + γ ϕ̄E + Y0 .
Using the previous expression of X, Y ∈ Γ(D), being D ϕ̄-invariant and
B(E, X) = 0 for any X ∈ Γ(T M ), we have
ḡ([X, Y ], ϕ̄E) = (γα − βδ)B(V, V ) − γB(V, ϕ̄X0 ) − αB(Y0 , V )
(3.4.31)
+ βB(V, ϕ̄Y0 ) + δB(X0 , V ) + B(Y0 , ϕ̄X0 ) − B(ϕ̄Y0 , X0 ).
87
So, if we suppose that D is integrable, being ϕ̄E, E, X0 and Y0 sections of D,
then
0 = ḡ([ϕ̄E, E], ϕ̄E) = −ḡ(ϕ̄E, ϕ̄E) = −B(V, V ).
Finally, if X ∈ Γ(D0 ) we find
0 = ḡ([X, E], ϕ̄E) = B(E, ϕ̄X) − B(X, ϕ̄E) = B(X, V ),
and if X, Y ∈ Γ(D0 ) we get
0 = ḡ([X, Y ], ϕ̄E) = B(ϕ̄Y, X) − B(Y, ϕ̄X).
Vice versa, using (3.4.31) and a), b), c), it is easy to check that [X, Y ] in
Γ(D).
Now, an immediate consequence is the following proposition.
Proposition 3.4.13. Let (M̄ , ϕ̄, ξ̄α , η̄α , ḡ) be an indefinite S-manifold and let
(M, g, S(T M )) be a lightlike hypersurface. If (M, g, S(T M )) is totally geodesic,
then the following statements hold:
a) the distribution D is integrable;
b) the distribution D is parallel with respect to the induced connection ∇;
c) M is locally a product M ∗ × C, where M ∗ is a leaf of D and C is a
lightlike curve tangent to the distribution ϕ̄(ltr(T M )).
Proof. Being (M, g, S(T M )) totally geodesic, a) follows from the previous
proposition.
To prove b), we need only to check g(∇X E, ϕ̄E) = 0, g(∇X ϕ̄E, ϕ̄E) = 0 and
g(∇X Y0 , ϕ̄E) = 0 for any X ∈ Γ(T M ) and Y0 ∈ Γ(D0 ). Hence, using Lemma
3.4.11, we get
¯ X E, ϕ̄E) = −ḡ(E, ∇
¯ X ϕ̄E) = −B(X, ϕ̄E) = 0,
g(∇X E, ϕ̄E) = ḡ(∇
¯ X ϕ̄E, ϕ̄E) = −ḡ(ϕ̄∇
¯ X ϕ̄E, E) = ḡ((∇
¯ X ϕ)ϕ̄E, E)
g(∇X ϕ̄E, ϕ̄E) = ḡ(∇
¯ X ϕ̄2 E, E) = ḡ(∇X E, E) = 0,
− ḡ(∇
¯ X Y0 , ϕ̄E) = −ḡ(ϕ̄∇
¯ X Y0 , E) = ḡ((∇
¯ X ϕ̄)Y0 , E)
g(∇X Y0 , ϕ̄E) = ḡ(∇
¯ X (ϕ̄Y0 ), E) = −B(X, ϕ̄Y0 ) = 0.
¯ X (ϕ̄Y0 ), E) = −ḡ(∇
− ḡ(∇
Finally, from a) we deduce that D determines a foliation. Being ϕ̄(ltr(T M ))
a 1-dimensional distribution, it defines a foliation.
So, being T M = D ⊕ ϕ̄(ltr(M )), we obtain c).
88
Chapter 3. Lightlike hypersurfaces
3.5
Totally umbilical lightlike hypersurface and totally umbilical screen distribution
We begin to prove the following Lemma that will be useful in this paragraph.
Lemma 3.5.1. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold with a lightlike
hypersurface (M, g, S(T M )). Then, for any X, Y, Z ∈ Γ(T M ), we find the
following relations between the Riemannian (0,4)-type curvature tensor field
of M̄ and that of M
R̄(X, Y, Z, E) = −{(∇X B)(Y, Z) − (∇Y B)(X, Z) + τ (X)B(Y, Z)
− τ (Y )B(X, Z)},
and
ḡ(R̄(X, Y, Z), N ) = ḡ(R(X, Y, Z), N ).
Proof. Using only the Gauss and Weingarten equations for lightlike hypersurface, we get
¯ X ∇Y Z, E) + X(B(Y, Z))
R̄(X, Y, Z, E) = −ḡ(R̄(X, Y, Z), E) = −{ḡ(∇
¯ Y ∇X Z, E) − Y (B(X, Z))
¯ X N, E) − ḡ(∇
+ B(Y, Z)ḡ(∇
¯ Y N, E) − ḡ(∇[X,Y ] Z, E) − B([X, Y ], Z)}
+ B(X, Z)ḡ(∇
= −{B(X, ∇Y Z) + X(B(Y, Z)) + τ (X)B(Y, Z) − B(Y, ∇X Z)
− Y (B(X, Z)) − τ (Y )B(X, Z) + B(∇Y X, Z) − B(∇X Y, Z)}
= −{(∇X B)(Y, Z) − (∇Y B)(X, Z) + τ (X)B(Y, Z)
− τ (Y )B(X, Z)},
and
¯ X ∇Y Z, N )
R̄(X, Y, Z, N ) = −ḡ(R̄(X, Y, Z), N ) = −{ḡ(∇
¯ Y ∇X Z, N )
¯ X N, N ) − ḡ(∇
+ B(Y, Z)ḡ(∇
¯ Y N, N ) − ḡ(∇[X,Y ] Z, N )}
+ B(X, Z)ḡ(∇
= −{ḡ(∇X ∇Y Z, N ) − ḡ(∇Y ∇X Z, N ) − ḡ(∇[X,Y ] Z, N )}
= R(X, Y, Z, N ).
89
Definition 3.5.2. Let (M̄ , ϕ̄, ξ¯α , η̄ α , ḡ) be an indefinite S-manifold and let
(M, g, S(T M )) be a lightlike hypersurface. Then M is called totally umbilical lightlike hypersurface if, for any coordinate neighbourhood U and
X, Y ∈ Γ(T M|U )
B(X, Y ) = ρg(X, Y )
where ρ ∈ F(U ).
Theorem 3.5.3. Let (M̄ (c), ϕ̄, ξ̄α , η̄ α , ḡ) be an indefinite S-space form and
(M, g, S(T M ))
a lightlike hypersurface. If (M, g, S(T M )) is totally umbilical
then c = ε = rα=1 εα .
Proof. From (2.3.14) for the Riemaniann curvature tensor field, for any X, Y, Z
in Γ(T M ) we get
c + 3ε
{ḡ(ϕ̄Y, ϕ̄Z)ḡ(ϕ̄X, ϕ̄E)
(3.5.1)
4
c−ε
{Φ(E, X)Φ(Z, Y )
− ḡ(ϕ̄X, ϕ̄Z)ḡ(ϕ̄Y, ϕ̄E)} −
4
− Φ(Z, X)Φ(E, Y ) + 2Φ(X, Y )Φ(E, Z)}
R̄(X, Y, Z, E) = −
− {η̄(E)η̄(X)ḡ(ϕ̄Z, ϕ̄Y ) − η̄(E)η̄(Y )ḡ(ϕ̄Z, ϕ̄X)
+ η̄(Y )η̄(Z)ḡ(ϕ̄E, ϕ̄X) − η̄(Z)η̄(X)ḡ(ϕ̄E, ϕ̄Y )}
c + 3ε
{ḡ(ϕ̄Y, ϕ̄Z)ḡ(ϕ̄X, ϕ̄E)
=−
4
c−ε
{Φ(E, X)Φ(Z, Y )
− ḡ(ϕ̄X, ϕ̄Z)ḡ(ϕ̄Y, ϕ̄E)} −
4
− Φ(Z, X)Φ(E, Y ) + 2Φ(X, Y )Φ(E, Z)}
− η̄(Y )η̄(Z)ḡ(ϕ̄E, ϕ̄X) + η̄(Z)η̄(X)ḡ(ϕ̄E, ϕ̄Y ).
We note that ḡ(ϕ̄E, ϕ̄X) = 0, for any X ∈ Γ(T M ) because
ḡ(ϕ̄E, ϕ̄X) = ḡ(E, X) −
r
εα η̄ α (X)η̄ α (E) = 0.
α=1
After this remark, (3.5.1) becomes:
c−ε
{Φ(E, X)Φ(Z, Y )
R̄(X, Y, Z, E) = −
4
− Φ(Z, X)Φ(E, Y ) + 2Φ(X, Y )Φ(E, Z)}
c−ε
{ḡ(V, X)Φ(Z, Y )
=−
4
− Φ(Z, X)ḡ(V, Y ) + 2Φ(X, Y )ḡ(V, Z)}.
(3.5.2)
90
Chapter 3. Lightlike hypersurfaces
On the other hand, from Lemma 3.5.1 we deduce
R̄(X, Y, Z, E) = −{(∇X B)(Y, Z) − (∇Y B)(X, Z) + τ (X)B(Y, Z)
(3.5.3)
− τ (Y )B(X, Z)}.
So, replacing X, Y, Z by P X, E, P Z in (3.5.2) and in (3.5.3), P being the
projection on S(T M ) with respect to the decomposition (3.1.2), on the one
hand we find
R̄(X, Y, Z, E) = −
=
c−ε
{−ḡ(V, P X)ḡ(P Z, V ) − 2ḡ(X, V )ḡ(V, Z)}
4
3
(c − ε)u(P Z)u(P X),
4
and on the other hand we have
R̄(X, Y, Z, E) = −B(∇P X E, P Z) − E(B(P X, P Z))
+ B(∇E P X, P Z) + B(P X, ∇E P Z)
− τ (E)B(P X, P Z).
Then, comparing the two corresponding equations and using Gauss and Weingarten equations, we get
3
(c − ε)u(P Z)u(P X) = ρB(P X, P Z) − E(ρ)g(P X, P Z)
4
− ρ(B(E, P X)ḡ(P Z, N ) − B(E, P Z)ḡ(P X, N ))
− ρτ (E)g(P X, P Z).
Therefore, being B(X, E) = 0 for any X ∈ Γ(T M ), we still find
3
(c − ε)u(P Z)u(P X) = (ρ2 − E(ρ) − ρτ (E))ḡ(P X, P Z).
4
Choosing X = Z = U ∈ S(T M ), we have P X = P Z = U and being u(U ) = 1
and g(U, U ) = 0, from the above equation we obtain
c = ε.
Corollary 3.5.4. Let (M̄ (c), ϕ̄, ξ¯α , η̄α , ḡ) be an indefinite S-space form. If
c = ε, then there exists no totally umbilical lightlike hypersurface.
91
Remark 3.5.5. If M̄ (c) is an indefinite S-space form with c = 0, r is an
even number and the number of timelike characteristic vector fields is equal
to the number of spacelike characteristic vector fields, i.e. ε vanish. Then it
is possible that there exists a totally umbilical lightlike hypersurface.
Definition 3.5.6. Let (M̄ , ϕ̄, ξ¯α , η̄α , ḡ) be an indefinite S-manifold and let
(M, g, S(T M )) be a lightlike hypersurface. The screen distribution S(T M ) is
called totally umbilical if for any coordinate neighbourhood U and for any
X, Y ∈ Γ(T M|U )
C(X, P Y ) = λg(X, P Y ),
where λ ∈ F(U).
Proposition 3.5.7. Let (M, g, S(T M )) be a lightlike hypersurface of an indefinite S-space form. Suppose that S(T M ) is totally umbilical, then S(T M ) is
totally geodesic.
Proof. From Lemma 3.5.1 we have, for any X, Y, Z ∈ Γ(T M ),
ḡ(R̄(X, Y, Z), N ) = ḡ(R(X, Y, Z), N ).
Then, replacing Z by P Z, we have
ḡ(R̄(X, Y, P Z), N ) = ḡ(R(X, Y, P Z), N )
= ḡ(∇X ∇Y P Z, N ) − ḡ(∇Y ∇X P Z, N ) − ḡ(∇[X,Y ] P Z, N )
∗
∗
∗
= ḡ(∇X ∇Y P Z, N ) + C(X, ∇Y P Z) + X(C(Y, P Z))
∗
− C(Y, P Z)ḡ(AE X, N ) − C(Y, P Z)τ (X)
∗
∗
∗
− ḡ(∇Y ∇X P Z, N ) − C(Y, ∇Y P Z) − Y (C(X, P Z))
∗
+ C(X, P Z)ḡ(AE Y, N ) + C(X, P Z)τ (Y )
+ C(∇Y X, P Z) − C(∇X Y, P Z)
∗
= X(C(Y, P Z)) − C(Y, ∇X P Z) − C(∇X Y, P Z)
∗
− τ (X)C(Y, P Z) − Y (C(X, P Z)) + C(X, ∇Y P Z)
+ C(∇Y X, P Z) + τ (Y )C(X, P Z).
92
Chapter 3. Lightlike hypersurfaces
Now, replacing X by E and Y, Z by U in the previous equation, we obtain
∗
ḡ(R̄(E, U, U ), N ) = E(C(U, U )) − C(U, ∇E U ) − C(∇E U, U )
∗
− τ (E)C(U, U ) − U (C(E, U )) + C(E, ∇U U )
+ C(∇U E, U ) + τ (U )C(E, U )
∗
= −λ(g(U, ∇E U ) + g(∇E U, U ) + g(∇U E, U )).
Note that
∗
∇E U = ∇E U − C(E, U )E = ∇E U,
∗
∇U U = ∇U U − C(U, U )E = ∇U U.
So, we find
¯ U E)
ḡ(R̄(E, U, U ), N ) = −λ(g(U, ∇U E) + 2g(∇E U, U )) = −λ(ḡ(U, ∇
¯ U E) + E(ḡ(U, U )))
¯ E U, U )) = −λ(ḡ(U, ∇
+ 2ḡ(∇
¯ U E, N )
¯ U E, U ) = −λḡ(ϕ̄∇
= −λḡ(∇
¯ U ϕ̄)E, N )}
¯ U ϕ̄E, N ) − ḡ((∇
= −λ{ḡ(∇
¯ U N ) = λτ (U )ḡ(ϕ̄E, N ) − λḡ(ϕ̄E, AN U )
= λḡ(ϕ̄E, ∇
= −λg(ϕ̄E, AN U ) = −λC(U, ϕ̄E) = λ2 g(U, V ) = λ2 .
Using (2.3.14) for the Riemannian curvature tensor field, we have
ḡ(R̄(E, U, U ), N ) =
c + 3ε
{ḡ(ϕ̄U, ϕ̄U )ḡ(ϕ̄E, ϕ̄N )
4
c−ε
{Φ(N, E)Φ(U, U )
− ḡ(ϕ̄N, ϕ̄U )ḡ(ϕ̄U, ϕ̄E)} +
4
− Φ(U, E)Φ(N, U ) + 2Φ(E, U )Φ(N, U )}
+ {η̄(N )η̄(E)ḡ(ϕ̄U, ϕ̄U ) − η̄(N )η̄(U )ḡ(ϕ̄U, ϕ̄E)
+ η̄(U )η̄(U )ḡ(ϕ̄N, ϕ̄E) − η̄(U )η̄(E)ḡ(ϕ̄N, ϕ̄U )}
c−ε
{−Φ(N, E)ḡ(ϕ̄N, N ) − Φ(U, E)ḡ(U, U )
=
4
+ 2Φ(E, U )ḡ(U, U )} = 0.
So, we obtain λ = 0 and S(T M ) is totally geodesic.
Chapter 4
Examples of indefinite
S-manifolds
We describe some examples of indefinite S-manifolds, where the characteristic
vector fields are either timelike or spacelike or of both types. Finally, we
discuss the existence of lightlike hypersurfaces of the given examples.
4.1
A first indefinite S-manifold M1 on R62
We consider R6 with its standard coordinates {x1 , x2 , y 1 , y 2 , z 1 , z 2 }. We introduce on R6 an indefinite g.f.f -structure (ϕ, ξ1 , ξ2 , η 1 , η 2 , g) by setting
ξα =
∂
,
∂z α
η α = dz α −
2
y i dxi ,
i=1
for any α ∈ {1, 2},
g=−
2
2
ηα ⊗ ηα +
α=1
1
((dxi )2 + (dy i )2 ),
2
i=1
and ϕ given, with respect to the frame { ∂x∂ 1 , ∂x∂ 2 , ∂y∂ 1 , ∂y∂ 2 , ξ1 , ξ2 }, by the matrix

0 I2 0
F =  −I2 0 0  ,
0
Y 0

Chapter 4. Examples of indefinite S-manifolds
94
where Y is a matrix 2 × 2 given by
1 2 y y
.
Y =
y1 y2
We put M1 = (R62 , ϕ, ξ1 , ξ2 , η 1 , η 2 , g). A straightforward computation shows
that g is a metric tensor field. Firstly we check that g is non-degenerate and
then we compute its index. The matrix G of g is given by

 1
1 2
−2y 1 y 2
0 0 y1 y1
2 − 2(y )
1
2 2 0 0
 −2y 1 y 2
y2 y2 
2 − 2(y )


1

0
0
0 0
0 
2
,

G=
0 
0
0
0 12 0



y2
0 0 −1 0 
y1
y1
y2
0
0
0
−1
hence,
1 detG = 4
1
2
− 2(y 1 )2
−2y 1 y 2
y1 y1
1
1
2
2
2
−2y y
y2 y2
2 − 2(y )
y1
y2
−1 0
1
y2
0 −1
y
1
1
1
= 0.
− (y 2 )2 − (y 1 )2 ) =
2
2
16
1 1 1 2 1 2 2 1
= ( (y ) + (y ) +
4 2
2
4
Now, to determine the index of g, we compute the eigenvalues of the matrix
G
1
− 2(y 1 )2 − λ
−2y 1 y 2
y1
y1
2
1
1
2
2
2
2
2
1
−2y y
−
2(y
)
−
λ
y
y
2
2
|G − λI| = ( − λ) 1
2
y
y
−1 − λ
0
2
1
2
y
0
−1 − λ y
1
1
3
1
= ( − λ)2 ( − ((y 1 )2 + (y 2 )2 + )λ + ((y 1 )2 + (y 2 )2 − )λ2
2
4
2
4
+ (2(y 1 )2 + 2(y 2 )2 + 1)λ3 + λ4 )
1
1
1
= −( − λ)3 (1 + λ)(λ2 + (2(y 1 )2 + 2(y 2 )2 + )λ − ),
2
2
2
and we find that the index of g is two; therefore g is a semi-Riemannian metric
of the index 2 on R6 . We remark that ξ1 and ξ2 are timelike vector fields. Now,
we show that (ϕ, ξα , η α ) is a g.f.f -structure and then we verify that the metric
95
g is compatible. First of all, we check that ϕ defines an f -structure on R62 ,
that is
ϕ3 + ϕ = 0,
and rankϕ is constant. Obviously rankϕ = 4 and
∂
∂
∂
∂
) + ϕ( i ) =
− i = 0,
i
i
∂x
∂x
∂y
∂y
2
2
∂
∂
∂
∂
ϕ3 ( i ) + ϕ( i ) = − i −
y i ξα + i +
y i ξα = 0.
∂y
∂y
∂x
∂x
α=1
α=1
ϕ3 (
Such structure is a g.f.f -structure, in fact we find, for any α, β ∈ {1, 2} and
i ∈ {1, 2}
ϕ2 (
2
∂
∂
∂
∂
)=− i−
y i ξα = − i + η α ( i )ξα ,
i
∂x
∂x
∂x
∂x
α=1
∂
∂
∂
∂
) = − i = − i + η α ( i )ξα ,
i
∂y
∂y
∂y
∂y
ϕ2 (ξβ ) = 0 = −ξβ + η α (ξβ )ξα .
ϕ2 (
Finally g is a compatible metric. Namely, for any i, j ∈ {1, 2} and α, β ∈ {1, 2}
we have
∂
∂
∂
1
1
∂
g(ϕ( i ), ϕ( j )) = g( i , j ) = δij = δij − 2(y i y j ) + y i y j + y i y j
∂x
∂x
∂y ∂y
2
2
2
∂
∂
∂
∂
= g( i , j ) −
εα η α ( i )η α ( j ),
∂x ∂x
∂x
∂x
g(ϕ(
α=1
2
∂
∂
∂
∂
), ϕ( j )) = −g( i , j ) −
i
∂x
∂y
∂y ∂x
= g(
g(ϕ(
∂
∂
,
)−
∂xi ∂y j
y j g(
α=1
2
εα η α (
α=1
2
∂
, ξα ) = 0
∂y i
∂
∂
)η α ( j ),
∂xi
∂y
∂
∂
∂
∂
∂
), ϕ( j )) = g( i , j ) +
y j g( i , ξα )
i
∂y
∂y
∂x ∂x
∂x
α=1
+
2
β=1
y i g(ξβ ,
2
∂
)
+
y i y j g(ξβ , ξα )
∂xj
α,β=1
1
1
= δij − 2(y i y j ) + 2y i y j + 2y i y j − 2y i y j = δij
2
2
Chapter 4. Examples of indefinite S-manifolds
96
2
= g(
∂
∂
∂
∂
, j)−
εα η α ( i )η α ( j ),
i
∂y ∂y
∂y
∂y
α=1
and for any i ∈ {1, 2} and α ∈ {1, 2}
∂
∂
∂
, ξα ) = y i = −η α ( i ) = εα η α ( i ),
∂xi
∂x
∂x
∂
∂
g( i , ξα ) = 0 = εα η α ( i ).
∂y
∂y
g(
Now, we prove that the indefinite g.f.f -manifold M1 is an indefinite Smanifold. For any α ∈ {1, 2} and i, j ∈ {1, 2} we have
∂
∂
∂
∂
∂
∂
, j ) = g( i , ϕ( j )) = 0 = dη α ( i , j ),
i
∂x ∂x
∂x
∂x
∂x ∂x
∂
∂
∂
1
∂
∂
∂
Φ( i , j ) = g( i , ϕ( j )) = δij = dη α ( i , j ),
∂x ∂y
∂x
∂y
2
∂x ∂y
∂
∂
∂
∂
∂
∂
Φ( i , j ) = g( i , ϕ( j )) = 0 = dη α ( i , j ),
∂y ∂y
∂y
∂y
∂y ∂y
Φ(
and, moreover, we observe
Φ(X, ξα ) = 0 = dη β (X, ξα ),
for any X ∈ Γ(T R62 ) and for any α, β ∈ {1, 2}.
It remains to show that this structure is normal. For any i, j ∈ {1, 2} we
have
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
N ( i , j ) = [ϕ i , ϕ j ] + ϕ2 [ i , j ] − ϕ[ϕ i , j ] − ϕ[ i , ϕ j ]
∂x ∂x
∂x
∂x
∂x ∂x
∂x ∂x
∂x
∂x
∂
∂
∂
∂
∂
∂
+ 2dη α ( i , j )ξα = ϕ[ i , j ] + ϕ[ i , j ] = 0,
∂x ∂x
∂y ∂x
∂x ∂y
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
N ( i , j ) = [ϕ i , ϕ j ] + ϕ2 [ i , j ] − ϕ[ϕ i , j ] − ϕ[ i , ϕ j ]
∂x ∂y
∂x
∂y
∂x ∂y
∂x ∂y
∂x
∂y
∂
∂
+ 2dη α ( i , j )ξα
∂x ∂y
2
2
2
∂
∂
∂
∂
j
j
= −[ i , j + y
ξα ] − ϕ[ i , j + y
ξα ] + δij
ξα
∂y ∂x
∂x ∂x
α=1
α=1
α=1
= −δij
2
α=1
ξα + δij
2
α=1
ξα = 0,
97
N(
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
, j ) = [ϕ i , ϕ j ] + ϕ2 [ i , j ] − ϕ[ϕ i , j ] − ϕ[ i , ϕ j ]
i
∂y ∂y
∂y
∂y
∂y ∂y
∂y ∂y
∂y
∂y
∂
∂
+ 2dη α ( i , j )ξα
∂y ∂y
2
2
2
∂
∂
∂
∂
i
j
j
= [ i +y
ξα , j + y
ξα ] − ϕ[ i , j + y
ξα ]
∂x
∂x
∂y ∂x
α=1
α=1
α=1
− ϕ[
2
2
2
∂
∂
i
+
y
ξ
,
]
=
−ϕ(δ
ξ
)
+
ϕ(δ
ξα ) = 0,
α
ij
α
ij
∂xi
∂y j
α=1
α=1
α=1
and, moreover, for any i ∈ {1, 2} and α, β ∈ {1, 2}
N(
4.2
∂
, ξα ) = 0,
∂xi
N(
∂
, ξα ) = 0,
∂y i
N (ξβ , ξα ) = 0.
A second indefinite S-manifold M2 on R62
The second example of an indefinite S-manifold is M2 = (R62 , ϕ, ξα , η α , g),
where ϕ and g are given by

0 I2 0
F =  −I2 0 0  ,
0
Y 0

where Y is a matrix 2 × 2 given by
−y 1 y 2
,
Y =
−y 1 y 2
and
g=
2
α=1
ηα ⊗ ηα +
1 2
τi ((dxi )2 + (dy i )2 ),
i=1
2
respectively, where τi = ∓1 according to whether i = 1 or i = 2. Moreover,
for any α ∈ {1, 2} we put
ξα :=
∂
,
∂z α
η α := dz α −
2
i=1
τi y i dxi .
Chapter 4. Examples of indefinite S-manifolds
98
The symmetric (0, 2)-type tensor field g is a semi-Riemannian metric because
1
− + 2(y 1 )2
−2y 1 y 2
y1
y 1 2
1
2 2 −y 2 −y 2 1
−2y 1 y 2
2 + 2(y )
= − 1 (− 1 (y 1 )2 + 1 (y 2 )2
detG = − 1
2
y
−y
1
0 4
4 2
2
y1
−y 2
0
1 1 1
1
1
= 0,
− − (y 2 )2 + (y 1 )2 ) =
4 2
2
16
therefore g is non degenerate, and
1
− + 2(y 1 )2 − λ
−2y 1 y 2
y1
y 1 2
1
2 2
1
−2y 1 y 2
−y 2
−y 2 2 + 2(y ) − λ
|G − λI| = −( − λ2 ) 1
2
y
−y
1−λ
0 4
1
2
−y
0
1−λ y
1
1
= (− − λ)( − λ)(λ4 − 2(1 + (y 1 )2 + (y 2 )2 )λ3
2
2
3
1
1
1 2
+ ( + (y ) + (y 2 )2 )λ2 + ( + (y 1 )2 + (y 2 )2 )λ − )
4
2
4
1
1
1
= (− − λ)( − λ)(λ − 1)(λ3 − (1 + 2(y 1 )2 + 2(y 2 )2 )λ2 − ( + (y 1 )2
2
2
4
1
2 2
+ (y ) )λ + )
4
1
3
1
1
= −( + λ)2 ( − λ)(λ − 1)(λ2 − ( + 2(y 1 )2 + 2(y 2 )2 )λ + ),
2
2
2
2
so the signs of eigenvalues are independent from the coordinates hence the
index of g is constant. We note that in this example ξ1 and ξ2 are spacelike.
We prove that ϕ is an f -structure, to this purpose we note that rank(ϕ) = 4
and we prove that ϕ3 + ϕ = 0. For any i ∈ {1, 2} we compute
∂
∂
∂
) = −ϕ2 ( i ) = −ϕ( i )
∂xi
∂y
∂x
2
∂
3 ∂
2 ∂
εi y i ξα ) = −ϕ( i )
ϕ ( i) = ϕ ( i +
∂y
∂x
∂y
ϕ3 (
α=1
and, being ϕ(ξα ) = 0 for any α ∈ {1, 2}, this is sufficient to obtain ϕ3 + ϕ = 0.
In the same way one can check that
ϕ2 = −I + η α ⊗ ξα .
99
Now, it remains to show that g is a compatible metric. For any i, j ∈ {1, 2}
and α, β ∈ {1, 2} we compute
g(ϕ(
∂
∂
∂
1
1
∂
), ϕ( j )) = g( i , j ) = εi δij = εi δij + 2εi εj (y i y j ) − εi εj y i y j
∂xi
∂x
∂y ∂y
2
2
2
∂
∂
∂
∂
i j
− εi εj y y = g( i , j ) −
εα η α ( i )η α ( j ),
∂x ∂x
∂x
∂x
α=1
g(ϕ(
∂
∂
∂
∂
), ϕ( j )) = −g( i , j ) −
i
∂x
∂y
∂y ∂x
= g(
g(ϕ(
2
εj y j g(
α=1
∂
, ξα ) = 0
∂y i
2
∂
∂
∂
∂
, j)−
εα η α ( i )η α ( j ),
i
∂x ∂y
∂x
∂y
α=1
2
∂
∂
∂
∂
∂
),
ϕ(
))
=
g(
,
)
+
εj y j g( i , ξα )
i
j
i
j
∂y
∂y
∂x ∂x
∂x
α=1
+
2
β=1
2
∂
εi y g(ξβ , j ) +
εi εj y i y j g(ξβ , ξα )
∂x
i
α,β=1
1
= εi δij + 2εi εj (y i y j ) − 2εi y i εj y j − 2εi y i εj y j + 2εi εj y i y j
2
2
1
∂
∂
∂
∂
= εi δij = g( i , j ) −
εα η α ( i )η α ( j ).
2
∂y ∂y
∂y
∂y
α=1
Moreover, for any i ∈ {1, 2} and α ∈ {1, 2} we find
∂
∂
∂
, ξα ) = −εi y i = η α ( i ) = εα η α ( i ),
i
∂x
∂x
∂x
∂
∂
g( i , ξα ) = 0 = εα η α ( i ).
∂y
∂y
g(
So, we have obtained that M2 is an indefinite g.f.f -manifold and our next
aim is to prove that it is an indefinite S-manifold. In fact, first of all note
that, for any α ∈ {1, 2} and i, j ∈ {1, 2}, we have
∂
∂
∂
∂
∂
∂
,
) = −g( i , j ) = 0 = dη α ( i , j ),
∂xi ∂xj
∂x ∂y
∂x ∂x
2
∂
∂
∂
1
∂
εj y j ξα ) = εi δij + 2εi εj y i y j − 2εi y i εj y j
Φ( i , j ) = g( i , j +
∂x ∂y
∂x ∂x
2
α=1
Φ(
Chapter 4. Examples of indefinite S-manifolds
100
1
∂
∂
εi δij = dη α ( i , j ),
2
∂x ∂y
2
∂
∂
∂
∂
∂
∂
Φ( i , j ) = g( i , j +
εj y j ξα ) = 0 = dη α ( i , j ),
∂y ∂y
∂y ∂x
∂y ∂y
=
α=1
and we note
Φ(X, ξα ) = 0 = dη β (X, ξα ),
for any X ∈ Γ(T R62 ) and for any α, β ∈ {1, 2}.
Now, it remains us to show that this structure is normal. Hence, for any
i, j ∈ {1, 2}, we compute
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
, j ) = [ϕ i , ϕ j ] + ϕ2 [ i , j ] − ϕ[ϕ i , j ] − ϕ[ i , ϕ j ]
i
∂x ∂x
∂x
∂x
∂x ∂x
∂x ∂x
∂x
∂x
∂
∂
∂
∂
∂
α ∂
+ 2dη ( i , j )ξα = ϕ[ i , j ] + ϕ[ i , j ] = 0,
∂x ∂x
∂y ∂x
∂x ∂y
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
N ( i , j ) = [ϕ i , ϕ j ] + ϕ2 [ i , j ] − ϕ[ϕ i , j ] − ϕ[ i , ϕ j ]
∂x ∂y
∂x
∂y
∂x ∂y
∂x ∂y
∂x
∂y
∂
∂
+ 2dη α ( i , j )ξα
∂x ∂y
2
2
∂
∂
∂
∂
= −[ i , j + εj y j
ξα ] − ϕ[ i , j + εj y j
ξα ]
∂y ∂x
∂x ∂x
α=1
α=1
N(
+ εi δij
2
ξα = −εj δij
α=1
N(
2
ξα + εj δij
α=1
2
ξα = 0,
α=1
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
, j ) = [ϕ i , ϕ j ] + ϕ2 [ i , j ] − ϕ[ϕ i , j ] − ϕ[ i , ϕ j ]
i
∂y ∂y
∂y
∂y
∂y ∂y
∂y ∂y
∂y
∂y
∂
∂
+ 2dη α ( i , j )ξα
∂y ∂y
2
2
∂
∂
= [ i + εi y i
ξα , j + εj y j
ξα ]
∂x
∂x
α=1
α=1
2
2
∂
∂
∂
∂
j
i
ξα ] − ϕ[ i + εi y
ξα , j ]
− ϕ[ i , j + εj y
∂y ∂x
∂x
∂y
α=1
α=1
= −ϕ(−εi δij
2
α=1
ξα ) − ϕ(εi δij
2
α=1
ξα ) = 0,
101
and for any i ∈ {1, 2} and α, β ∈ {1, 2}
N(
4.3
∂
, ξα ) = 0,
∂xi
N(
∂
, ξα ) = 0,
∂y i
N (ξβ , ξα ) = 0.
A special indefinite S-manifold M3 on R41
The third example of a indefinite S-manifold is M3 = (R41 , ϕ, ξ1 , ξ2 , η 1 , η 2 , g)
and we denote its standard coordinates with {x, y, z 1 , z 2 }. We endow R4 with
the structure (ϕ, ξ1 , ξ2 , η 1 , η 2 , g) where the tensor fields ϕ and g are given by


 1
0 −1 0 0
0 y −y
2
 1 0 0 0 
 0 1 0 0
2


F := 
 0 y 0 0  and G :=  y 0 1 0
0 y 0 0
−y 0 0 −1




respectively and where
ξα =
∂
,
∂z α
η α = dz α + ydx,
for any α ∈ {1, 2}. A immediate computation shows that g is non-degenerate
and its index is constant. In fact, we have
detG =
1
1 1
(− + y 2 − y 2 ) = − ,
2 2
4
and
1
|G−λI| = ( −λ) 2
1
2
−λ
y
−y
y
1−λ
0
−y
0
−1 − λ
= ( 1 −λ)(λ3 − 1 λ2 −(2y 2 +1)λ+ 1 ),
2
2
2
hence detG = 0 and, using Cartesio’s rule, we deduce that the index is 1.
Therefore, the tensor field g is a semi-Riemannian metric. Now, we observe
that ξ1 is a spacelike vector field while ξ2 is a timelike vector field and that
M3 is Lorentzian manifold. Having
∂
∂
∂
) = ϕ2 ( ) = −ϕ( ),
∂x
∂y
∂x
∂
∂
∂
+ yξ1 + yξ2 ) = −ϕ( ),
ϕ3 ( ) = ϕ2 (−
∂y
∂x
∂y
ϕ3 (
Chapter 4. Examples of indefinite S-manifolds
102
and being rankϕ = 2, we deduce that ϕ is an f -structure and this f -structure
is an indefinite g.f.f -structure. In fact we find
ϕ2 (
2
∂
∂
∂
∂
ξα = (−I + η α ⊗ ξα )( ),
) = ϕ( ) = −
+y
∂x
∂y
∂x
∂x
α=1
ϕ2 (
2
∂
∂
∂
∂
) = ϕ(−
+y
= (−I + η α ⊗ ξα ))( ),
ξα ) = −
∂y
∂x
∂y
∂y
α=1
moreover, we prove
g(ϕ(
∂
1
∂ ∂
∂
∂
∂
∂
∂
), ϕ( )) = = g( ,
) − η 1 ( )η 1 ( ) + η 2 ( )η 2 ( )
∂x
∂x
2
∂x ∂x
∂x
∂x
∂x
∂x
2
∂
∂
∂ ∂
)−
εα η α ( )η α ( ),
= g( ,
∂x ∂x
∂x
∂x
α=1
2
∂
∂ ∂
∂
∂
∂
εα η α ( )η α ( ),
g(ϕ( ), ϕ( )) = 0 = g( , ) −
∂x
∂y
∂x ∂y
∂x
∂y
α=1
g(ϕ(
∂
∂
∂
), ϕ( )) = g(−
+
∂y
∂y
∂x
= g(
2
2
yξα , −
α=1
∂
1
+
yξα ) = − 2y 2 + 2y 2
∂x
2
α=1
2
∂ ∂
∂
∂
,
)−
εα η α ( )η α ( ),
∂y ∂y
∂y
∂y
α=1
and, for any α ∈ {1, 2},
∂
∂
, ξα ) = 0 = εα η α ( ),
∂y
∂y
∂
∂
g( , ξ1 ) = y = ε1 η 1 ( ),
∂x
∂x
∂
2 ∂
g( , ξ2 ) = −y = ε2 η ( ).
∂x
∂x
Then, we show that the indefinite g.f.f -manifold M3 is an indefinite almost
S-manifold. Namely, for any α ∈ {1, 2}, we get
g(
∂ ∂
∂ ∂
∂ ∂
,
) = g( , ) = 0 = dη α ( ,
),
∂x ∂x
∂x ∂y
∂x ∂x
∂
∂
1
1
∂ ∂
∂ ∂
+ yξ1 + yξ2 ) = − + y 2 − y 2 = − = dη α ( ,
),
Φ( , ) = g( , −
∂x ∂y
∂x ∂x
2
2
∂x ∂y
∂
∂
∂ ∂
∂ ∂
+ yξ1 + yξ2 ) = 0 = dη α ( ,
).
Φ( , ) = g( , −
∂y ∂y
∂y ∂x
∂y ∂y
Φ(
103
Finally, it is possible to show that this structure is normal. In fact we obtain
∂
∂
∂ ∂
∂ ∂
∂ ∂
∂ ∂
,
) = [ , ϕ ] + ϕ2 [ ,
] − ϕ[ ,
] − ϕ[ ,
]
∂x ∂x
∂y ∂y
∂x ∂x
∂y ∂x
∂x ∂y
∂ ∂
+ 2dη α ( ,
)ξα = 0,
∂x ∂x
∂ ∂
∂
∂
∂ ∂
∂
∂
∂ ∂
N ( , ) = [ϕ , ϕ ] + ϕ2 [ ,
] − ϕ[ϕ , ] − ϕ[ , ϕ ]
∂x ∂y
∂x ∂y
∂x ∂y
∂x ∂y
∂x ∂y
∂ ∂
+ 2dη α ( , )ξα
∂x ∂y
2
2
∂
∂
∂
∂
+y
+y
= [ ,−
ξα ] − ϕ[ , −
ξα ] − ξ1 − ξ2
∂y ∂x
∂x ∂x
N(
α=1
α=1
= ξ1 + ξ2 − ξ1 − ξ2 = 0,
∂
∂
∂ ∂
∂
∂
∂ ∂
∂ ∂
] − ϕ[ϕ ,
] − ϕ[ , ϕ ]
N ( , ) = [ϕ , ϕ ] + ϕ2 [ ,
∂y ∂y
∂y ∂y
∂y ∂y
∂y ∂y
∂y ∂y
∂ ∂
+ 2dη α ( , )ξα
∂y ∂y
2
2
∂
∂
+y
+y
= [−
ξα , −
ξα ]
∂x
∂x
α=1
α=1
− ϕ[
2
2
α=1
α=1
∂
∂
∂
∂
,−
+y
+y
ξα ] − ϕ[−
ξα , ]
∂y ∂x
∂x
∂y
= −ϕ(−ξ1 − ξ2 ) − ϕ(ξ1 + ξ2 ) = 0,
and for α, β ∈ {1, 2}
N(
∂
, ξα ) = 0,
∂x
N(
∂
, ξα ) = 0,
∂y
N (ξβ , ξα ) = 0.
Finally since ε = 0, the structure is special indefinite S-structure.
4.4
Lightlike hypersufaces of M1 and M2
We begin with a general remark. Consider a hypersurface M of a semiRiemannian manifold (M̄ 6 , ḡ) given by
xA = f A (u1 , u2 , u3 , u4 , u5 ),
rank(
∂f A
) = 5,
∂ua
Chapter 4. Examples of indefinite S-manifolds
104
where A ∈ {1, . . . , 6} and a ∈ {1, . . . , 5}. A natural basis of the tangent bundle
of M is
∂f A ∂
∂
=
,
∂ua
∂ua ∂xA
the induced metric on M , denoted by g, has components given by
gab = ḡAB
∂f A ∂f B
.
∂ua ∂ub
Proposition 4.4.1. Let M be a hypersurface of (M̄ 6 , ḡ). M is lightlike if and
only if rank(gab ) ≤ 4, i.e.
∆ = det(
∂f B
∂f A
ḡ
) = 0.
AB
∂ua
∂ub
Proposition 4.4.2 ([32]). Let A = (aij ) ∈ Mm,n (R)
If we consider C = AB, then we have
a1j1 a2j1 . . . anj1 a1j2 a2j2 . . . anj2 detC =
...
.
.
.
.
.
.
.
.
.
1≤j1 ≤j2 ...≤jn ≤m a1jn a2jn . . . anjn (4.4.1)
and B = (bjk ) ∈ Mn,m (R).
bj1 1 bj1 2
bj1 1 bj1 1
... ...
bj1 1 bj1 1
. . . bj1 1
. . . bj1 1
... ...
. . . bj1 1
.
Now, considering the two indefinite g.f.f -structures on R62 given in example 1 and example 2, we look for a lightlike hypersurface. We begin with
M2 .
We compute all the non null determinants MAB , obtained by take the line
A and the column B out of G,
1
M11 = M33 = − ,
8
1
M22 = M44 = ,
8
M55 = M66 =
1
1
1
+ (y 2 )2 − (y 1 )2 ,
16 8
8
and
M15 =
1 1
y ,
8
1
M16 = − y 1 ,
8
1
M25 = − y 2 ,
8
1
M26 = y 2 ,
8
1
1
M56 = − (y 2 )2 + (y 1 )2 .
8
8
We observe that obviously MAB = MBA . Now, we find a condition which
is equivalent to (4.4.1) for a lightlike hypersurface in M2 in terms of DA
A
and MAB , where D = ( ∂f
) and, for any A ∈ {1, . . . , 6}, DA denotes the
∂ui
determinant of the matrix obtained by D deleting the column of index A.
105
Proposition 4.4.3. Let M be a hypersurface of M2 . Then, M is lightlike if
and only if
1
1
( − (D1 )2 + (D2 )2 − (D3 )2 + (D4 )2 + ( + (y 2 )2 − (y 1 )2 )((D5 )2 + (D6 )2 ))
2
2
+ y 1 (D1 D5 − D1 D6 ) − y 2 (D2 D5 − D2 D6 ) + ((y 1 )2 − (y 2 )2 )D5 D6 = 0.
Proof. From Proposition 4.4.1 we know that M is a lightlike hypersurface if
A
∂f B
and only if ∆ = det( ∂f
∂ua gAB ∂ub ) = 0, thus, using Proposition 4.4.2, we have
∆=
6
A,B=1
1
1
DA MAB DB = − ((D1 )2 + (D3 )2 ) + y 1 (D1 D5 − D1 D6 )
8
4
1
1
1
+ ((D2 )2 + (D4 )2 ) − y 2 (D2 D5 − D2 D6 ) + ((y 1 )2 − (y 2 )2 )D5 D6
8
4
4
1 1
2 2
1 2
5 2
6 2
+ ( + (y ) − (y ) )((D ) + (D ) ).
8 2
This ends the proof.
Now, we want to find a condition which ensures that the characteristic
vector fields belong to T M .
Proposition 4.4.4. Let M be a hypersurface of M2 . Then, the characteristic
vector fields are tangent to M if and only if D5 = D6 = 0.
Proof. We suppose that, for any α ∈ {1, 2}, ξα is tangent to M , then we can
write
∂
ξα = Xαa a .
∂u
Being
∂
∂ua
=
∂f A ∂
∂ua ∂xA ,
we have
ξα = Xαa
∂f A ∂
,
∂ua ∂xA
on the other hand we know
ξα =
So, we obtain

∂f A

 X1a ∂ua = 0

 X a ∂f 5 = 1
1 ∂ua
∂
.
∂z α
A = 5
(4.4.2)
Chapter 4. Examples of indefinite S-manifolds
106
and

∂f A

 X2a ∂ua = 0
A = 6
.

 X a ∂f 6 = 1
2 ∂ua
(4.4.3)
We observe that the first equations in (4.4.2) state that (X1a )a∈{1,...,5} is a non
zero solution of a homogeneous system whose matrix has to be degenerate, i.e.
D 5 = 0. Analogously from (4.4.3) D6 = 0 follows.
Vice versa, if we suppose that D5 = 0, therefore (4.4.2) admits non trivial
solutions and ξ1 = X1a ∂u∂ a . Analogously, supposing D6 = 0, from (4.4.3) we
obtain ξ2 = X2a ∂u∂ a .
The following theorem is an immediate consequence of Proposition 4.4.3
and Proposition 4.4.4.
Proposition 4.4.5. Let M be a hypersurface of M2 . M is a lightlike hypersurface, with ξ1 , ξ2 ∈ Γ(T M ), if and only if
−(D 1 )2 + (D2 )2 − (D3 )2 + (D4 )2 = 0
D5 = D6 = 0.
Now, after these general remarks, we shall give an example of a lightlike
hypersurface M in M2 such that ξ1 , ξ2 and ϕ(E) belong to S(T M ). We
consider
f (u1 , u2 , u3 , u4 , u5 ) = (u1 + u5 , u2 , u3 , u1 + u5 , u4 , u5 ).
This map describes a hypersurface in M2 ; in fact its Jacobian is



D=


1
0
0
0
1
0
1
0
0
0
0
0
1
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
1



,


with rank five. A direct computation gives:
D 1 = D4 = 1,
D2 = D3 = D5 = D6 = 0,
hence such a hypersurface verifies Proposition 4.4.5. Then M is a lightlike
hypersurface and ξ1 and ξ2 belong to Γ(T M ). The natural basis of T M is
107
given by
U1 =
U2 =
U3 =
U4 =
U5 =
∂
∂u1
∂
∂u2
∂
∂u3
∂
∂u4
∂
∂u5
= (1, 0, 0, 1, 0, 0) =
= (0, 1, 0, 0, 0, 0) =
= (0, 0, 1, 0, 0, 0) =
= (0, 0, 0, 0, 1, 0) =
= (1, 0, 0, 1, 0, 1) =
∂
∂
+ 2,
1
∂x
∂y
∂
,
∂x2
∂
,
∂y 1
∂
= ξ1 ,
∂z 1
∂
∂
+
+ ξ2 .
∂x1 ∂y 2
We deduce that ξ2 = U5 − U1 . Considering the vector field of M2
E=−
∂
∂
− 2 + y 1 ξ1 + y 1 ξ2 ,
1
∂x
∂y
it is easy to check that E ∈ Γ(T M ⊥ ) and that E is a lightlike vector field. In
fact we have
∀a ∈ {1, . . . , 5} g(E, Ua ) = 0,
g(E, E) = 0,
and obviously E belongs to Rad(T M ). We construct N ∈ Γ(T M2 ) such that
g(N, N ) = 0,
g(N, E) = 1.
To this aim, we consider Z = (1, 0, 0, 0, −y 1 , −y 1 ) = ∂x∂ 1 − y 1 ξ1 − y 1 ξ2 such
that g(Z, E) = 12 = 0, and we put
g(Z, Z)
1
Z−
E ,
N=
g(Z, E)
2g(Z, E)
then we have N = (1, 0, 0, −1, −y 1 , −y 1 ) = ∂x∂ 1 − ∂y∂ 2 − y 1 ξ1 − y 1 ξ2 . Now we
want to give a basis of the screen distribution S(T M ). To this purpose we
compute both ϕ(E) and ϕ(N ) and we have
ϕ(E) = (0, −1, 1, 0, −y 2 , −y 2 ) = −
∂
∂
+ 1 − y 2 ξ1 − y 2 ξ2 ,
2
∂x
∂y
and
ϕ(N ) = (0, −1, −1, 0, −y 2 , −y 2 ) = −
∂
∂
−
− y 2 ξ1 − y 2 ξ2 ,
∂x2 ∂y 1
Chapter 4. Examples of indefinite S-manifolds
108
being ϕ(E) = −U2 + U3 − y 2 U4 − y 2 (U5 − U1 ) and ϕ(N ) = −U2 − U3 − y 2 U4
− y 2 (U5 − U1 ) a linear combination of Uα , they are tangent to M . Thus, we
define the screen distribution
S(T M ) =< ξ1 , ξ2 , ϕ(E), ϕ(N ) >,
and it is easy to check that the matrix GS of the restriction gS of ḡ (or g) to
S(T M ) is


1 0 0 0
 0 1 0 0 

GS = 
 0 0 0 1 .
0 0 1 0
So, the index of gS is one and therefore gS is a metric tensor field of S(T M ).
Moreover, we compute
S(T M )⊥ = {X ∈ T M2 | ∀S ∈ S(T M ) g(S, X) = 0},
and we find
S(T M )⊥ =< (1, 0, 0, 0, −y 1 , −y 1 ), (0, 0, 0, 1, 0, 0) > .
Finally, being Z ∈ Γ(S(T M )⊥ ), we deduce that N is the vector field provided
by Theorem 3.1.3.
Finally, we consider the indefinite S-manifold M1 in the example 1. In this
case, we obtain
M11 = M22 = M33 = M44 =
M15 =
1 1
y ,
8
1
,
8
M55 = M66 = −
1
M16 = − y 1 ,
8
1
1
1
+ (y 1 )2 + (y 1 )2 ,
16 8
8
1
M25 = − y 2 ,
8
1
M26 = y 2 ,
8
and
1
1
M56 = − (y 1 )2 − (y 2 )2 ,
8
8
while the other MAB vanish. Using Proposition 4.4.1 and Proposition 4.4.2, we
can prove the following proposition omitting its proof because it is analogous
to that of Proposition 4.4.3.
109
Proposition 4.4.6. Let M be a hypersurface of M1 . Then, M is a lightlike
hypersurface if and only if
1
1
((D1 )2 + (D2 )2 + (D3 )2 + (D4 )2 + (− + (y 2 )2 + (y 1 )2 )((D5 )2 + (D6 )2 ))
2
2
− y 2 (D2 D5 − D2 D6 ) + ((y 1 )2 + (y 2 )2 )D5 D6 + y 1 (D1 D5 − D1 D6 ) = 0.
Now, we note that Proposition 4.4.4 does not depend on the metric, therefore Proposition 4.4.4 also holds in this case. Thus, omitting the proof, we
have
Proposition 4.4.7. Let M be a hypersurface of M1 . M is a lightlike hypersurface, with ξ1 , ξ2 ∈ Γ(T M ), if and only if
(D 1 )2 + (D2 )2 + (D3 )2 + (D4 )2 = 0,
D5 = D6 = 0.
Theorem 4.4.8. There are not lightlike hypersurfaces with ξα ∈ Γ(T M ), for
any α ∈ {1, 2}, in M1 .
4.5
Induced geometrical object
In the case of the described lightlike hypersurface of M2 we compute its geometrical object and we write the Gauss and Weingarten equations for such
¯ be the Levi-Civita
hypersurface and for the screen distribution S(T M ). Let ∇
connection of M2 and {E, ξ1 , ξ2 , U = −ϕE, V = −ϕN } a basis of T M . Then,
reminding that B(X, E) = 0 for any X ∈ Γ(T M ) and that B is symmetric,
we compute B with respect to the aforesaid basis and we get
¯ X ξα , E) = −εα ḡ(ϕX, E) = −ḡ(ϕX, E)
B(X, ξα ) = ḡ(∇

X = ξβ for any β ∈ {1, 2}
 0
.
X=U
ḡ(ϕ2 E, E) = 0
=

2
ḡ(ϕ N, E) = −1 X = V
Now, we compute the Christoffel symbols of the Levi-Civita connection
order to find B(U, U ), B(U, V ) and B(V, V ). So, we find

−2 0
0 0
2y 1
2y 1
2
 0
2
0 0
2y
2y 2

 0
0 −2 0
0
0
G−1 = 
 0
0
0
2
0
0

 2y 1 2y 2 0 0 1 − 2(y 1 )2 + 2(y 2 )2
2(y 2 )2 − 2(y 1 )2
2y 1 2y 2 0 0
2(y 2 )2 − 2(y 1 )2
1 − 2(y 1 )2 + 2(y 2 )2
¯ in
∇








Chapter 4. Examples of indefinite S-manifolds
110
and, with respect to { ∂x∂ 1 , ∂x∂ 2 , ∂y∂ 1 , ∂y∂ 2 , ∂z∂ 1 , ∂z∂ 2 }, it is easy to check
Γ311 = 4y 1 ,
Γ513 = Γ613
Γ123 = Γ132
Γ514 = Γ614
Γ312 = Γ321 = −2y 2 , Γ412 = Γ421 = 2y 1 , Γ422 = −4y 2 ,
1
= Γ531 = Γ631 = + 2(y 1 )2 , Γ113 = Γ131 = Γ214 = Γ241 = −2y 1 ,
2
1
2
2
= Γ24 = Γ42 = 2y 2 , Γ524 = Γ542 = Γ624 = Γ642 = − + 2(y 2 )2 ,
2
= Γ541 = Γ641 = Γ523 = Γ623 = Γ532 = Γ632 = −2y 1 y 2 ,
Γ315 = Γ351 = Γ316 = Γ361 = Γ425 = Γ452 = Γ426 = Γ462 = 1,
Γ535 = Γ553 = Γ536 = Γ563 = Γ635 = Γ653 = Γ636 = Γ663 = y 1 ,
Γ245 = Γ254 = Γ246 = Γ264 = Γ135 = Γ153 = Γ136 = Γ163 = −1,
Γ545 = Γ554 = Γ546 = Γ564 = Γ645 = Γ654 = Γ646 = Γ664 = −y 2 ,
while the other Γhij vanish. Being U =
∂
∂x2
−
∂
∂y 1
+ y 2 ( ∂z∂ 1 +
∂
),
∂z 2
we have
¯U ∂ − ∇
¯ U ∂ + y 2 (∇
¯U ∂ + ∇
¯U ∂ )
¯ UU = ∇
∇
2
1
1
∂x
∂y
∂z
∂z 2
= [Γh22 − 2Γh23 + 2y 2 (Γh52 + Γh62 − Γh53 − Γh63 )]
∂
∂xh
∂
∂
∂
∂
∂
∂
− 4y 2 1 + 4y 1 y 2 ( 1 + 2 ) + 2y 2 (2 2 + 2 1
2
∂y
∂x
∂z
∂z
∂y
∂x
∂
∂
+ 2y 1 1 + 2y 1 2 ) = 0,
∂z
∂z
= −4y 2
hence, we deduce
¯ U U, E) = 0.
B(U, U ) = ḡ(∇
Being U =
find
∂
∂x2
−
∂
∂y 1
+ y 2 ( ∂z∂ 1 +
∂
∂z 2 )
and V =
∂
∂x2
+
∂
∂y 1
+ y 2 ( ∂z∂ 1 +
¯V ∂ − ∇
¯ V ∂ + y 2 (∇
¯V ∂ + ∇
¯V ∂ )
¯V U = ∇
∇
2
1
1
∂x
∂y
∂z
∂z 2
∂
= [Γh22 + 2y 2 (Γh52 + Γh62 )] h
∂x
2 ∂
2 ∂
= −4y
+ 4y
= 0,
∂y 2
∂y 2
therefore we have
¯ V U, E) = 0.
B(V, U ) = ḡ(∇
∂
∂z 2 ),
we
111
Finally, being V =
∂
∂x2
+
∂
∂y 1
+ y 2 ( ∂z∂ 1 +
∂
∂z 2 ),
we obtain
¯V ∂ +∇
¯ V ∂ + y 2 (∇
¯V ∂ + ∇
¯V ∂ )
¯VV = ∇
∇
2
1
1
∂x
∂y
∂z
∂z 2
= [Γh22 + 2Γh23 + 2y 2 (Γh52 + Γh62 + Γh53 + Γh63 )]
∂
∂xh
∂
∂
∂
∂
+ 4y 2 1 − 4y 1 y 2 ( 1 + 2 )
∂y 2
∂x
∂z
∂z
∂
∂
∂
∂
+ 2y 2 (2 2 − 2 1 + 2y 1 1 + 2y 1 2 ) = 0,
∂y
∂x
∂z
∂z
= −4y 2
we get
¯ V V, E) = 0.
B(V, V ) = ḡ(∇
Then M is neither totally umbilical nor totally geodesic but it is minimal
because
1
µ = (B(ξ1 , ξ1 ) + B(ξ2 , ξ2 ) + B(E, E) + B(U, U ) + B(V, V )) = 0.
5
¯ X N for any
Using (3.2.6), we compute τ . To this aim, we find the value of ∇
X ∈ Γ(T M ). For any α ∈ {1, 2}, it is easy to check
¯E ∂ − ∇
¯ E ∂ − y 1 (∇
¯E ∂ + ∇
¯E ∂ )
¯ EN = ∇
∇
1
2
1
∂x
∂y
∂z
∂z 2
∂
∂
∂
= [−Γh11 + 2y 1 (Γh15 + Γh16 )] h = −4y 1 1 + 4y 1 1 = 0,
∂x
∂y
∂y
∂
∂
∂
∂
¯ξ
¯ξ
¯ξ
¯ξ
¯ξ N = ∇
−∇
− y 1 (∇
+∇
)
∇
α
α
α
α
α
∂x1
∂y 2
∂z 1
∂z 2
∂
∂
∂
∂
∂
= [Γh4+α1 − Γh4+α4 ] h = 1 + 2 + y 2 ( 1 + 2 ),
∂x
∂y
∂x
∂z
∂z
¯U ∂ − ∇
¯ U ∂ − y 1 (∇
¯U ∂ + ∇
¯ U ∂ ) − U (y 1 )( ∂ + ∂ )
¯ UN = ∇
∇
∂x1
∂y 2
∂z 1
∂z 2
∂z 1 ∂z 2
h
h
h
2 h
h
h
h
1 h
= [Γ21 − Γ31 − Γ24 + y (Γ15 + Γ16 − Γ45 − Γ46 ) − y (Γ25 + Γh26 − Γh35
∂
∂
∂
∂
∂
∂
− Γh36 )] h + 1 + 2 = −2y 2 1 + 2y 1 2 + 2y 1 1
∂x
∂z
∂z
∂y
∂y
∂x
∂
1 ∂
∂
∂
∂
1
∂
∂
− ( 1 + 2 ) − 2(y 1 )2 ( 1 + 2 ) − 2y 2 2 + ( 1 + 2 )
2 ∂z
∂z
∂z
∂z
∂x
2 ∂z
∂z
∂
∂
∂
∂
∂
∂
− 2(y 2 )2 ( 1 + 2 ) + 2y 2 1 + 2y 2 2 + 2(y 2 )2 ( 1 + 2 )
∂z
∂z
∂y
∂x
∂z
∂z
∂
∂
∂
∂
∂
∂
− 2y 1 2 − 2y 1 1 + 2(y 1 )2 ( 1 + 2 ) + 1 + 2
∂y
∂x
∂z
∂z
∂z
∂z
Chapter 4. Examples of indefinite S-manifolds
112
∂
∂
+ 2,
1
∂z
∂z
∂
¯VN = ∇
¯V
¯ V ∂ − y 1 (∇
¯V ∂ + ∇
¯ V ∂ ) − V (y 1 )( ∂ + ∂ )
∇
−∇
1
2
1
∂x
∂y
∂z
∂z 2
∂z 1
∂z 2
h
h
h
2 h
h
h
h
1 h
h
= [Γ21 + Γ31 − Γ24 + y (Γ15 + Γ16 − Γ45 − Γ46 ) − y (Γ25 + Γ26 + Γh35
∂
∂
∂
∂
∂
∂
+ Γh36 )] h − 1 − 2 = −2y 2 1 + 2y 1 2 − 2y 1 1
∂x
∂z
∂z
∂y
∂y
∂x
1 ∂
∂
∂
∂
1 ∂
∂
∂
+ ( 1 + 2 ) + 2(y 1 )2 ( 1 + 2 ) − 2y 2 2 + ( 1 + 2 )
2 ∂z
∂z
∂z
∂z
∂x
2 ∂z
∂z
∂
∂
∂
∂
∂
∂
− 2(y 2 )2 ( 1 + 2 ) + 2y 2 1 + 2y 2 2 + 2(y 2 )2 ( 1 + 2 )
∂z
∂z
∂y
∂x
∂z
∂z
∂
∂
∂
∂
∂
∂
+ 2y 1 1 − 2(y 1 )2 ( 1 + 2 ) − 2y 1 2 − 1 − 2 = 0.
∂x
∂z
∂z
∂y
∂z
∂z
=
Immediately we find
¯ ξ N, E) = 0, for any α ∈ {1, 2}
τ (ξα ) = ḡ(∇
α
¯ E N, E) = 0,
τ (E) = ḡ(∇
¯ U N, E) = 0,
τ (U ) = ḡ(∇
¯ V N, E) = 0.
τ (V ) = ḡ(∇
So, vanishing τ , we have from (3.2.6)
¯ X N,
AN X = −∇
for any X ∈ Γ(T M ).
Again using (3.2.5), we compute ∇X E for any X ∈ Γ(T M ) and we have
∂
h
1 h
h
¯ ξ E = [−Γh
∇ξα E = ∇
α
4+α1 − Γ4+α4 + y (Γ4+α5 + Γ4+α6 )]
∂xh
∂
∂
∂
∂
= − 1 + 2 + y 2 ( 1 + 2 ),
∂y
∂x
∂z
∂z
¯ E E = [Γh + 2Γh − 2y 1 (Γh + Γh + Γh + Γh )] ∂ = 0
∇E E = ∇
11
41
15
16
45
46
∂xh
¯ U E = [−Γh + Γh − Γh − y 2 (Γh + Γh + Γh + Γh )
∇U E = ∇
21
31
24
15
16
45
46
∂
∂
∂
+ y 1 (Γh25 + Γh26 − Γh35 − Γh36 )] h − 1 − 2 = 0,
∂z
∂z
∂x
¯ V E = [−Γh − Γh − Γh − y 2 (Γh + Γh + Γh + Γh )
∇V E = ∇
21
31
24
15
16
45
46
∂
∂
∂
∂
∂
1 h
h
h
h
+ y (Γ25 + Γ26 − Γ35 − Γ36 )] h + 1 + 2 = 1 + 2 .
∂x
∂z
∂z
∂z
∂z
113
Using (3.2.17) and being τ (X) = 0 for any X ∈ Γ(T M ), we have
A∗E X = −∇X E,
for any X ∈ Γ(T M ). Finally, recalling that S(T M ) =< ξ1 , ξ2 , U, V > and
C(X, P Y ) = ḡ(h∗ (X, P Y ), N ), we value C and we find
¯ X P Y, N )
C(X, P Y ) = ḡ(h∗ (X, P Y ), N ) = ḡ(∇X P Y, N ) = ḡ(∇
For any α ∈ {1, 2}, we have
¯ X ξα , N ) = −εα ḡ(ϕX, N ) = ḡ(X, ϕN )
C(X, ξα ) = ḡ(∇

0
for X = ξβ and any β ∈ {1, 2}



0
for X = V
.
=
−1
for X = U



0
for X = E
¯ξ V =
Being ∇
α
∂
∂y 2
−
∂
∂x1
+ y 1 ( ∂z∂ 1 +
∂
∂z 2 ),
(4.5.1)
we obtain
¯ ξ V, N ) = 0.
C(ξα , V ) = ḡ(∇
α
¯ V V = 0, we have
¯ U V = 0 and ∇
Analogously, being ∇
¯ V V, N ) = 0.
C(V, V ) = ḡ(∇
¯ U V, N ) = 0,
C(U, V ) = ḡ(∇
¯ E V = − ∂1 −
As ∇
∂z
∂
,
∂z 2
we get
¯ E V, N ) = 0.
C(E, V ) = ḡ(∇
¯ U U = 0, we have
¯ V U = 0 and ∇
Since ∇
¯ V U, N ) = 0,
C(V, U ) = ḡ(∇
¯ U U, N ) = 0,
C(U, U ) = ḡ(∇
¯ξ U =
¯ E U = 0 and ∇
moreover, computing ∇
α
α ∈ {1, 2}, we find
¯ E U, N ) = 0,
C(E, U ) = ḡ(∇
∂
∂y 2
+ ∂x∂ 1 − y 1 ( ∂z∂ 1 + ∂z∂ 2 ) for any
¯ ξ U, N ) = −1.
C(ξα , U ) = ḡ(∇
α
Chapter 4. Examples of indefinite S-manifolds
114
Distribution D0 of the lightlike hypersurface of
M2
4.6
We check some properties of D0 . To this end, we note that the distribution
D0 =< ξ1 , ξ2 > coincides with ker(ϕ̄) so, by Lemma 2.3.1, it is integrable,
¯ It is also easy to check that D0
totally geodesic and flat with respect to ∇.
verifies the conditions of Theorem 3.4.5. Moreover, from (3.4.18), we obtain
◦
h(ξα , ξβ ) = −C(ξα , ϕξβ )ϕE − B(ξα , ϕξβ )ϕN + C(ξα , ξβ )E = 0,
◦
and we deduce that h|D0 ×D0 vanishes, while the second fundamental form
◦
h : Γ(T M ) × Γ(D0 ) → Γ(F) does not vanish. In fact, for any α ∈ {1, 2},
¯ E ξα = −ϕE, ∇U ξα = ∇
¯ U ξα − B(U, ξα )E = −N and
we have ∇E ξα = ∇
¯
∇V ξα = ∇V ξα − B(V, ξα )E = −E + E = 0, we obtain
◦
h(E, ξα ) = g(∇E ξα , ϕN )ϕE + g(∇E ξα , ϕE)ϕN + g(∇E ξα , N )E
= g(−ϕE, ϕN )ϕE = −ϕE,
◦
h(U, ξα ) = g(∇U ξα , ϕN )ϕE + g(∇U ξα , ϕE)ϕN + g(∇U ξα , N )E
= g(−N, ϕN )ϕE = 0,
◦
h(V, ξα ) = g(∇V ξα , ϕN )ϕE + g(∇V ξα , ϕE)ϕN + g(∇V ξα , N )E = 0.
◦
◦
Furthermore, for any α ∈ {1, 2}, h(ξα , ξα ) = 0, then we have trace(h) = 0 i.e.
D0 is a minimal distribution in M , with respect to ∇.
4.7
Lightlike hypersufaces of M3
We start this paragraph considering a hypersurface M of a semi-Riemannian
manifold (M̄ 4 , ḡ) given by
xA = f A (u1 , u2 , u3 ),
rank(
∂f A
) = 3,
∂ua
where A ∈ {1, . . . , 4} and a ∈ {1, 2, 3}. Obviously, a natural basis of T M is
∂f A ∂
∂
=
,
∂ua
∂ua ∂xA
115
and denoting by g the induced metric on M , its components are given by
gab = ḡAB
∂f A ∂f B
.
∂ua ∂ub
As in the previous section, we can prove the following proposition
Proposition 4.7.1. Let M be a hypersurface of (M̄ 4 , ḡ). M is lightlike if and
only if rank(gab ) ≤ 2, i.e.
∆ = det(
∂f A
∂f B
g
) = 0.
AB
∂ua
∂ub
(4.7.1)
Now, considering the indefinite g.f.f -structure on R41 given in example 3, we
look for a lightlike hypersurface.
We compute all the non null determinants MAB , obtained by take the line
A and the column B out of G,
1
M11 = M22 = − ,
2
1 1
M33 = − − y 2 ,
4 2
M44 =
1 1 2
− y
4 2
and
1
M13 = y,
2
1
M14 = − y,
2
1
M34 = y 2 .
2
We observe that obviously MAB = MBA . We interpret (4.7.1) for a lightlike
hypersurface in M3 in terms of DA and MAB .
Proposition 4.7.2. Let M be a hypersurface of M3 . Then, M is lightlike if
and only if
1 1
1
− ((D1 )2 + (D2 )2 ) + y(D1 D3 − D1 D4 ) − ( + y 2 )(D3 )2 + y 2 D3 D4
2
4 2
1 1
+( − y 2 )(D4 )2 = 0.
4 2
Again, as in the case of lightlike hypersurfaces of M1 and M2 , we prove a
condition which ensures that the characteristic vector fields belong to T M .
Proposition 4.7.3. Let M be a hypersurface of M3 . Then, the characteristic
vector fields are tangent to M if and only if D3 = D4 = 0.
An immediate consequence of Proposition 4.7.2 and Proposition 4.7.3 is
given by the following proposition.
Chapter 4. Examples of indefinite S-manifolds
116
Proposition 4.7.4. Let M be a hypersurface of M2 . M is a lightlike hypersurface, with ξ1 , ξ2 ∈ Γ(T M ), if and only if
(D 1 )2 + (D2 )2 = 0
D3 = D4 = 0.
Theorem 4.7.5. There are not lightlike hypersurfaces in M3 with ξα ∈ Γ(T M ),
for any α ∈ {1, 2}.
4.8
The curvature of M3
Now, we want to compute the tensor field Q, the sectional curvature and ϕsectional curvature on M3 . We begin with the following remark. The tensor
field Q vanishes. In fact, for any X, Y, Z, W ∈ Γ(D), we know that
Q(X, Y ; Z, W ) = 0,
moreover we have
Q(ξ1 , Y ; Z, W ) = −Q(ξ2 , Y ; Z, W ) = −ḡ(W, ϕY )η(Z) + ḡ(Z, ϕY )η(W ) = 0,
for any Y, Z, W ∈ Γ(D). So, from its symmetry’s properties, we deduce that
Q vanishes.
Now, we continue with the computation of G−1 and we find

G−1
2
 0
=
 −2y
−2y

0
−2y
−2y

2
0
0
,
2
2

2y
0 1 + 2y
2
2
0
2y
−1 + 2y
and we obtain
1
Γ312 = Γ412 = , Γ213 = −Γ214 = −Γ123 = Γ124 = −1,
2
Γ323 = Γ423 = −Γ324 = −Γ424 = −y,
the other Γkij vanish. Now, we compute the ϕ-sectional curvature on M3 . To
∂
∂
− yξ1 − yξ2 and Y = ϕX = ∂y
,
this end, being D =< X, Y > where X = ∂x
117
we value H(X). So, we have
R(X, ϕX, X) = ∇X ∇ϕX X − ∇ϕX ∇X X − ∇[X,ϕX] X
∂
h
h
h
= ∇X Γ21 − y(Γ23 + Γ24 ) h − ξ1 − ξ2 − ∇ξ1 X − ∇ξ2 X
∂x
1
= − ∇X (ξ1 + ξ2 ) − (Γh31 − y(Γh33 + Γh34 ) + Γh41
2
∂
− y(Γh43 + Γh44 )) h
∂x
= [Γh11 − y(Γh31 + Γh41 ) − y(Γh13 − y(Γh33 + Γh43 )
∂
+ Γh14 − y(Γh34 + Γh44 ))] h = 0,
∂x
and we note
∂
∂
∂ ∂
) − 2y(g( , ξ1 ) + g( , ξ2 ))
g(X, X) = g( ,
∂x ∂x
∂x
∂x
1
+ y 2 (g(ξ1 , ξ1 ) + g(ξ1 , ξ2 ) + g(ξ2 , ξ2 )) = .
2
It follows that
1
g(R(X, ϕX, X), ϕX) = 0.
H(X) = −
g(X, X)2
Supposing that Z ∈ D, then it can be written
Z = αX + βϕX,
and using (2.3.8) where
we find
X
= αX, X = βϕX, Y = αϕX and Y = −βX,
R(Z, ϕZ, Z, ϕZ) = α4 R(X, ϕX, X, ϕX) − 2α3 βR(X, ϕX, X, X)
(4.8.1)
+ 2α3 βR(X, ϕX, ϕX, ϕX) − 2α2 β 2 R(X, ϕX, ϕX, X)
+ α2 β 2 R(X, X, X, X) − 2α2 β 2 R(X, X, ϕX, ϕX)
− 2α2 β 2 R(X, X, ϕX, ϕX) + α2 β 2 R(ϕX, ϕX, ϕX, ϕX)
− 2αβ 3 R(ϕX, ϕX, ϕX, X) + β 4 R(ϕX, X, ϕX, X)
= (α2 + β 2 )2 R(X, ϕX, X, ϕX).
Using (4.8.1) and being H(X) = 0, we get
1
R(Z, ϕZ, Z, ϕZ) = H(X) = 0,
H(Z) = −
g(Z, Z)2
for any non lightlike vector field Z ∈ D. Then, M3 is an indefinite S-space
form with c = 0 = ε.
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