Chapter 3
THE AXIOM OF SPHERES IN
SEMI-RIEMANNIAN
GEOMETRY WITH
LIGHTLIKE SUBMANIFOLDS
3.1
Introduction
The notion of axiom of planes for Riemannian manifolds was originally introduced
by Elie Cartan [10] as: A Riemannian manifold M of dimension m ≥ 3 satisfies
the axiom of r-planes, if for each point p in M and for every r-dimensional linear
subspace T of Tp (M ), there exists a r-dimensional totaly geodesic submanifold N
of M containing p such that Tp (N ) = T . He proved the following:
Theorem (3.1.1): A Riemannian manifold of dimension m ≥ 3 satisfies the
axiom of r-planes, for some r, 2 ≤ r < m, if and only if, it is a real space form.
In 1971, D. S. Leung and K. Nomizu [9], generalized this notion by introducing the axiom of r-spheres as: A Riemannian manifold M of dimension
m ≥ 3 satisfies the axiom of r-spheres, if for each point p in M and for every
linear subspace T of Tp (M ), there exists a r-dimensional totaly umbilical sub43
manifold N of M with parallel mean curvature vector field such that p ∈ N and
Tp (N ) = T . They proved the following result to characterize a real space form:
Theorem (3.1.2): A Riemannian manifold of dimension m ≥ 3 is a real space
form, if and only if, it satisfies the axiom of r-spheres, for some r, 2 ≤ r < m.
L. Graves and K. Nomizu [20], generalized these notions of axioms of
planes and spheres, for indefinite Riemannian manifolds.
3.2
Axiom of spheres and planes in lightlike geometry
The growing importance of lightlike submanifolds in Mathematical Physics, especially in relativity, motivated us to study lightlike submanifolds extensively. Here,
in particular, the axioms of planes and spheres for semi-Riemannian manifolds
with lightlike submanifolds have been studied. These axioms for semi-Riemannian
manifolds have been proposed as follows:
Axiom of r-planes (r-spheres): A semi-Riemannian manifold M̄ of dimension m + n ≥ 3, satisfies the axiom of r-planes (r-spheres) if for each point
p ∈ M̄ and for every r-dimensional linear subspace T of Tp (M̄ ), there exists
a r-dimensional totally geodesic lightlike submanifold (totally umbilical lightlike
submanifold with parallel transversal curvature vector field) M , such that p ∈ M
and Tp (M ) = T, 2 ≤ r < m + n and following theorems have been proved.
Theorem (3.2.1): Let (M̄ , ḡ) be a semi-Riemannian manifold of dimension
m + n ≥ 3 and (M, g, S(T M ), S(T M ⊥ )) be a lightlike submanifold of M̄ . If M̄
satisfies the axiom of r-spheres, 2 ≤ r < m + n, then M̄ is a real space form.
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Theorem (3.2.2): Let (M̄ , ḡ) be a semi-Riemannian manifold of dimension
m + n ≥ 3 and (M, g, S(T M ), S(T M ⊥ )) be a lightlike submanifold of M̄ . If M̄
satisfies the axiom of r-planes, 2 ≤ r < m + n, then M̄ is a real space form.
Definition:([16]) A lightlike submanifold (M, g, S(T M ), S(T M ⊥ )) of a semiRiemannian manifold (M̄ , ḡ) is totally umbilical if there exists a smooth transversal vector field H ∈ Γ(tr(T M )) on M , called the transversal curvature vector field
of M , such that for all X, Y ∈ Γ(T M )
h(X, Y ) = Hg(X, Y ).
(3.1)
Using (1.41)-(2.66) and (3.1), it is easy to see that M is totally umbilical if
and only if on each coordinate neighborhood u, there exist smooth vector fields
Hl ∈ Γ(ltr(T M )) and Hs ∈ Γ(S(T M ⊥ )) such that
hl (X, Y ) = Hl g(X, Y ),
Dl (X, W ) = 0,
hs (X, Y ) = Hs g(X, Y ).
(3.2)
(3.3)
The transversal curvature vector field of M is parallel (in tr(T M )) if
∇tX H = 0 or equivalently
∇lX Hl = 0 and ∇sX Hs = 0.
(3.4)
Also, in case of totally umbilical submanifolds, we have [16]
hl (X, ξ) = 0,
hs (X, ξ) = 0,
A∗ξ ξ´ = 0,
AW ξ = 0.
(3.5)
Then, from (1.50) by replacing X by ξ we have
Ds (ξ, N ) = 0.
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(3.6)
Definition:([14]) A lightlike submanifold M of M̄ is said to be totally geodesic
if any geodesic of M with respect to an induced connections ∇ is a geodesic of
¯
M̄ with respect to the Levi-Civita connection ∇.
Theorem (3.2.3):([14]) Let (M, g, S(T M ), S(T M ⊥ )) be a lightlike submanifold
of (M̄ , ḡ). Then the following assertions are equivalent:
(i) M is totally geodesic.
(ii) hl and hs vanish identically on M .
(iii) A∗ξ vanishes identically on M , for any ξ ∈ Γ(Rad(T M )), AW is Γ(Rad(T M ))valued for any W ∈ Γ(S(T M )) and Dl (X, SV ) = 0 for any X ∈ Γ(T M ) and
V ∈ Γ(tr(T M )) .
To prove the theorems, the following lemmas are required.
Lemma (3.2.4): Let (M, g, S(T M ), S(T M ⊥ )) be a totally umbilical submanifold of a semi-Riemannian manifold M̄ . Then ∇lX Hl = 0 if and only if ∇X hl = 0.
Proof Let X, Y, Z ∈ Γ(T M ). Then by(1.58) and (3.2), we have
(∇X hl )(Y, Z) = ∇lX (g(Y, Z)Hl ) − g(∇X Y, Z)Hl − g(Y, ∇X Z)Hl
= Xg(Y, Z)Hl + g(Y, Z)∇lX Hl − g(∇X Y, Z)Hl − g(Y, ∇X Z)Hl
= (∇X g)(Y, Z)Hl + g(Y, Z)∇lX Hl ,
(3.7)
(∇X hl )(Y, Z) = g(Y, Z)∇lX Hl .
(3.8)
or
Lemma (3.2.5): Let (M, g, S(T M ), S(T M ⊥ )) be a totally umbilical submanifold of a semi-Riemannian manifold M̄ . Then ∇sX Hs = 0 if and only if ∇X hs = 0.
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Proof Let X, Y, Z ∈ Γ(T M ). Then by(1.59) and (3.3), we have
(∇X hs )(Y, Z) = ∇sX (g(Y, Z)Hs ) − g(∇X Y, Z)Hs − g(Y, ∇X Z)Hs
= Xg(Y, Z)Hs + g(Y, Z)∇sX Hs − g(∇X Y, Z)Hs − g(Y, ∇X Z)Hs
= (∇X g)(Y, Z)Hs + g(Y, Z)∇sX Hs ,
(3.9)
or
(∇X hs )(Y, Z) = g(Y, Z)∇sX Hs .
(3.10)
Lemma (3.2.6):([20]) Let (M̄ , ḡ) be a semi-Riemannian manifold.
If
g(R̄(X, Y )Z, X) = 0 for every orthonormal triple X, Y, Z ∈ Γ(T M ), then M̄
has constant sectional curvature.
Proof of theorem (3.2.1): At an arbitrary point p ∈ M let X, ξ, V be orthonormal at p. Let T be a r-dimensional subspace of Tp (M̄ )|M containing X
and ξ, transversal to V . By the axiom there exists an r-dimensional totally
umbilical lightlike submanifold M with parallel transversal curvature H such
that Tp (M ) = T . Since the transversal curvature vector field is parallel, i. e.,
∇tX H = 0, by (3.4) and Lemma 3.2.1 and Lemma 3.2.2, we have (∇X hl ) = 0 and
(∇X hs ) = 0 or in particular
(∇X hl )(X, X) = 0, (∇X hl )(ξ, X) = 0, (∇X hs )(X, X) = 0, (∇X hs )(ξ, X) = 0.
(3.11)
Now form (1.60), for X, ξ ∈ Γ(T M ) the transversal form of R̄(X, ξ)X is as below
(R̄(X, ξ)X)N = (∇X hl )(ξ, X) − (∇ξ hl )(X, X) + (∇X hs )(ξ, X)
−(∇ξ hs )(X, X) + Dl (X, hs (ξ, X)) − Dl (ξ, hs (X, X))
+Ds (X, hl (ξ, X)) − Ds (ξ, hl (X, X)).
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(3.12)
Since M is totally umbilical therefore by using (3.2), (3.6) and (3.11) in (3.12), we
obtain (R̄(X, ξ)X)N = 0. Hence g(R̄(X, ξ)X, V ) = 0 then the Theorem follows
from the Lemma 3.2.3.
Proof of theorem (3.2.2): Since there exists a r-dimensional totally geodesic
lightlike submanifold on M therefore from Theorem (E) and (3.12) we have
(R̄(X, ξ)X)N = 0. Hence, g(R̄(X, ξ)X, V ) = 0, then the Theorem follows from
the Lemma 3.2.3.
Remark: From (1.61), (1.62) and (1.63) it is clear that a semi-Riemannian
manifold with coisotropic, isotropic or totally lightlike submanifolds, is a real
space form if it satisfies the axiom of r-planes and spheres.
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