GEOMETRY OF SLANT
SUBMANIFOLDS
BY
HANAN RABEH ALOHALI
A THESIS
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
IN THE DEPARTMENT OF MATHEMATICS AT THE COLLEGE OF
SCIENCE
KING SAUD UNIVERSITY
1426-2005
ACKNOWLEDGEMENTS
To my wonderful parents and my husband who set me on my path
First of all, I thank my God who gave me all what I have. I would like
to express my great appreciation and gratefulness to Dr. Sharief Deshmukh
who guided me throughout my studies and devoted much of this precious time
to help me in preparing this thesis. I wish to thank King Saud University,
particularly the Mathematics Department of the Faculty of Science, for giving
me the opportunity for graduate study. I’m greatly indebted and grateful to
my parents for their invaluable help and support. To say thank you to my
husband is the least thing I can do to show my appreciation for his great
e¤orts. He, a winter cloud, was always there to help, support, encourage and
bless my steps.
I must record my sincere thanks to my brothers and sisters for encouragement and the assistance they extended to me as well as my friends for their
advice, my colleagues were all kindly helpful to me.
Table Of Contents
Preface
Chapter-1, Introduction
i
1
1.1 Riemannian Manifolds
1
1.2 Complex Manifolds
7
1.3 Submanifolds
Chapter-2, Slant Submanifolds
10
15
2.1 Preliminaries
15
2.2 Examples
25
2.3 Properties of operators P and F
29
Chapter-3, Existence theorem for slant Submanifolds
3.1 Preliminaries
3.2 Existence Theorem
40
40
46
Chapter-4, H-umbilical Slant Submanifolds
4.1 Some Lemmas
54
54
4.2 Classi…cation of H-umblical slant sumbmanifolds
59
4.3 A general inequality for Kaehlerian slant submanifolds
69
References
73
PREFACE
For a given Riemannian manifold (M; g), to study its geometry, some times
it is convenient to immerse it in some Riemannian manifold with known geometry and then analyze the induced geometry. This lead to the study of geometry
of submanifold. In that, most interesting is the geometry of submanifold of
a complex manifold owing to the existence of complex structure which sends
each vector to a vector which is orthogonal to the original vector. This leads
to the study of two important classes of submanifolds of a complex manifold
namely invariant and anti-invariant submanifolds. These submanifolds are
generalized by B.Y. Chen by introducing what are known as Slant submanifolds [3]. This thesis is devoted to the study of Slant submanifolds and all the
results are taken from the papers cited in [4], [5], [6], [8] and [9].
The thesis is divided into four chapters and each chapter is divided into
subsections and the results in each section are numbered as (a.b.c), for instance
Theorem a.b.c, means Theorem number c in the section b of chapter a.
The …rst chapter is introductory and is basically intended to make the
thesis as self-contained as possible. In this chapter we gave basic de…nitions
and summarized the basic formulae and results on Riemannian manifolds,
complex manifolds and submanifolds, which are essential for the other three
chapters.
The second chapter is devoted to the study of the slant submanifolds In the
i
…rst preliminary section, we introduce the notion of a slant submanifold of an
almost Hermitian manifold and the operators P and F which arise naturally
from the de…nition of a slant manifold as well as prepare the basic fundamental
equations of submanifold theory in terms of moving frame. In section two we
give some examples of slant submanifolds and in the last section we study
properties of the operators P and F and the results on the geometry of slant
submanifolds with restrictions on these operators.
In chapter three we are interested in slant submanifolds of a complex space
form (A Kaehler manifold of constant holomorphic sectional curvature c). In
…rst section we prepare for basic equations of submanifold theory for slant
submanifolds of complex space form and in then in second section we study
the existence theorem and uniqueness theorem for slant submanifolds in these
spaces.
In last chapter, we are interested in deriving an inequality satis…ed by the
mean curvature vector of a slant submanifold of a complex space form. First we
prepare some Lemmas in the …rst section for slant submanifolds of a complex
space form. In second section we use the Lemmas in …rst section to prove
a classi…cation theorem for H-umbilical slant submanifolds of complex space
forms. In last section we prove and inequality for the length of mean curvature
vector of a slant submanifold of a complex space form. This inequality is an
import result in it self as it gives an estimate for the length of mean curvature
vector, however it is used in classifying cylindrical slant submanifolds of a
complex space form which we do not consider here as it requires some more
preparation not intended in this thesis.
ii
The thesis ends with a list of references, which by no means is exhaustive
on the subject, but lists only those references which have either been directly
used in the thesis or have relevance to our work.
iii
1
SLANT SUBMANIFOLDS
CHAPTER I
INTRODUCTION
In this introductory chapter, we will describe basic de…nitions, results and
formulas which are related to our subsequent chapters. Throughout this thesis,
we will denote by Tp M the tangent space to M at p 2 M; by X (M ) the liealgebra of the smooth vector …eld on M and by C 1 (M ) the ring of smooth
functions on M: The di¤erential of the map f : M ! N at p 2 M is denoted
by dfp which is the linear map dfp : Tp M ! Tf (p) N:
1.1 RIEMANNIAN MANIFOLDS
In this section we will discuss the connection on a manifold, a Riemannian
metric, the Riemannian connection on a Riemannian manifold and the properties of curvature tensor, Ricci tensor and scalar curvature. Moreover we
will discuss some special types of spaces such as space of constant curvature,
followed by some examples.
De…nition 1.1.1 Let f : M ! N be a smooth map. Then
1) f is called an immersion if dfp : Tp M ! Tf (p) N is one-to-one map for
all p 2 M .
2) f is called imbedding if f is one-to-one immersion.
3) f is called di¤eomorphism if f is a bijection and f
1
is smooth.
2
HANAN ALOHALI
Now we state the Implicit function theorem.
Theorem 1.1.2 Suppose f : M ! N is a smooth map and that dfq :
Tq M ! Tp N
is onto for all q 2 M with f (q) = p. Then the set F =
fq 2 M : f (q) = pg is a smooth manifold and dim F = dim M
dim N , more-
over the inclusion i : F ! M is an imbedding.
De…nition 1.1.3 Let M be an n-dimensional smooth manifold. A Riemannian metric on M is a tensor …eld g of type (0; 2) which satis…es:
(i) g is symmetric that is g(X; Y ) = g(Y; X); 8 X; Y 2 X(M ):
(ii) for each p 2 M , gp is positive de…nite non-degenerate bilinear form on
Tp M . That is gp (Xp ; Yp )
0; 8 Xp 2 Tp M; gp (Yp ; Yp ) = 0 implies Yp = 0 and
if gp (Xp ; Yp ) = 0; 8Yp 2 Tp M then Xp = 0:
A smooth manifold M together with a given Riemannian metric g is called
a Riemannian manifold, and denoted by (M; g).
Example 1.1.4 Let h : Rn ! R be map given by h(x1 ;
1). Then 0 is a regular value of h and h
Sn
1
1 (0)
= xi 2 Rn : x21 +
; xn ) =
+
is the unit sphere of Rn . The metric induced from Rn on S n
the canonical metric of S n
1,
n
P
(x2i
i=1
x2n =
1
1 =
is called
which is a Riemannian metric.
De…nition1.1.5 (i) Let M be a Riemannian manifold of dimension m =
n + k, and let us suppose that to each p 2 M is assigned an n-dimensional
subspace Dp of Tp M . Suppose moreover that in a neighborhood U of each
p 2 M there are n-linearly independent smooth vector …elds X1 ;
; Xn ,
3
SLANT SUBMANIFOLDS
whose values at the point q form a basis of Dq for every q 2 U . Then we shall
say that D is a smooth distribution of dimension n on M , and X1 ;
; Xn is a
local basis of D, and for a vector …eld X 2 X(M ), if Xp 2 Dp for each p 2 M ,
we say X 2 D:
(ii) We shall say that the distribution D is involutive if [X; Y ] 2 D ; 8
X; Y 2 D.
(iii) Finally, if D is a smooth distribution on M , we say that D is integrable,
if for each p 2 M , there exists an n-dimensional submanifold N containing p
of M such that Tq N = Dq ; 8q 2 N , and then N is called a leaf of D passing
through the point p.
Now we state the following theorem of Frobenius.
Theorem 1.1.6 A distribution D on a manifold M is integrable if and
only if it is involutive.
Next we introduce a connection on a smooth manifold.
De…nition 1.1.7 A connection r on a smooth manifold M is a map
r : X (M )
X(M ) ! X (M ), (X; Y ) ! rX Y , which satisfy the following:
(i) rX (Y + Z) = rX Y + rX Z
(ii) r(X+Y ) Z = rX Z + rY Z
(iii)rf X Y = f rX Y
4
HANAN ALOHALI
(iv)rX (f Y ) = X (f ) Y + f rX Y; X; Y; Z 2 X (M ), f 2 C 1 (M ).
; xn ) be the coordinate system on Rn . Then for
n
n
P
P
@
@
g i @x
f i @x
;
Y
=
; f i ; g i 2 C 1 (Rn ). De…ne
X,Y 2 X (Rn ) ; we have X =
i
i
Example 1.1.8 Let (x1 ;
i=1
r : X (Rn )
X (Rn ) ! X (Rn ) by rX Y =
i=1
n
P
X gi
i=1
@
@xi ,
then it can be easily
veri…ed that r satis…es the requirement for a connection. This connection r
on Rn is known as the Euclidean connection.
Remark 1.1.9 On a smooth manifold M there could be several Riemannian metrics (once existence of one is known), for instance if g is a Riemannian metric on M and f : M ! M is a non-singular smooth map (that is
the matrix (dfp ) is non-singular of each p 2 M ), then g (X; Y ) = f g (X; Y ) =
g (df (X) ; df (Y )) is also a Riemannian metric on M .
De…nition 1.1.10 (i) A connection r on Riemannian manifold (M; g)
with Xg (Y; Z) = g (rX Y; Z) + g (Y; rX Z) ; 8X; Y; Z 2 X (M ) is called connection compatible to the metric g.
(ii) A connection r on a smooth manifold M is said to be symmetric, if
rX Y
rY X = [X; Y ], X; Y 2 X (M ).
Theorem(Levi-civita) 1.1.11 Given a Riemannian manifold (M; g) there
exists a unique connection r on M satisfying:
(i) r is symmetric
(ii) r is compatible with g.
5
SLANT SUBMANIFOLDS
The unique connection r on the Riemannian manifold (M; g) satisfying
(i) & (ii) in (1.1.11) is called the Riemannian connection.
De…nition 1.1.12 For a connection r on a smooth manifold M , there is
associated a tensor …eld R of type (1; 3) called the curvature tensor …eld of
the connection r de…ned by
R (X; Y ) Z = rX rY Z
rY rX Z
r[X;Y ] Z,
X; Y; Z 2 X (M )
where [X; Y ] is the Lie-bracket of the vector …elds X, Y .
The properties of the curvature tensor R of the Riemannian connection r
on (M; g) are summarized in the following :
Theorem 1.1.13 The curvature tensor R of the Riemannian connection
r on a Riemannian manifold (M; g) satis…es:
(i) R (X; Y ) Z + R (Y; X) Z = 0.
(ii) R (X; Y ) Z + R (Y; Z) X + R (Z; X) Y = 0
(iii) R (X; Y ; Z; W ) = R (Z; W ; X; Y ),
X; Y; Z; W 2 X (M ),
where R (X; Y ; Z; W ) = g (R (X; Y ) Z; W ).
De…nition 1.1.14 Let P be a plane section on (M; g): The sectional curvature of the plane P is de…ned by
K (P ) =
R (X; Y ; Y; X)
X ^Y
6
HANAN ALOHALI
where X, Y 2 X(M ) are vector …elds which span P , and X^Y = g (X; X) g (Y; Y )
g(X; Y )2 .
If K (P ) is a constant c for all plane sections P on M , then M is said to
have constant sectional curvature c.
For constant sectional curvature manifolds, we will have a simple formula
for R given in the following theorem.
Theorem 1.1.15 If the Riemannian manifold (M; g) is of constant sectional curvature c, then its curvature tensor …eld is given by
R (X; Y ) Z = c fg (Y; Z) X
De…nition 1.1.16 Let fe1 ;
g (X; Z) Y g ,
X; Y; Z 2 X (M )
; en g be a local orthonormal frame on a
Riemannian manifold (M; g) i.e. it satis…es g (ei ; ej ) = ij , where ij =
1, i = j
: If R is the tensor of type (0; 4) described in 1.1.13, then the
0, i 6= j
Ricci tensor …eld Ric of M is de…ned by:
Ric (X; Y ) =
n
X
i=1
R (ei ; X; Y; ei ) ; X; Y 2 X (M )
and the scaler curvature S of M is the trace of the Ricci tensor, that is , S is
n
P
de…ned by : S =
Ric (ei ; ei ) :
i=1
Example 1.1.17 Consider (Rn ; g) where g is the Euclidean inner product
n
n
n
P
P
P
@
@
,
Y
=
g i @x
on Rn de…ned by g (X; Y ) =
g i f i , where X =
f i @x
i,
i
i=1
i=1
i=1
f i ; g i 2 C 1 (Rn ). Then according to the Euclidean connection de…ned in
7
SLANT SUBMANIFOLDS
example 1.1.8 it can be easily veri…ed that r is a Riemannian connection
with respect to g. The curvature tensor …eld R will given by R (X; Y ) Z =
n
n
n
n
P
P
P
P
@
@
@
@
Y X hi @x
[X; Y ] hi @x
hi @x
XY hi @x
i
i
i = 0 , where Z =
i.
i=1
i=1
i=1
i=1
Thus we get that g (R (X; Y ) Z; W ) = 0, X; Y; Z; W 2 X (Rn ), which means
that K (P ) = 0, for any plane section P of Rn and thus Rn is of constant
sectional curvature zero.
1.2 COMPLEX MANIFOLDS
In this section we discuss di¤erent classes of almost complex manifolds.
De…nition 1.2.1 An almost complex structure on a smooth manifold M is
a tensor …eld J of type (1; 1) which is, at every point p of M , an endomorphism
of Tp M such that Jp2 =
Ip , where Ip denotes the identity transformation of
Tp M . A smooth manifold M together with an almost complex structure J
is called an almost complex manifold (M; J). A complex manifold M is an
analytic manifold where in the de…nition of analytic structure we replace the
real Euclidean space Rn with the complex Euclidean space C n .
Remark 1.2.2 Clearly from de…nition 1.2.1, it follows that an almost
complex manifold is of even dimension and that an n-dimensional complex
manifold is a 2n-dimensional real analytic manifold.
Example 1.2.3 R2n , S 2 , CP n and T M the tangent bundle of a smooth
manifold M with a given connection are almost complex manifolds, of which
R2n = C n , S 2 and CP n are complex manifolds. The natural almost complex structure J of R2n is de…ned by J
@
@xi
=
@
;
@y i
J
@
@y i
=
@
,
@xi
where
8
HANAN ALOHALI
x1 ; :::; xn ; y 1 ; :::; y n are Euclidean coordinates on R2n .
De…nition 1.2.4 A Hermitian metric on an almost complex manifold
(M; J) is a Riemannian metric g such that g (JX; JY ) = g (X; Y ), X; Y 2
X (M ), that is, the Riemannian metric g and J are compatible.
An almost complex manifold (respectively a complex manifold) with Hermitian metric is called an almost Hermitian manifold (M; J; g) (respectively
a Hermitian manifold). Notice that every almost complex manifold M with
a Riemannian metric g admits an almost Hermitian metric. Indeed, for any
almost complex structure J on M with a Riemannian metric g, putting
h (X; Y ) = g (X; Y ) + g (JX; JY ) ; X; Y 2 X (M )
we obtain an almost Hermitian metric h.
De…nition 1.2.5 An almost Hermitian manifold (M; J; g) is called a Kaehler
manifold if the almost complex structure J of M is parallel, that is,
(rX J) (Y ) = 0; X; Y 2 X (M )
It is easy to verify that the Euclidean metric on R2n is Hermitian metric
with respect to the natural almost complex structure J de…ned above and that
R2n is a Kaehler manifold.
Let (M; J; g) be a Kaehler manifold. Let K (P ) be the sectional curvature
9
SLANT SUBMANIFOLDS
of M for a plane P
Tq M spanned by orthonormal vectors X, Y . If P is
invariant under J; that is, Jq P = P , then K (P ) is called holomorphic sectional
curvature. If P is invariant under J and X is a unit vector in P , then fX; JXg
is orthonormal basis for P and hence K (P ) = R (X; JX; JX; X) ; we denote
it by H (X). If K (P ) is constant for all J-invariant planes P
Tq M , for
all points q of M , then M is called space of constant holomorphic sectional
curvature.
We have the following characterization for the spaces of constant holomorphic sectional curvature.
A Kaehler manifold (M; J; g) of constant holomorphic sectional curvature c
is called a complex space form and is denoted by M (c). The curvature tensor
R of M (c) is given by
R (X; Y ) Z =
c
fg (Y; Z) X
4
g (X; Z) Y + g (JY; Z) JX
g (JX; Z) JY
+ 2g(X; JY )JZg
We denoted by C n the complex space, CP n the complex projective space
and Dn the open unit ball in C n of complex dimension n. It is known that a
simply connected complete Kaehler manifold of constant holomorphic sectional
curvature c can be identi…ed with CP n , Dn or C n according as c > 0, c < 0 or
c = 0.
10
HANAN ALOHALI
1.3 SUBMANIFOLDS
Given two smooth manifolds M and M , if there exists a smooth immersion
: M ! M , then we say that M is a submanifold of M . If
is imbedding,
then M is said to be an imbedded submanifold of M . The geometry of submanifolds is a very useful branch in geometry, for some times it is convenient
to immerse the given manifold into a known Riemannian manifold (if it is possible1 ) and then study the induced geometry on this manifold as submanifold
of the known Riemannian. In fact the study of Di¤erential Geometry started
with the geometry of the submanifolds. For instance the regular smooth curves
in R3 are 1-dimensional submanifolds of R3 as well as the regular surfaces in
R3 are its 2-dimensional submanifolds. Let M ; g be a Riemannian manifold
and
: M ! M be a smooth immersion of a smooth manifold M into M .
Then the submanifold M receives the induced Riemannian metric g de…ned
by
gp (Xp ; Yp ) = g d
p Xp ; d p Yp
, X; Y 2 X (M ) , p 2 M
Since an immersion is a local imbedding, when we are dealing with local expressions on a submanifold M of M we shall identify d
, X 2 X (M ) and
p Xp
by Xp
(p) by p, p 2 M , and distinguish the entries on the
manifolds M and M from the context. Thus we shall also denote by g the
induced Riemannian metric on M ; and we shall say M is a submanifold of a
Riemannian manifold M without referring to the immersion.
1
It is known that the real projective space RP 3 dose not admit any smooth immersion
in R4 which induces the usual metric on RP 3 of constant positive curvature.
11
SLANT SUBMANIFOLDS
Let M be a submanifold of a Riemannian manifold M ; g . Then for each
p 2 M , the tangent space Tp M of M is the direct sum Tp M = Tp M
Tp? M ,
where Tp M is the tangent space of M and
Tp? M = Xp 2 Tp M = gp (Xp ; Yp ) = 0; Yp 2 Tp M
is the orthogonal complement of Tp M in Tp M . the subspace Tp? M of Tp M is
said to be the normal space of M at p 2 M . the bundle T ? M = [ Tp? M
p2M
over M is called the normal bundle of M . We denote by
sections of the normal bundle T ? M . Thus N 2
the space of smooth
implies N is a smooth vector
…eld in X M and satis…es g (X; N ) = 0, X 2 X (M ). Though it is abuse of
term, but
it self is referred to as normal bundle (instead of space of sections
on the normal bundle) in the literature on the geometry of submanifolds and
so in this thesis.
Let r be the Riemannian connection on the Riemannian manifold M ; g
and M be a submanifold of M . Then using X M jM = X (M )
, where
X M jM is the restriction of X M to M , for X; Y 2 X (M ) ; as rX Y 2
X M , we express rX Y as
rX Y = rX Y + h (X; Y )
where rX Y 2 X (M ) and h (X; Y ) 2
(1.3.1)
are respectively the tangential and
normal components of rX Y . At this stage the terms rX Y and h (X; Y ) are
merely the notations till their properties are revealed to distinguish them geometrically . Utilizing the properties of the Riemannian connection r as given
in de…nition 1.1.7, it easily follows that the symbol r appearing in equation
12
HANAN ALOHALI
(1.3.1) satis…es the properties required by the Riemannian connection on M .
In this veri…cation it also turns out that the normal component in equation
(1.3.1) gives rise to the bilinear symmetric mapping h : X (M )
X (M ) !
called the second fundamental form of the submanifold M . Similarly, for
X 2 X (M ) ; N 2
as rX N 2 X(M ), we can express it as
rX N =
where
AN X + DX N
AN X 2 X (M ) and DX N 2
(1.3.2)
are respectively the tangential and
normal components of rX N . Using the properties of r, it turns out that
the tangential component
AN X in equation (1.3.2) gives rise to the linear
mapping AN : X (M ) ! X (M ) satisfying g (AN X; Y ) = g (X; AN Y ); and
D : X (M )
!
satis…es the properties similar to those for a connection.
The mapping AN is called the Weingarten map with respect to the normal
N2
and D is called the connection in the normal bundle .
Taking inner product by Y 2 X (M ) in the equation (1.3.2) and using
g (Y; N ) = 0, that is, g rX Y; N =
g Y; rX N , and equation (1.3.1), we
obtain
g (h (X; Y ) ; N ) = g (AN X; Y ) ; X; Y 2 X (M ) ; N 2
The fundamental equations (1.3.1) and (1.3.2) for a submanifold M of a
Riemannian manifold M are called the Gauss and Weingarten formulae.
Let R and R be the curvature tensors of M and M respectively. then using
13
SLANT SUBMANIFOLDS
the Gauss and Weingarten formulae we obtain the following equation relating
the curvature tensors R and R.
R (X; Y ; Z; W ) = R (X; Y ; Z; W ) + g (h (X; Z) ; h (Y; W ))
g (h (Y; Z) ; h (X; W ))
(1.3.3)
X; Y; Z; W 2 X (M ) :
Example 1.3.1: Let S n be a unit sphere centered at origin in Rn+1 . Then
the inclusion i : S n ! Rn+1 is an immersion, and thus S n is a submanifold
of Rn+1 with each normal space is 1-dimensional and thus the normal bundle
is spanned by a single unit normal vector …eld N . Since N represents the
position vector …eld of each point on S n and the Riemannian connection r on
Rn+1 is de…ned in example 1.1.8, we have, for each X 2 X (S n ), that
rX N =
that is, AN X =
n
X
X (xi )
i=1
@
=X
@xi
X and DX N = 0, where N =
n
P
i=1
…eld on S n .
@
is the position vector
xi @x
i
Also, h (X; Y ) = g (h (X; Y ) ; N ) N = g (AN X; Y ) N =
g (X; Y ) N , X; Y 2
X (S n ). Thus the Gauss and Weingarten formulae for S n are
rX Y = rX Y
g (X; Y ) N
and rX N = X. Since the curvature tensor of Rn+1 , R = 0, the curvature
14
HANAN ALOHALI
tensor R of S n as given in equation 1.3.2 is
R (X; Y ; Z; W ) = g (Y; Z) g (X; W )
g (X; Z) g (Y; W ) ; X; Y; Z; W 2 X (S n ) .
15
SLANT SUBMANIFOLDS
CHAPTER 2
In this chapter we introduce the notion of a slant submanifold in an
almost Hermitian manifold. In section 2.1 we formulate the basic equations
of submanifold in terms of local frame and state the fundamental equations.
In section 2.2 we give some examples of slant submanifolds and in the last
section 2.3 we obtain some fundamental properties of the tensors P and F for
the submanifolds of an almost Hermitian manifolds and use them to study the
geometry of slant submanifolds. Section 2.3 is taken mostly from [3] and [4].
2.1 PRILIMINARIES
let N be an n-dimensional Riemannian manifold isometrically immersed
in an almost Hermitian manifold M with almost complex structure J and
almost Hermitian metric g . We denote by h; i the inner product for N as well
as for M . For any vector X tangent to N we put
JX = P X + F X
(2.1.1)
where P X and F X are the tangential and the normal components of JX
respectively . Thus, P is an endomorphism of the tangent bundle T N and
F a normal-bundle-valued 1-form on T N . The submanifold N is called a
complex submanifold if F = 0 and is called a totally real submanifold if
P = 0, and called proper if it is neither a complex submanifold nor a totally
real submanifold.
16
HANAN ALOHALI
De…nition 2.1.1 Let M ; J; g be an almost Hermitian manifold and M
be a submanifold of M , if for each p 2 M and Xp 2 Tp M the angle
between JXp and the tangent space Tp M is constant
(Xp )
then M is called a
slant submanifold.
Remark 2.1.2 If M is a slant submanifold of M ; J; g and p 2 M , then
for a local orthonormal frame fe1 ; ::::::; en g of M we should have jhJei ; ej ij =
cos
=constant.
> 0, consider f : R2 ! R4 de…ned by
Example 2.1.3 For any
f (u; v) = (u cos ; u sin ; v; 0)
then at any point p of R2 , we have
3
0
0 7
7
1 5
0
2
cos
6 sin
dfp = 6
4 0
0
Let fe1 ; e2 g be a local orthonormal frame on R2 , then we can choose it as
e1 =
@
@u
@
@u
dfp
dfp
=
cos
sin
0 0
and
e2 =
where dfp
@
@u
= dfp
dfp
dfp
@
@v
@
@v
@
@v
=
0 0 1 0
= 1. Let J0 be the natural almost complex
structure of R4 as de…ned in chapter 1. Then we have
J0 e1 =
0 0 cos
sin
17
SLANT SUBMANIFOLDS
1 0 0 0
J0 e2 =
Then we see that hJ0 e1 ; e2 i = cos , hJ0 e2 ; e1 i =
cos , jhJ0 ei :ej ij = cos
>
0, which is a constant. This implies that R2 is slant submanifold of R4 , so f
in R4 .
is de…nes a slant plane with slant angle
Let M be an n-dimensional submanifold in an m-dimensional Riemannian
manifold M . Choose a local orthonormal frames e1 ; :::; en ; en+1 ; :::em such
that, restricted to N , the vectors e1 ; :::; en are tangent to M and en+1 ; :::; em
are normal to M . Let 1
A; B; C; :::
m and 1
i; j; k; :::
n. Suppose
A =
that w1 ; :::; wn ; wn+1 ; :::; wm be the dual frame. Then we have wB
A + w B = 0 where w A (X) = hX; e i, X 2 X M and 1
wB
A
A
A
B )
wA
m: De…ne
A by
the connection forms wB
rX eB =
m
X
C
wB
(X) eC
C=1
where r is Riemannian connection on M .
For any vector …eld X 2 X(M ) these forms are also given by
rX ei = T + N
0
0
where T 2 X(M ) is the tangential component and N 2 ( ) is the norP j
P r
mal component. If we write rX ei =
ej +
er , then we obtain j =
r
j
rX ei ; ej = wji (X) and
r
rX ei =
= rX ei ; er = wri (X). Thus we have
n
X
j=1
wji
(X) ej +
X
r=n+1
m
wri (X) er
18
HANAN ALOHALI
and
rX er =
These 1-forms
wji ; wri
and
X
wjr (X) ej +
wsr (X) es
s=n+1
j=1
wsr
X
are called the connection forms of M in M .
Lemma 2.1.4
dwA =
X
B
A
wB
^ wB
Proof : For X; Y 2 X M we have
dwA (X; Y ) = X wA (Y )
Y wA (X)
wA ([X; Y ])
and
A
A
wB ^ wB
(X; Y ) = wB (X) wB
(Y )
A
wB (Y ) wB
(X)
Thus we arrive at
X
B
A
wB ^ wB
(X; Y ) =
X
B
A
wB
^ wB (X; Y )
and
X
B
A
wB ^ wB
(X; Y ) = dwA (X; Y )
which proves
dwA =
X
B
A
B
Now, suppose that
A
wB
^ wB
be the curvature 2-form on M , then the structure
equations of M are given by
A
B
(X; Y ) =
1X A
KBCD wC ^ wD (X; Y )
2
C;D
19
SLANT SUBMANIFOLDS
A
where, KBCD
= R (eA ; eB ; eC ; eD ). Thus we have
A
B
(X; Y ) =
1X
R (eA ; eB ; eC ; eD ) wC (X) wD (Y )
2
wC (Y ) wD (X)
C;D
= R (X; Y; eA ; eB )
Next, we prove
Lemma 2.1.5
A
dwB
=
X
C
A
C
wC
^ wB
+
A
B
Proof : For X; Y 2 X(M ), we compute
X
C
A
C
wC
^ wB
(X; Y ) +
A
B
(X; Y ) =
X
A
C
wC
(X) wB
(Y )
C
X
A
C
wC
(Y ) wB
(X)
C
+R (X; Y; eA ; eB )
X
=
rX eA ; eC rY eC ; eB
C
X
C
rY eA ; eC
rX eC ; eB
+R (X; Y; eA ; eB )
X
=
rX eA ; eC Y heC ; eB i
C
X
C
rY eA ; eC
+R (X; Y; eA ; eB )
X heC ; eB i
eC ; rY eB
eC ; rX eB
20
HANAN ALOHALI
X
C
As heC ; eB i =constant, we get Y heC ; eB i = 0, consequently that
A
C
wC
^ wB
(X; Y ) +
A
B
(X; Y ) =
hrX eA ; rY eB i + hrY eA ; rX eB i
+R (X; Y; eA ; eB )
=
hrX rY eA
rY rX eA ; eB i
A
Y wB
(X)
A
+XwB
(Y ) + R (X; Y; eA ; eB )
=
R (X; Y ) eA + r[X;Y ] eA ; eB
A
Y wB
(X)
A
+XwB
(Y ) + R (X; Y; eA ; eB )
A
= XwB
(Y )
A
Y wB
(X)
A
wB
([X; Y ])
A
= dwB
[X; Y ]
This proves that
A
dwB
=
X
C
A
C
wC
^ wB
+
A
B
A
A
where KBCD
+ KBDC
= 0.
Now, we restrict these forms to the submanifold M . Then we have wr = 0,
which gives
0 = dwr =
X
i
P
P
r
wir ^ wi
Let fN1 ; :::Nr gbe the local frame for the normals. Then 8 2
h ;N iN
as en+1 ; :::; em are normal to M , so for any
h ; er i er , n + 1
r
2
( ),
( ),
=
=
m where en+1 ; :::; em are normal to M . For the
second fundamental form h : X(M ) X(M ) ! ( ), we have h (ei ; ej ) 2 ( ),
P
and consequently h (ei ; ej ) = hrij er , which gives hh (ei ; ej ) ; er i = hrij . Also
r
we have
21
SLANT SUBMANIFOLDS
P r j
P
hij w (X) = hh (ei ; ej ) ; er i hX; ej i
j
=
j
*
h ei ;
P
j
hX; ej i ej
!
; er
+
= hh (X; ei ) ; er i = hei ; Aer Xi,
where Aer is Weingarten map and as DX er 2
( ), rX er =
Aer X + DX er
So hei ; DX er i = 0 , where r be connection on M . Thus we have
X
j
hrij wj (X) = hei ; Aer Xi
=
rX ei ; er
hei ; DX er i =
ei ; rX er
X hei ; er i = rX ei ; er
= wir (X)
This proves wir =
wir (X) =
=
P r j
hij w , where h is symmetric hrij = hrji . Also we have
j
rX ei ; er = X hei ; er i
ei ; rX er
hei ; Aer X + DX er i = hei ; Aer Xi = hAer ei ; Xi
where we have used the fact that the Weingarten maps are symmetric.
Now , for any X 2 X (M ), if r; r are the Riemannian connections on M
and M respectively. Then the components of curvature tensor …elds R and K
corresponding to r and r satisfy the relation given in the following
Lemma 2.1.6
Rijkl = Kijkl
X
r
hrik hrjl
hril hrjk
22
HANAN ALOHALI
Proof : We have with the choice of above local orthonormal frame
X
hrik hrjl
hril hrjk
=
r
X
r
hh (ei ; ek ) ; er i hh (ej ; el ) ; er i
X
r
hh (ei ; el ) ; er i hh (ej ; ek ) ; er i
= hh (ei ; ek ) ; h (ej ; el )i
hh (ei ; el ) ; h (ej ; ek )i
Also,
K (ei ; ej ) ek = rei rej ek
rej rei ek
Then for any X; Y 2 X (M ) and N 2
r[ei ;ej ] ek
( ), using the equations (1.3.1)
and (1.3.2),
we compute
K (ei ; ej ) ek = rei rej ek
rej rei ek
+Dei h (ej ; ek )
h (ej ; rei ek )
r[ei ;ej ] ek + Ah(ei ;ek ) ej
Ah(ej ;ek ) ei
Dej h (ei ; ek ) + h ei ; rej ek
h ([ei ; ej ] ; ek )
Thus the components of K are given by
Kijkl = K (ei ; ej ; ek ; el ) = hK (ei ; ej ) ek ; el i = hR (ei ; ej ) ek ; el i
D
E
Ah(ej ;ek ) ei ; el
+ Ah(ei ;ek ) ej ; el
= R (ei ; ej ; ek ; el ) + hh (ei ; ek ) ; h (ej ; el )i
X
hrik hrjl hril hrjk
= Rijkl +
hh (ei ; el ) ; h (ej ; ek )i
r
Lemma 2.1.7
r
r
Kskl
= Rskl
+
X
i
(hrik hsil
hril hsik )
23
SLANT SUBMANIFOLDS
Proof : Continuing with the same local orthonormal frame we have wir =
P r j
P A
P
A =
C + A, 1
hij w , and dwB
wC ^ wB
C m. Thus dwir = wjr ^ wij +
B
j
j
C
P
wsr ^ wjs + ri
s
P P r k
hjk w
=
j
=
k
^
wij
P
+ wsr ^
s
P s j
hij w
j
P r
P
hjk wk ^ wij + hsij wsr ^ wj +
dwir =
+
r
i
r
i
j;s
j;k
!
P r j
P
P r
w ^ wk :
hjk wk ^ wij + hsij wsr ^ wj + 12 Kijk
j;s
j;k
and dwsr =
P r
wt ^ wst +
t
r,
s
where
j;k
r
s
=
1P r
Rskl wk
2
k;l
^ wt :
Now
K (er ; es ; ek ; el ) = K (ek ; el ; er ; es )
=
rek rel er
rel rek er
r[ek ;el ] er ; es
and
rek rel er = rek ( Aer el + Del er ) =
h (ek ; Aer el ) + Dek Del er
+ (tangential part)
rel rek er =
h (el ; Aer ek ) + Del Dek er + (tangential part)
r[ek ;el ] er = D[ek ;el ] er + (tangential part)
Thus
K (er ; es ; ek ; el ) = R? (ek ; el ; er ; es ) + hh (el ; Aer ek ) ; es i
= R? (er ; es ; ek ; el ) + hAes el ; Aer ek i
hh (ek ; Aer el ) ; es i
hAes ek ; Aer el i
24
HANAN ALOHALI
where R? is the curvature tensor of the normal connection. Since, Aer ek is
tangential,
Aer ek =
X
i
hAer ek ; ei i ei =
X
i
hh (ei ; ek ) ; er i ei =
X
X
X
hrik ei
i
we get
K (er ; es ; ek ; el ) = R? (er ; es ; ek ; el ) +
i
= R? (er ; es ; ek ; el ) +
X
hrik hAes el ; ei i
hrik hsil
i
X
i
hril hAes ek ; ei i
hril hsik
i
Thus we have
r
r
+
= Rskl
Kskl
X
(hrik hsil
hril hsik )
i
Finally we have the following Lemma which related the curvature tensors
R; R and R? of the connections r; r and D, the …rst equation is called the
equation of Gauss the second is called the equation of Codazzi and the third
is called equation of Ricci
Lemma 2.1.8
R (X; Y ; Z; W ) = R (X; Y ; Z; W ) + hh (X; Z) ; h (Y; W )i
hh (X; W ) ; h (Y; Z)i
(i)
R (X; Y ) Z
?
= rX h (Y; Z)
rY h (X; Z)
RD (X; Y ; ; ) = R (X; Y ; ; ) + h[A ; A ] (X) ; Y i
(ii)
(iii)
where rX h (Y; Z) = DX (h (Y; Z)) h (rX Y; Z) h (Y; rX Z), R (X; Y ) Z
?
denotes the normal component of R (X; Y ) Z, X; Y; Z; W 2 X(M ), , 2 ( ).
25
SLANT SUBMANIFOLDS
De…nition 2.1.9 A submanifold M of a Riemannian manifold M is called
a parallel submanifold if the second fundamental form h is parallel , that is
rh = 0 identically.
2.2 EXAMPLES
In the following , R2m denotes the Euclidean 2m-space with the standard
metric . An almost complex structure J on R2m is said to be compatible if
R2m ; J is complex analytically isometric to the complex number space C m
with standard ‡at Kaehlerian metric. We denote by J0 (the almost complex
structure introduced in chapter 1) and J1 (when m is even ) the compatible
almost complex structure on R2m de…ned respectively by
J0 (a1 ; :::am ; b1 ; :::; bm ) = ( b1 ; :::; bm ; a1 ; :::; am )
J1 (a1 ; :::; am ; b1 ; :::; bm ) = ( a2 ; a1 ; :::; am ; am
1 ; b2 ;
b1 ; :::; bm ; bm
1)
In this section we give some examples of proper slant submanifolds in
C 2 = R4 ; J0 .
Example 2.2.1 Let M be a complex surface in C 2 . Then for any constant
,0<
2,
M is slant surface in R4 ; J
with slant angle , where J is
the compatible almost complex structure on R4 de…ned by
J (a; b; c; d) = (cos ) ( c; d; a; b) + (sin ) ( b; a; d:
c)
Since complex submanifolds in a Kaehler manifold are minimal. This ex-
26
HANAN ALOHALI
ample shows that there exist in…nitely many proper slant minimal surfaces in
C 2 . It is clear that
J (a; b; c; d) = (cos ) J0 (a; b; c; d) + (sin ) J1 (a; b; c; d)
and it is easy to show that J is an almost complex structure on C 2 .
The following example provides us non-minimal proper slant surfaces in
C 2.
Example 2.2.2 For any positive constant k, de…ne f : R2 ! R4 , by
f (u; v) = (u; k cos v; v; k sin v). Then for any point p = (a; b) 2 R2 , we get
2
3
1
0
6 0
k sin b 7
7
dfp = 6
4 0
5
1
0 k cos b
We compute fe1 ; e2 g a local orthonormal frame on R2 as follows: We have
dfp
@
@u p
= 1 , dfp
Thus we choose e1 =
e2 =
@
df ( @v
)
@
kdf ( @v
)k
=
@
@v p
@
df ( @u
)
@
kdf ( @u
)k
p 1
k2 +1
=
p
p
k 2 sin2 b + k 2 cos2 b + 1 = k 2 + 1
= (1; 0; 0; 0) and
(0; k sin v; 1; k cos v).
As J0 e1 = (0; 0; 1; 0), J0 e2 =
hJ0 e1 ; e2 i = p
p 1
k2 +1
1
k2
+1
( 1; k cos v; 0; k sin v) we have
(0 + 0 + 1 + 0) = p
and
hJ0 e2 ; e1 i = p
1
k2 + 1
1
k2
+1
27
SLANT SUBMANIFOLDS
thus
jhJ0 ei ; ej ij = p
1
k2
+1
; 1
i 6= j
2
Choosing e2 = J0 e1 , we conclude that f de…nes a slant surface with slant
angle cos
1
p 1
k2 +1
. It is easy to verify that this surface is also ‡at and
non-minimal.
Example2.2.3 Let
: (a; b) ! R2 ,
(s) = (g (s) ; h (s)) be a unit speed
curve in R2 and k any positive number. Then it can be veri…ed that f :
R
(a; b) ! R4 de…ned by f (u; s) = ( ks sin u; g (s) ; ks cos u; h (s)) is a non-
minimal, ‡at, proper slant surface in R4 .
28
HANAN ALOHALI
2.3 PROPERTIES OF OPERATORS P AND F
Let f : N ! M be isometric immersion of an n-dimensional Riemannian
manifold into an almost Hermitian manifold. Let P and F be the endomorphism and the normal-bundle-valued 1-form on the tangent bundle de…ned in
equation 2.1.1. Since M is almost Hermitian, we have hP X; Y i =
hX; P Y i,
X; Y 2 X(N ). Hence if we put Q = P 2 then Q is a symmetric (self-adjoint)
endomorphism of X(N ).
Therefore, each tangent space Tx N of N at x 2 N admits an orthogonal
direct decomposition of eigenspaces of Q
Tx N = Dx1
Since P is skew-symmetric and J 2 =
and; moreover , if
i
:::
Dxk(x)
I , each eigenvalue
i
of Q lies in [ 1; 0]
6= 0 ,then corresponding eigenspace Dxi is of even dimen-
sion and it invariant under the endomorphism P , that is P Dxi = Dxi . Furthermore, for each
i
6=
1, dim F Dxi = dim Dxi and the normal subspaces
F Dxi ; i = 1; :::; k (x) , are mutually perpendicular. From these arguments,
we have dim M
2 dim N
corresponding to eigenvalue
dim
x,
where
x
denotes the eigenspace of Q
1.
The following lemma follows from the de…nition of rQ which is de…ned by
(rX Q) Y = rX (QY )
for X; Y 2 X(N ).
Q (rX Y )
29
SLANT SUBMANIFOLDS
Lemma 2.3.1 Let N be a submanifold of an almost Hermitian manifold
M . Then the symmetric endomorphism Q is parallel, that is rQ = 0, if and
only if
(i) each eigenvalue
i
of Q is constant on N .
(ii) each distribution Di (associated with the eigenvalue
i)
is completely
integrable and
(iii) N is locally the Riemannian product N1
:::
Nk of the leaves of the
distributions.
Proof. Since Q is a self-adjoint endomorphism of the tangent bundle
T N , there exist n continuous functions
1
2
:::
n
such that
i
; i = 1; :::; n, are the eigenvalues of Q at each point p 2 N . Let e1 ; :::; en
be a local orthonormal frame given by eigenvectors of Q . If Q is parallel rX QY = Q (rX Y ), 8X; Y 2 X (N ), we get rX Qei = Q (rX ei ), that
is X ( i ) ei +
X ( i) +
i rX ei
= Q (rX ei ) take inner product with ei , we arrive at
i hrX ei ; ei i
i hrX ei ; ei i
= hQ (rX ei ) ; ei i. Thus we conclude that X ( i ) +
= hrX ei ; Qei i =
and this proves that
i
i hrX ei ; ei i,
that is X ( i ) = 0 ; X 2 X (N )
is a constant.
For statements (ii) and (iii) we let
1 ; :::;
k
denote the distinct eigenvalues
of Q. For each i = 1; :::; k let Di be the distribution given by the eigenspaces of
Q corresponding to the eigenvalue
Q (X) =
i X;
Q (Y ) =
iY
i.
Then any two vector …elds X; Y 2 Di ,
and consequently we get rX QY = Q (rX Y ) and
rY QX = Q (rY X), that is rX ( i Y )
rY ( i X) = Q ([X; Y ]) or
i [X; Y
]=
30
HANAN ALOHALI
Q ([X; Y ]), which proves that [X; Y ] 2 Di , so Di is completely integrable. Also
we can similarly show that rX Y 2 Di . Therefore each maximal integrable
submanifold Ni of Di is totally geodesic in N . Consequently , N = N1 ::: Nk
For the converse of this statement, let X; Y 2 X (N ) and Y 2 Di 1
. Thus we have QY =
iY
and rX (QY ) = rX ( i Y ) = X ( i ) Y +
i
k
i rX Y
=
0 + Q (rX Y ), that is (rX Q) Y = 0, which proves Q is parallel.
Lemma 2.3.2 Let N be a submanifold of an almost Hermitian manifold
M . Then rP = 0 if and only if N is locally the Riemannian product N1
:::
Nk where each Ni is either a complex submanifold, a totally real submanifold
,or a Kaehlerian slant submanifold of M .
Proof. If P is parallel then Q is parallel. Thus, by applying lemma 2.3.1,
we see that N is locally the Riemannian product N1
of distributions Di corresponding to eigenvalues
constant. If
i
Nk of leaves Ni
Moreover if each
i
is a
= 0, then QX = 0, X 2 X (N ) would imply P X = 0, that is
N is totally real submanifold. If
P2 =
i.
:::
i
=
1 , then P : X (Ni ) ! X (Ni ), satis…es
I, that is P is almost complex structure on Ni and this would imply
Ni is almost complex manifold .
6= 0; 1, then because Di is invariant under the endomorphism P and
p
i
hP X; P Y i =
i jXj :
i hX; Y i for any X; Y 2 D we would havejP Xj =
If
i
Thus the Wirtinger angle (X) satis…es
cos (X) =
p
i
which is a constant 6= 0; 1, therefore ,Ni is a proper slant submanifold.
SLANT SUBMANIFOLDS
Assume
i
31
6= 0, we put Pi = P jT Ni : Then Pi is nothing but the endomor-
phism of T Ni induced from the almost complex structure on totally geodesic
submanifold in N , we have riX Pi Y = (rX P ) Y = 0 for any X; Y 2 X (Ni ),
this shows that if Ni is a complex submanifold. Thus Ni is a Kaehlerian slant
submanifold of M by de…nition.
From above two lemmas we conclude that
Proposition 2.3.3 Let N be an irreducible submanifold of an almost
Hermitian manifold M . If N is neither complex nor totally real, then N is
a Kaehlerian slant submanifold if and only if the endomorphism P is parallel,
that is rP = 0.
Next we prove the following theorem for surfaces in an almost Hermitian
manifold.
Theorem 2.3.4 Let N be a surface in an almost Hermitian manifold M
. Then the following three statement are equivalent
(i) N is neither totally real nor complex in M and rP = 0 that is P is
parallel.
(ii) N is a Kaehlerian slant surface.
(iii) N is a proper slant surface.
Proof. Since every proper slant submanifold is of even dimension, lemma
2.3.2 implies that if the endomorphism P is parallel then N is a Kaehlerian
32
HANAN ALOHALI
surface, or a totally real surface, or a Kaehlerian slant surface. Thus if N is
neither totally real nor complex, then statement (i) and (ii) are equivalent by
de…nition.
It obvious that (ii) implies (iii). Now we prove that (iii) implies (ii):
Let N be a proper slant surface in M with slant angle . If we choose
an orthonormal frame e1 ; e2 tangent to N such that, P e1 = (cos ) e2 , P e2 =
(cos ) e1 . Then we know that rX e1 = w11 (X) e1 + w12 (X) e2 and rX e2 =
w21 (X) e1 + w22 (X) e2 , which implies (rX P ) e1 = 0 and (rX P ) e2 = 0. Thus
rP = 0, that is P is parallel and this implies N is a Kaehlerian slant surface.
For submanifolds of a Kaehlerian manifold we have the following general
lemma.
Lemma 2.3.5 Let N be a submanifold of a Kaehlerian manifold M . Then
(i) For vectors X; Y 2 X (N ), we have
(rX P ) Y = th (X; Y ) + AF Y X
and hence rP = 0 if and only if AF X Y = AF Y X, X; Y 2 X (N )
(ii) For any X; Y 2 X (N ), we have
(rX F ) Y = f h (X; Y )
and hence rF = 0 if and only if Af X =
X 2 X (N ),
h (X; P Y )
A P X, for any
2
( ) and
33
SLANT SUBMANIFOLDS
such that
Jh (X; Y ) = th (X; Y ) + f h (X; Y )
where th (X; Y ) and f h (X; Y ) is the tangential and the normal components of Jh (X; Y )
Proof. Since M is a Kaehlerian, J is parallel. Thus for X; Y 2 X (N )
0 = rX JY
JrX Y = rX (P Y + F Y )
= rX P Y + h (X; P Y )
F (rX Y )
J (rX Y + h (X; Y ))
AF Y X + DX F Y
th (X; Y )
P (rX Y )
f h (X; Y )
Equating tangential and normal components we arrive at (rX P ) Y =
th (X; Y ) + AF Y X and (rX F ) Y = f h (X; Y )
h (X; P Y ). Thus P is parallel
if and only if hth (X; Y ) + AF Y X; Zi = 0, X; Y; Z 2 X (N ), which is equivalent
to hAF Y X; Zi =
hth (X; Y ) ; Zi = hAF X Y; Zi.
Also rF = 0 if and only if hf h (X; Y )
hf h (X; Y ) ; i =
h (X; P Y ) ; i = 0 , hh (X; P Y ) ; i =
hAf Y; Xi
, hh (P Y; X) ; i =
hAf Y; Xi , hA P Y; Xi =
hAf Y; Xi ,
A PY =
Af Y .
Remark 2.3.1 If N is either a totally real or complex submanifold of a
Kaehlerian manifold, then rP = rF = 0, automatically.
Corollary 2.3.6 Let N be a surface in a Kaehlerian manifold M . Then
N is slant if and only if AF Y X = AF X Y , X; Y 2 X (N ).
34
HANAN ALOHALI
Let N be a slant surface in the complex number space C 2 with slant angle
. For a unit tangent vector …eld e1 to N we put
e2 = (sec ) P e1 ; e3 = (csc ) F e1 ; e4 = (csc ) F e2
Then we get e1 =
(sec ) P e2 , and e1 ; e2 ; e3 ; e4 is an orthonormal frame such
that e1 ; e2 2 X (N ) and e3 ; e4 2
( ). As before we put hrij = hh (ei ; ej ) ; er i,
i; j = 1; 2; r = 3; 4. Let R and RD denote the Gauss and normal curvature of
N in C 2 respectively. Then we have
R = h311 h322
h312
2
+ h411 h422
RD = h311 h412 + h312 h422
h312 h411
h412
2
h322 h412
Theorem 2.3.7 If N is a slant surface in C 2 , then R = RD , identically.
Proof. Let N be a slant surface in C 2 . Then corollary 2.3.6 implies
AF Y X = AF X Y , X; Y 2 X (N ). Let e1 ; e2 ; e3 ; e4 be an orthonormal frame
as in the proof of theorem 2.3.4. Then we have
h312 = hh (e1 ; e2 ) ; e3 i = hh (e1 ; e2 ) ; (csc )F e1 i
= (csc ) hh (e1 ; e2 ) ; F e1 i = (csc ) hAF e1 e1 ; e2 i
= (csc ) hAF e1 e2 ; e1 i = (csc ) hAF e2 e1 ; e1 i
= (csc ) hh (e1 ; e1 ) ; F e2 i = hh (e1 ; e1 ) ; e4 i = h411
Similarly, we can prove that h322 = h412 . Therefore, by (3.6) and (3.7), we
obtain R = RD .
35
SLANT SUBMANIFOLDS
In the remaining part of this section we mention some properties of the
normal-bundle valued 1-form F . In order to do so, we recall the following
de…nition .
De…nition 2.3.1 Let N be a submanifold of a Riemannian manifold M
. Then N is called a minimal submanifold if trh = 0. And it is called auster
if for each normal vector
multiplication by
the set of eigenvalues of A is invariant under
1 this is equivalent to the condition that all the invariants of
odd order of the Weingarten map at each normal vector of N vanish identically.
Of course every auster submanifold is a minimal submanifold.
Theorem 2.3.8 Let N be a proper slant submanifold of a Kaehlerian
manifold M . If rF = 0, then N is auster.
Proof. Let N be a proper slant submanifold of a Kaehlerian manifold M .
If rF = 0, then we have from Lemma 2.3.5 that f h (X; Y ) = h (X; P Y ). Let
X be any unit eigenvector of Q = P 2 with eigenvalue
6= 0. Then X =
PX
p
is a unit vector perpendicular to X. Thus, we have
h (X ; X ) = h
PX PX
p ;p
=
1
h P 2 X; X =
1
h ( X; X) =
which gives for any normal vector
hA X; Xi = hh (X; X) ; i =
Now, suppose that
hh (X ; X ) ; i =
hA X ; X i
is an eigenvalue of A , then
=
hX; Xi = h X; Xi = hA X; Xi =
=
h X ;X i =
hX ; X i =
hA X ; X i
h (X; X)
36
HANAN ALOHALI
Thus
is invariant under multiplication by
1, which implies that N is auster.
If M (c) is a complex-space-form, then we have the following reduction
theorem.
Theorem 2.3.9 Let N be an n-dimensional proper slant submanifold of
a complex m-dimensional complex space form M m (c) with constant holomorphic sectional curvature c. If rF = 0, then N is contained in a complex
n-dimensional complex totally geodesic submanifold of M m (c) as an auster
submanifold.
Proof. Let N be an n-dimensional proper slant submanifold of C m . Assume
that rF = 0 Then the normal bundle
of N has the following orthogonal
direct decomposition
= F (T N )
such that
p
? F (Tp N ), for any point p 2 N . For any vector …eld
2
,
and any N 2 F (T N ) there exists X 2 X(N ), such that N = F X that is
JN = J (F X). Thus tN + f N = t (F X) + f (F X), that is tN = t (F X) and
f N = 0, since f F = 0. This proves JN = tN . Since N and
are orthogonal,
0 = hN; i = hJN; J i = htN; t i, and as tN 6= 0, we get t = 0 that is J = f
proving that J 2 . Thus we have shown that
2
)J 2 .
37
SLANT SUBMANIFOLDS
Now for
2 , and X; Y 2 X(N )
hAJ X; Y i = hh (X; Y ) ; J i = rX Y; J
=
JY; rX
JrX Y;
=
rX JY;
= hJY; A X + DX i
=
hP Y; A Xi + hF Y; DX i
=
hP Y; A Xi + hDX F Y; i
Thus we have hDX F Y; i =
any
=
hA P Y + AJ Y; Xi. On the other hand, for
normal to N
J =t +f
and the lemma 2.3.5 gives Af Y + A P Y = 0. Since f = J on the normal
subbundle , formulas in Lemma 2.3.5 imply hDX F Y; i = 0,
2 . From this
we conclude that the normal subbundle F (T N ) is a parallel normal subbundle.
Next, we claim the …rst normal subbundle Imh is contained in F (T N ) :This
can be proved as follows:
Since rF = 0, statement (ii) of lemma 2.3.5 implies hh (X; Y ) ; J i =
hh (X; P Y ) ; i for any normal vector
2
. Thus, for any eigenvector Y
of the self-adjoint endomorphism Q with eigenvalue
and any normal vector
2 , we have
h (X; Y ) ; J 2
hh (X; P Y ) ; J i = h X; P 2 Y ;
=
since QY = P 2 Y =
hh (X; Y ) ; i =
1 <
Y.
Thus
hh (X; Y ) ; i.
0 . Thus
6=
hh (X; Y ) ; i =
=
hh (X; Y ) ; i
hh (X; Y ) ; i that is
Since N is a proper slant submanifold,
1 and this gives hh (X; Y ) ; i = 0, 2
h (X; Y ) 2 F (T N ) and consequently Imh
F (T N ) :
that is
38
HANAN ALOHALI
39
SLANT SUBMANIFOLDS
CHAPTER 3
In this chapter we are interested in proving an existence theorem [5] for
slant submanifold in a complex projective space and prove that these submanifolds are unique up to an isometry required by the data of the slant
submanifold. There are two sections in this chapter. In section 3.1, we collect
basic facts about slant submanifold which are required to prove the existence
theorem in section 3.2. Here we would like to point out that we use the notations of [5], where some times the smooth vector …elds are taken as elements
of tangent bundle but it is understood from the context that they are smooth
vector …elds.
3.1 PRILIMNAIRIES
We denote by M
m
(4c) the complete simply -connected Kaehlerian m-
manifold with constant holomorphic sectional curvature 4c. The curvature
tensor R of M
m
(4c) is given by
R X; Y Z = c
for X; Y ; Z tangent to M
Let f : M ! M
manifold into M
m
m
Y ;Z X
X; Z Y + JY ; Z JX
JX; Z JY + 2 X; JY JZ ;
m
(3.1.1)
(4c) :
(4c) be an isometric immersion of a Riemannian n-
(4c). We denote by h and A the second fundamental form
40
HANAN ALOHALI
and the Weingarten map of f and by r and r the Levi-Civita connections of
M and M .
For X; Y 2 X (M ) and
2
( ), the second fundamental form h and the
Weingarten map A are related by
hA X; Y i = hh (X; Y ) ; i :
(3.1.2)
The mean curvature vector H of the immersion is
n
1
1X
H = trh =
h (ei ; ei )
n
n
(3.1.3)
i=1
where fe1 ; :::; en g is a local orthonormal frame …eld on M .
Denote by R the curvature tensor of M and by RD the curvature tensor
of the normal connection D . Then the equation of Gauss and the equation
of Ricci are given respectively by
R (X; Y ; Z; W ) = R (X; Y ; Z; W ) + hh (X; Z) ; h (Y; W )i
hh (X; W ) ; h (Y; Z)i ;
(3.1.4)
RD (X; Y ; ; ) = R (X; Y ; ; ) + h[A ; A ] (X) ; Y i
(3.1.5)
41
SLANT SUBMANIFOLDS
for vectors X; Y; Z; W 2 X (M ) and ; 2
( ):
For the second fundamental form h, we de…ne the covariant derivative rh
of h with respect to the connection on T M
rX h (Y; Z) = DX (h (Y; Z))
T ? M by
h (rX Y; Z)
h (Y; rX Z) :
(3.1.6)
The equation of Codazzi is given by
R (X; Y ) Z
where R (X; Y ) Z
?
?
= rX h (Y; Z)
rY h (X; Z) ;
(3.1.7)
denotes the normal component of R (X; Y ) Z:
For an endomorphism Q on the tangent bundle of the submanifold, we
de…ne rQ by
(rX Q) Y = rX (QY )
Q (rX Y ) :
(3.1.8)
n
Now , suppose that M is -slant in M (4c), then we have
P2 =
cos2
I; hP X; Y i + hX; P Y i = 0;
(rX P ) Y = th (X; Y ) + AF Y X;
(3.1.9)
(3.1.10)
42
HANAN ALOHALI
DX (F Y )
F (rX Y ) = f h (X; Y )
h (X; P Y ) ;
(3.1.11)
where I is the identity map. For simplicity for each X 2 X(M ), we put
X =
1
F X:
sin
(3.1.12)
We de…ne a symmetric bilinear T M -valued form
on M by
(X; Y ) = th (X; Y )
)
(X; Y ) =
1
sin
(3.1.13)
F (X; Y )
and
J (X; Y ) = P (X; Y ) + F (X; Y )
) J (X; Y ) = P (X; Y ) + (sin )
(X; Y ) :
(3.1.14)
Since Jh (X; Y ) = th (X; Y ) + f h (X; Y ), put
Jh (X; Y ) =
where
(X; Y ) +
(X; Y ) ;
(3.1.15)
is also a symmetric bilinear T M -valued on M . From (3.1.12),(3.1.14)
and (3.1.15) we have
43
SLANT SUBMANIFOLDS
J 2 h (X; Y )
h (X; Y )
)
= h (X; Y ) = J (X; Y ) + J (X; Y )
= P (X; Y ) + (sin ) (X; Y ) + sin1 JF (X; Y )
= P (X; Y ) + (sin ) (X; Y )
(csc ) (X; Y ) (csc ) J (P (X; Y ))
= P (X; Y ) + (sin ) (X; Y ) (csc ) (X; Y )
(csc ) P 2 (X; Y ) (csc ) (F (P (X; Y )))
= P (X; Y ) + (sin ) (X; Y )
2
1
(X; Y ) + cos
(X; Y ) (P (X; Y ))
sin
sin
= P (X; Y ) + (sin ) (X; Y )
1 cos2
(X; Y ) (P (X; Y ))
sin
h (X; Y )
= P (X; Y ) (sin ) (X; Y )
+ (sin ) (X; Y ) (P (X; Y ))
h (X; Y ) = P (X; Y )
+ (sin )
(sin ) (X; Y )
(X; Y )
(3.1.16)
(P (X; Y ))
Thus
P (X; Y ) = (sin ) (X; Y ) ;
and
h (X; Y ) = (sin )
(X; Y )
(P (X; Y ))
Thus
(X; Y ) = (csc ) P (X; Y )
) P (X; Y ) = (csc ) P 2 (X; Y ) =
(csc ) cos2
(X; Y )
and
h (X; Y ) = (P (X; Y ))
(sin )
(X; Y ) =
(csc )
(X; Y )
44
HANAN ALOHALI
So
h (X; Y )
= (csc ) (X; Y )
= (csc ) sin1 (J (X; Y ) P (X; Y ))
= csc2 [P (X; Y ) J (X; Y )]
) h (X; Y ) = csc2 [P (X; Y )
J (X; Y )] :
By (3.1.10)
(rX P ) Y = th (X; Y ) + AF Y X =
(X; Y ) + AF Y X;
so
h(rX P ) Y; Zi = h (X; Y ) ; Zi
h (X; Z) ; Y i
n
For an n-dimensional -slant submanifold in M (4c) with
(3.1.17)
6= 0 (3.1.1),
(3.1.7), (3.1.8), (3.1.10) and (3.1.17) imply that the equation of Gauss and
n
Codazzi of M in M (4c) are
R(X; Y ; Z; W ) = c fhY; Zi hX; W i hX; Zi hY; W i + hP Y; Zi hP X; W i
hP X; Zi hP Y; W i +2 hX; P Y i hP Z; W ig
+ csc2 fh (X; W ) ; (Y; Z)i
h (X; Z) ; (Y; W )ig
(3.1.18)
(rX ) (Y; Z) + csc2 fP (X; (Y; Z)) + (X; P (Y; Z))g
+(sin2 )c fhX; P Y i Z + hX; P Zi Y g
= (rY ) (X; Z) + csc2 fP (Y; (X; Z)) + (Y; P (X; Z))g
+ sin2 c fhY; P Xi Z + hY; P Zi Xg :
(3.1.19)
45
SLANT SUBMANIFOLDS
3.2 EXISTANCE THEOREM
Theorem(Existence) 3.2.1 Let c;
be two constants with 0 <
2
and M a simply connected Riemannian n-manifold with inner product h; i.
Suppose there exist an endomorphism P of the tangent bundle T M and a
symmetric bilinear T M -valued form
on M such that for X; Y; Z; W 2 T M
, we have
P2 =
cos2
I;
(3.2.1)
hP X; Y i + hX; P Y i = 0;
h(rX P ) Y; Zi = h (X; Y ) ; Zi
h (X; Z) ; Y i ;
R (X; Y; Z; W ) = csc2 fh (X; W ) ; (Y; Z)i
+c fhX; W i hY; Zi
(3.2.2)
(3.2.3)
h (X; Z) ; (Y; W )ig
hX; Zi hY; W i + hP X; W i hP Y; Zi
hP X; Zi hP Y; W i + 2 hX; P Y i hP Z; W ig
(rX ) (Y; Z) = csc2 fP (X; (Y; Z)) + (X; P (Y; Z))g
+ sin2 c fhX; P Zi Y + hX; P Y i Zg
(3.2.4)
(3.2.5)
46
HANAN ALOHALI
is totally symmetric. Then there exists a -slant isometric immersion from M
n
into M (4c) whose second fundamental form h is given by
h (X; Y ) = csc2 (P (X; Y )
Proof : Let c;
J (X; Y )) :
be two constants with 0 <
2
(3.2.6)
and M a simply-
connected Riemannian n-manifold equipped with an endomorphism P and a
symmetric bilinear T M -valued form
satisfying the …ve conditions stated in
the theorem.
Consider the Witney sum T M
T M . For each X 2 T M , we identify
(X; 0) with X; and also we denote (0; X) by X . We de…ne the inner product
T M by using the product metric. Let Jb be the endomorphism
h; i on T M
on T M
T M de…ned by
b = P X + (sin )X , JX
b =
JX
(sin ) X
PX ;
for X 2 T M , then we have
Jb2 X = Jb2 ((X; 0)) = Jb (P X; 0) + Jb (0; (sin ) X)
=
=
P 2 X; (sin ) P X
(X; 0) =
sin2
X
which gives
Jb2 X =
X:
X; (sin ) P X
(3.2.7)
47
SLANT SUBMANIFOLDS
Similarly, we have
Jb2 X =
Thus,
Jb2 =
Now as
X :
I:
h(X; Y ) ; hZ; W ii = hX; Zi + hY; W i
we have
D
b JY
b
JX;
E
=
D
E
Jb (X; 0) ; Jb (Y; 0)
= h(P X; (sin ) X) ; (P Y; (sin ) Y )i
= hP X; P Y i + h(sin ) X; (sin ) Y i
= cos2 hX; Y i + sin2 hX; Y i = hX; Y i
b h; i is a Hermitian structure on T M
Thus, J;
TM.
Now , we de…ne A; h and D by
AY X = csc f(rX P ) Y
h (X; Y ) =
DX Y = (rX Y ) + csc2 fP
(csc )
(X; Y )g ;
(X; Y ) ;
(X; Y ) +
(X; P Y )g ;
(3.2.8)
(3.2.9)
(3.2.10)
48
HANAN ALOHALI
for vector …elds X; Y 2 T M . It is easy to verify that each AY is an endomorphism on T M .
Now,
h (X; Y ) =
(csc )
(X; Y ) ;
as we see
b = P X + (sin ) X = P X + F X
JX
) (sin ) X = F X ) X =
1
FX
sin
thus we conclude
(X; Y ) =
and as
1
F (X; Y ) ;
sin
is a symmetric bilinear T M -valued,
is also a symmetric bilinear
(T M ) -valued form, that is, h is also a (T M ) -valued symmetric bilinear form
on T M , and D is a metric connection of the vector bundle (T M ) over M .
b denote the canonical connection on T M
Let r
T M induced from the
Levi-Civita connection on T M . Then from (3.2.7)-(3.2.10), we have on using
Kozul’s formula that :
b X Jb Y = r
b X Jb Y = 0
r
for vector …elds X; Y tangent to M .
(3.2.11)
49
SLANT SUBMANIFOLDS
Let RD denote the curvature tensor associated with the connection D on
(T M ) , that is,
RD (X; Y ) Z = DX DY Z
DY DX Z
D[X;Y ] Z ;
(3.2.12)
for X; Y tangent to M . Then by (3.2.1),(3.2.5),(3.2.10),(3.2.12) and a simple
computation, we obtain
RD (X; Y ) Z = (R (X; Y ) Z) + fcP [hY; P Zi X hX; P Zi Y 2 hX; P Y i Z]
c Y; P 2 Z X
X; P 2 Z Y 2 hX; P Y i P Z
2
+ csc [(rX P ) (Y; Z) (rY P ) (X; Z)
(X; (rY P ) Z) + (Y; (rX P ) Z)]g
(3.2.13)
Also, (3.2.8) yields
sin2 h[AZ ; AW ] X; Y i = h(rY P ) Z; (rX P ) W i h(rX P ) Z; (rY P ) W i
+ h(rX P ) Z; (Y; W )i + h(rY P ) W; (X; Z)i
h(rY P ) Z; (X; W )i h(rX P ) W; (Y; Z)i
+ h (X; W ) ; (Y; Z)i h (X; Z) ; (Y; W )i :
(3.2.14)
From (3.2.2) we have
h (Y; Z) ; P W i + hP (Y; Z) ; W i = 0
(3.2.15)
By taking the derivative of (3.2.15) with respect to X and using (3.1.8) and
50
HANAN ALOHALI
(3.2.2), we …nd
X h (Y; Z) ; P W i + X hP (Y; Z) ; W i = 0
hrX (Y; Z) ; P W i + h (Y; Z) ; rX P W i + hrX P (Y; Z) ; W i +
hP (Y; Z) ; rX W i = 0
) h (Y; Z) ; (rX P ) W i + h(rX P ) (Y; Z) ; W i = 0
(3.2.16)
Also by (3.2.3) we obtain
h(rX P ) Z; (rY P ) W i = h(rX P ) Z; (Y; W )i
h (Y; (rX P ) Z) ; W i
(3.2.17)
Hence on using (3.2.13),(3.2.14),(3.2.16)-(3.2.17) and a direct computation, we
arrive at
2
c sin
RD (X; Y ) Z ; W
h[AZ ; AW ] X; Y i =
(hY; Zi hX; W i hX; Zi hY; W i) 2 hX; P Y i hP Z; W i
(3.2.18)
Equation (3.1.1),(3.2.1),(3.2.2) and (3.2.18) imply that (M; A; D) satis…es
n
the equation of Ricci for an n-dimensional -slant submanifold in M (4c).
Also, (3.2.4) and (3.2.5) imply that (M; h) satis…es the equation of Gauss
n
and Codazzi for a -slant submanifold in M (4c). Hence, the vector bundle
51
SLANT SUBMANIFOLDS
TM
T M over M equipped with the product metric, the Weingarten map
A, the second fundamental form h, and the connections D and r satisfy the
n
structure equations of n-dimensional -slant submanifolds in M (4c). This
n
proves that there exists a -slant isometric immersion of M into M (4c) with
h = csc2 (P
J ) as its second fundamental form A as its Weingarten
map, and D as its normal connection.
n
Theorem(Uniqueness) 3.2.2 Let x1 ; x2 : M ! M (4c) be two -slant
0<
<
2
isometric immersions of a connected Riemannian n-manifold M
n
into the complex -spacef orm M (4c) with second fundamental form h1 and
h2 . If
h1 (X; Y ) ; Jx1 Z = h2 (X; Y ) ; Jx2
for all vector …elds X; Y; Z tangent to M , then there exists an isometry
n
M (4c) such that x1 =
of
x2 .
Proof : Let p any point of M . If necessary by applying an isometry of
n
M (4c), we may assume that x1 (p) = x2 (p) and dx1p = dx1p . Let us then take
a geodesic
1
through the point p that is p =
= x1 ( ) and
(0). It is su¢ cient to prove that
= x2 ( ) coincide. As x1 (p) = x2 (p) and dx1p = dx2p we
2
have
0
1 (0)
where
0
2 (0).
1 (0)
=d
1
j0
= x1 (p) and
d
j0
dt
2 (0)
= dx1 d
= x2 (p), and
d
j0
dt
0
0
2 (0)
= dx1p
= dx2p
0
0
(0)
(0) )
0
1 (0)
=
52
HANAN ALOHALI
then
1
=
2.
53
SLANT SUBMANIFOLDS
CHAPTER 4
In this chapter we are interested in deriving an inequality satis…ed by the
mean curvature of a slant submanifold. First we prepare lemmas in section
4.1, then in section 4.2 we use these lemmas to classify H-umbilical slant
submanifolds, and in the section 4.3, we obtain an inequality which can be
used classify a cylindrical slant submanifold in C n that requires some material
which is out of realm of present thesis therefore not included in this thesis.
4.1 SOME LEMMAS
If M is an n-dimensional proper -slant submanifold of a complex space
n
form M (4") of constant holomorphic sectional curvature 4", then n is even;
say n = 2m . We choose a canonical orthonormal frame e1 ; ::::; en ; e1 ; ::::; en
in such way that
e2 = (sec ) P e1 ; ::::; e2m = (sec ) P e2m 1 ;
e1 = (csc ) F e1 ; ::::; e2m = (csc ) F e2m ;
where
(4.1.1)
is the slant angle. We call such an orthonormal frame an adapted
frame.
Now we claim
tei =
(sin ) ei ; i = 1; :::; 2m
54
HANAN ALOHALI
To prove it observe that
Jei
= J [(csc ) F ei ] = (csc ) J [Jei
P ei ]
= (csc ) [ ei
J ((cos ) ei+1 )]
=
(csc ) ei
(cot ) Jei+1
=
(csc ) ei
(cot ) P ei+1
(cot ) (sin ) e(i+1)
=
(csc ) ei
(cot ) P ei+1
(cos ) e(i+1)
Thus we have
tei =
(csc ) ei
(cot ) P ei+1
and
f ei =
(cos ) e(i+1)
that is
tei =
(csc ) ei
(cot ) ( cos ) ei =
Hence tei =
(sin ) ei and that
) f e(2j
(cos ) e(2j) :
=
1)
cos2
sin
1
sin
ei =
(sin ) ei
(cos ) e(i+1) = f ei put i = 2j
By direct computation we also have
f e(2j) = (cos ) e(2j
1)
; P e2j =
(cos ) e2j
1; j
For any vector X tangent to M we put
rX ei =
n
P
j=1
rX ei =
! ji (X) ej +
n
P
j=1
n
P
j=1
! ji (X) ej +
! ji (X) ej ;
n
P
j=1
! ji (X) ej , i; j = 1; :::; n:
= 1; :::; m
1
55
SLANT SUBMANIFOLDS
Then we have ! ji =
! ij , to prove this, we take inner product of (1.4) with ej
then with ej to arrive at
rX ei ; ej = ! ji (X)
rX ei ; ej
= ! ji (X)
! ji (X) = rX ei ; ej = X hei ; ej i
) ! ji =
rX ej ; ei =
! ij (X) :
! ij :
Similarly we …nd ! ji =
! ij , and taking inner product of (1.5) with ej we
…nd rX ei ; ej = ! ji (X)
=
ei ; rX ej =
ei ; rX ej
! ij (X), which proves ! ji =
! ji
=
n
X
k=1
! ij . Moreover , we also have
hjik ! k ; hjik = hh (ei ; ek ) ; ej i :
(4.1.2)
where ! 1 ; :::; ! n is the dual frame of e1 ; :::; en :
We need the following lemmas for later use.
Lemma 4.1.1 Let M be an n-dimensional (n = 2m) proper -slant submanifold of a Kahlerian n-manifold. Then with respect to an adapted frame,
we have
56
HANAN ALOHALI
(2i 1)
1
(2i 1)
! 2j
(2j)
! 2i
(2i 1)
! 2j 1
(2j)
! 2i
(2j 1)
! (2i 1)
(2i 1)
! (2j)
(2j)
! (2i)
(2i)
! 2j 1
! 2j
(2j 1)
1
(2j)
! 2i 1
(2i)
! 2j
(2j 1)
! 2i 1
(2i)
! 2j
1
! 2j
2i 1
1
! 2i
2j
! 2j
2i
(2j 1)
! 2i
! 2i
= cot
! 2j
2i
1
= cot ! 2j
2i
2j 1
= cot ! 2i
(2j)
= cot ! (2i 1)
(2j 1)
= cot ! (2i)
(2j)
= cot ! 2i 1
(2j 1)
= cot ! 2i 1
(2j)
= cot ! 2i 1
(2j)
= cot ! (2i)
! 2i
2j
1 ;
2j 1
! 2i 1 ;
1
! 2i
;
2j
(2i)
! (2j 1)
(2i 1)
! (2j)
(2j 1)
! 2i
(2j)
! 2i
(2j 1)
! 2i
(2j 1)
! (2i 1)
;
;
;
;
;
;
for any i; j = 1; :::; m:
Lemma 4.1.2 Let M be an n-dimensional proper -slant submanifold
of a complex space form M
m
(4"). Then the curvature tensor R of M
m
(4")
satis…es
R (X; Y ) Z
?
= " fhJY; Zi F X
hJX; Zi F Y + 2 hX; JY i F Zg
for X; Y; Z tangent to M , where R (X; Y ) Z
?
denotes the normal component
of R (X; Y ) Z
This lemma follows easily from the curvature formula of complex space
form.
For Kahlerian slant submanifolds we have the following.
Lemma 4.1.3 Let M be an n-dimensional (n = 2m) proper -slant submanifold of a Kahlerian slant submanifold. If M is Kahlerian slant, then, with
57
SLANT SUBMANIFOLDS
respect to an adapted frame, we have
! 12j
1
2j
= ! 2j
2 ; !1 =
! 2j
2
1
; j = 1; :::; m
Proof : Since M is Kahlerian slant , rX (P Y ) = P (rX Y ) for X; Y tangent
to M . Thus we have
! 2j
1
1
(X) =
rX e1 ; e2j
= sec
=
that is ! 2j
1
! 2j
2
1
1
1
= rX e1 ; (sec )P e2j = sec
e1 ; P rX e2j =
sec
P e1 ; rX e2j =
e2 ; rX e2j
rX e2 ; e2j = ! 2j
2 (X)
= ! 2j
2 . Also we have
(X) =
rX e2 ; e2j
1
= rX (sec ) P e1 ;
(sec ) P e2j
= sec2
P e1 ; rX (P e2j ) = sec2
= sec2
P 2 e1 ; rX e2j = e1 ; rX e2j =
that is ! 2j
1 =
e1 ; rX (P e2j )
! 22j
1
P e1 ; P rX e2j
rX e1 ; e2j =
! 2j
1 (X)
; j = 1; :::; m.
De…nition 4.1.4: An n-dimensional submanifold M of a Kahlerian manifold is called H-umbilical if its second fundamental form h takes the following
simple form:
h (e1 ; e1 ) = e1
, h (e2 ; e2 ) = ::: = h (en ; en ) = e1
h (e1 ; ej ) = ej , h (ej ; ek ) = 0 , j 6= k and j; k = 2; :::; n
for some suitable functions
and
with respect to some suitable ortho-
normal local frame …eld e1 ; :::; en where e1 ; :::; en are unit vectors in the
directions of F e1 ; :::; F en respectively.
58
HANAN ALOHALI
De…nition 4.1.5: An H-umbilical slant submanifold with
= 0 is simply
called cylindrical slant.
Lemma 4.1.6 Every n-dimensional H-umbilical proper slant submanifolds
n
of a Kahlerian n-manifold M is Kahlerian slant.
Proof: Follows from the de…nition of H-umbilical submanifolds and the
fact that a proper slant submanifold M of a Kahlerian manifold is a Kahlerian
slant if and only if the shape operator A of M satis…es AF X Y = AF Y X for
X; Y tangent to M .
4.2 ClASSIFICATION OF H-UMBLICAL SLANT SUBMANIFOLDS
We introduce the notion of slant space forms as follows. We call a Riemannian manifold (N; g) a slant space form if there exist a
2 0;
2
and an
endomorphism P on the tangent bundle T N so that:
(1) rP = 0; P 2 =
cos2
I and g (P X; Y ) + g (X; P Y ) = 0 for X; Y
tangent to N .
(2) N has constant slant sectional curvature, i.e., the slant sectional curvature K (X; P Y ) of the slant 2-plane spanned by X; P X is independent of
the choice of the vector X 2 T N , X 6= 0.
Clearly, a Kahlerian slant submanifold M of a Kahlerian manifold is a
slant space form if M; g; Je is a complex space form where Je = (sec ) P:
The purpose of this section to prove the following classi…cation theorem.
59
SLANT SUBMANIFOLDS
Theorem 4.2.1 Let M be an n-dimensional H-umbilical proper slant
n
submanifold of a complete simply-connected complex space form M (4c), n >
2. Then one of the following three statements holds:
(1) M is ‡at and is immersed as an open part of a slant n-plane in the
complex Euclidean n-space C n ;
(2) M is ‡at and is immersed as a cylindrical slant submanifold in C n ;
(3) c < 0, M is a slant space form of constant slant sectional curvature
4c cos2 , and M is immersed as an H-umbilical submanifold in the complex
p
hyperbolic n-space CH n (4c) satisfying 4.1.4 with = 2 = 2
c sin :
n
Proof . If M is an H-umbilical proper slant submanifold of M (4c), then
the second fundamental form of M takes the following form:
h (e1 ; e1 ) = e1 ; h (e2 ; e2 ) = ::: = h (en ; en ) = e1 ;
h (e1 ; ej ) = ej ; h (ej ; ek ) = 0; j 6= k; j; k = 2; :::; n
for some suitable functions
and
(4.2.1)
with respect to some adapted local frame
…eld e1 ; :::; en ; e1 ; :::; en : From lemma 4.1.6 we know that M is Kahlerian
slant.
Using 4.1.2 and 4.2.1, we have
! 11 = ! 1 ; ! j1 = ! 1j = ! j ; ! jj = ! 1 ; 2
! kj = 0; 2 j =
6 k n:
j
n;
(4.2.2)
60
HANAN ALOHALI
! 11 =
n
P
k=1
h11k ! k
h11k = g (h (e1 ; ek ) ; e1 ) =
g (e1 ; e1 ) = ; k = 1
g (ek ; e1 ) = 0; k = 2; 3; ::
since,
h (e1 ; ek ) =
ek
e1 , k = 1
; k = 2; 3; ::
) ! 11 = ! 1 :
From Lemma 4.1.1 and 4.2.2 we …nd
! 21 = ! 21 ( + ) cot ! 1 ;
1
cot ! 2k ; k
! 2k 1 = ! 12k 1
! 12k = ! 12k + cot ! 2k 1 ; k
! 2j = ! 2j
cot ! j ; j 3;
! jl = ! jl ; j; l 3:
If
=
2;
2;
(4.2.3)
= 0, then M is a totally geodesic -slant submanifolds, since only
n
totally geodesic submanifolds of a complex space form M (4c) with c 6= 0 are
either complex submanifolds or totally real submanifolds, but M is proper so
neither complex nor totally real ) c = 0:
Now, we assume that M non totally geodesic. From Equation of Codazzi
R (X; Y ) Z
?
= rX h (Y; Z)
rY h (X; Z)
with X = e1 ; Y = Z = e2 ; and using (4.1.1), (4.2.1-3) and lemma (4.1.2) we
get
61
SLANT SUBMANIFOLDS
2 ) ! 21 (e2 ) ;
e1 = (
e2 = 3 ! 21 (e1 )
(4.2.4)
( + ) cot + 3c sin cos ;
(4.2.5)
! j1 (e1 ) = 0; j = 3; :::; n
(4.2.6)
to prove these using
R (X; Y ) Z
?
= c fhJY; Zi F X
hJX; Zi F Y + 2 hX; JY i F Zg
we have
?
R (e1 ; e2 ) e2
= c fhJe2 ; e2 i F e1
hJe1 ; e2 i F e2 + 2 he1 ; Je2 i F e2 g
hJe2 ; e2 i = hP e2 ; e2 i = h cos e1 ; e2 i = 0
hJe1 ; e2 i = hP e1 ; e2 i = hcos e2 ; e2 i = cos
and hJe2 ; e1 i =
cos
R (e1 ; e2 ) e2
?
= c f0
=
cos F e2
2 cos F e2 g =
3c cos F e2
3c sin cos e2
Now,
(De1 h) (e2 ; e2 )
(De2 h) (e1 ; e2 ) = De1 h (e2 ; e2 )
2h (re1 e2 ; e2 )
+h (re2 e1 ; e2 ) + h (e1 ; re2 e2 )
De2 h (e1 ; e2 )
62
HANAN ALOHALI
with
re1 e2 =
re2 e1 =
re2 e2 =
n
P
k=1
n
P
k=1
n
P
k=1
! k2 (e1 ) ek = ! 12 (e1 ) e1 + ! 22 (e1 ) e2 + :::
! k1 (e2 ) ek = ! 11 (e2 ) e1 + ! 21 (e2 ) e2 + :::
! k2 (e2 ) ek = ! 12 (e2 ) e1 + ! 22 (e2 ) e2 + :::
since ! ii = 0; ) h (e2 ; re1 e2 ) = ! 12 (e1 ) e2
h (re2 e1 ; e2 ) = ! 21 (e2 ) e1
h (e1 ; re2 e2 ) =
n
P
k=1
! k2 (e2 ) ek :
and
De2 e2 = De2 e2 +e2 ( ) e2 =
n
P
j=1
De1 e1 = De1 e1 +e1 ( ) e1 =
n
P
j=1
! j2 (e2 ) ej +
! j1 (e1 ) ej +
n
P
j=1
n
P
j=1
! j2 (e2 ) ej +e2 ( ) e2
! j1 (e1 ) ej +e1 ( ) e1
we get
(De1 h) (e2 ; e2 ) (De2 h) (e1 ; e2 )
= De1 e1
2! 12 (e1 ) e2
De2 e2
n
P
! k2 (e2 ) ek = 3c sin cos e2
+! 21 (e2 ) e1 +
(4.2.7)
k=1
Now taking inner product in (4.2.7) with e1 we get hDe2 e2 ; e1 i =
! 12 (e2 ) and hDe1 e1 ; e1 i = e1 ( ).
So,
63
SLANT SUBMANIFOLDS
e1 ( )
! 12 (e2 ) + ! 21 (e2 ) + ! 12 (e2 ) = 0
! 21 (e2 ) =
) e1 ( ) =
! 21 (e2 )
( + ) cot ! 1 (e2 ) :
Next taking the inner product in (4.2.7) with e2
hDe2 e2 ; e2 i = e2 ( )
hDe1 e1 ; e2 i = ! 21 (e1 ) = ! 21 (e1 )
( + ) cot ;
and,
! 21 (e1 )
2 ! 12 (e1 )
( + ) cot
) e2 ( ) = 3 ! 21 (e1 )
e2 =
( + ) cot + 3c sin cos :
Taking inner product in (4.2.7) with ej ; j
hDe1 e1 ; ej i =
= ! j1 (e1 ) ;
n
P
k=1
3c sin cos
! j1 (e1 ) ek +
n
P
k=1
3; we get
! k1 (e1 ) ek + e1 ( ) e1 ; ej
hDe2 e2 ; ej i = ! j2 (e2 ) :
and,
! j1 (e1 )
2 (0)
! j2 (e2 ) + ! j2 (e2 ) = 0
) ! j1 (e1 ) + ! 2j (e2 )
) ! j1 (e1 ) = 0:
2 cot
! j (e2 )
! 2j (e2 ) = 0
64
HANAN ALOHALI
Similarly, from Equation of Codazzi with X = e1 ; fY; Zg = fe2 ; ej g for
j
3, and using (4.2.1), (4.2.2) and Lemmas (4.1.1) and (4.1.2), we …nd
e2 = ! 21 (e1 ) + 2c sin cos ;
(4.2.8)
! j1 (e1 ) = 0;
! k2 (e1 ) = (
2 ) ! k1 (e2 ) = (
(4.2.9)
2 ) ! 21 (ek ) ; k
3:
(4.2.10)
Combining (4.2.5) and (4.2.8) we get
3 ! 21 (e1 )
( + ) cot + 3c sin cos = ! 21 (e1 ) + 2c sin cos
) 2 ! 21 (e1 ) = ( + ) cot
c sin cos
From Equation of Codazzi with X = Z = e1 ; Y = e2j
(4.2.11)
1
for j > 1; and
using (4.1.1), (4.2.1-3) and Lemma (4.1.2), we …nd
e1 = (
e2j
1
=(
2 ) ! 2j
1
1
(e2j
2 ) ! 2j
1
1
1) ;
(4.2.12)
(e1 ) ;
(4.2.13)
65
SLANT SUBMANIFOLDS
2 ) ! 21 (e2j
(
1)
=(
2 ) ! k1 (e2j ) = 0; k 6= 2; 2j
2 )! 12j (e2j
(
1)
=(
1; 2j;
2 ) cot :
Similarly, from Equation of Codazzi with X = e2j
1;
(4.2.14)
(4.2.15)
Y = e1 ; Z = e2j for
j > 1; and using (4.1.1), (4.2.1-3) and Lemma (4.1.2), we …nd
) ! 2j
1 (e2j
(2
1)
= c sin cos +
2
cot
(4.2.16)
by comparing the coe¢ cients of e1 : Combining (4.2.15) and (4.2.16) yields
(2
) cot = c sin cos +
) 2
)
2
2
2
2 cot
cot = c sin cos
= c sin2
)(
)
= c sin2 :
(4.2.17)
Divide (4.2.11) by cot we …nd
2 ! 21 (e1 ) tan = ( + )
then using (4.2.17) we get
c sin2
66
HANAN ALOHALI
2 ! 21 (e1 ) tan = ( + )
(
)
=
) ! 21 (e1 ) =
2
+
2
+
=2
cot :
(4.2.18)
On the other hand, from Equation of Codazzi with X = e2 ; Y = Z = ej
for j
3; and using (4.1.1), (4.2.1-3) and Lemma (4.1.2), we …nd
e2 = 0;
(4.2.19)
! 21 (e2 ) = ! j1 (ej ) ; j
! k1 (e2 ) = 0; k
3;
(4.2.20)
3;
(4.2.21)
3 ! j1 (e2 ) = ! 21 (ej ) :
(4.2.22)
From (4.2.19) and (4.2.9) we obtain ! 21 (e1 ) =
2c sin cos : Therefore,
we …nd
=
2c sin2
by (4.2.18). By Combing this with (4.2.17) we get
=
2(
)
)
+2
2
2
=0)
2
2
+
=0
67
SLANT SUBMANIFOLDS
)
Consequently, either
(
= 0 or
2 ) = 0:
(4.2.23)
= 2 at each point on the slant submani-
fold.
If
= 0 at some point on the proper slant submanifold, then (4.2.17)
yields c = 0: Moreover, in this case, (4.2.17) and (4.2.23) imply that
=0
identically. Hence, we obtain statement (2) in this case.
If
= 2
and
6= 0, then (4.2.17) implies " < 0 and
=
p
c sin :
Hence, the curvature tensor R of M satis…es
R (X; Y ; Z; W ) = c cos2 fhX; W i hY; Zi hX; Zi hY; W ig +
c fhP X; W i hP Y; Zi hP X; Zi hP Y; W i +
2 hX; P Y i hP Z; W ig :
(4.2.24)
Thus M is a slant space form with slant sectional curvature 4c cos2 : Consequently, we obtain statement (3).
4.3 A GENERAL INEQUALITY FOR KAEHLERIAN SLANT SUBMANIFOLDS
Consider the complex number (m + 1)-space C1m+1 endowed with the
pseudo-Euclidean metric
g0 =
dz0 dz 0 +
n
X
j=1
dzj dz j
68
HANAN ALOHALI
Put
H12m+1 = fz = (z0 ; z1 ; :::; zm ) : hz; zi =
1g
where h; i denotes the inner product on C1m+1 induced from the metric g0 :
Let C =
2C:
= 1 : Then there is a C -action on H12m+1 de…ned by
z 7 ! z: At z 2 H12m+1 , iz is tangent to the ‡ow of the action. The orbit of
this action is given by zt = eit z with dzt =dt = izt which lies in the negativede…nite plane spanned by z and iz. The quotient space H12m+1 =
under
the C -action is nothing but the complex hyperbolic space CH m ( 4). The
canonical projection
: H12m+1 ! CH m ( 4) is called the hyperbolic Hopf
…bration.
c =
For each isometric immersion f : M ! CH m ( 4), the preimage M
1 (M )
is a principal circle bundle over M with totally geodesic …bers and
c ! H 2m+1 of f is an isometric immersion such that the diagram
the lift fb : M
1
b
f
c !
M
#
M
f
H12m+1
#
! CH m ( 4)
commutes.
Conversely, if
c ! H 2m+1 is an isometric immersion which is in: M
1
variant under the action of C , there is a unique isometric immersion
c
M
! CH m ( 4) ; called the projection of
:
, such that the associated
diagram commutes.
Theorem 4.3.1 Let
fn (4c) ; c 2 f 1; 0; 1g ; be a Kaehlerian
:M !M
-slant submanifold of dimension n in a complete simply-connected complex
69
SLANT SUBMANIFOLDS
fn (4") : Then we have
space form M
n+2
n
2 (n + 2)
n2 (n 1)
H2
1+
3 cos2
n 1
c
where H 2 is the squared mean curvature and is the scalar curvature de…ned
P
Kij ; Kij is the sectional curvature of the 2-plane spanned by ei and
by =
i<j
ej for a local orthonormal frame e1 ; :::en .
n
Proof : If M is a complex submanifold of M (4c), then
= 0 and M is
a minimal submanifold. Thus, in this case the above inequality reduces to
n(n+2)c
;
2
with the equality holding if and only if M is a totally geodesic
complex submanifold.
n
Now, assume that M is a Kaehlerian -slant submanifold of M (4c) with
6= 0. Then we know that the shape operator satis…es
AF X Y = AF Y X; for X; Y 2 T M:
(4.3.1)
From the de…nition of mean curvature function we have
n2 H 2 =
X
i
0
@
X
j
hijj
1
X
2
+ 2 hijj hikk A
From Equation (3.2.4) of Gauss, we get
j<k
(4.3.2)
70
HANAN ALOHALI
2 = n2 H 2
X
hijk
2
+ n (n
1) + 3n cos2
c;
(4.3.3)
i;j;k=1
where hijk = hh (ej ; ek ) ; ei i : Thus, by applying (4.3.1), (4.3.2) and (4.3.3), we
obtain
=
1
n (n
2
Let m =
n2 H 2
m 2
1) + 3n cos2
c+
XX
X
hijj hikk
i j<k
(n+2)
(n 1) :
2
hijj
3
X
2
hijk
:
i<j<k
i6=j
(4.3.4)
Then, from (4.3.2), (4.3.3) and (4.3.4), we get
n (n
1) c
3nc cos2
=
X
i
= +6m
=
X
2
hijj
+ (1 + 2m)
X
hijj
i6=j
X
2
hijk
i<j<k
2
hiii +
i
6m
2 (m
X
1)
1)
XX
hijj
hikk
(n
2) (m
1))
+ (1 + 2m
2 (m
X
1)
X
0
which implies our inequality.
1
1
hijj
XX
hijj
2
j6=i
j6=i
i 2
hjk
X
2
X
hiii hijj
+ (m
i<j<k
n
hijj hikk
i j<k
2
hijk
i6=j ;kj<k
+
XX
i<j<k
+ (m
= 6m
2
1)
hikk
2
i6=j;k j<k
hiii
(n
1) (m
1) hijj
2
j6=i
(4.3.5)
71
SLANT SUBMANIFOLDS
REFERENCES
[1] Boothoby, W. An Introduction to Di¤erentiable Manifolds and Riemannian Geometry, Academic Press, 1975.
[2] Chen, B.Y., Total Mean Curvature and Submanifolds of Finite Type,
World Scienti…c 1983.
[3] Chen, B.Y., Geometry of Slant Submanifolds, Katholieke Universiteit
Leuven, Belgium (1990).
[4] Chen, B.Y., A General Inequality for Kahlerian Slant Submanifolds and
related results, Geometriae Dedicata 85: 253-271, (2001).
[5] Chen, B.Y., and Vrancken, L., Existence and uniqueness theorem for
slant immersions and its applications, Results Math. 31 (1997), 28-39; addendum, ibid 39 (2001).
[6] Chen, B.Y., Flat slant surfaces in complex projective and complex
hyperbolic planes, Results Math. 2002.
[7] Chen, B.Y., Geometry of Submanifolds, Marcel Dekker. INC. New york
(1973).
[8] Chen, B.Y., and Tazawa, Y., Slant submanifolds of complex projective
and complex hyperbolic spaces, Glasgow Math. J. 42 (2000), 439-454.
72
HANAN ALOHALI
[9] Chen, B.Y., Special slant surfaces and a basic inequality, Results Math.
33 (1998), 65-78.
[10] Cheeger, J. and Ebin, D. Comparison Theorems in Riemannian Geometry, North-Holland Publ. Amsterdam 1975.
[11] DoCarmo, M. Riemannian Geometry, Brikhauser, 1992.
[12] Kobayashi, S. and Nomizu, K., Foundations in Di¤erential Geometry,
Wiley-Interscience (1969) N.Y.