1. Introduction and Main Results
A traditional topic in Riemannian geometry is to find the geometrical and topological structures of submanifolds; there has been much progress in this field. For instance, the rigidity theorem was proved by Berger [
1] for an even-dimensional complete simply connected manifold
M with sectional curvature
. Further, Gauhmen [
2] considered even
-dimensional submanifolds minimally immersed in the unit sphere
with a co-dimension equal to one, and showed that if
for any unit vector
u of
where
h is the second fundamental form
, then
is totally geodesic in
. If
, then
is
minimally embedded in
, as described. A very famous result in this respect was formulated by Poincare [
3], who stated that every simply connected closed 3-manifold is homeomorphic to a 3-sphere. Smale [
4] generalized the Poincare conjecture and proved that for a closed
-manifold
which has the homotopic types of an
n-dimensional sphere greater than five, the manifold
is homeomorphic to
. The differentiable sphere theorem was proven by Brendle and Schoen [
5] under Ricci flow. In recent years, much attention has been paid to the classification of geometric function theory, topological sphere theorems, and differentiable sphere theorems (see [
6,
7,
8,
9,
10,
11]). In the sequelae, the homology groups of a manifold are important topological invariants that provide algebraic information about the manifold. Federer-Fleming [
7] showed that any non-trivial integral homology class in
corresponds to a stable current. Motivated by the work of Federer and Fleming [
7], Lawson and Simon [
9], and Xin [
11] proved the nonexistence of stable integral currents in a submanifold
and vanishing homology groups of
with non-negative sectional curvature according to the following theorem.
Theorem 1 ([
9,
11]).
Let be a compact n-dimensional submanifold isometrically immersed in the space form of curvature with the second fundamental form h. Let be any positive integers such that and for any and an orthonormal frame of the tangent space . Then, there do not exist stable -currents in and where stands for i-the homology group of and is a finite abelian group with integer coefficients. Due to these previous studies on large scales, a particular case we consider here is that of warped product pointwise semi-slant submanifolds of complex space form where
is represented as a constant sectional curvature. In this regard, our motivation comes from the study of Sahin [
12], where he discussed the warped product pointwise semi-slant submanifolds in a Kaehler manifold and showed that a warped product pointwise semi-slant submanifold of type
is nontrivial when angle
is treated as a slant function. Furthermore, it was shown in [
12] that the warped product pointwise semi-slant submanifold
of a Kaehler manifold is a natural generalization of CR-warped products [
13]. Inspired by this notion, we define the extrinsic condition to prove nonexistence-stable integral
-currents and vanishing homology groups in a warped product pointwise semi-slant submanifold of complex space forms
. We use Theorem 1 on this basis to arrive at our first result.
Theorem 2. Let be a compact warped product pointwise semi-slant submanifold of a complex space form . If the following condition is satisfiedthen there do not exist stable integral -currents in andwhere stands for i-the homology group of with integer coefficients, and are the gradient and the Laplacian of the warped function f, respectively, and represents the components of the second fundamental form h in an invariant subspace μ. Our next result is in accordance with Lemma 3.1 in [
12], which states that the inner product of the second fundamental form of
and
F-components of
is equal to zero. To be precise, we have the following result.
Theorem 3. Let be a compact warped product pointwise semi-slant submanifold of a complex space form . If the inequalityholds, then there do not exist stable integral -currents in andThe notation is the same as in Theorem 2. To apply Theorems 2 and 3 in [
14], let the slant function
become globally constant, setting
in Theorems 2 and 3. Then, the pointwise slant submanifold
is turned into a totally real submanifold
. Thus, a warped product pointwise semi-slant submanifold
becomes CR-warped products in a Kaehler manifold of type
. Therefore, following to the motivation of Chen [
13], we deduce the following result from Theorem 2 for the nonexistence of stable integral
-currents and vanishing homology in a CR-warped product submanifold of complex space forms
.
Corollary 1. Let be a compact CR-warped product submanifold of complex space form . If the following condition is satisfiedthen there do not exist stable integral -currents in and As an immediate consequence of Theorem 3, we have
Corollary 2. Let be a compact CR-warped product submanifold of complex space form satisfying the following inequality Then, there do not exist stable integral -currents in and we have the trivial homology groups, i.e.,
Our next motivation comes from Calin [
15] who studied geometric mechanics on Riemannian manifolds and defined a positive differentiable function
(
) on a compact Riemannian manifold
. The
Dirichlet energy of a function
is defined in [
15] (see p. 41) as follows:
In view of the kinetic energy formula (
5) for a compact oriented manifold without boundary along with Theorem 2, we arrive at the following result.
Theorem 4. Let be a compact warped product pointwise semi-slant submanifold of a complex space form without boundary. If the following condition is satisfiedwhere is the Dirichlet energy of the warping function f with respect to the volume element , then there do not exist stable integral -currents in and An important concept relates to the geometrical and topological properties on Riemannian manifolds when considering the pinched condition on its metric. It is interesting to investigate the curvature and topology of submanifolds in a Riemannian manifold and the usual sphere theorems in Riemannian geometry. For instance, using the nonexistence of stable currents on compact submanifolds, Lawson and Simon [
9] obtained their striking sphere theorem, which proved that for an
n-dimensional compact-oriented submanifold
in a unit sphere
with the second fundamental form bounded above by a constant which depends on the dimension
n, then
is homeomorphic to a sphere
when
and
are homotopic to a sphere
.
Making use of Lawson and Simon [
9], Leung [
16] proved that for a compact connected oriented submanifold
in the unit sphere
such that
, when
and
are homotopic to a sphere
, then
is homeomorphic to a sphere
. Recently, it has been shown in [
17] that if the sectional curvature satisfies some pinching condition
for
n-dimensional compact oriented minimal submanifold
M in the unit sphere
with co-dimension
, then
M is either a totally geodesic sphere, one of the Clifford minimal hyper-surfaces
in
for
, or a Veronese surface in
. More recently, several results have been derived on topological and differentiable structures of submanifolds when imposing certain conditions on the second fundamental form, Ricci curvatures, and sectional curvatures in a series of articles [
4,
10,
11,
18,
19,
20,
21,
22,
23] by different geometers. For the warped product structure, we refer to [
20,
24,
25,
26,
27,
28,
29,
30].
The second target of note is to establish topological sphere theorems from the viewpoint of warped product submanifold geometry with positive constant sectional curvature and pinching conditions in terms of the squared norm of the warping function and Laplacian of the warped function as extrinsic invariants. In this sense, we work with conditions on the extrinsic curvature (second fundamental form, warping function), which have the advantage of being invariant under rigid motions. Motivated by Lawson and Simon [
9], (p. 441, Theorem 4), we consider a warped product pointwise semi-slant submanifold in a complex space form
such that the constant holomorphic sectional curvature is
, and state our main theorem of this paper.
Theorem 5. Let be a compact warped product pointwise semi-slant submanifold in a complex space form satisfying the condition (2). Then, is homeomorphic to sphere when while is homotopic to a sphere . Remark 1. As a consequence of Theorem 5, we obtain the following sphere theorem for a compact CR-warped product submanifold in a complex space form , thanks to Chen [13]. Corollary 3. Let be a compact CR-warped product submanifold in a complex space form satisfying the pinching condition (4). Then, is homeomorphic to a sphere when and is homotopic to a sphere . Using Theorem 4 and 5, we can now obtain an important result.
Corollary 4. Let be a compact warped product pointwise semi-slant submanifold of complex space form . If (6) is satisfied, then is homeomorphic to sphere when and is homotopic to a sphere . Remark 2. The principle behind Cheng’s eigenvalue comparison theorem (see [31]) forms the basis of the following finding. With the help of the first non-zero eigenvalue of the Laplacian operator, Cheng has demonstrated that if M is complete and isometric to the sphere of the standard unit then the following theorem can be inferred using the maximum principle for the first non-zero eigenvalue , provided that and . Theorem 6. Let be a compact warped product pointwise semi-slant submanifold of a complex space form with f being a non-constant eigenfunction of the first non-zero eigenvalue such that the following inequality is satisfied: Then, is homeomorphic to sphere when and is homotopic to a sphere when .
Motivated by Bochner’s formula [
32], we arrive at the following result.
Theorem 7. Let be a compact warped product pointwise semi-slant submanifold of a complex space form such that following inequality holds:where denotes the Hessian form of the warping function f and denotes the Ricci curvature along the base manifold . Then, is homeomorphic to sphere when and is homotopic to a sphere when . 2. Preliminaries
Let
be a complex space form with the complex dimension
. Then, the curvature tensor
R of
with constant holomorphic sectional curvature
is expressed as
The Gauss and Weingarten formulas for transforming submanifold
into an almost Hermitian manifold
are provided by
for each
and
such that the second fundamental form and the shape operator are denoted by
h and
. They are connected as
. Now, for any
and
, we have
where
and
are the tangential and normal components of
, respectively.
The Gauss equation for a submanifold
is defined as
for any
, where
and
R are the curvature tensors on
and
, respectively.
The norm of second fundamental form
h for an orthonormal frame
of the tangent space
on
is defined by
Let
be an local orthonormal frame of vector field
. Then, we have
where
and
are the gradient of function
and its squared norm.
The following classifications can be provided as:
- (i)
If for every , then is a holomorphic submanifold.
- (ii)
If for each , then is a totally real submanifold.
There are four types of submanifolds of a Kaehler manifold, namely, the CR-submanifold, slant submanifold, semi-slant submanifold, pointwise slant submanifold, and pointwise semi-slant submanifold. The definitions and classifications of such submanifolds are discussed in [
12,
13]. Moreover, for examples of a pointwise semi-slant submanifold in a Kaehler manifold and related problems, we refer to [
12]. It follows from Definition 3.1 in [
12] that if we denote as
and
the dimensions of a complex distribution
and pointwise slant distribution
of a pointwise semi-slant submanifold in a Kaehler manifold
, then the following remarks hold:
Remark 3. is invariant if and pointwise slant if .
Remark 4. If we consider the slant function as globally constant on and , then is a CR-submanifold.
Remark 5. An invariant subspace μ under J of normal bundle , is defined as .