1. Introduction
The study of warped product manifolds has attracted considerable attention in the field of research, particularly due to their applications in physics and the theory of relativity [
1]. These manifolds have proven to be valuable tools in providing essential solutions to the Einstein field equations [
1], which govern the behavior of gravity in spacetime. One of the most intriguing applications of warped product manifolds is their role in modeling the behavior of spacetime near black holes in the universe. By utilizing warped product manifolds, researchers are able to gain insights into the intricate nature of these astrophysical phenomena.
An important example of a warped product manifold is the Robertson–Walker model, which serves as a cosmological model for the structure and spacetime of the universe [
2]. This model describes the expanding universe, taking into account the curvature of spacetime and the distribution of matter and energy within it. The warped product structure captures the spatial geometry of the universe, providing a framework for understanding its evolution and dynamics.
In their investigations, Bishop and Neill delved into the geometry of Riemannian manifolds with negative curvature and introduced the concept of warped products for such manifolds [
3]. This notion, defined in
Section 2 of their work, extends the idea of Riemannian product manifolds and offers a more flexible and versatile framework for studying curved spaces. By utilizing the concept of warped products, researchers are able to explore and analyze a wide range of geometric structures and their associated properties.
The properties of warped product manifolds have been a subject of significant interest [
3]. Researchers have sought to understand the intrinsic characteristics and behavior of these manifolds, which differ from those of Riemannian product manifolds. By investigating the properties of warped product manifolds, researchers have been able to uncover unique geometric features and gain deeper insights into the underlying structures of curved spaces.
Overall, the study of warped product manifolds has emerged as a captivating topic of research, driven by their applications in physics and the theory of relativity. These manifolds provide valuable solutions to the Einstein field equations and offer a versatile framework for modeling various physical phenomena, such as the behavior of spacetime near black holes. The exploration of warped product manifolds, including their properties and geometric characteristics, contributes to our understanding of curved spaces and their implications in different fields of study.
In 1981, Chen made a significant contribution to the field by introducing the concept of warped products as a means to study CR-submanifolds of Kaehler manifolds [
4,
5]. This work focused on investigating the existence of CR-warped product submanifolds in the context of Kaehler manifolds. Chen demonstrated that it is possible to construct such submanifolds of the form
, where
represents the holomorphic submanifold and
represents the totally real submanifold.
Expanding on Chen’s pioneering work, Hasegawa and Mihai [
6] extended the study of CR-warped product submanifolds to Sasakian manifolds. They explored the contact CR-warped product submanifolds within the framework of Sasakian manifolds. By considering the interplay between the warping function and the squared norm of the second fundamental form, Mihai [
7] derived an estimate for the latter for contact CR-warped product submanifolds in Sasakian space forms.
Following these foundational contributions, numerous researchers have engaged in the study of warped product submanifolds in different settings of Riemannian manifolds. As a result, a wealth of existence results and findings have emerged in this field of research. For a comprehensive overview of these developments, one can refer to the survey article [
8], which provides a thorough exploration of the topic.
In 1999, Chen made a notable contribution regarding the connection between Ricci curvature and the squared mean curvature vector in any Riemannian manifold [
9]. This breakthrough led to a series of subsequent articles that aimed to formulate and explore the relationship between Ricci curvature and squared mean curvature within the context of various key structures on Riemannian manifolds [
7,
10,
11,
12,
13,
14,
15]. These works built upon Chen’s initial findings, delving into the intricate relationship between these two geometric quantities.
More recently, Ali et al. [
16] made significant contributions to this line of research by establishing a relation between Ricci curvature and squared mean curvature, specifically for warped product submanifolds of a sphere. Their work not only formulated this relationship but also provided numerous physical applications, highlighting the practical implications of this geometric connection. By studying the interplay between Ricci curvature and squared mean curvature in the context of warped product submanifolds, Ali et al. shed light on the underlying geometric structures and their relevance in various physical phenomena.
On the other hand, the introduction of the idea of a semi-symmetric linear connection on a Riemannian manifold can be attributed to Friedmann and Schouten [
17]. Subsequently, Hayden [
18] defined a semi-symmetric connection as a linear connection, ∇, existing on an
n-dimensional Riemannian manifold
, where the torsion tensor,
T, satisfies
, with
representing a 1-form and
. The properties of semi-symmetric metric connections were further explored by K. Yano [
19]. He demonstrated that a conformally flat Riemannian manifold admitting a semi-symmetric connection exhibits a vanishing curvature tensor. Sular and Ozgur [
20] delved into the investigation of warped product manifolds equipped with a semi-symmetric metric connection, focusing specifically on Einstein warped product manifolds with such connections. However, in their work [
21], they obtained additional results pertaining to warped product manifolds with a semi-symmetric metric connection. Motivated by these previous studies, our interest lies in examining the influence of a semi-symmetric metric connection on Ricci curvature of contact CR-warped product submanifolds and their geometry within an odd dimensional sphere.
2. Preliminaries
Suppose
is an odd dimensional Riemannian manifold. We define
as an almost contact metric manifold if there exists a tensor field,
, of type
and a global vector field,
, on
satisfying the following conditions
The dual 1-form of
is denoted as
. It is a well-known fact that an almost contact metric manifold can be classified as a Sasakian manifold if and only if the following tensorial equation holds
It is straightforward to observe that on a Sasakian manifold,
, the following can be readily deduced
here,
, and
represents the Riemannian connection associated with the metric
g on
.
Now, defining a connection,
, as
such that
for any
where
is the Riemannian connection with respect to
g. The connection
is semi-symmetric because
Using (
5) in (
3), we have
and
If a S-M has a constant -holomorphic sectional curvature, c, it is called a S-S-F and denoted as .
The expression for the curvature tensor,
, corresponding to the S-S-M connection,
, can be expressed as
Likewise, the curvature tensor,
, can be defined for the Riemannian connection,
.
Now, by the application of (
5), (
8), and (
9), we obtain
By employing the computed value of
, as detailed in [
22], by computation, we obtain the following expression for the curvature tensor,
, of a S-S-F:
for all
For a submanifold,
M, isometrically immersed in a Riemannian manifold,
, admitting a S-S-M connection, it is easy to derive the Gauss and Weingarten formula as follows
and
where ∇ is the induced s-s-m connection on
M,
, and
The relationship between the second fundamental form,
ℏ, and the shape operator,
, can be expressed by the following formula
For the vector fields
and
, we can decompose the expression as follows
and
where
and
are the tangential and normal components of
, correspondingly.
The equation of Gauss for a S-S-M connection, for the Riemannian curvature tensor,
R, can be expressed as follows [
22]
for
In their article [
20], Sular and Ozgur investigated the warped products denoted as
, where a S-S-M connection is defined on the manifold
, along with an associated vector field,
P. Here,
and
are Riemannian manifolds, and the warping function,
f, is a positive differentiable function defined on
. We present several important findings from [
20] as a lemma, which will be relevant for our subsequent analysis.
Lemma 1. Let be a W-P manifold with S-S-M connection , we have
- (i)
If the associated vector field , then - (ii)
where , and π is the 1-form associated with the vector field P.
Let us examine the relationship between the curvature tensors,
R and
, associated with the warped product submanifold,
, of a Sasakian manifold,
, corresponding to the induced S-S-M connection, ∇, and the induced Riemannian connection,
, on
M. Expressing this relationship, we have
for any vector field
on
M [
20].
According to part (ii) of Lemma 3.2 in reference [
20], for the warped product submanifold
, the following relationship holds
where
, respectively, and
is the Hessian of the warping function.
By taking into account Equations (
15) and (
16), we can infer the following
for the vector fields
,
, and
By substituting
into Equation (
5), we introduce the S-S-M connection. As a result, for a W-P submanifold,
, of the Riemannian manifold,
, we can deduce the following relationship using part
of Lemma 1.
and
In addition, Equation (
21) with (
7) yields
for
, and
.
Let us define the mean curvature vector,
, and its squared norm for a given point,
x, on the manifold,
M, considering an orthonormal basis
of the tangent space,
, as follows
Here, the dimension of
M is denoted by
n, and we can define certain properties based on the mean curvature vector,
, and its squared norm. When
, the submanifold is referred to as totally geodesic, and, if
, it is said to be minimal. Additionally, if
holds for all
, the submanifold,
M, is known as totally umbilical.
Let us denote the scalar curvature of an
m-dimensional Riemannian manifold,
, as
. Its mathematical expression is given by
where
. In this study, we will adopt an equivalent formulation of the aforementioned equation, which can be expressed as follows
Similarly, we can express the scalar curvature,
, of an
L-plane as follows
Consider an orthonormal basis
of the tangent space,
. If
is an element of the orthonormal basis
of the normal space,
, we can express the relationship as follows
and
Consider the sectional curvatures,
and
, associated with the plane sections generated by
and
at a point
x in the
n-dimensional manifold,
M, and the
m-dimensional Riemannian space form,
, respectively. Applying the Gauss equation, we obtain the following relationship
Let us define the global tensor field for an orthonormal frame of vector fields
on the
n-dimensional manifold,
M, as
for all
The tensor mentioned above is known as the Ricci tensor. In the case where we choose a specific vector,
, from the orthonormal frame
on the
n-dimensional manifold,
M, denoted by
, the Ricci curvature is defined as follows
Let us denote the gradient of a scalar function,
f, as
. The gradient is defined as follows
for all
Consider an orthogonal basis
of the tangent space,
, of an
n-dimensional Riemannian manifold,
M. Then, Equation (
25) yields the following result
The Laplacian of
f is defined by
The Hessian tensor, denoted as
, is a symmetric covariant tensor of rank 2. It is defined for a differentiable function,
f, as
where
is the Hessian of
For the warped product submanifolds of the type
with a S-S-M connection, using Formula (
20) we can derive the following result
3. Contact CR-Warped Product Submanifolds
In 1981, A. Bejancu [
23] introduced the concept of semi-invariant submanifolds in almost contact metric manifolds. An
m-dimensional Riemannian submanifold,
M, of a S-M
is classified as a semi-invariant submanifold if the vector field,
, is tangent to
M and there exists a differentiable distribution
on
M, such that
is invariant under the action of the structure vector field,
. The orthogonal complementary distribution
to
on
M is anti-invariant, meaning that
, where
and
represent the tangent space and normal space at
, respectively.
In a subsequent work by Hesigawa and Mihai [
6], they considered a specific type of submanifold known as a warped product submanifold of the form
in a Sasakian manifold,
. Here,
denotes an invariant submanifold,
represents an anti-invariant submanifold, and
belongs to
. These submanifolds were referred to as contact CR-submanifolds, and the authors provided some fundamental results related to them.
Our analysis commences by investigating a specific class of submanifolds known as contact CR-w-p submanifolds in a S-M endowed with a S-S-M connection. These submanifolds are of the form , where is an anti-invariant submanifold and represents an invariant submanifold, and satisfying the condition .
Theorem 1. Let be a S-M with a S-S-M connection. Then there does not exist a W-P submanifold of the type , such that .
Proof. For any
and
, then by (
19), the Gauss formula, and (
2), we have
Equivalently
By substituting
for both
and
in the preceding equation, we obtain
. This indicates that
f must be a constant, thereby establishing the desired result. □
In this study, we focus on analyzing warped product submanifolds of the form in a Sasakian manifold, , where these submanifolds are equipped with a semi-symmetric metric connection and belongs to . We refer to these specific submanifolds as contact CR-warped product submanifolds. Furthermore, we denote the invariant subspace of the normal bundle as .
Now, we commence with the following preliminary findings
Lemma 2. Let be a contact CR-W-P submanifold of a S-M endowed with a S-S-M connection, then
- (i)
- (ii)
- (iii)
and
Proof. By employing the Gauss formula and Equation (
6), we obtain the following expression
Now, utilizing Formula (
19), we obtain the following:
which is part
Again, using (
6), (
19), and the Gauss formula, part
(ii) is proven straightforwardly. Now, using the formula
and applying Equation (
19), we have
or
which is part
. □
Lemma 3. Let be a contact CR-W-P submanifold of a S-M endowed with a S-S-M connection, thenfor all and Proof. From the Gauss formula along with Equation (
6), we have
taking the inner product with
, we find
replacing
with
and using Equation (
1), we have
By considering Equations (
31) and (
32), we can deduce the following
which proves the assertion. □
Based on Lemma 3, it is clear that the isometric immersion of into a Sasakian manifold is characterized as being D-minimal. This D-minimal property establishes a valuable connection between the contact CR-warped product submanifold and the equation of Gauss.
Definition 1. An isometric immersion of the warped product into a Riemannian manifold, , is referred to as totally geodesic if the partial second fundamental form, , is identically zero. If the partial mean curvature vector, , becomes zero for , it is termed -minimal.
Consider a local orthonormal frame of vector fields on the contact CR-W-P submanifold given by . Here, are tangent to , and are tangent to . Additionally, forms a local orthonormal frame of the normal space, .
By considering Lemma 3, it is possible to observe
Hence, based on Lemma 3, it can be deduced that the trace of
ℏ with respect to
is zero. Therefore, in light of Definition 1, we obtain the following significant result.
Theorem 2. Consider a contact CR-W-P submanifold that is isometrically immersed in a S-M admitting a S-S-M connection. It can be concluded that possesses the property of being D-minimal.
Therefore, it can be readily concluded that the following statement holds.
where
is the squared mean curvature.
4. Inequalities for Ricci curvature
This section focuses on deriving the Ricci curvature for a contact CR-W-P isometrically immersed in an odd dimensional unit sphere within the context of the mean curvature vector and warping function, f.
Theorem 3. Let be a contact CR-W-P submanifold isometrically immersed in an odd dimensional unit sphere, , admitting a semi-symmetric metric connection. If for each orthogonal unit vector field orthogonal to ξ, either tangent to or , then
- (1)
The Ricci curvature is subject to the following inequalities
- (2)
In the case where for all points , there exists a unit vector field, χ that satisfies the equality condition of if and only if is a mixed T-G submanifold and χ belongs to the relative null space, , at x.
- (3)
In the equality case, we have the following
- (a)
The equality case of (34) holds true for all unit vector fields tangent to at each point if and only if is a mixed T-G submanifold and a D-T-G contact CR-W-P submanifold in . - (b)
The equality case of (1) holds true for all unit tangent vectors to at each if and only if either is T-G submanifold or is a mixed T-G T-U and -T-G submanifold with dim
Here and represent the dimensions of and , respectively.
Proof. Let us consider
as a contact CR-W-P submanifold of an odd dimensional sphere
. By utilizing the Gauss equation, we can obtain the following expression
Let
be a set of orthonormal vector fields on
such that the frame
is tangent to
and
is tangent to
. So, the unit tangent vector
can be expanded (
36) as follows
□
The aforementioned expression can be expanded in the following manner.
In light of Lemma 3, the preceding expression can be written as follows.
According to Equation (
23), we obtain the following
By substituting the values from Equation (
38) into Equation (
37), we uncover the following
For the submanifold
, we can define the scalar curvature of
based on Equation (
21) as follows:
By utilizing Equations (
17), (
21) and (
26), we can derive the following expression:
By combining Equation (
41) with Equation (
5) in Equation (
39), we obtain the following
When considering , we have two possible scenarios: either is tangent to the submanifold or it is tangent to the fiber .
Case 1: Assuming that
is tangent to
, let us consider a unit tangent vector from the set
and suppose
. By using Equation (
42) and Equation (
24), we can determine the following
By combining Equations (
5), (
21) and (
22), we obtain the following expression
Substituting into Equation (
43), we obtain
Moreover, we can express the seventh and eighth terms on the right-hand side of Equation (
47) as follows
Similarly, we obtain
By substituting the above two values into Equation (
47), we obtain
Since
is
-minimal then we can observe the following
and
Furthermore, we can reach the following conclusion
By substituting Equations (
49) and (
50) into Equation (
48), and evaluating Equation (
51), we obtain the final expression:
Moreover, by utilizing the fact that
, we obtain
From the inequality mentioned earlier, we can deduce the inequality stated in Equation (
34).
Case 2: Assuming that
is tangent to
, we choose the unit vector from the set
and suppose it is
, i.e.,
. Then, using Equations (
5), (
21) and (
22), we obtain the following expressions
Now, following a similar approach as in case 1, and utilizing Equation (
52), we obtain
By following similar steps as in case
i, the above inequality can be expressed as
Observing Lemma 3, it can be seen that
By utilizing this result in Equation (
56), we obtain
The final term in the aforementioned inequality can be expressed as
Additionally, we can expand the fifth term on the right-hand side of Equation (
58) as follows
By substituting the last two values into Equation (
58), we have
or equivalently
Applying similar techniques as in the proof of case 1, we reach the following expression
which gives the inequality (
36).
Let us now investigate the equality cases of the inequality (
34). Firstly, we redefine the concept of the relative null space,
, of the submanifold,
, within the odd dimensional sphere,
, at any given point,
. The notion of the relative null space was originally introduced by B. Y. Chen [
9] and can be defined as follows
For a unit vector field,
, tangent to
at point
x, the equality in (
34) holds true if and only if
When and condition is satisfied, it indicates that is a mixed T-G contact CR-W-P submanifold. By combining statements and with the fact that is a contact CR-W-P submanifold, we can conclude that the unit vector field belongs to the relative null space, . The converse of this statement is straightforward, thus proving statement (2).
For a contact CR-W-P submanifold, the equality satisfies in (
34) for all unit tangent vectors belonging to
at
x if
where
and
Considering that
is a contact CR-W-P submanifold, the third condition implies that
for
. By incorporating this information into condition
, we can assert that
is a
D-T-G contact CR-W-P submanifold in
, and its mixed T-G nature arises from condition
. This establishes statement
in
.
The equality sign in (
34) holds identically for all unit tangent vector fields tangent to
at
x in the case of a contact CR-W-P submanifold if and only if
such that
and
Two cases arise from condition
, namely
If the first case of Equation (
60) is satisfied, it can be easily concluded, based on condition
, that
is a
-T-G contact CR-W-P submanifold in
. This corresponds to the first case of part
in statement
.
Alternatively, if
is not a
-T-G contact CR-warped product submanifold and
, condition
of Equation (
60) implies that
is a
-T-U contact CR-W-P submanifold in
. This corresponds to the second case in part
of statement
. Thus, part
of statement
is confirmed.
To establish
using parts
and
of
, we combine Equations (
59) and (
60). In the first case of this part, let us assume that
. Based on parts
and
of
, we can conclude that
is both a
D-T-G and
-T-G submanifold in
. Consequently,
is a T-G submanifold in
.
In the case where the first case does not satisfy, we can conclude from parts and that is a mixed T-G and D-T-G submanifold of with . Based on condition , it is evident that is a -T-U contact CR-W-P submanifold, and from part , it is a D-T-G. This satisfies part of the theorem. Thus, the theorem is proven.
5. Conclusions
This paper has delved into the study of contact CR-warped product submanifolds within the framework of Sasakian space forms endowed with a semi-symmetric metric connection. Through our comprehensive investigation, we have uncovered several important results and made significant contributions to the understanding of these submanifolds.
One of the main achievements of this paper is the establishment of various key findings regarding contact CR-warped product submanifolds. We have explored their geometric properties, such as the characterization of their induced metric, and the determination of the necessary and sufficient conditions for a submanifold to be contact CR-warped. Additionally, we have investigated the behavior of the mean curvature vector and the warping function on these submanifolds.
Moreover, we have derived an inequality that relates the Ricci curvature to the mean curvature vector and the warping function. This inequality provides a valuable geometric constraint on contact CR-warped product submanifolds in Sasakian space forms with a semi-symmetric metric connection. It deepens our understanding of the interplay between the intrinsic curvature of the submanifold and its extrinsic mean curvature vector.
The findings presented in this paper have implications for various areas of differential geometry and mathematical physics. They contribute to the broader field of Riemannian geometry, particularly in the study of Sasakian space forms and submanifold theory. Furthermore, these results can potentially be applied in other mathematical and physical contexts where contact CR-warped product submanifolds arise.
In conclusion, this paper expands our knowledge of contact CR-warped product submanifolds in Sasakian space forms equipped with a semi-symmetric metric connection. The insights gained from this research, along with the derived inequality, provide a solid foundation for further investigations and applications in related fields. In future research, we will explore how our results can be applied with soliton theory, submanifold theory, and related fields presented in the papers [
1,
6,
7,
14,
24,
25,
26,
27,
28,
29,
30,
31,
32] to obtain new results.