Application of Differential Equations on the Ricci Curvature of Contact CR-Warped Product Submanifolds of S²ⁿ⁺¹(1) with Semi-Symmetric Metric Connection

Abstract

In this paper, we explore the uses of Obata’s differential equation in relation to the Ricci curvature of an odd-dimensional sphere that possesses a semi-symmetric metric connection. Specifically, we establish that, given certain conditions, the underlying submanifold can be identified as an isometric sphere. Additionally, we investigate the impact of specific differential equations on these submanifolds and demonstrate that, when certain geometric conditions are met, the base submanifold can be characterized as a special type of warped product.

1. Introduction

Semi-symmetric connections on Riemannian manifolds have been the subject of significant study in differential geometry. These connections are characterized by a specific property of their torsion tensor. The torsion tensor measures the failure of a connection to be symmetric, and in the case of semi-symmetric connections, it exhibits a particular structure.

In the referenced paper [1], the author provided a definition for a semi-symmetric connection on an n-dimensional Riemannian manifold $(M,g)$, while Hayeden [2] further explored the semi-symmetric connection. The connection, denoted by ∇, is a linear connection that satisfies the condition

$$T({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)=\pi \left({\mathsf{\Lambda }}_2\right){\mathsf{\Lambda }}_1-\pi \left({\mathsf{\Lambda }}_1\right){\mathsf{\Lambda }}_2,$$

where T is the torsion tensor, $\pi$ represents a 1-form, and ${\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2$ are vector fields on the tangent bundle $TM$. This equation essentially describes how the torsion tensor acts on pairs of vector fields.

The study of semi-symmetric connections gained further insights through the work of K. Yano [3]. Yano investigated the properties of semi-symmetric metric connections, particularly focusing on conformally flat Riemannian manifolds. A Riemannian manifold is conformally flat if the metric can be locally transformed to a constant multiple of the Euclidean metric. Yano demonstrated that in the case of a conformally flat manifold equipped with a semi-symmetric connection, the curvature tensor vanishes.

The vanishing curvature tensor in such situations is an interesting result, as it implies that the manifold exhibits certain geometric simplifications. In particular, it suggests a close relationship between the semi-symmetric connection, the conformal flatness of the metric, and the underlying geometry of the manifold. These findings have been explored in various applications and have contributed to our understanding of the interplay between curvature, symmetry, and geometric structures in Riemannian manifolds.

Warped product manifolds provide an intriguing geometric structure for investigating the dynamics of spacetime near black holes and regions with intense gravitational fields. Bishop and O’Neill [4] originally introduced these manifolds to examine spaces with negative curvature, but they have since progressed to include warping functions, expanding upon the idea of Riemannian product manifolds.

The study of warped products in submanifold theory was originally pioneered by B. Y. Chen. Chen’s seminal investigations delved into CR-warped product submanifolds within the context of almost Hermitian manifolds [5]. In his research, Chen not only introduced the notion of CR-warped product submanifolds but also formulated a method to estimate the norm of the second fundamental form by integrating a warping function.

Expanding upon Chen’s contributions, Hesigawa and Mihai [6] delved deeper into the study of contact forms associated with these submanifolds. Specifically, they investigated contact CR-warped product submanifolds that are embedded in Sasakian space forms. Similar to Chen’s approach, they derived an approximation for the second fundamental form of these submanifolds, taking into account the underlying warping function and the geometry of the ambient Sasakian space form.

The works of Chen, Hesigawa, and Mihai have significantly advanced our understanding of the geometric properties and structures of CR-warped product submanifolds. Their research provides valuable insights into the interplay between submanifold theory, almost Hermitian manifolds, contact forms, and Sasakian space forms, shedding light on the intricate relationships and approximations that govern these geometric objects.

In the realm of warped product manifolds, Sular and Ozgur [7] conducted an in-depth exploration of Einstein warped product manifolds equipped with a semi-symmetric metric connection. Their primary objective was to investigate the properties and behaviors exhibited by such manifolds. Subsequently, in their follow-up research [8], they obtained further results pertaining to warped product manifolds possessing a semi-symmetric metric connection, thereby expanding our understanding of this geometric framework.

Chen’s groundbreaking work in 1999 revealed a significant correlation between Ricci curvature and the squared mean curvature vector in all Riemannian manifolds [9]. This discovery triggered a series of subsequent studies that sought to explore and understand the intricate relationship between Ricci curvature and squared mean curvature in diverse structural contexts within Riemannian manifolds [10,11,12,13,14,15,16]. These subsequent research endeavors built upon Chen’s initial breakthrough and delved deeper into the complex interplay between these two geometric measures.

The investigation technique introduced by Obata [17] has garnered significant attention and importance in the field of geometric analysis. Obata’s contribution revolves around the presentation of the Obata equation as a theorem that provides a characterization of a regular sphere through a differential equation.

In essence, Obata’s findings establish that in the context of a complete Riemannian manifold $(M^k,g)$, a non-constant function f defined on $M^k$ satisfies the differential equation ${\nabla }^{2}f+cfg=0$ or equivalently $Hessian\left(f\right)+cfg=0$ if and only if the manifold $M^k$ is isometric to an n-dimensional sphere with a radius of c. This result has paved the way for numerous in-depth studies in this area, shedding light on the significance of various domains in the analysis of differential geometry of manifolds, including the Euclidean space, Euclidean sphere, and complex projective space. These domains have been extensively explored and examined in relation to the Obata equation, with a substantial body of research [17,18,19,20,21,22,23,24,25,26,27]. Of particular note is the special case when the differential equation reduces to ${\nabla }^{2}f=cg$, where c is a constant. This equation precisely characterizes the Euclidean space, wherein the metric tensor g corresponds to the standard Euclidean metric. This notable result was established by Tashiro [27], further solidifying the connection between the Obata equation and the geometric properties of the Euclidean space.

In a significant contribution, Lichnerowicz [28] demonstrated that, subject to certain geometric conditions, an isometry exists between the Riemannian manifold $(M^k,g)$ and the n-dimensional sphere $S^k$. Building upon this result, the authors of [23] employed Obata’s differential equation to establish a connection between connected Riemannian manifolds and n-dimensional spheres of radius c. They showed that if the Ricci curvature of $(M^k,g)$ satisfies the inequality $0<Ric=(k-1)(2-(kc/{\gamma }_1)c)$, where c is a constant and ${\gamma }_1$ represents the first eigenvalue of the Laplacian, then $(M^k,g)$ is isometric to an k-dimensional sphere.

In a recent study by Khan et al. [29], they investigated the Ricci curvature inequalities for contact CR-warped product submanifolds within the context of an odd-dimensional sphere, employing a semi-symmetric metric connection.

In 2018, Jamali and Shahid [21] delved into the application of Obata’s differential equation within Sasakian space forms, employing the Bochner formula. Building upon this work, Ali et al. [18] investigated Ricci curvature inequalities for warped product submanifolds within a sphere, exploring Obata’s differential equation through the lens of Ricci curvature with a foundation in the Levi–Civita connection. Contrasting this, the rich geometric properties inherent in semi-symmetric metric connections led Khan et al. [29] to establish Ricci curvature inequalities for contact CR-warped product submanifolds within odd-dimensional spheres using the semi-symmetric metric connection. Although their study lacked applications, this present paper endeavors to offer the applications of Obata’s differential equation through Ricci curvature.

The main objective of our paper is to establish a relationship between the geometry of the semi-symmetric connection and the differential equations presented by Obata. The aim of this article may be summarized as follows.

2. Definitions and Basic Results

Let $(\overline{M},g)$ be an Riemannian manifold of odd dimensions. We characterize $\overline{M}$ as an almost contact metric manifold if it includes a type $(1,1)$ tensor field $\varphi$ and a global vector field $\xi$ that adhere to the following criteria:

$${\varphi }^2{\mathsf{\Lambda }}_1=-{\mathsf{\Lambda }}_1+\eta \left({\mathsf{\Lambda }}_1\right)\xi ,g({\mathsf{\Lambda }}_1,\xi )=\eta \left({\mathsf{\Lambda }}_1\right),$$

(1)

$$g(\varphi {\mathsf{\Lambda }}_1,\varphi {\mathsf{\Lambda }}_2)=g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)-\eta \left({\mathsf{\Lambda }}_1\right)\eta \left({\mathsf{\Lambda }}_2\right).$$

(2)

The dual 1-form of $\xi$ is represented as $\eta$. It is widely recognized that an almost contact metric manifold can be categorized as a Sasakian manifold if and only if the following conditions are met:

$$\left({\stackrel{=}{\nabla }}_{{\mathsf{\Lambda }}_1}\varphi \right){\mathsf{\Lambda }}_2=g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)\xi -\eta \left({\mathsf{\Lambda }}_2\right){\mathsf{\Lambda }}_1.$$

(3)

Next, we shall introduce a connection $\overline{\nabla }$ in the following manner:

$${\overline{\nabla }}_{{\mathsf{\Lambda }}_1}{\mathsf{\Lambda }}_2={\stackrel{=}{\nabla }}_{{\mathsf{\Lambda }}_1}{\mathsf{\Lambda }}_2+\eta \left({\mathsf{\Lambda }}_2\right){\mathsf{\Lambda }}_1-g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)\xi$$

(4)

such that $\overline{\nabla }g=0$ for any ${\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2\in TM$. In the context where $\stackrel{=}{\nabla }$ represents the Riemannian connection corresponding to g, the connection $\overline{\nabla }$ exhibits semi-symmetry due to the relationship $T({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)=\eta \left({\mathsf{\Lambda }}_2\right){\mathsf{\Lambda }}_1-\eta \left({\mathsf{\Lambda }}_1\right){\mathsf{\Lambda }}_2$. By applying (3) within the scope of (1), we obtain

$$\left({\overline{\nabla }}_{{\mathsf{\Lambda }}_1}\varphi \right){\mathsf{\Lambda }}_2=g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)\xi -g({\mathsf{\Lambda }}_1,\varphi {\mathsf{\Lambda }}_2)\xi -\eta \left({\mathsf{\Lambda }}_2\right){\mathsf{\Lambda }}_1-\eta \left({\mathsf{\Lambda }}_2\right)\varphi {\mathsf{\Lambda }}_1$$

(5)

and

$${\overline{\nabla }}_{{\mathsf{\Lambda }}_1}\xi ={\mathsf{\Lambda }}_1-\eta \left({\mathsf{\Lambda }}_1\right)\xi -\varphi {\mathsf{\Lambda }}_1.$$

(6)

The curvature tensor $\overline{R}$ of Sasakian space form with semi-symmetric connection is given by

$$\begin{array}{ll}\overline{R}({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3,{\mathsf{\Lambda }}_4)=& {\displaystyle \frac{c+3}{4}}\{g({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3)g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_4)-g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_3)g({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_4)\}\hfill \\ & +{\displaystyle \frac{c-1}{4}}\{\eta \left({\mathsf{\Lambda }}_1\right)\eta \left({\mathsf{\Lambda }}_3\right)g({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_4)-\eta \left({\mathsf{\Lambda }}_2\right)\eta \left({\mathsf{\Lambda }}_3\right)g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_4)\hfill \\ & +g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_3)\eta \left({\mathsf{\Lambda }}_2\right)\eta \left({\mathsf{\Lambda }}_4\right)-g({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3)\eta \left({\mathsf{\Lambda }}_1\right)\eta \left({\mathsf{\Lambda }}_4\right)\hfill \\ & +g(\varphi {\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3)g(\varphi {\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_4)+g(\varphi {\mathsf{\Lambda }}_3,{\mathsf{\Lambda }}_1)g(\varphi {\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_4)\hfill \\ & -2g(\varphi {\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)g(\varphi {\mathsf{\Lambda }}_3,{\mathsf{\Lambda }}_4)\}+\beta ({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_3)g({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_4)\hfill \\ & -\beta ({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3)g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_4)+\beta ({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_4)g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_3)\hfill \\ & -\beta ({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_4)g({\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3),\hfill \end{array}$$

(7)

for ${\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3,{\mathsf{\Lambda }}_4\in \mathsf{\Gamma }\left(T\overline{M}\right),$ where $\beta =\left({\overline{\nabla }}_{{\mathsf{\Lambda }}_1}\eta \right){\mathsf{\Lambda }}_2-\eta \left({\mathsf{\Lambda }}_1\right)\eta \left({\mathsf{\Lambda }}_2\right)+{\displaystyle \frac{1}{2}}g({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)$; more details can be seen in [30].

Let M be a k-dimensional submanifold of an m-dimensional almost contact metric manifold $\overline{M}$. For an orthonormal basis $\{{\omega }_1,{\omega }_2,\dots ,{\omega }_k\}$ of the tangent space $T_{x}M$, the mean curvature vecotor $H\left(x\right)$ and its squared norm, respectively, are given by

$$H\left(x\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^k}\delta ({\omega }_i,{\omega }_i)$$

$${\parallel H\parallel }^2={\displaystyle \frac{1}{k^2}}{\displaystyle \sum _{i,j=1}^k}g(\delta ({\omega }_i,{\omega }_i),\delta ({\omega }_j,{\omega }_j)),$$

where $\delta$ represents the second fundamental form of M.

The scalar curvature of $\overline{M}$ is denoted by $\overline{\tau }\left(\overline{M}\right)$ and is defined as

$$\overline{\tau }\left(\overline{M}\right)={\displaystyle \sum _{1\le \alpha <\beta \le m}}{\kappa }_{\alpha \beta },$$

where ${\kappa }_{\alpha \beta }=\kappa ({\omega }_{\alpha }\wedge {\omega }_{\beta })$ represents the sectional curvature of the 2-plane section formed by ${\omega }_{\alpha },{\omega }_{\beta }.$

The global tensor field associated with the orthonormal frame of vector fields $\{{\omega }_1,\dots ,{\omega }_n\}$ on $\overline{M}$ is given by

$$S({\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2)={\displaystyle \sum _{i=1}^k}\left\{g(R({\omega }_i,{\mathsf{\Lambda }}_1){\mathsf{\Lambda }}_2,{\omega }_i)\right\}$$

for ${\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2\in T_{x}M$, where R denotes the Riemannian curvature tensor. This tensor is commonly known as the Ricci tensor. If we select a distinct vector ${\omega }_u$ from $\{{\omega }_1,\dots ,{\omega }_n\}$ on $M^k$, which is determined by $\upsilon$, then the Ricci curvature is defined as

$$Ric\left(\upsilon \right)={\displaystyle \sum _{\alpha =1,\alpha \ne u}^k}\kappa ({\omega }_{\alpha }\wedge {\omega }_u).$$

Let us consider a contact CR-warped product (CR-W-P) submanifold denoted by $M^k=N_T^{k_1}{\times }_f{N}_{\perp }^{k_2}$, which is a part of an almost contact metric manifold. Here, $N_T^{k_1}$ represents the $k_1$-dimensional invariant submanifold, while $N_{\perp }^{k_2}$ represents the $k_2$-dimensional anti-invariant submanifold of the almost contact metric manifold.

To describe this submanifold, we can employ a local orthonormal frame of vector fields, denoted as $\{{\omega }_1,\dots ,{\omega }_{\beta },{\omega }_{\beta +1}=\varphi {\omega }_1,\dots ,{\omega }_{k_1-1}=\varphi {\omega }_{\beta },{\omega }_{k_1}=\xi ,{\omega }_{k_1+1},\dots ,{\omega }_n\}$. In this frame, $\{\xi ,{\omega }_1,\dots ,{\omega }_{k_1}\}$ are tangent vectors to $N_T$, whereas $\{{\omega }_{k_1+1},\dots {\omega }_n\}$ are tangent vectors to $N_{\perp }$. Additionally, $\{{\omega }_1^{*}=\varphi {\omega }_{k_1+1},\dots ,{\omega }_k^{*}=\varphi {\omega }_n,{\omega }_{n+1}^{*},\dots ,{\omega }_m^{*}\}$ form a local orthonormal frame for the normal space $T^{\perp }M$.

Recently, Khan et al. [29] studied the Ricci curvature for contact CR-warped product submanifolds of an odd dimensional sphere admitting a semi-symmetric metric connection and obtained some inequalities for Ricci curvature [29]. Moreover, for contact CR-warped product submanifold $M=N_T{\times }_f{N}_{\perp }$, we have

$$k_2{\displaystyle \frac{\Delta f}{f}}=k_2{(\Delta lnf-\parallel \nabla lnf\parallel }^2),$$

(8)

where $k_1$ and $k_2$ are the dimensions of the invariant and anti-invariant submanifold, respectively. Using Equation (8), we obtained the following result, given in [29], Theorem 3.

Theorem 1

(cf. [29], Theorem 3). Suppose $M=N_T^{k_1}{\times }_f{N}_{\perp }^{k_2}$ represents a contact CR-W-P submanifold that is isometrically embedded in an odd-dimensional unit sphere $S^{2n+1}\left(1\right)$ and features a semi-symmetric metric connection. If, for every orthogonal unit vector field $\upsilon \in T_{x}M$ that is orthogonal to ξ and lies tangent to either $N_T^{k_1}$ or $N_{\perp }^{k_2}$, then the Ricci curvature is constrained by the following set of inequalities:

(i)

if $\upsilon \in T{N}_T$

$$\begin{array}{ll}Ric\left(\upsilon \right)+k_2\Delta lnf\le & {\displaystyle \frac{1}{4}}{k}^2{\parallel H\parallel }^2+k_2{\parallel \nabla lnf\parallel }^2+(k-k_1{k}_2+2k_2-1)\hfill \\ & -(1+k_1){\displaystyle \sum _{i=k_1+1}^n}\alpha ({\omega }_i,{\omega }_i)-(1+k_2){\displaystyle \sum _{i=1}^{k_1}}\alpha ({\omega }_i,{\omega }_i)\hfill \\ & -(k-2)\alpha ({\omega }_1,{\omega }_1).\hfill \end{array}$$

(9)

(ii)

if $\upsilon \in T{N}_{\perp }$

$$\begin{array}{cc}Ric\left(\upsilon \right)+k_2\Delta lnf\le & {\displaystyle \frac{1}{4}}{k}^2{\parallel H\parallel }^2+k_2{\parallel \nabla lnf|}^2+(k-k_1{k}_2+2k_2-1)\hfill \\ & -(1+k_1){\displaystyle \sum _{i=k_1+1}^n}\alpha ({\omega }_i,{\omega }_i)-(1+k_2){\displaystyle \sum _{i=1}^{k_1}}\alpha ({\omega }_i,{\omega }_i)\hfill \\ & -(k-2)\alpha ({\omega }_n,{\omega }_k).\hfill \end{array}$$

(10)

The instances of equality can be observed in [29], Theorem 3.

3. Main Results

In this section, we study the application of Obata’s differential equation on contact CR-W-P submanifold $M^k=N_T^{k_1}{\times }_f{N}_{\theta }^{k_2}$ in an odd-dimensional sphere $S^{2m+1}\left(1\right)$ by using the Ricci curvature. Now, we have the following result.

Theorem 2.

Let $M^k=N_T^{k_1}{\times }_f{N}_{\perp }^{k_2}$ be a compact orientable contact CR-W-P submanifold immersed isometrically in $S^{2m+1}\left(1\right)$ with positive curvature $Ric\left(\upsilon \right)\ge 0,$$\upsilon \in T{N}_T^{k_1}$, satisfying the following relation:

$$\begin{array}{cc}{\displaystyle \frac{k^2}{4}}{\parallel H\parallel }^2+{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\parallel {\nabla }^2\rho \parallel }^2+& (k-k_1{k}_2-2k_2-1)=(1+k_1){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)\hfill \\ & +(1+k_2){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)+(k-2)\alpha ({\omega }_1,{\omega }_1),\hfill \end{array}$$

(11)

where ${\gamma }_1>0$ represents an eigenvalue of the warping function $\rho =lnf$. Consequently, the base manifold $N_T^{k_1}$ is isometric to the sphere $S^{k_1}\left({\displaystyle \frac{{\gamma }_1}{k_1}}\right)$ with a constant sectional curvature of ${\displaystyle \frac{{\gamma }_1}{k_1}}$.

Proof. 

Let $\upsilon \in T{N}_T$, and considering that $\rho =lnf$, define the following relation as

$$\parallel {\nabla }^2{\rho -t\rho I\parallel }^2=\parallel {\nabla }^2{\rho \parallel }^2+t^2{\rho }^2{\parallel I\parallel }^2-2t\rho g({\nabla }^2\rho ,I),$$

(12)

where I denotes the identity operator on the submanifold $T{N}_T^{k_1}$, and it is known that ${\parallel I\parallel }^2=\mathrm{trace}(I,I^{*})=k_1$ and

$$g({\nabla }^2\rho ,I^{*})=trace({\nabla }^2\rho ,I^{*})=trace{\nabla }^2\rho .$$

(13)

Thus, Equation (12) transforms into

$$\parallel {\nabla }^2{\rho -t\rho I\parallel }^2={\parallel {\nabla }^2\rho \parallel }^2+k_1{t}^2{\rho }^2-2t\rho \Delta \rho ,$$

(14)

If ${\gamma }_1$ is an eigenvalue of the eigenfunction $\rho$, then $\Delta \rho ={\gamma }_1\rho$. Consequently, we derive

$$\parallel {\nabla }^2{\rho -t\rho I\parallel }^2={\parallel {\nabla }^2\rho \parallel }^2+(k_1{t}^2-2t\rho ){\rho }^2.$$

(15)

On the other hand, we obtain $\Delta {\rho }^2=2\rho \Delta \rho +{\parallel \nabla \rho \parallel }^2$ or ${\gamma }_1{\rho }^2=2{\gamma }_1{\rho }^2+{\parallel \nabla \rho \parallel }^2$, which implies that ${\rho }^2=-{\displaystyle \frac{1}{{\gamma }_1}}{\parallel \nabla \rho \parallel }^2$, and using this in Equation (15), we obtain

$$\parallel {\nabla }^2{\rho -t\rho I\parallel }^2=\parallel {\nabla }^2{\rho \parallel }^2+(2t-{\displaystyle \frac{k_1{t}^2}{{\gamma }_1}}){\parallel \nabla \rho \parallel }^2.$$

(16)

Specifically, by setting $t=-{\displaystyle \frac{{\gamma }_1}{k_1}}$ in Equation (16) and integrating with respect to the volume element, we obtain

$${\int }_{M^k}\parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}{\rho I\parallel }^{2}dV={\int }_{M^k}\parallel {\nabla }^2{\rho \parallel }^{2}dV-{\displaystyle \frac{3{\mu }_1}{k_1}}{\int }_{M^k}{\parallel \nabla \rho \parallel }^{2}dV$$

(17)

By integrating inequality (9) and considering the fact that ${\int }_{M^k}\Delta \varphi dV=0$, we obtain

$$\begin{array}{ll}{\int }_{M^k}Ric\left(\upsilon \right)dV\le & {\displaystyle \frac{k^2}{4}}{\int }_{M^k}{\parallel H\parallel }^{2}dV+k_2{\int }_{M^k}{\parallel \nabla lnf|}^{2}dV+(k-k_1{k}_2+2k_2-1)Vol\left(M^k\right)\hfill \\ & -(1+k_1){\displaystyle \sum _{i=k_1+1}^n}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)-(1+k_2){\displaystyle \sum _{i=1}^{k_1}}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)\hfill \\ & -(k-2)\alpha ({\omega }_1,{\omega }_1)Vol\left(M^k\right).\hfill \end{array}$$

(18)

From (17) and (18), we obtain

$$\begin{array}{ll}{\int }_{M^k}Ric\left(\upsilon \right)dV\le & {\displaystyle \frac{k^2}{4}}{\int }_{M^k}{\parallel H\parallel }^{2}dV+{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\int }_{M^k}\parallel {\nabla }^2{\rho \parallel }^{2}dV-{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\int }_{M^k}{\parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}\rho I\parallel }^2\hfill \\ & +(k-k_1{k}_2+2k_2-1)Vol\left(M^k\right)-(1+k_1){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)\hfill \\ & -(1+k_2){\displaystyle \sum _{i=1}^{k_1}}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)-(k-2)\alpha ({\omega }_1,{\omega }_1)Vol\left(M^k\right).\hfill \end{array}$$

(19)

Under the assumption that $Ric\left(\upsilon \right)\ge 0$, the aforementioned inequality yields

$$\begin{array}{ll}{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\int }_{M^k}& \parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}{\rho I\parallel }^{2}dV\le {\displaystyle \frac{k^2}{4}}{\int }_{M^k}{\parallel H\parallel }^{2}dV+{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\int }_{M^k}{\parallel {\nabla }^2\rho \parallel }^{2}dV\hfill \\ & +(k-k_1{k}_2+2k_2-1)Vol\left(M^k\right)-(1+k_1){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)\hfill \\ & -(1+k_2){\displaystyle \sum _{i=1}^{k_1}}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)-(k-2)\alpha ({\omega }_1,{\omega }_1)Vol\left(M^k\right),\hfill \end{array}$$

(20)

or

$$\begin{array}{cc}\hfill {\int }_{M^k}& \parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}{\rho I\parallel }^{2}dV\le {\displaystyle \frac{3k^2{\mu }_1}{4k_1{k}_2}}{\int }_{M^k}{\parallel H\parallel }^{2}dV+{\int }_{M^k}{\parallel {\nabla }^2\rho \parallel }^{2}dV\hfill \\ & +{\displaystyle \frac{3{\mu }_1}{k_1{k}_2}}(k-k_1{k}_2+2k_2-1)Vol\left(M^k\right)-{\displaystyle \frac{3(1+k_1){\gamma }_1}{k_1{k}_2}}{\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)\hfill \\ & -{\displaystyle \frac{3(1+k_2){\gamma }_1}{k_1{k}_2}}{\displaystyle \sum _{i=1}^{k_1}}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)-{\displaystyle \frac{3(k-2){\gamma }_1}{k_1{k}_2}}\alpha ({\omega }_1,{\omega }_1)Vol\left(M^k\right).\hfill \end{array}$$

(21)

From Equation (11), we obtain

$${\int }_{M^k}{\parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}\rho I\parallel }^{2}dV\le 0,$$

(22)

but we know that

$${\int }_{M^k}{\parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}\rho I\parallel }^{2}dV\ge 0.$$

(23)

By combining the preceding two statements, we derive

$${\int }_{M^k}{\parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}\rho I\parallel }^{2}dV=0,$$

(24)

which implies that

$${\nabla }^2\rho =-{\displaystyle \frac{{\gamma }_1}{k_1}}\rho I.$$

(25)

Given that the warping function $\rho =lnf$ is not a constant function on $M^k$, Equation (25) represents Obata’s equation [17] with a positive constant $c={\displaystyle \frac{{\gamma }_1}{k_1}}$. Since ${\gamma }_1>0$, the base manifold $N_T^{k_1}$ is therefore isometric to the sphere $S^{k_1}\left({\displaystyle \frac{{\gamma }_1}{k_1}}\right)$ with a constant sectional curvature of ${\displaystyle \frac{{\gamma }_1}{k_1}}$. This conclusion establishes the validity of the theorem. □

Moreover, considering the vector field $\upsilon$ tangent to $N_{\perp }^{k_2}$ and following similar steps as in the preceding theorem, we arrive at the following result.

Theorem 3.

Consider $M^k=N_T^{k_1}{\times }_f{N}_{\perp }^{k_2}$ as a compact orientable contact CR-W-P submanifold, embedded isometrically in $S^{2m+1}\left(1\right)$ with positive Ricci curvature $Ric\left(\upsilon \right)\ge 0$, where $\upsilon \in T{N}_{\perp }^{k_2}$, and fulfilling the following condition:

$$\begin{array}{ll}\hfill {\displaystyle \frac{k^2}{4}}{\parallel H\parallel }^2+{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\parallel {\nabla }^2\rho \parallel }^2+& (k-k_1{k}_2-2k_2-1)=(1+k_1){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)\hfill \\ & +(1+k_2){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)+(k-2)\alpha ({\omega }_n,{\omega }_k)\hfill \end{array}$$

(26)

Given that ${\gamma }_1>0$ is an eigenvalue of the warping function $\rho =lnf$, the base manifold $N_T^{k_1}$ is isometric to the sphere $S^{k_1}\left({\displaystyle \frac{{\gamma }_1}{k_1}}\right)$ with a constant sectional curvature of ${\displaystyle \frac{{\gamma }_1}{k_1}}$.

In the investigation carried out by Garcia-Rio and colleagues as detailed in [26], they delved into an alternative form of Obata’s differential equation to characterize the Euclidean sphere. Essentially, this study showcased that if $\rho$ represents a non-constant real-valued function on a Riemannian manifold that satisfies $\Delta \rho +{\gamma }_1\rho =0$ and $\rho <0$, then $M^k$ can be isometrically depicted as a warped product of the Euclidean line and a comprehensive Riemannian manifold. In this context, the warping function $\rho$ serves as the solution to the following differential equation:

$${\displaystyle \frac{d^2\rho }{d{t}^2}}+{\gamma }_1\rho =0.$$

Expanding on the research conducted by Garcia-Rio and colleagues [26] and Ali and team [18], we formulate the subsequent theorem.

Theorem 4.

Consider $M^k=N_T^{k_1}{\times }_f{N}_{\perp }^{k_2}$, a compact orientable contact CR-W-P submanifold that is immersed isometrically in $S^{2m+1}\left(1\right)$ with positive curvature $Ric\left(\upsilon \right)>0,$ where $\upsilon \in T{N}_T^{k_1}$, and satisfying the following condition:

$$\begin{array}{ll}\hfill {\displaystyle \frac{k^2}{4}}{\parallel H\parallel }^2+{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\parallel {\nabla }^2\rho \parallel }^2+& (k-k_1{k}_2-2k_2-1)=(1+k_1){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)\hfill \\ & +(1+k_2){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)+(k-2)\alpha ({\omega }_1,{\omega }_1),\hfill \end{array}$$

(27)

In this scenario, where ${\gamma }_1<0$ signifies a negative eigenvalue of the eigenfunction $\rho =lnf$, $N_T^{k_1}$ is analogous to a warped product of the Euclidean line and a complete Riemannian manifold. The warping function $\rho =lnf$ satisfies the following differential equation:

$${\displaystyle \frac{d^2\rho }{d{t}^2}}+{\gamma }_1\rho =0.$$

(28)

Proof. 

Given our assumption of positive Ricci curvature, Myers’s theorem states that a complete Riemannian manifold with positive Ricci curvature is compact. Consequently, $M^k$ represents a compact contact CR-warped product submanifold with open boundary conditions [31]. By applying the assumption $Ric\left(\upsilon \right)>0$ to Equation (19), we derive

$$\begin{array}{ll}\hfill {\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\int }_{M^k}\parallel {\nabla }^2\rho & +{\displaystyle \frac{{\gamma }_1}{k_1}}{\rho I\parallel }^{2}dV<{\displaystyle \frac{k^2}{4}}{\int }_{M^k}{\parallel H\parallel }^{2}dV+{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\int }_{M^k}{\parallel {\nabla }^2\rho \parallel }^{2}dV\hfill \\ & +(k-k_1{k}_2+2k_2-1)Vol\left(M^k\right)-(1+k_1){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)\hfill \\ & -(1+k_2){\displaystyle \sum _{i=1}^{k_1}}\alpha ({\omega }_i,{\omega }_i)Vol\left(M^k\right)-(k-2)\alpha ({\omega }_1,{\omega }_1)Vol\left(M^k\right).\hfill \end{array}$$

(29)

If Equation (27) is satisfied, then based on the aforementioned inequality, we obtain $\parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}{\rho I\parallel }^{2}dV<0$, which is untenable. Therefore, we conclude that $\parallel {\nabla }^2\rho +{\displaystyle \frac{{\gamma }_1}{k_1}}{\rho I\parallel }^{2}dV=0$. Given that ${\gamma }_1<0$, as per the findings in [26], the submanifold $N_T^{k_1}$ is akin to a warped product of the Euclidean line and a full Riemannian manifold, where the warping function on R represents the solution to the differential Equation (28), thereby validating the theorem. □

Furthermore, when we take into account the vector field $\upsilon$ tangent to $N_{\perp }^{k_2}$, by following analogous procedures as in the previous theorem, we obtain the subsequent outcome.

Theorem 5.

Consider $M^k=N_T^{k_1}{\times }_f{N}_{\perp }^{k_2}$, a compact orientable contact CR-W-P submanifold that is immersed isometrically in $S^{2m+1}\left(1\right)$ with positive curvature $Ric\left(\upsilon \right)>0,$ where $\upsilon \in T{N}_{\perp }^{k_2}$, satisfying the following condition:

$$\begin{array}{ll}\hfill {\displaystyle \frac{k^2}{4}}{\parallel H\parallel }^2+{\displaystyle \frac{k_1{k}_2}{3{\mu }_1}}{\parallel {\nabla }^2\rho \parallel }^2+& (k-k_1{k}_2-2k_2-1)=(1+k_1){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)\hfill \\ & +(1+k_2){\displaystyle \sum _{i=k_1+1}^k}\alpha ({\omega }_i,{\omega }_i)+(k-2)\alpha ({\omega }_n,{\omega }_k),\hfill \end{array}$$

(30)

In this context, where ${\gamma }_1<0$ denotes a negative eigenvalue of the eigenfunction $\rho =lnf$, $N_T^{k_1}$ is equivalent to a warped product of the Euclidean line and a complete Riemannian manifold. The warping function $\rho =lnf$ satisfies the following differential equation:

$${\displaystyle \frac{d^2\rho }{d{t}^2}}+{\gamma }_1\rho =0.$$

(31)

4. Conclusions

In conclusion, this study delves into the applications of Obata’s differential equation concerning the Ricci curvature of an odd-dimensional sphere endowed with a semi-symmetric metric connection. Through rigorous analysis, it is shown that under specific conditions, the submanifold can be recognized as an isometric sphere. Furthermore, this research explores the influence of distinct differential equations on these submanifolds, showcasing that under certain geometric criteria, the base submanifold can be distinguished as a unique form of warped product. This investigation not only deepens our understanding of differential geometry but also sheds light on the intricate interplay between curvature properties and geometric structures within the realm of semi-symmetric metric connections on odd-dimensional spheres.