Eigenvalues for the Generalized Laplace Operator of Slant Submanifolds in the Sasakian Space Forms Admitting Semi-Symmetric Metric Connection

Abstract

This study is focused on pioneering new upper bounds on mean curvature and constant sectional curvature relative to the first positive eigenvalue of the generalized Laplacian operator in the differentiable manifolds with a semi-symmetric metric connection. Multiple approaches are being explored to determine the principal eigenvalue for the generalized-Laplacian operator in closed oriented-slant submanifolds within a Sasakian space form (ssf) with a semi-symmetric metric (ssm) connection. By utilizing our findings on the Laplacian, we extend several Reilly-type inequalities to the generalized Laplacian on slant submanifolds within a unit sphere with a semi-symmetric metric (ssm) connection. The research is concluded with a detailed examination of specific scenarios.

1. Introduction

Friedmann and Schouten [1] made significant contributions by introducing the concept of a semi-symmetric linear connection within the realm of Riemannian manifolds. Later, Hayden [2] provided a precise definition of a semi-symmetric metric (ssm) connection as a linear connection ∇ defined on an n-dimensional Riemannian manifold $(\mathrm{Y},g)$. In this context, the torsion tensor T satisfies the equation $T({\upsilon }_1,{\upsilon }_2)=\pi ({\upsilon }_2){\upsilon }_1-\pi ({\upsilon }_1){\upsilon }_2$, where $\pi$ is a 1-form, and ${\upsilon }_1,{\upsilon }_2\in T\mathrm{Y}$.

Expanding on this groundwork, K. Yano [3] extensively investigated the properties and implications of semi-symmetric metric connections. One significant finding by Yano was that in a conformally flat Riemannian manifold equipped with a semi-symmetric connection, the curvature tensor uniformly vanishes.

Applications of semi-symmetric connections extend to various areas of differential geometry and theoretical physics. These connections play a crucial role in the study of special classes of Riemannian manifolds and their geometric structures. They are particularly relevant in the investigation of spacetime metrics in general relativity and in the formulation of field theories in theoretical physics.

Furthermore, the study of semi-symmetric connections provides insights into the geometric properties of manifolds with specific curvature characteristics. Understanding these connections can lead to advancements in fields such as cosmology, where the geometric properties of spacetime are of paramount importance in modeling the universe’s evolution and structure. The concepts of semi-symmetric connections, product manifolds, and other relevant theories, thus, serves as a fundamental tool for exploring the intricate interplay between geometry and physics at a deeper level [4,5,6].

On the other hand, within the domain of Riemannian geometry, a fundamental concern revolves around the precise delineation of the boundaries that constrain the Laplacian operator on a given manifold. One of the primary objectives is to ascertain the eigenvalue that arises as a solution to the Dirichlet or Neumann boundary value problems associated with curvature functions. These boundary value problems play a crucial role in understanding the behavior of functions on the manifold, especially in relation to how they interact with the manifold’s geometric properties. When dealing with various boundary conditions on a manifold, some theoretical approaches and methods from Li [7,8,9], Chen [10,11], Özgür [12,13], etc., may be adopted to tackle the Dirichlet boundary condition. In this context, leveraging the upper bound for the eigenvalue serves as a powerful analytical tool. This upper bound not only aids in determining the eigenvalue itself, but also helps in establishing the appropriate limits of the Laplacian operator on the manifold, shedding light on the intrinsic properties of the manifold under consideration.

The calculation of eigenvalues for both the Laplacian and $ϵ$-Laplacian operators has garnered increased attention and recognition over an extended period. The extension of the conventional Laplacian operator to incorporate anisotropic mean curvature represents a significant advancement in the field of differential geometry. By exploring this extension, researchers aim to deepen their understanding of how curvature functions behave in manifold settings with varying geometric and boundary conditions.

The referenced work [14] delves into this extension, offering valuable insights into the dynamics of anisotropic mean curvature within the context of the Laplacian operator. Such investigations not only contribute to the theoretical foundations of Riemannian geometry, but also have practical implications in diverse fields, including physics and engineering, where the analysis of geometric properties and boundary value conditions is of paramount importance. Let ${\mathrm{Y}}^m$ denote a complete non-compact Riemannian manifold, and D represent the compact domain within $\mathrm{Y}$. Suppose ${\mu }_1(D)>0$ denotes the first eigenvalue of the Dirichlet boundary value problem.

$$\Delta ϵ+{\mu }_1ϵ=0,\mathrm{in}D\mathrm{and}ϵ\mathrm{on}\partial D$$

Here, $\Delta$ denotes the Laplacian operator on the Riemannian manifold ${\mathrm{Y}}^m$. Reilly’s formula specifically addresses the fundamental geometric properties of a given manifold, a fact widely recognized by the following assertion.

Consider a compact m-dimensional Riemannian manifold $({\mathrm{Y}}^m,g)$, where ${\mu }_1$ represents the first non-zero eigenvalue of the Neumann problem, given by:

$$\Delta ϵ+{\mu }_1ϵ=0,\mathrm{on}\mathrm{Y}\mathrm{and}{\displaystyle \frac{\partial \psi }{\partial \mathrm{Y}}}=0,\mathrm{on}\partial \mathrm{Y},$$

where $\mathrm{Y}$ represents the outward normal on $\partial {\mathrm{Y}}^m$.

According to Reilly [15], the inequality for a manifold ${\mathrm{Y}}^m$ embedded in a Euclidean space with $\partial {\mathrm{Y}}^m=0$ can be expressed as follows:

$${\mu }_1^{\nabla }\le {\displaystyle \frac{1}{\mathrm{Vol}({\mathrm{Y}}^m)}}{\int }_{{\mathrm{Y}}^m}{\parallel H\parallel }^{2}dV,$$

(1)

Here, H represents the mean curvature vector of the immersion of ${\mathrm{Y}}^m$ into ${\mathbb{R}}^n$, ${\mu }_1^{\nabla }$ denotes the first non-zero eigenvalue of the Laplacian on ${\mathrm{Y}}^m$, and $dV$ signifies the volume element of ${\mathrm{Y}}^m$.

Papers [16,17] elucidate the first non-trivial Laplacian eigenvalue, an extension of Reilly’s work [18]. Results on distinct classes of Riemannian submanifolds in diverse ambient spaces reveal that the first non-zero eigenvalues exhibit similar inequalities and share identical upper bounds [16,19]. In the context of the ambient manifold, previous research has highlighted the significant role played by Laplace and Laplace operators on Riemannian manifolds in various accomplishments within Riemannian geometry (refer to [18,20,21,22,23,24,25,26,27]). More motivations for the paper refer to some recent results related to classical differential geometry [28,29,30], submanifolds theory [31,32], and so on.

The $ϵ$-Laplacian on an m-dimensional Riemannian manifold ${\mathrm{Y}}^m$ is defined as follows:

$${\Delta }_{ϵ}={\mathrm{div}(|\nabla G|}^{ϵ-2}\nabla G),$$

(2)

where $ϵ>1$. When $ϵ=2$, the formula above simplifies to the conventional Laplacian operator.

The eigenvector of $\Delta G$ behaves akin to the Laplacian. For a function $G\ne 0$ satisfying the equation under Dirichlet or Neumann boundary conditions, as previously discussed:

$${\Delta }_{ϵ}G=-\mu {|G|}^{ϵ-2}G,$$

where $\mu$ represents a real number known as the Dirichlet eigenvalue. The same conditions apply to the Neumann boundary condition.

In the context of Riemannian manifolds without boundaries, the Reilly-type inequality for the first non-zero eigenvalue ${\mu }_{1,ϵ}$ for the $ϵ$-Laplacian was computed in [33] as follows:

$${\mu }_{1,ϵ}=inf\left\{{\displaystyle \frac{{\int }_{\mathrm{Y}}{|\nabla G|}^q}{{\int }_{\mathrm{Y}}{|G|}^q}}:G\in W^{1,ϵ}({\mathrm{Y}}^1)\{0\},{\int }_{\mathrm{Y}}{|G|}^{ϵ-2}G=0\right\}.$$

(3)

Chen [34] initially introduced the concept of slant immersions, which served as a natural extension encompassing both holomorphic and totally real immersions. Subsequently, Lotta [35] defined slant submanifolds within the framework of almost contact metric manifolds, and J. L. Cabrerizo et al. [36] further delved into the study of these submanifolds. Particularly, J. L. Cabrerizo et al. explored slant submanifolds within Sasakian manifolds.

Upon reviewing the existing literature, a compelling question arises: Is it possible to derive Reilly-type inequalities for submanifolds of spheres admitting semi-symmetric metric connection using almost contact metric manifolds, as discussed in [16,17,37]? To address this question, we investigate the Reilly-type inequalities for slant submanifolds isometrically immersed in a Sasakian space form $\overline{\mathrm{Y}}(c)$ admitting semi-symmetric metric connection. Our aim is to ascertain the bounds for the first non-zero eigenvalue using the $ϵ$-Laplacian. This study is informed by the application of the Gauss equation and previous research findings outlined in [16,19,38].

2. Preliminaries

In the scenario where $(\overline{\mathrm{Y}},g)$ represents an odd-dimensional Riemannian manifold, we characterize $\overline{\mathrm{Y}}$ as an almost contact metric manifold if it features a tensor field $\varphi$ of type $(1,1)$ and a global vector field $\xi$ that adhere to the subsequent conditions:

$${\varphi }^2{\upsilon }_1=-{\upsilon }_1+\eta ({\upsilon }_1)\xi ,g({\upsilon }_1,\xi )=\eta ({\upsilon }_1)$$

(4)

$$g(\varphi {\upsilon }_1,\varphi {\upsilon }_2)=g({\upsilon }_1,{\upsilon }_2)-\eta ({\upsilon }_1)\eta ({\upsilon }_2).$$

(5)

The 1-form dual to $\xi$ is symbolized as $\eta$. It is a recognized truth that an almost contact metric manifold can be categorized as a Sasakian manifold only if the ensuing conditions are fulfilled:

$$({\overline{\overline{\nabla }}}_{{\upsilon }_1}\varphi ){\upsilon }_2=g({\upsilon }_1,{\upsilon }_2)\xi -\eta ({\upsilon }_2){\upsilon }_1.$$

(6)

In a Sasakian manifold $\overline{\mathrm{Y}},$ the following can be easily deduced

$${\overline{\overline{\nabla }}}_{{\upsilon }_1}\xi =-\varphi {\upsilon }_1.$$

(7)

In this context, ${\upsilon }_1$ and ${\upsilon }_2$ are elements of the tangent space of $\overline{\mathrm{Y}},$ and $\overline{\overline{\nabla }}$ represents the Riemannian connection linked to the metric g on $\overline{\mathrm{Y}}.$

Let us now establish a connection denoted as $\overline{\nabla }$, in the subsequent manner:

$${\overline{\nabla }}_{{\upsilon }_1}{\upsilon }_2={\overline{\overline{\nabla }}}_{{\upsilon }_1}{\upsilon }_2+\eta ({\upsilon }_2){\upsilon }_1-g({\upsilon }_1,{\upsilon }_2)\xi$$

(8)

such that $\overline{\nabla }g=0$ for any ${\upsilon }_1,{\upsilon }_2\in T\overline{\mathrm{Y}},$ where $\overline{\overline{\nabla }}$ is the Riemannian connection with respect to g. The connection $\overline{\nabla }$ is semi-symmetric because $T({\upsilon }_1,{\upsilon }_2)=\eta ({\upsilon }_2){\upsilon }_1-\eta ({\upsilon }_1){\upsilon }_2.$ Using (8) in (6), we have

$$({\overline{\nabla }}_{{\upsilon }_1}\varphi ){\upsilon }_2=g({\upsilon }_1,{\upsilon }_2)\xi -g({\upsilon }_1,\varphi {\upsilon }_2)\xi -\eta ({\upsilon }_2){\upsilon }_1-\eta ({\upsilon }_2)\varphi {\upsilon }_1$$

(9)

and

$${\overline{\nabla }}_{{\upsilon }_1}\xi ={\upsilon }_1-\eta ({\upsilon }_1)\xi -\varphi {\upsilon }_1.$$

(10)

When a Sasakian manifold $\overline{\mathrm{Y}}$ exhibits a constant $\varphi$-holomorphic sectional curvature c, it is termed a Sasakian space form and is symbolized as $\overline{\mathrm{Y}}(c)$.

The formula for the curvature tensor $\overline{R}$ associated with the ssm connection $\overline{\nabla }$ can be expressed as follows:

$$\overline{R}({\upsilon }_1,{\upsilon }_2){\upsilon }_3={\overline{\nabla }}_{{\upsilon }_1}{\overline{\nabla }}_{{\upsilon }_2}{\upsilon }_3-{\overline{\nabla }}_{{\upsilon }_2}{\overline{\nabla }}_{{\upsilon }_1}{\upsilon }_3-{\overline{\nabla }}_{[{\upsilon }_1,{\upsilon }_2]}{\upsilon }_3.$$

(11)

Similarly, the curvature tensor $\overline{\overline{R}}$ can be determined for the Riemannian connection $\overline{\overline{\nabla }}$.

Let

$$\beta ({\upsilon }_1,{\upsilon }_2)=({\overline{\nabla }}_{{\upsilon }_1}\eta ){\upsilon }_2-\eta ({\upsilon }_1)\eta ({\upsilon }_2)+{\displaystyle \frac{1}{2}}g({\upsilon }_1,{\upsilon }_2).$$

(12)

Upon utilizing Equations (8), (11) and (12), we derive

$$\begin{array}{cc}\hfill \overline{R}({\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4)& =\overline{\overline{R}}({\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4)+\beta ({\upsilon }_1,{\upsilon }_3)g({\upsilon }_2,{\upsilon }_4)\hfill \\ \hfill & -\beta ({\upsilon }_2,{\upsilon }_3)g({\upsilon }_1,{\upsilon }_4)+\beta ({\upsilon }_2,{\upsilon }_4)g({\upsilon }_1,{\upsilon }_3)\hfill \\ \hfill & -\beta ({\upsilon }_1,{\upsilon }_4)g({\upsilon }_2,{\upsilon }_3).\hfill \end{array}$$

(13)

Using the calculated value of $\overline{\overline{R}}({\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4)$, as outlined in [39], we can determine the following formula for the curvature tensor $\overline{R}$ of a Sasakian space form:

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

(14)

for all ${\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4\in T\overline{\mathrm{Y}}.$

When considering a submanifold $\mathrm{Y}$ that is isometrically immersed in a differentiable manifold $\overline{\mathrm{Y}}$ with ssm connection. Then, it is easy to observed that the Gauss and Weingarten formula are given by

$${\overline{\nabla }}_{{\upsilon }_1}{\upsilon }_2={\nabla }_{{\upsilon }_1}{\upsilon }_2+h({\upsilon }_1,{\upsilon }_2)$$

and

$${\overline{\nabla }}_{{\upsilon }_1}N=-A_N{\upsilon }_1+{\nabla }_{{\upsilon }_1}^{\perp }N+\eta (N){\upsilon }_1,$$

respectively. In this context, $\overline{\nabla }$ signifies the covariant derivative concerning the ssm connection on $\overline{\mathrm{Y}}$, ∇ indicates the induced S-S-M connection on $\mathrm{Y}$, and ${\upsilon }_1$ and ${\upsilon }_2$ denote tangent vectors on $\mathrm{Y}$. Additionally, N represents a normal vector to the submanifold $\mathrm{Y}$, ${\nabla }^{\perp }$ denotes the covariant derivative along the normal bundle $T^{\perp }\mathrm{Y}$, and $\eta (N)$ is a scalar function.

The relationship between the second fundamental form h and the shape operator $A_N$ can be described by the following expression:

$$g(h({\upsilon }_1,{\upsilon }_2),N)=g(A_N{\upsilon }_1,{\upsilon }_2).$$

For the vector fields ${\upsilon }_1\in T\mathrm{Y}$ and ${\upsilon }_3\in T^{\perp }\mathrm{Y}$, we can decompose the expression as follows

$$\varphi {\upsilon }_1=P{\upsilon }_1+F{\upsilon }_1$$

(15)

and

$$\varphi {\upsilon }_3=t{\upsilon }_3+f{\upsilon }_3$$

(16)

Here, $P{\upsilon }_1(t{\upsilon }_3)$ and $F{\upsilon }_1(f{\upsilon }_3)$ represent the tangential and normal components of $\varphi {\upsilon }_1(\varphi {\upsilon }_3)$, respectively.

The Gauss equation for an ssm connection, concerning the Riemannian curvature tensor R, is detailed as follows [39]:

$$\overline{R}({\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4)=R({\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4)-g(h({\upsilon }_1,{\upsilon }_4),h({\upsilon }_2,{\upsilon }_3))+g(h({\upsilon }_2,{\upsilon }_4),h({\upsilon }_1,{\upsilon }_3))$$

(17)

for ${\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4\in T\mathrm{Y}.$

The concept of slant submanifolds within contact geometry was initially introduced by A. Lotta [35]. Subsequently, Cabrerizo et al. [36] delved into the study of these submanifolds, leading to the following precise definition of slant submanifolds:

Definition 1. 

A submanifold Y of an almost contact metric manifold $\overline{\mathrm{Y}}$ is termed a slant submanifold if, for every $x\in \mathrm{Y}$ and $\upsilon \in T_x\mathrm{Y}-\langle \xi \rangle$, the angle between υ and $\varphi \upsilon$ remains constant. The fixed angle $\alpha \in [0,\pi /2]$ is referred to as the slant angle of Y within $\overline{\mathrm{Y}}$. When $\alpha =0$, the submanifold is considered an invariant submanifold, and when $\alpha =\pi /2$, it is termed an anti-invariant submanifold. If $\alpha \ne 0,\pi /2$, it is classified as a proper slant submanifold.

Furthermore, Cabrerizo et al. [36] derived the tensorial equation for slant submanifolds. In particular, they illustrated that a submanifold ${\mathrm{Y}}^m$ is recognized as a slant submanifold if there exists a constant $\gamma \in [0,\pi /2]$ and a $(1,1)$ tensor field P that fulfills the following relation:

$$P^2=\gamma (I-\eta \otimes \xi ),$$

(18)

where $\gamma =-{cos}^2\alpha$.

From Equation (18), one can readily deduce the following:

$$g(P{\upsilon }_1,P{\upsilon }_2)={cos}^2\alpha \{g({\upsilon }_1,{\upsilon }_2)-\eta ({\upsilon }_1)\eta ({\upsilon }_2)\}$$

(19)

∀ ${\upsilon }_1,{\upsilon }_2\in \mathrm{Y}.$

Suppose ${\mathrm{Y}}^{m=2p+1}$ is a slant submanifold of dimension m in which $2p$ is the dimension of the slant distributions $S_{\alpha }$. Moreover, let $\{u_1,u_2=sec{\alpha }_{1}T{u}_1,u_3,u_4=sec{\alpha }_{1}T{u}_3,\dots ,u_{2p}=sec{\alpha }_{1}T{u}_{2p-1}\}$ is an orthonormal frame of vectors, which forms a basis for the slant submanifold. By the Equation (9), the curvature tensor $\overline{R}$ for slant submanifold ${\mathrm{Y}}^{2p+1}$ is given by

$$\begin{array}{cc}\hfill \overline{R}(u_i,u_j,u_i,u_j)=& {\displaystyle \frac{c+3}{4}}\{g(u_j,u_i)g(u_i,u_j)-g(u_i,u_i)g(u_j,u_j)\}\hfill \\ \hfill & +{\displaystyle \frac{c-1}{4}}\{\eta (u_i)\eta (u_i)g(u_j,u_j)-\eta (u_j)\eta (u_i)g(u_i,u_j)\hfill \\ \hfill & +g(u_i,u_i)\eta (u_j)\eta (u_j)-g(u_j,u_i)\eta (u_i)\eta (u_j)\hfill \\ \hfill & +g(\varphi u_j,u_i)g(\varphi u_i,u_j)+g(\varphi u_j,u_i)g(\varphi u_j,u_j)\hfill \\ \hfill & -2g(\varphi u_i,u_j)g(\varphi u_j,u_j)\}+\beta (u_i,u_i)g(u_j,u_j)\hfill \\ \hfill & -\beta (u_j,u_i)g(u_i,u_j)+\beta (u_j,u_j)g(u_i,u_i)\hfill \\ \hfill & -\beta (u_i,u_j)g(u_j,u_i),\hfill \end{array}$$

(20)

After a routine calculation, we obtain the following.

$$\overline{R}(u_i,u_j,u_i,u_j)={\displaystyle \frac{c+3}{4}}(m^2-m)+{\displaystyle \frac{c-1}{4}}\left(3\sum _{i,j=1}^m{g}^2(\varphi u_i,u_j)-2(m-1)\right)+2mtrace\beta ,$$

(21)

using the formula (18) for slant distribution, we have

$$g^2(\varphi u_i,u_{i+1})={cos}^2\alpha ,fori\in \{1,\dots ,2p-1\}.$$

Then

$$\sum _{i,j=1}^m{g}^2(\varphi u_i,u_j)=m{cos}^2\alpha .$$

The relation (21) implies that

$$\overline{R}(u_i,u_j,u_i,u_j)={\displaystyle \frac{c+3}{4}}(m^2-m)+{\displaystyle \frac{c-1}{4}}\left(3m{cos}^2\alpha -2(m-1)\right)+2mtrace\beta .$$

(22)

From the relation (22) and Gauss equation, one has

$${\displaystyle \frac{c+3}{4}}m(m-1)+{\displaystyle \frac{c-1}{4}}\left(3m{cos}^2\alpha -2(m-1)\right)+2mtrace\beta =2\tau -n^2{\parallel H\parallel }^2+{\parallel \sigma \parallel }^2$$

or

$$2\tau =n^2{\parallel H\parallel }^2-{\parallel \sigma \parallel }^2+{\displaystyle \frac{c+3}{4}}m(m-1)+{\displaystyle \frac{c-1}{4}}\left(3m{cos}^2\alpha -2(m-1)\right)+2mtrace\beta .$$

(23)

In the paper by Ali et al. [37], an investigation was conducted on the impact of conformal transformations on curvature and the second fundamental form. Specifically, considering that ${\overline{\mathrm{Y}}}^{2t+1}$ contains a conformal metric $g=e^{2\omega }\overline{g},$ where $\omega \in C^{\infty }(\overline{\mathrm{Y}}),$ the expressions ${\overline{\Gamma }}_a=e^{\omega }{\Gamma }_a$ denote the dual co-frame of $(\overline{\mathrm{Y}},\overline{g}),$ and ${\overline{\upsilon }}_a=e^{\omega }{\upsilon }_a$ represents the orthogonal frame of $(\overline{\mathrm{Y}},\overline{g})$. Furthermore, the following relation can be established:

$${\overline{\Gamma }}_{ab}={\Gamma }_{ab}+{\omega }_a{\Gamma }_b-{\omega }_b{\Gamma }_a,$$

(24)

where ${\omega }_a$ is the covariant derivative of $\omega$ along the vector ${\upsilon }_a$, i.e., $d\omega ={\sum }_a{\omega }_a{\upsilon }_a.$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{2\omega }{\overline{R}}_{pqrs}=& R_{pqrs}-({\omega }_{pr}{\delta }_{qs}+{\omega }_{qs}{\delta }_{pr}-{\omega }_{ps}{\delta }_{qr}-{\omega }_{qr}{\delta }_{ps})\hfill \\ \hfill & +({\omega }_p{\omega }_r{\delta }_{qs}+{\omega }_q{\omega }_s{\delta }_{pr}-{\omega }_q{\omega }_t{\delta }_{ps}-{\omega }_p{\omega }_s{\delta }_{qr})\hfill \\ \hfill & -|{\nabla }_{ϵ}{|}^2({\delta }_{pr}{\delta }_{qs}-{\delta }_{il}{\delta }_{qr}).\hfill \end{array}\end{array}$$

(25)

Utilizing the pullback property in Equation (24) on ${\mathrm{Y}}^m$ at point x, we obtain

$${\overline{\sigma }}_{pq}^{ϵ}=e^{-\omega }({\sigma }_{pq}^{ϵ}-{\omega }_{ϵ}{\delta }_{qp})$$

(26)

$${\overline{H}}^{ϵ}=e^{ϵ}(H^{ϵ}-{\omega }_{ϵ}),$$

(27)

Here, ${\overline{\sigma }}_{pq}^{ϵ}$ and ${\overline{H}}^{ϵ}$ represent the components of the second fundamental form and the mean curvature vector, respectively.

The subsequent crucial connection was demonstrated in [37].

$$e^{2\omega }(\parallel \overline{\sigma }{\parallel }^2-m\parallel \overline{H}{\parallel }^2{)+m\parallel H\parallel }^2={\parallel \sigma \parallel }^2$$

(28)

3. Main Results

Firstly, we will delve into fundamental results and formulas that align with the works in the papers [25,37].

A widely recognized fact is that a simply connected ssf ${\overline{\mathrm{Y}}}^{2t+1}$ corresponds to a $(2t+1)$-sphere $S^{2t+1}$ and Euclidean space $R^{2t+1}$ with constant sectional curvature $c=1$ and $c=-3$, respectively.

Now, the ensuing outcome stems from the preceding arguments.

Lemma 1 

([37]). Consider ${\mathrm{Y}}^m$ as a slant submanifold of a closed and oriented 2t + 1-dimensional ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ with dimension $\ge 2$. Let $f:{\mathrm{Y}}^m\to {\overline{\mathrm{Y}}}^{2t+1}(c)$ be an embedding from ${\mathrm{Y}}^m$ into ${\overline{\mathrm{Y}}}^{2t+1}(c)$. Then, there exists a standard conformal map $x:{\overline{\mathrm{Y}}}^{2t+1}(c)\to S^{2t+1}(1)\subset {\mathbb{R}}^{2t+2}$, such that the composition $\Gamma =x\circ f=({\Gamma }^1,\dots ,{\Gamma }^{2t+2})$ satisfies:

$${\int }_{{\mathrm{Y}}^m}{|{\Gamma }^a|}^{ϵ-2}{\Gamma }^{a}d{V}_{\mathrm{Y}}=0,a=1,\dots ,2(t+1),$$

(29)

for $ϵ>1.$

Remark 1. 

The statement in Lemma 1 holds for slant submanifolds with a semi-symmetric metric connection as well. Its proof follows the same approach as outlined in [37].

In the following outcome, we derive a result that mirrors Lemma 2.7 from [25]. Specifically, in Lemma 1, through the utilization of a test function, we establish an upper bound for ${\mu }_{1,ϵ}$ relative to a conformal function.

Proposition 1. 

Consider ${\mathrm{Y}}^m$ as an m-dimensional slant submanifold, closed, orientable, isometrically immersed in a 2t + 1-dimensional ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ equipped with a ssm connection. In this scenario, we obtain:

$${\mu }_{1,ϵ}Vol({\mathrm{Y}}^m)\le 2^{|1-{\displaystyle \frac{ϵ}{2}}|}{(t+1)}^{|1-{\displaystyle \frac{ϵ}{2}}|}{m}^{{\displaystyle \frac{ϵ}{2}}}{\int }_{{\mathrm{Y}}^m}{(e^{2\omega })}^{{\displaystyle \frac{ϵ}{2}}}dV,$$

(30)

Here, x represents the conformal mapping utilized in Lemma 3.1, with $ϵ>1$. The standard metric denoted by $L_c$ is associated with $x^{*}{L}_1=e^{2p}{L}_c$.

Proof. 

Consider ${\Gamma }^a$ as a test function along with Lemma 3.1, we have

$${\mu }_{1,ϵ}{\int }_{{\mathrm{Y}}^m}|{\Gamma }^a{|}^{ϵ}\le {|\nabla {\Gamma }^a|}^{ϵ}dV,1\le a\le 2(t+1),$$

(31)

observing that ${\sum }_{a=1}^{2t+2}{|{\Gamma }^a|}^2=1,$ then $|{\Gamma }^a|\le 1$, we obtain

$$\sum _{a=1}^{2t+2}|\nabla {\Gamma }^a{|}^2=\sum _{i=1}^m{|{\nabla }_{{\mu }_i}\Gamma |}^2=m{e}^{2\omega }.$$

(32)

Upon using $1<ϵ\le 2,$ we conclude

$$|{\Gamma }^a{|}^2\le {|{\Gamma }^a|}^{ϵ}.$$

(33)

Utilizing Holder’s inequality in conjunction with (31)–(33), we obtain:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}Vol({\mathrm{Y}}^m)& ={\mu }_{1,ϵ}\sum _{a=1}^{2t+2}{\int }_{{\mathrm{Y}}^m}|{\Gamma }^a{|}^{2}dV\le {\mu }_{1,ϵ}\sum _{a=1}^{2t+2}\sum _{{\mathrm{Y}}^m}{\int }_{{\mathrm{Y}}^m}{|{\Gamma }^a|}^{ϵ}dV\hfill \\ \hfill & \le {\mu }_{1,ϵ}{\int }_{{\mathrm{Y}}^m}\sum _{a=1}^{2t+2}{|\nabla {\Gamma }^a|}^{ϵ}dV\le {(2t+2)}^{1-ϵ/2}{\int }_{{\mathrm{Y}}^m}(\sum _{a=1}^m|\nabla {\Gamma }^a{{|}^2)}^{ϵ/2}dV\hfill \\ \hfill & =2^{1-{\displaystyle \frac{ϵ}{2}}}{(t+1)}^{1-{\displaystyle \frac{ϵ}{2}}}{\int }_{{\mathrm{Y}}^m}{(m{e}^{2\omega })}^{{\displaystyle \frac{ϵ}{2}}}dV.\hfill \end{array}\end{array}$$

(34)

This corresponds to (30). Alternatively, assuming $ϵ\ge 2$, we can apply Holder’s inequality to derive:

$$I=\sum _{a=1}^{2t+2}{|{\Gamma }^a|}^2\le {(2t+2)}^{1-{\displaystyle \frac{2}{ϵ}}}{\left(\sum _{a=1}^{2t+2}{|{\Gamma }^a|}^{ϵ}\right)}^{{\displaystyle \frac{2}{ϵ}}}.$$

(35)

As a result, we obtain

$${\mu }_{1,ϵ}Vol(N^m)\le {(2t+2)}^{{\displaystyle \frac{ϵ}{2}}-1}\left(\sum _{a=1}^{2t+2}{\mu }_{1,ϵ}{\int }_{N^m}{|{\Gamma }^a|}^{ϵ}dV\right).$$

(36)

The Minkowski inequality provides

$$\sum _{a=1}^{2t+2}{|\nabla {\Gamma }^a|}^{ϵ}\le {\left(\sum _{a=1}^{2t+2}{|\nabla {\Gamma }^a|}^2\right)}^{{\displaystyle \frac{ϵ}{2}}}={(m{e}^{2\omega })}^{{\displaystyle \frac{ϵ}{2}}}.$$

(37)

By utilizing Equations (30), (35) and (36), we readily derive Equation (30). □

In the upcoming theorem, we will present a precise estimation for the first eigenvalue of the $ϵ$-Laplace operator on a slant submanifold within the ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$.

Theorem 1. 

Suppose ${\mathrm{Y}}^m$ is an m-dimensional slant submanifold of a 2t + 1-dimensional ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ with a semi-symmetric metric connection, then

1.

The first non-null eigenvalue ${\mu }_{1,ϵ}$ of the $ϵ$-Laplacian satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}& \le {\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\hfill \\ \hfill & \times {\left\{{\int }_{{\mathrm{Y}}^m}\left({\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}\left({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}}\right)-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)\right\}}^{ϵ/2}dV\hfill \end{array}\end{array}$$

(38)

for $1<ϵ\le 2,$ and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}& \le {\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\hfill \\ \hfill & \times {\left\{{\int }_{{\mathrm{Y}}^m}\left({\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}\left({\displaystyle \frac{3{cos}^2\alpha }{(m-1)}}-{\displaystyle \frac{2}{m-1}}\right)-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)\right\}}^{ϵ/2}dV\hfill \end{array}\end{array}$$

(39)

for $2<ϵ\le {\displaystyle \frac{m}{2}}+1$.

2.

The equalities hold true in (38) and (39), if and only if $ϵ=2$ and ${\mathrm{Y}}^m$ is minimally immersed in a geodesic sphere of radius ${\rho }_c$ within ${\overline{\mathrm{Y}}}^{2t+1}(c)$ with the following relationships:

$${\rho }_0={\left({\displaystyle \frac{m}{{\mu }_1^{\Delta }}}\right)}^{1/2},{\rho }_1={sin}^{-1}{\rho }_0,{\rho }_{-1}={sinh}^{-1}{\rho }_0.$$

Proof. 

$1<ϵ\le 2$⇒${\displaystyle \frac{ϵ}{2}}\le 1.$ Proposition 1, together with the Holder inequality, provides

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}Vol({\mathrm{Y}}^m)& \le 2^{1-{\displaystyle \frac{ϵ}{2}}}{(t+1)}^{1-{\displaystyle \frac{ϵ}{2}}}{m}^{{\displaystyle \frac{ϵ}{2}}}{\int }_{{\mathrm{Y}}^m}{(e^{2\omega })}^{{\displaystyle \frac{ϵ}{2}}}dV\hfill \\ \hfill & \le 2^{1-{\displaystyle \frac{ϵ}{2}}}{(t+1)}^{|1-{\displaystyle \frac{ϵ}{2}}|}{m}^{{\displaystyle \frac{ϵ}{2}}}{(Vol({\mathrm{Y}}^m))}^{1-{\displaystyle \frac{ϵ}{2}}}{\left({\int }_{{\mathrm{Y}}^m}{e}^{2\omega }dV\right)}^{{\displaystyle \frac{ϵ}{2}}}.\hfill \end{array}\end{array}$$

(40)

By utilizing conformal relations and the Gauss equation, we can determine $e^{2\omega }$. Consider ${\overline{\mathrm{Y}}}^{2t+1}={\overline{\mathrm{Y}}}^{2t+1}(c)$, $\overline{g}=e^{-2\omega }{L}_c$, and $\overline{g}=c^{*}{L}_1$. Referring to (23), the Gauss equation for the embedding f and the slant embedding $\Gamma =x\circ f$, we obtain:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R={\displaystyle \frac{c+3}{4}}m(m-1)& +{\displaystyle \frac{c-1}{4}}\{3m{cos}^2\alpha -2(m-1)\}+2mtrace\beta \hfill \\ \hfill & +{m(m-1)\parallel H\parallel }^2+{m\parallel H\parallel }^2-{\parallel \sigma \parallel }^2\hfill \end{array}\end{array}$$

(41)

$$\overline{R}-m(m-1)=m(m-1)\parallel \overline{H}{\parallel }^2+(m\parallel \overline{H}{\parallel }^2-\parallel \overline{\sigma }{\parallel }^2).$$

(42)

Upon tracing (25), we have

$$e^{2\omega }\overline{R}=R-(m-2)(m-1){|{\nabla }_{\omega }|}^2-2(m-1){\Delta }_{\omega },$$

(43)

using (41) and (42) in (43), we obtain □

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{2\omega }(m(m-1)+m(m-1)& \parallel \overline{H}{\parallel }^2+(m\parallel \overline{H}{\parallel }^2-\parallel \overline{\sigma }{|}^2))={\displaystyle \frac{c+3}{4}}m(m-1)\hfill \\ \hfill & +{\displaystyle \frac{c-1}{4}}\{3m{cos}^2\alpha -2(m-1)\}+2mtrace\beta \hfill \\ \hfill & +{m(m-1)\parallel H\parallel }^2+{(m\parallel H\parallel }^2-{\parallel \sigma |}^2)\hfill \\ \hfill & -{(m-2)(m-1)\parallel \nabla \omega \parallel }^2-2(m-1){\Delta }_{\omega }.\hfill \end{array}\end{array}$$

(44)

The above relation implies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{2\omega }{\parallel \overline{\sigma }|}^2-& (m-2)(m-1)|{\nabla }_{\omega }{|}^2-2(m-1){\Delta }_{\omega }\hfill \\ \hfill & =m(m-1)[\{e^{2\omega }-{\displaystyle \frac{c+3}{4}}-{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}})+{\displaystyle \frac{2}{m-1}}trace\beta \}\hfill \\ \hfill & +(e^{2\omega }\parallel \overline{H}{\parallel }^2-{\parallel H\parallel }^2)]+m(e^{2\omega }\parallel \overline{H}{\parallel }^2-{\parallel H\parallel }^2).\hfill \end{array}\end{array}$$

(45)

By combining (27) and (28), we can deduce

$$\begin{array}{cc}\hfill m(m-1)& \{e^{2\omega }-{\displaystyle \frac{c+3}{4}}-{\displaystyle \frac{c-1}{4}}\left({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}}\right)+{\displaystyle \frac{2}{m-1}}trace\beta )\}+m(m-1)\sum _{ϵ}{(H^{ϵ}-\omega \psi )}^2\hfill \\ \hfill & ={m(m-1)\parallel H\parallel }^2-(m-2)(m-1){|{\nabla }_{\omega }|}^2-2(m-1){\Delta }_{\omega }.\hfill \end{array}$$

(46)

Further, upon simplification we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e^{2\omega }=\{\left({\displaystyle \frac{c+3}{4}}\right)& +{\displaystyle \frac{c-1}{4}}\left({\displaystyle \frac{3{cos}^2\alpha }{m-1}}\right)-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\}\hfill \\ \hfill & -{\displaystyle \frac{2}{m}}{\Delta }_{\omega }-{\displaystyle \frac{m-2}{m}}|{\Delta }_{\omega }{|}^2-{\parallel {({\nabla }_{\omega })}^{\perp }-H\parallel }^2.\hfill \end{array}\end{array}$$

(47)

Upon integrating along $dV,$ it is easy to see that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}Vol({\mathrm{Y}}^m)& \le 2^{|1-{\displaystyle \frac{ϵ}{2}}|}{(t+1)}^{|1-{\displaystyle \frac{ϵ}{2}}|}{m}^{{\displaystyle \frac{ϵ}{2}}}{(Vol({\mathrm{Y}}^m))}^{1-{\displaystyle \frac{ϵ}{2}}}{\left({\int }_{{\mathrm{Y}}^m}{e}^{2\omega }dV\right)}^{{\displaystyle \frac{ϵ}{2}}}.\hfill \\ \hfill & \le {\displaystyle \frac{2^{|1-\frac{ϵ}{2}|}{(t+1)}^{|1-\frac{ϵ}{2}|}{m}^{\frac{ϵ}{2}}}{{(Vol({\mathrm{Y}}^m))}^{\frac{ϵ}{2}-1}}}\{{\int }_{{\mathrm{Y}}^m}\{{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta \hfill \\ \hfill & +{\parallel H\parallel }^2\}dV{\}}^{ϵ/2}.\hfill \end{array}\end{array}$$

(48)

This equivalence corresponds to (38). If $ϵ>2$, applying Holder’s inequality to control ${\int }_{{\mathrm{Y}}^m}{(e^{2\omega }dV)}^{{\displaystyle \frac{ϵ}{2}}}$ using ${\int }_{{\mathrm{Y}}^m}(e^{2\omega })$ is not feasible. Next, multiply both sides of (47) by $e^{(ϵ-2)\omega }$ and integrate over ${\mathrm{Y}}^m$.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\mathrm{Y}}^m}{e}^{ϵ\omega }dV& \le {\int }_{{\mathrm{Y}}^m}\left\{{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right\}{e}^{(ϵ-2)\omega }dV\hfill \\ \hfill & -\left({\displaystyle \frac{m-2-2\psi +4}{m}}\right){\int }_{{\mathrm{Y}}^m}{e}^{(ϵ-2)}{|\Delta \omega |}^{2}dV\hfill \\ \hfill & \le {\int }_{{\mathrm{Y}}^m}\left\{{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right\}{e}^{(ϵ-2)\omega }dV.\hfill \end{array}\end{array}$$

(49)

Given the assumption, it is clear that $m\ge 2\psi -2$. By employing Young’s inequality, we reach

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\mathrm{Y}}^m}& \left\{{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right\}{e}^{(ϵ-2)\omega }dV\hfill \\ \hfill & \le {\displaystyle \frac{2}{ϵ}}{\int }_{{\mathrm{Y}}^m}{\left\{|{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2|\right\}}^{ϵ/2}dV\hfill \\ \hfill & +{\displaystyle \frac{ϵ-2}{ϵ}}{\int }_{{\mathrm{Y}}^m}{e}^{{\displaystyle \frac{ϵ}{\omega }}}dV.\hfill \end{array}\end{array}$$

(50)

Based on (49) and (50), we can draw the following conclusion

$${\int }_{{\mathrm{Y}}^m}{e}^{ϵ\omega }dV\le {\int }_{{\mathrm{Y}}^m}{\left\{|{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2{\alpha }_2}{m-1}}-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2|\right\}}^{ϵ/2}dV.$$

(51)

Substituting Equation (51) into Equation (30) yields Equation (39). For slant submanifolds with a semi-symmetric metric connection, equality holds in Equation (38). The equality cases of Equations (31) and (33) imply that:

$$|{\Gamma }^a{|}^2={|{\Gamma }^a|}^{ϵ}$$

$${\Delta }_{ϵ}{\Gamma }^a={\mu }_{1,ϵ}{|{\Gamma }^a|}^{ϵ-2}{\Gamma }^a$$

For $a=1,\dots ,2t+2$. When $1<ϵ<2$, we have $|{\Gamma }^a|=0$ or 1. Consequently, there exists only one a for which $|{\Gamma }^a|=1$ and ${\mu }_{i,ϵ}=0$, which is not possible as the eigenvalue ${\mu }_{i,ϵ}\ne 0$. Thus, we set $ϵ=2$ to apply Theorem 1.5 from [17].

For $ϵ>2$, the equality in Equation (39) still holds, indicating that the equalities in Equations (36) and (37) are satisfied, leading to:

$$|{\Gamma }^1{|}^{ϵ}=\cdots ={|{\Gamma }^{2t+2}|}^{ϵ}$$

Moreover, there exists an a, such that $|\nabla {\Gamma }^a|=0$, demonstrating that ${\Gamma }^a$ is constant and ${\mu }_{1,ϵ}=0$. However, this contradicts the fact that ${\mu }_{1,ϵ}\ne 0$, thereby completing the proof.

• Note 3.1: If $ϵ=2$, the $ϵ$-Laplacian operator simplifies to the Laplacian operator. Thus, we obtain the following corollary:

Corollary 1. 

Consider ${\mathrm{Y}}^m$ as an m-dimensional slant submanifold of a ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ admitting a semi-symmetric metric connection. In this context, the first non-null eigenvalue ${\mu }_1^{\Delta }$ of the Laplacian fulfills the condition:

$${\mu }_1^{\Delta }\le {\displaystyle \frac{m}{(Vol(\mathrm{Y}))}}{\int }_{{\mathrm{Y}}^m}\left\{{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}\left({\displaystyle \frac{3{cos}^2{\alpha }_2}{m-1}}-{\displaystyle \frac{2}{m}}\right)-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)\}dV$$

(52)

Using Theorem 1 for $1<ϵ\le 2$, we derive the following outcome:

Theorem 2. 

Let ${\mathrm{Y}}^m$ be a $m$-dimensional slant submanifold of a 2t + 1-dimensional ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ admitting semi-symmetric metric connection, then the first non-null eigenvalue ${\mu }_{1,ϵ}$ of the $ϵ$-Laplacian satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}& \le ({\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\hfill \\ \hfill & \times {\left[{\int }_{{\mathrm{Y}}^m}{\left({\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}\left({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}}\right)-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)}^{{\displaystyle \frac{ϵ}{2(ϵ-1)}}}\right]}^{ϵ-1}dV\hfill \end{array}\end{array}$$

(53)

for $1<ϵ\le 2.$

Proof. 

Suppose $1<ϵ\le 2,$ we have ${\displaystyle \frac{ϵ}{2(ϵ-1)}}\ge 1,$ then the Holder inequality provides

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_{{\mathrm{Y}}^m}\{& {\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}({\displaystyle \frac{3{cos}^2\alpha }{m-1}}-{\displaystyle \frac{2}{m}})-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\}dV\hfill \\ \hfill & \le \left({(Vol({\mathrm{Y}}^m))}^{1-{\displaystyle \frac{2(ϵ-1)}{ϵ}}}\right)\times \hfill \\ \hfill & {\left[{\int }_{{\mathrm{Y}}^m}{\left({\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}\left({\displaystyle \frac{3{cos}^2{\alpha }_2}{m-1}}-{\displaystyle \frac{2}{m}}\right)-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)}^{{\displaystyle \frac{ϵ}{2(ϵ-1)}}}\right]}^{{\displaystyle \frac{2(ϵ-1)}{ϵ}}}.\hfill \end{array}\end{array}$$

(54)

By merging Equations (38) and (54), we derive the desired inequality, thereby finalizing the proof. □

• Note 3.2: When $c=1$, the simply connected ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ transforms into an odd-dimensional sphere $B^{2t+1}(1)$. Moreover, if $c=-3$, ${\overline{M}}^{2t+1}(c)$ transitions to a $(2t+1)$-dimensional Euclidean space.

Based on the aforementioned arguments, we can infer

Corollary 2. 

If ${\mathrm{Y}}^m$ represents an m-dimensional slant submanifold within a 2t + 1-dimensional ssf $B^{2t+1}(1)$ (an odd-dimensional sphere) equipped with a semi-symmetric metric connection, then

1.

The first non-null eigenvalue ${\mu }_{1,ϵ}$ of the $ϵ$-Laplacian satisfies

$$\begin{array}{c}\hfill {\mu }_{1,ϵ}\le {\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\times {\left\{{\int }_{{\mathrm{Y}}^m}\left(1+{\parallel H\parallel }^2-{\displaystyle \frac{2}{m-1}}trace\beta \right)\right\}}^{ϵ/2}dV\end{array}$$

(55)

for $1<ϵ\le 2,$ and

$$\begin{array}{c}\hfill {\mu }_{1,ϵ}\le {\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\times {\left\{{\int }_{{\mathrm{Y}}^m}\left(1+{\parallel H\parallel }^2-{\displaystyle \frac{2}{m-1}}trace\beta \right)\right\}}^{ϵ/2}dV\end{array}$$

(56)

for $2<ϵ\le {\displaystyle \frac{m}{2}}+1.$

• Note 3.3: When $\alpha =0$ or $\alpha ={\displaystyle \frac{\pi }{2}}$, the slant submanifolds transform into invariant or totally real submanifolds.

Applying the above results leads us to derive the subsequent outcomes concerning invariant and anti-invariant submanifolds in the context of Sasakian manifolds.

Corollary 3. 

If ${\mathrm{Y}}^m$ denotes an m-dimensional invariant submanifold of a 2t + 1-dimensional ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ featuring a semi-symmetric metric connection, then

1.

The first non-null eigenvalue ${\mu }_{1,ϵ}$ of the $ϵ$-Laplacian satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}& \le ({\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\hfill \\ \hfill & \times {\left\{{\int }_{{\mathrm{Y}}^m}\left({\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}\times {\displaystyle \frac{m+2}{m^2-m}}-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)\right\}}^{ϵ/2}dV\hfill \end{array}\end{array}$$

(57)

for $1<ϵ\le 2,$ and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}& \le ({\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\hfill \\ \hfill & \times {\left\{{\int }_{{\mathrm{Y}}^m}\left({\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{c-1}{4}}\times {\displaystyle \frac{m+2}{m^2-m}}-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)\right\}}^{ϵ/2}dV\hfill \end{array}\end{array}$$

(58)

for $2<ϵ\le {\displaystyle \frac{m}{2}}+1.$

2.

The equalities hold true in (59) and (60), if and only if $ϵ=2$ and ${\mathrm{Y}}^m$ is minimally immersed in a geodesic sphere of radius ${\rho }_c$ within ${\overline{\mathrm{Y}}}^{2t+1}(c)$, subject to the following relationships:

$${\delta }_0={\left({\displaystyle \frac{m}{{\mu }_1^{\Delta }}}\right)}^{1/2},{\rho }_1={sin}^{-1}{\rho }_0,{\rho }_{-1}={sinh}^{-1}{\rho }_0.$$

Corollary 4. 

If ${\mathrm{Y}}^m$ represents an m-dimensional anti-invariant submanifold of a ssf ${\overline{\mathrm{Y}}}^{2t+1}(c)$ equipped with a semi-symmetric metric connection, then

1.

The first non-null ev ${\mu }_{1,ϵ}$ of the $ϵ$-Laplacian satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}& \le ({\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\hfill \\ \hfill & \times {\left\{{\int }_{{\mathrm{Y}}^m}\left({\displaystyle \frac{c+3}{4}}-{\displaystyle \frac{c-1}{2m}}-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)\right\}}^{ϵ/2}dV\hfill \end{array}\end{array}$$

(59)

for $1<ϵ\le 2,$ and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_{1,ϵ}& \le ({\displaystyle \frac{2^{(1-\frac{ϵ}{2})}{(t+1)}^{(1-\frac{ϵ}{2})}{m}^{\frac{ϵ}{2}}}{{(Vol(\mathrm{Y}))}^{ϵ/2}}}\hfill \\ \hfill & \times {\left\{{\int }_{{\mathrm{Y}}^m}\left({\displaystyle \frac{c+3}{4}}-{\displaystyle \frac{c-1}{2m}}-{\displaystyle \frac{2}{m-1}}trace\beta +{\parallel H\parallel }^2\right)\right\}}^{ϵ/2}dV\hfill \end{array}\end{array}$$

(60)

for $2<ϵ\le {\displaystyle \frac{m}{2}}+1.$

2.

The equalities are met in (59) and (60), if and only if $ϵ=2$ and ${\mathrm{Y}}^m$ is minimally immersed in a geodesic sphere of radius ${\rho }_c$ within ${\overline{\mathrm{Y}}}^{2t+1}(c)$, as per the following relations:

$${\rho }_0={\left({\displaystyle \frac{m}{{\mu }_1^{\Delta }}}\right)}^{1/2},{\rho }_1={sin}^{-1}{\rho }_0,{\rho }_{-1}={sinh}^{-1}{\rho }_0.$$

4. Conclusions

A key focus of the research involves exploring a range of methodologies to determine the primary eigenvalue for the generalized-Laplacian operator within closed-oriented slant submanifolds situated in a Sasakian space form that admits a semi-symmetric metric connection. This intricate investigation delves into the geometric properties and interplay of curvature measures within these complex manifold settings.

Through the course of the research, significant findings have been uncovered, particularly in relation to extending various Reilly-like inequalities to the generalized Laplacian operator on the slant submanifolds within a unit sphere that also admits a semi-symmetric metric connection. These extensions provide deeper insights into the geometric relationships and inequalities that govern these specialized spaces, shedding light on the interplay between curvature measures and eigenvalues within this framework.

The study concludes by delving into specific scenarios with a keen analytical lens, providing a detailed examination of the implications and applications of the research findings. By examining these scenarios in depth, the research not only contributes to the theoretical understanding of Riemannian manifolds and Laplacian operators, but also offers practical insights that can potentially be applied in various mathematical and geometric contexts.