Curvature Inequalities in Golden-like Statistical Manifolds Admitting Semi-Symmetric Metric Connection

Abstract

This article investigates fundamental inequalities within a golden-like statistical manifold (GLSM) equipped with a semi-symmetric metric connection (SSMC). We explore key geometric and analytical properties, including curvature relations and inequalities analogous to those in classical information geometry. The interplay between the golden-like structure and the SSMC yields new insights into the underlying differential geometric framework. Our results extend known inequalities in the statistical manifold (SM), providing a foundation for further studies in optimization and divergence theory within this generalized framework.

1. Introduction

Statistical manifolds, introduced by Amari [1], feature dual, torsion-free connections resembling affine dual connections [2]. Defining sectional curvature (SC) for these manifolds is non-trivial, but Opozda [3] resolved this, enabling studies of inequalities between intrinsic and extrinsic invariants. For instance, Euler’s inequality $\varkappa \le {|\mu |}^2$ (relating Gaussian curvature $\varkappa$ and mean curvature $\mu$) and Chen’s extensions [4,5] exemplify such relations. Recent advances include SM with complex, contact, and quaternionic structures [6,7,8,9,10] and inequalities for submanifolds in Hessian, Kähler-like, and holomorphic statistical spaces [9,10,11].

The study of semi-symmetric linear connections has evolved significantly since their introduction in 1924 [12]. Hayden [13] pioneered the definition of SSMC on Reimannian manifold (RM), while Imai [14] and Yano [15] explored their properties. Nakao [16] extended Imai’s work by establishing Gauss-like and Codazzi–Mainardi-like equations. In [17,18], they investigated semi-symmetric non-metric connections in 1925.

Recent research has shown a growing interest in geometric inequalities on statistical and golden Riemannian manifolds (GRM), particularly with specialized structures such as cosymplectic, Kenmotsu, and Sasakian statistical manifolds. Kazaz et al. [19] studied geometric inequalities for statistical submanifolds in cosymplectic statistical manifolds, while Malek and Fazlollahi [20] explored key results concerning Kenmotsu and Sasakian statistical manifolds. Siddiqui et al. [21] investigated CR $\delta$-invariants on generic statistical submanifolds and established optimal estimates for Sasakian statistical manifolds [22]. In parallel, Blaga and Vilcu [23] developed a framework connecting Ricci and Hessian metrics to gradient solitons on statistical structures. On the GRM side, Chen et al. [24] presented a comprehensive review of GRM, and several works by Choudhary and collaborators [25,26] have extended fundamental inequalities on golden product manifolds endowed with semi-symmetric connections. Recent contributions by Lee et al. [27] have also addressed Casorati curvature bounds in golden Riemannian space forms (GRSF) with SSMC. These studies collectively highlight a unifying theme of extending classical geometric inequalities to new settings involving statistical and golden structures, providing motivation and context for the present work. However, to the best of our knowledge, no previous work has addressed inequalities on GLSM equipped with SSMC involving scalar and SC.

The study of GLSM equipped with SSMC is motivated by their ability to generalize both statistical and golden structures, providing a unified framework for analyzing curvature inequalities. While classical Chen-type inequalities have been widely explored in Riemannian and statistical geometry, their counterparts in the context of GLSM with SSMC have not been thoroughly investigated. Our work fills this gap by deriving new curvature inequalities that extend the known results to a broader class of manifolds, thus offering new insights into the intrinsic and extrinsic geometry of GLSM.

The article is organized as follows. Section 2 presents the fundamental preliminaries, including golden semi-Riemannian manifolds (GSRM), and their extension to GLSM with SSMC. Section 3 establishes our main results, deriving key curvature inequalities for statistical submanifolds in this framework and analyzing their equality conditions. Section 4 develops the $\delta (2,2)$ Chen-type inequality for GLSM with SSMC, while Section 5 concludes with a discussion of potential applications in information geometry and related fields, along with future research directions.

2. Preliminaries

A manifold $\tilde{M}$ is said to admit a polynomial structure if it possesses a $(1,1)$ tensor field ($(1,1)$-TF) $\mathrm{\nu }$ that satisfies a certain algebraic condition.

$$P\left(u\right)=u^k+a_k{u}^{k-1}+\cdots +a_{2}u+a_{1}I=0.$$

Here, I represents the identity transformation, and $a_1,\dots ,a_m$ are real numbers. The transformation I is linearly independent at every point $w\in \tilde{M}$. The expression $P\left(u\right)$ is known as the structure polynomial. Some important special cases include:

• Almost complex structure ($P\left(\mathrm{\nu }\right)={\mathrm{\nu }}^2+I$),
• Almost product structure ($P\left(\mathrm{\nu }\right)={\mathrm{\nu }}^2-I$),
• Almost tangent structure ($P\left(\mathrm{\nu }\right)={\mathrm{\nu }}^2$).

Note: The presence of an almost complex structure confines $\tilde{M}$ to having even dimensions.

2.1. Golden Semi-Riemannian Manifold (GSRM) and GLSM

Suppose $(\tilde{M},g)$ be an semi-Riemannian manifold (SRM), and $\mathrm{\nu }$ be a $(1,1)-TF$ on $\tilde{M}$ for which [28,29,30]:

$${\mathrm{\nu }}^2=\mathrm{\nu }+I,$$

(1)

I being the identity map. We refer to $\mathrm{\nu }$ as a golden structure (GS) on $\tilde{M}$. If g satisfies

$$g(\mathrm{\nu }{\alpha }_1,{\alpha }_2)=g({\alpha }_1,\mathrm{\nu }{\alpha }_2),$$

(2)

for all vector fields ${\alpha }_1,{\alpha }_2$, then $(\tilde{M},g,\mathrm{\nu })$ is called a GSRM.

Given a $\mathrm{\nu }$-compatible metric g, we have:

$$\begin{array}{c}g(\mathrm{\nu }{\alpha }_1,{\alpha }_2)=g({\alpha }_1,\mathrm{\nu }{\alpha }_2),\end{array}$$

(3)

$$\begin{array}{c}g(\mathrm{\nu }{\alpha }_1,\mathrm{\nu }{\alpha }_2)=g\left({\mathrm{\nu }}^2{\alpha }_1,{\alpha }_2\right)=g(\mathrm{\nu }{\alpha }_1,{\alpha }_2)+g({\alpha }_1,{\alpha }_2),{\alpha }_1,{\alpha }_2\in T\tilde{M}.\end{array}$$

(4)

Definition 1. 

Take $(\tilde{M},g,\mathrm{\nu })$ to be a GSRM together with a $(1,1)-TF$${\mathrm{\nu }}^{}$ that fulfills:

$$g(\mathrm{\nu }{\alpha }_1,{\alpha }_2)=g\left({\alpha }_1,{\mathrm{\nu }}^{\ast }{\alpha }_2\right),$$

(5)

 for any vector field ${\alpha }_1$ and ${\alpha }_2$ on $\tilde{M}$. We derive

$$\begin{array}{cc}\hfill {\left({\mathrm{\nu }}^{\ast }\right)}^2{\alpha }_1& ={\mathrm{\nu }}^{\ast }{\alpha }_1+{\alpha }_1,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill g\left(\mathrm{\nu }{\alpha }_1,{\mathrm{\nu }}^{\ast }{\alpha }_2\right)& =g(\mathrm{\nu }{\alpha }_1,{\alpha }_2)+g({\alpha }_1,{\alpha }_2).\hfill \end{array}$$

(7)

Here, $(\tilde{M},g,\mathrm{\nu })$ is termed a GLSM.

Suppose $\tilde{M}$ is an RM equipped with a torsion-free affine connection ▽. When the covariant derivative $▽g$ is symmetric in all three arguments, $(\tilde{M},▽,g)$ defines an SM. The corresponding conjugate connection ${▽}^{\ast }$ is given by

$${\alpha }_{1}g({\alpha }_2,{\alpha }_3)=g\left({▽}_{{\alpha }_1}{\alpha }_2,{\alpha }_3\right)+g\left({▽}_{{\alpha }_1}^{\ast }{\alpha }_3,{\alpha }_2\right),$$

(8)

for vector fields ${\alpha }_1,{\alpha }_2$, and ${\alpha }_3$ defined on $\tilde{M}$, the connection ${▽}^{\ast }$ is termed the dual of ▽ relative g. ${▽}^{\ast }$ is torsion-free, and the covariant derivative ${▽}^{\ast }g$ is symmetric. Moreover, the connection ${▽}^0$, defined by

$${▽}^0={\displaystyle \frac{1}{2}}(▽+{▽}^{\ast }),$$

(9)

satisfies the compatibility conditions of a statistical structure. Consequently, the triplet $(\tilde{M},{▽}^{\ast },g)$ also forms an SM.

The curvature of $\tilde{M}$ with respect to ▽ and ${▽}^{\ast }$ is denoted by $\mathsf{R}$ and ${\mathsf{R}}^{\ast }$, respectively. The curvature of ${▽}^0$, called the Riemannian curvature tensor (RCT), is denoted by ${\mathsf{R}}^0$. Then, we have the relation:

$$g(\mathsf{R}({\alpha }_1,{\alpha }_2){\alpha }_3,{\alpha }_4)=-g\left({\alpha }_3,{\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2){\alpha }_4\right),$$

for vector fields ${\alpha }_1,{\alpha }_2,{\alpha }_3$, and ${\alpha }_4$ on $\tilde{M}$, where

$$\mathsf{R}({\alpha }_1,{\alpha }_2){\alpha }_3=\left[{▽}_{{\alpha }_1},{▽}_{{\alpha }_2}\right]{\alpha }_3-{▽}_{[{\alpha }_1,{\alpha }_2]}{\alpha }_3.$$

Dual connections usually do not preserve the metric, meaning that SC cannot be defined as in semi-Riemannian geometry. As a result, Opozda found two different types of SC specifically for SM (see [3,31]).

Suppose $\tilde{M}$ is an SM and $\pi$ is a section of the plane in its tangent bundle $T\tilde{M}$, generated by an orthonormal basis ${\alpha }_1,{\alpha }_2$. According to [3], the sectional $\varkappa$-curvature is given by:

$$\varkappa (\pi )={\displaystyle \frac{1}{2}}\left[g(\mathsf{R}({\alpha }_1,{\alpha }_2){\alpha }_2,{\alpha }_1)+g\left({\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2){\alpha }_2,{\alpha }_1\right)-g\left({\mathsf{R}}^0({\alpha }_1,{\alpha }_2){\alpha }_2,{\alpha }_1\right)\right].$$

(10)

The scalar curvature associated with the sectional $\varkappa$-curvature is

$$\top ={\displaystyle \frac{1}{2}}\sum _{\begin{array}{c}1\le \mathsf{ı}\le n\\ \mathsf{ı}<\mathsf{ȷ}\le n\end{array}}\left[g\left(\mathsf{R}\left({\mathsf{ϵ}}_{\mathsf{ı}},{\mathsf{ϵ}}_{\mathsf{ȷ}}\right){\mathsf{ϵ}}_{\mathsf{ȷ}},{\mathsf{ϵ}}_{\mathsf{ı}}\right)+g\left({\mathsf{R}}^{\ast }\left({\mathsf{ϵ}}_{\mathsf{ı}},{\mathsf{ϵ}}_{\mathsf{ȷ}}\right){\mathsf{ϵ}}_{\mathsf{ȷ}},{\mathsf{ϵ}}_{\mathsf{ı}}\right)-2g\left({\mathsf{R}}^0\left({\mathsf{ϵ}}_{\mathsf{ı}},{\mathsf{ϵ}}_{\mathsf{ȷ}}\right){\mathsf{ϵ}}_{\mathsf{ȷ}},{\mathsf{ϵ}}_{\mathsf{ı}}\right)\right].$$

(11)

An illustrative example of a GLSM is now presented.

Example 1. 

Let ${\mathsf{R}}_n^{2n+m}$ be a $(2n+m)$-dimensional affine space with coordinate system 

$$(x_1,\dots ,x_n,y_1,\dots ,y_n,z_1,\dots ,z_m).$$

We define a semi-Riemannian metric g and a $(1,1)$-tensor field $\mathrm{\nu }$ by

$$g=\left(\begin{array}{ccc}-\kappa {\delta }_{ij}& 0& 0\\ 0& \kappa {\delta }_{ij}& 0\\ 0& 0& (1-\kappa ){\delta }_{ij}\end{array}\right),\mathrm{\nu }={\displaystyle \frac{1}{2}}\left(\begin{array}{ccc}{\delta }_{ij}& (2\kappa -1){\delta }_{ij}& 0\\ (2\kappa -1){\delta }_{ij}& {\delta }_{ij}& 0\\ 0& 0& \kappa {\delta }_{ij}\end{array}\right),$$

where ${\delta }_{ij}$ is the Kronecker delta, and κ is the golden ratio:

$$\kappa ={\displaystyle \frac{1+\sqrt{5}}{2}}.$$

Define the conjugate tensor field ${\mathrm{\nu }}^{\ast }$ as

$${\mathrm{\nu }}^{\ast }={\displaystyle \frac{1}{2}}\left(\begin{array}{ccc}{\delta }_{ij}& (1-2\kappa ){\delta }_{ij}& 0\\ (1-2\kappa ){\delta }_{ij}& {\delta }_{ij}& 0\\ 0& 0& \kappa {\delta }_{ij}\end{array}\right).$$

We now verify the three defining conditions of a GLSM:

Condition 1: $g(\mathrm{\nu }{\alpha }_1,{\alpha }_2)=g({\alpha }_1,{\mathrm{\nu }}^{\ast }{\alpha }_2)$.

Take ${\alpha }_1={\partial }_{x_i}$, ${\alpha }_2={\partial }_{y_j}$. Then,

$$\begin{array}{cc}\hfill \mathrm{\nu }\left({\partial }_{x_i}\right)& ={\displaystyle \frac{1}{2}}\left({\partial }_{x_i}+(2\kappa -1){\partial }_{y_i}\right),\hfill \\ \hfill {\mathrm{\nu }}^{\ast }\left({\partial }_{y_j}\right)& ={\displaystyle \frac{1}{2}}\left((1-2\kappa ){\partial }_{x_j}+{\partial }_{y_j}\right).\hfill \end{array}$$

Now compute both sides:

$$\begin{array}{cc}\hfill g(\mathrm{\nu }{\partial }_{x_i},{\partial }_{y_j})& ={\displaystyle \frac{1}{2}}g({\partial }_{x_i},{\partial }_{y_j})+{\displaystyle \frac{2\kappa -1}{2}}g({\partial }_{y_i},{\partial }_{y_j})={\displaystyle \frac{2\kappa -1}{2}}\kappa {\delta }_{ij},\hfill \\ \hfill g({\partial }_{x_i},{\mathrm{\nu }}^{\ast }{\partial }_{y_j})& ={\displaystyle \frac{1-2\kappa }{2}}g({\partial }_{x_i},{\partial }_{x_j})+{\displaystyle \frac{1}{2}}g({\partial }_{x_i},{\partial }_{y_j})={\displaystyle \frac{1-2\kappa }{2}}(-\kappa ){\delta }_{ij}={\displaystyle \frac{2\kappa -1}{2}}\kappa {\delta }_{ij}.\hfill \end{array}$$

So the condition is satisfied.

Condition 2: ${\left({\mathrm{\nu }}^{\ast }\right)}^2\alpha ={\mathrm{\nu }}^{\ast }\alpha +\alpha$, take $\alpha ={\partial }_{x_i}$.

$$\begin{array}{cc}\hfill {\mathrm{\nu }}^{\ast }{\partial }_{x_i}& ={\displaystyle \frac{1}{2}}\left({\partial }_{x_i}+(1-2\kappa ){\partial }_{y_i}\right),\hfill \\ \hfill {\mathrm{\nu }}^{\ast }{\mathrm{\nu }}^{\ast }{\partial }_{x_i}& ={\mathrm{\nu }}^{\ast }\left({\displaystyle \frac{1}{2}}{\partial }_{x_i}+{\displaystyle \frac{1-2\kappa }{2}}{\partial }_{y_i}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\mathrm{\nu }}^{\ast }{\partial }_{x_i}+{\displaystyle \frac{1-2\kappa }{2}}{\mathrm{\nu }}^{\ast }{\partial }_{y_i}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}{\partial }_{x_i}+{\displaystyle \frac{1-2\kappa }{2}}{\partial }_{y_i}\right)+{\displaystyle \frac{1-2\kappa }{2}}\left({\displaystyle \frac{1-2\kappa }{2}}{\partial }_{x_i}+{\displaystyle \frac{1}{2}}{\partial }_{y_i}\right)\hfill \\ \hfill & =\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{{(1-2\kappa )}^2}{4}}\right){\partial }_{x_i}+\left({\displaystyle \frac{1-2\kappa }{4}}+{\displaystyle \frac{1-2\kappa }{4}}\right){\partial }_{y_i}\hfill \\ \hfill & =\left({\displaystyle \frac{1+{(1-2\kappa )}^2}{4}}\right){\partial }_{x_i}+{\displaystyle \frac{1-2\kappa }{2}}{\partial }_{y_i}.\hfill \end{array}$$

Now compute ${\mathrm{\nu }}^{\ast }{\partial }_{x_i}+{\partial }_{x_i}$:

$$\begin{array}{c}\hfill {\mathrm{\nu }}^{\ast }{\partial }_{x_i}+{\partial }_{x_i}=\left({\displaystyle \frac{1}{2}}{\partial }_{x_i}+{\displaystyle \frac{1-2\kappa }{2}}{\partial }_{y_i}\right)+{\partial }_{x_i}=\left({\displaystyle \frac{3}{2}}{\partial }_{x_i}+{\displaystyle \frac{1-2\kappa }{2}}{\partial }_{y_i}\right).\end{array}$$

To match, compute ${\left({\mathrm{\nu }}^{\ast }\right)}^2{\partial }_{x_i}$ with numerical value $\kappa ={\displaystyle \frac{1+\sqrt{5}}{2}}\approx 1.618$:

$${(1-2\kappa )}^2\approx {(-2.236)}^2=5,\Rightarrow {\displaystyle \frac{1+5}{4}}={\displaystyle \frac{6}{4}}={\displaystyle \frac{3}{2}}.$$

Thus,

$${\left({\mathrm{\nu }}^{\ast }\right)}^2{\partial }_{x_i}={\displaystyle \frac{3}{2}}{\partial }_{x_i}+{\displaystyle \frac{1-2\kappa }{2}}{\partial }_{y_i}={\mathrm{\nu }}^{\ast }{\partial }_{x_i}+{\partial }_{x_i}.$$

So, Condition 2 is verified.

Condition 3: $g(\mathrm{\nu }{\alpha }_1,{\mathrm{\nu }}^{\ast }{\alpha }_2)=g(\mathrm{\nu }{\alpha }_1,{\alpha }_2)+g({\alpha }_1,{\alpha }_2)$

Take ${\alpha }_1={\partial }_{x_i}$, ${\alpha }_2={\partial }_{x_j}$. Then:

$$\begin{array}{cc}\hfill \mathrm{\nu }{\partial }_{x_i}& ={\displaystyle \frac{1}{2}}\left({\partial }_{x_i}+(2\kappa -1){\partial }_{y_i}\right),\hfill \\ \hfill {\mathrm{\nu }}^{\ast }{\partial }_{x_j}& ={\displaystyle \frac{1}{2}}\left({\partial }_{x_j}+(1-2\kappa ){\partial }_{y_j}\right).\hfill \end{array}$$

Compute:

$$\begin{array}{cc}\hfill g(\mathrm{\nu }{\partial }_{x_i},{\mathrm{\nu }}^{\ast }{\partial }_{x_j})& =〈{\displaystyle \frac{1}{2}}{\partial }_{x_i}+{\displaystyle \frac{2\kappa -1}{2}}{\partial }_{y_i},{\displaystyle \frac{1}{2}}{\partial }_{x_j}+{\displaystyle \frac{1-2\kappa }{2}}{\partial }_{y_j}〉\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}g({\partial }_{x_i},{\partial }_{x_j})+{\displaystyle \frac{(2\kappa -1)(1-2\kappa )}{4}}g({\partial }_{y_i},{\partial }_{y_j})\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}(-\kappa {\delta }_{ij})+{\displaystyle \frac{-5}{4}}\kappa {\delta }_{ij}=-{\displaystyle \frac{6\kappa }{4}}{\delta }_{ij}=-{\displaystyle \frac{3\kappa }{2}}{\delta }_{ij}.\hfill \end{array}$$

Now compute RHS:

$$\begin{array}{cc}\hfill g(\mathrm{\nu }{\partial }_{x_i},{\partial }_{x_j})& ={\displaystyle \frac{1}{2}}g({\partial }_{x_i},{\partial }_{x_j})=-{\displaystyle \frac{\kappa }{2}}{\delta }_{ij},\hfill \\ \hfill g({\partial }_{x_i},{\partial }_{x_j})& =-\kappa {\delta }_{ij},\hfill \\ \hfill g(\mathrm{\nu }{\partial }_{x_i},{\partial }_{x_j})+g({\partial }_{x_i},{\partial }_{x_j})& =-{\displaystyle \frac{\kappa }{2}}{\delta }_{ij}-\kappa {\delta }_{ij}=-{\displaystyle \frac{3\kappa }{2}}{\delta }_{ij}.\hfill \end{array}$$

Therefore, Condition 3 is satisfied.

Conclusion: The triple $\left({\mathsf{R}}_n^{2n+m},g,\mathrm{\nu }\right)$ satisfies all three conditions and hence is a GLSM.

To derive the inequalities, we use the following.

Theorem 1 

([32]). Assume that $(\tilde{M},g)$ is a locally decomposable golden Riemannian manifold (LDGRM) possessing constant golden SC $\tilde{c}$. Under this assumption:

$$\begin{array}{cc}\hfill \tilde{\mathsf{R}}\left({\alpha }_1,{\alpha }_2\right){\alpha }_3=& {\displaystyle \frac{\tilde{c}}{3}}\left\{g\left({\alpha }_2,{\alpha }_3\right){\alpha }_1-g\left({\alpha }_1,{\alpha }_3\right){\alpha }_2\right.\hfill \\ \hfill & -g\left({\alpha }_2,\mathrm{\nu }{\alpha }_3\right){\alpha }_1-g\left({\alpha }_2,{\alpha }_3\right)\mathrm{\nu }{\alpha }_1+2g\left({\alpha }_2,\mathrm{\nu }{\alpha }_3\right)\mathrm{\nu }{\alpha }_1\hfill \\ \hfill & \left.+g\left({\alpha }_1,\mathrm{\nu }{\alpha }_3\right){\alpha }_2+g\left({\alpha }_1,{\alpha }_3\right)\mathrm{\nu }{\alpha }_2-2g\left({\alpha }_1,\mathrm{\nu }{\alpha }_3\right)\mathrm{\nu }{\alpha }_2\right\},\hfill \end{array}$$

(12)

$\forall {\alpha }_1,{\alpha }_2,{\alpha }_3$ on $\tilde{M}$.

2.2. Statistical Manifold and Statistical Submanifold (SSBM)

Definition 2. 

Let $(\tilde{M},g,▽,{▽}^{\ast })$ be a statistical manifold, where g is a semi-Riemannian metric and $▽,{▽}^{\ast }$ are dual torsion-free affine connections satisfying

$${\alpha }_3·g({\alpha }_1,{\alpha }_2)=g({▽}_{{\alpha }_3}{\alpha }_1,{\alpha }_2)+g({\alpha }_1,{▽}_{{\alpha }_3}^{\ast }{\alpha }_2),$$

for all vector fields ${\alpha }_1,{\alpha }_2,{\alpha }_3\in \mathsf{\Gamma }\left(T\tilde{M}\right)$. A submanifold $M\subset \tilde{M}$ is called a statistical submanifold if the induced metric g, together with the induced affine connections ▽ and ${▽}^{\ast }$, form a dualistic structure on M; that is:

1. 

The induced connections ▽ and ${▽}^{\ast }$ on M are torsion-free;

2. 

They satisfy the compatibility condition:

$${\alpha }_3·g({\alpha }_1,{\alpha }_2)=g({▽}_{{\alpha }_3}{\alpha }_1,{\alpha }_2)+g({\alpha }_1,{▽}_{{\alpha }_3}^{\ast }{\alpha }_2),$$

for all vector field ${\alpha }_1,{\alpha }_2$ and ${\alpha }_3$ on M.

When $(\tilde{M},g,\nu )$ is a GLSM, the submanifold M is called an SSBM of $\tilde{M}$.

Consider $M^n$ as an SSBM of $({\tilde{M}}^m,g,\mathrm{\nu })$. The corresponding Gauss and Weingarten formulas are expressed as follows [33]:

$$\begin{array}{cccc}{▽}_{{\alpha }_1}{\alpha }_2& ={\widetilde{▽}}_{{\alpha }_1}{\alpha }_2+\beta ({\alpha }_1,{\alpha }_2),\hfill & {▽}_{{\alpha }_1}^{\ast }{\alpha }_2& ={\widetilde{▽}}_{{\alpha }_1}^{\ast }{\alpha }_2+{\beta }^{\ast }({\alpha }_1,{\alpha }_2),\end{array}$$

(13)

$$\begin{array}{cccc}{▽}_{{\alpha }_1}\alpha & =-{\mathsf{A}}_{\alpha }{\alpha }_1+{▽}_{{\alpha }_1}^{\perp }\alpha ,& {▽}_{{\alpha }_1}^{\ast }\alpha & =-{\mathsf{A}}_{\alpha }^{\ast }{\alpha }_1+{▽}_{{\alpha }_1}^{\ast \perp }\alpha ,\hfill \end{array}$$

(14)

$\forall {\alpha }_1,{\alpha }_2\in \mathsf{\Gamma }\left(TM\right)$ and $\alpha ,\eta \in \mathsf{\Gamma }\left(T^{\perp }M\right)$. In addition to Equation (8), the following relation holds:

$$\begin{array}{cc}& g(\beta ({\alpha }_1,{\alpha }_2),\alpha )=g\left({\mathsf{A}}_{\alpha }^{\ast }{\alpha }_1,{\alpha }_2\right),\hfill \\ & g\left({\beta }^{\ast }({\alpha }_1,{\alpha }_2),\alpha \right)=g\left({\mathsf{A}}_{\alpha }{\alpha }_1,{\alpha }_2\right),\hfill \\ & {\alpha }_{1}g(\alpha ,\eta )=g\left({▽}_{{\alpha }_1}^{\perp }\alpha ,\eta \right)+g\left(\alpha ,{▽}_{{\alpha }_1}^{\ast \perp }\eta \right).\hfill \end{array}$$

Consider an orthonormal frame ${\alpha }_1,\dots ,{\alpha }_k$ spanning the tangent space and a frame ${\alpha }_{k+1},\dots ,{\alpha }_m$ spanning the normal space; mean curvature vector fields are

$$\mu ={\displaystyle \frac{1}{k}}\sum _{i=1}^k\beta ({\alpha }_i,{\alpha }_i)={\displaystyle \frac{1}{k}}\sum _{{\rm Y}=k+1}^m\left(\sum _{i=1}^k{\beta }_{ii}^{{\rm Y}}\right){\alpha }_{{\rm Y}},{\beta }_{ij}^{{\rm Y}}=g\left(\beta ({\alpha }_i,{\alpha }_j),{\alpha }_{{\rm Y}}\right),$$

$${\mu }^{\ast }={\displaystyle \frac{1}{k}}\sum _{i=1}^k{\beta }^{\ast }({\alpha }_i,{\alpha }_i)={\displaystyle \frac{1}{k}}\sum _{{\rm Y}=k+1}^m\left(\sum _{i=1}^k{\beta }_{ii}^{\ast {\rm Y}}\right){\alpha }_{{\rm Y}},{\beta }_{ij}^{\ast {\rm Y}}=g\left({\beta }^{\ast }({\alpha }_i,{\alpha }_j),{\alpha }_{{\rm Y}}\right),$$

where $1\le i,j\le k$ and $k+1\le {\rm Y}\le m$. We recall the relations $2{\beta }^0=\beta +{\beta }^{\ast }$ and $2{\mu }^0=\mu +{\mu }^{\ast }$, where ${\beta }^0$ and ${\mu }^0$ are computed with respect to the Levi–Civita connection (LCC) ${▽}^0$. For the submanifold, take $k=n$.

The squared mean curvatures are

$${\parallel \mu \parallel }^2={\displaystyle \frac{1}{k^2}}\sum _{{\rm Y}=k+1}^m{\left(\sum _{i=1}^k{\beta }_{ii}^{{\rm Y}}\right)}^2,{\parallel {\mu }^{\ast }\parallel }^2={\displaystyle \frac{1}{k^2}}\sum _{{\rm Y}=k+1}^m{\left(\sum _{i=1}^k{\beta }_{ii}^{\ast {\rm Y}}\right)}^2.$$

(15)

Proposition 1 

([34]). Let M be a statistical submanifold of a GLSM $(\tilde{M},g,\nu )$. Let $\mathsf{R}$ and ${\mathsf{R}}^{\ast }$ be the curvature tensors corresponding to the dual connections ▽ and ${▽}^{\ast }$ on $\tilde{M}$, respectively. Then the following relations hold:

$$\begin{array}{cc}\hfill g\left({\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2){\alpha }_3,{\alpha }_4\right)& =g\left(\mathsf{R}({\alpha }_1,{\alpha }_2){\alpha }_3,{\alpha }_4\right)+g\left(\beta ({\alpha }_1,{\alpha }_3),{\beta }^{\ast }({\alpha }_2,{\alpha }_4)\right)\hfill \\ & -g\left({\beta }^{\ast }({\alpha }_1,{\alpha }_4),\beta ({\alpha }_2,{\alpha }_3)\right),\hfill \\ \hfill g\left(\mathsf{R}({\alpha }_1,{\alpha }_2){\alpha }_3,{\alpha }_4\right)& =g\left({\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2){\alpha }_3,{\alpha }_4\right)+g\left({\beta }^{\ast }({\alpha }_1,{\alpha }_3),\beta ({\alpha }_2,{\alpha }_4)\right)\hfill \\ & -g\left(\beta ({\alpha }_1,{\alpha }_4),{\beta }^{\ast }({\alpha }_2,{\alpha }_3)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill g\left({\mathsf{R}}^{\perp }({\alpha }_1,{\alpha }_2)\xi ,\eta \right)& =g\left(\mathsf{R}({\alpha }_1,{\alpha }_2)\xi ,\eta \right)+g\left([{\mathsf{A}}_{\xi }^{\ast },{\mathsf{A}}_{\eta }]{\alpha }_1,{\alpha }_2\right),\hfill \\ \hfill g\left({\mathsf{R}}^{\ast \perp }({\alpha }_1,{\alpha }_2)\xi ,\eta \right)& =g\left({\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2)\xi ,\eta \right)+g\left([{\mathsf{A}}_{\xi },{\mathsf{A}}_{\eta }^{\ast }]{\alpha }_1,{\alpha }_2\right),\hfill \end{array}$$

 where the commutators are defined by

$$[{\mathsf{A}}_{\xi },{\mathsf{A}}_{\eta }^{\ast }]={\mathsf{A}}_{\xi }{\mathsf{A}}_{\eta }^{\ast }-{\mathsf{A}}_{\eta }^{\ast }{\mathsf{A}}_{\xi },[{\mathsf{A}}_{\xi }^{\ast },{\mathsf{A}}_{\eta }]={\mathsf{A}}_{\xi }^{\ast }{\mathsf{A}}_{\eta }-{\mathsf{A}}_{\eta }{\mathsf{A}}_{\xi }^{\ast },$$

 for all ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\in \mathsf{\Gamma }\left(TM\right)$ and $\xi ,\eta \in \mathsf{\Gamma }\left(T^{\perp }M\right)$.

We now present two key lemmas that will be instrumental in proving the inequalities in the subsequent sections.

Lemma 1 

([9]). Assume that $\mathsf{r}\ge 3$ is an integer and that ${\mathsf{q}}_1,{\mathsf{q}}_2,\dots ,{\mathsf{q}}_{\mathsf{r}}$ are $\mathsf{r}$ real parameters. Then, the following geometric relation holds:

$$\sum _{1\le i<j\le \mathsf{r}}^{\mathsf{r}}{\mathsf{q}}_i{\mathsf{q}}_j-{\mathsf{q}}_1{\mathsf{q}}_2\le {\displaystyle \frac{\mathsf{r}-2}{2(\mathsf{r}-1)}}{\left(\sum _{i=1}^{\mathsf{r}}{\mathsf{q}}_i\right)}^2.$$

Moreover, the equality holds iff ${\mathsf{q}}_1+{\mathsf{q}}_2={\mathsf{q}}_3=\dots ={\mathsf{q}}_{\mathsf{r}}$.

Remark 1. 

Lemma 1 is often used to control mixed terms in curvature computations, particularly those involving shape operators or second fundamental forms, making it valuable for deriving SC estimates.

Lemma 2 

([35]). Let $\mathsf{r}\ge 4$ be an integer, and let ${\mathsf{q}}_1,{\mathsf{q}}_2,\dots ,{\mathsf{q}}_{\mathsf{r}}$ represent $\mathsf{r}$ real numbers. Then, the following holds:

$$\sum _{1\le i<j\le \mathsf{r}}^{\mathsf{r}}{\mathsf{q}}_i{\mathsf{q}}_j-{\mathsf{q}}_1{\mathsf{q}}_2-{\mathsf{q}}_3{\mathsf{q}}_4\le {\displaystyle \frac{\mathsf{r}-3}{2(\mathsf{r}-2)}}{\left(\sum _{i=1}^{\mathsf{r}}{\mathsf{q}}_i\right)}^2.$$

Moreover, the equality holds iff ${\mathsf{q}}_1+{\mathsf{q}}_2={\mathsf{q}}_3+{\mathsf{q}}_4={\mathsf{q}}_5=\cdots ={\mathsf{q}}_{\mathsf{r}}$.

Remark 2. 

Lemma 2 generalizes Lemma 1 and plays a key role in bounding higher-order mixed terms, especially in curvature pinching and geometric inequalities.

2.3. Semi Symmetric Metric Connection

We consider a RM $({\tilde{M}}^m,g)$ equipped with ${▽}^{\ast }$, for which the torsion ⊤ satisfies the condition given in [36]:

$$\top ({\alpha }_2,{\alpha }_3)={\rm Y}\left({\alpha }_3\right){\alpha }_2-{\rm Y}\left({\alpha }_2\right){\alpha }_3,$$

(16)

where ${▽}^{\ast }$ is called a semi-symmetric connection (SSC). Given a vector field $\overline{\eta }$ and its associated 1-form ${\rm Y}$ defined by

$${\rm Y}\left({\alpha }_1\right)=g({\alpha }_1,\overline{\eta }),$$

the connection ${▽}^{\ast }$ is called

• a SSMC when

  $${▽}^{\ast }g=0,$$

  (17)
• a semi-symmetric not metric connection (SSNMC) when

  $${▽}^{\ast }g\ne 0.$$

  (18)

Following [36], the SSMC on $\tilde{M}$ is given by

$${▽}_{{\alpha }_1}^{\ast }{\alpha }_2={\rm Y}\left({\alpha }_2\right){\alpha }_1-g({\alpha }_1,{\alpha }_2)\overline{\eta }+{▽}_{{\alpha }_1}{\alpha }_2.$$

Here, ▽ represents the LCC on $\tilde{M}$. Consider ${\mathsf{R}}^{\ast }$ and $\mathsf{R}$ to be the curvatures corresponding to ${▽}^{\ast }$ and ▽, respectively. Then, as stated in [5]:

$$\begin{array}{cc}\hfill {\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2){\alpha }_3& =\mathsf{R}({\alpha }_1,{\alpha }_2){\alpha }_3+g({\alpha }_1,{\alpha }_3)\kappa {\alpha }_2-\varrho ({\alpha }_2,{\alpha }_3){\alpha }_1\hfill \\ \hfill & -g({\alpha }_2,{\alpha }_3)\kappa {\alpha }_1+\varrho ({\alpha }_1,{\alpha }_3){\alpha }_2,\forall {\alpha }_i\in T\tilde{M},\hfill \end{array}$$

(19)

where $\varrho$ represents a $(0,2)$-TF described as

$$\varrho ({\alpha }_1,{\alpha }_2)={\displaystyle \frac{1}{2}}{\rm Y}\left(\overline{\eta }\right)g({\alpha }_1,{\alpha }_2)+({▽}_{{\alpha }_1}{\rm Y}){\alpha }_2-{\rm Y}\left({\alpha }_1\right){\rm Y}\left({\alpha }_2\right),$$

and $\kappa$ satisfies

$$g(\kappa {\alpha }_1,{\alpha }_2)=\varrho ({\alpha }_1,{\alpha }_2).$$

Now consider ${\tilde{M}}^m$ as a RM with SSMC and $M^n$ as its submanifold. Let ▽ and $\overline{▽}$ denote the LCC in M and $\tilde{M}$, respectively, and we denote by ${\mathbb{S}}_N$ the shape operator of the submanifold M relative to the normal vector field $N\in \mathsf{\Gamma }\left(T^{\perp }M\right)$. We have the fundamental equations:

$${\overline{▽}}_{{\alpha }_1}{\alpha }_2={▽}_{{\alpha }_1}{\alpha }_2+h({\alpha }_1,{\alpha }_2),$$

$${\overline{▽}}_{{\alpha }_1}N=-{\mathbb{S}}_N{\alpha }_1+{▽}_{{\alpha }_1}^{\perp }N,$$

where ${▽}^{\perp }$ is the normal connection with the relation:

$$g({\mathbb{S}}_N{\alpha }_1,{\alpha }_2)=g(h({\alpha }_1,{\alpha }_2),N).$$

The Gauss equation for the curvatures $\mathsf{R}$ of M and $\overline{\mathsf{R}}$ of $\tilde{M}$ is [37]:

$$\begin{array}{cc}\hfill \mathsf{R}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4)& =\overline{\mathsf{R}}({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4)-g(h({\alpha }_1,{\alpha }_4),h({\alpha }_2,{\alpha }_3))\hfill \\ \hfill & +g(h({\alpha }_1,{\alpha }_3),h({\alpha }_2,{\alpha }_4)),\hfill \end{array}$$

(20)

$\forall {\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\in \mathsf{\Gamma }\left(TM\right)$. For normal vector fields ${\alpha }_1,{\alpha }_2$, we have [36]

$$g(\overline{\mathsf{R}}({\alpha }_1,{\alpha }_2){\alpha }_1,{\alpha }_2)=g({\mathsf{R}}^{\perp }({\alpha }_1,{\alpha }_2){\alpha }_1,{\alpha }_2)+g([{\mathbb{S}}_{{\alpha }_1},{\mathbb{S}}_{{\alpha }_2}]{\alpha }_1,{\alpha }_2),$$

(21)

where $[{\mathbb{S}}_{{\alpha }_1},{\mathbb{S}}_{{\alpha }_2}]={\mathbb{S}}_{{\alpha }_1}{\mathbb{S}}_{{\alpha }_2}-{\mathbb{S}}_{{\alpha }_2}{\mathbb{S}}_{{\alpha }_1}$.

Finally, the curvature ${\mathsf{R}}^{\ast }$ of $\tilde{M}$ with SSMC ${▽}^{\ast }$ can be expressed as

$$\begin{array}{cc}\hfill {\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2){\alpha }_3& =\mathsf{R}({\alpha }_1,{\alpha }_2){\alpha }_3-\varrho ({\alpha }_2,{\alpha }_3){\alpha }_1+\varrho ({\alpha }_1,{\alpha }_3){\alpha }_2\hfill \\ \hfill & -g({\alpha }_2,{\alpha }_3)\kappa {\alpha }_1+g({\alpha }_1,{\alpha }_3)\kappa {\alpha }_2.\hfill \end{array}$$

(22)

3. LDGLSM (Locally Decomposable GLSM) Endowed with SSMC

Theorem 2. 

Let $\tilde{M}$ be an LDGLSM endowed with SSMC and let M be its SSBM. Under these conditions, we have the following:

$$\begin{array}{cc}\hfill (\top -\varkappa (\pi ))-\left({\top }_0-{\varkappa }_0(\pi )\right)\ge & {\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n+{2\parallel \mathrm{\nu }\parallel }^2-2n\parallel \mathrm{\nu }\parallel \right]-2\left(n-1\right)tr\varrho \hfill \\ \hfill & -\left[{\displaystyle \frac{\tilde{c}}{3}}\left\{1-tr\left({\mathrm{\nu }}_{\mid \pi }\right)\right\}+2\theta (\pi )-3tr\left({\beta }_{\mid \pi }\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \mu \parallel }^2+{∥{\mu }^{\ast }∥}^2\right]+2{\widehat{\varkappa }}_0(\pi )-2{\widehat{\top }}_0.\hfill \end{array}$$

(23)

Proof. 

Consider orthonormal frames $\{{\mathsf{ϵ}}_1,\dots ,{\mathsf{ϵ}}_n\}\subset T\mathrm{M}$ and $\{{\mathsf{ϵ}}_{n+1},\dots ,{\mathsf{ϵ}}_m\}\subset T^{\perp }\mathrm{M}$. Using Theorem (1), Equation (22) and Gauss equations, we have

$$\begin{array}{cc}\hfill \tilde{\mathsf{R}}({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_j,{\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i)={\displaystyle \frac{\tilde{c}}{3}}& [g({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_j)g({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_i)-g({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_j)g({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i)\hfill \\ & -g({\mathsf{ϵ}}_j,\mathrm{\nu }{\mathsf{ϵ}}_j)g({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_i)-g({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_j)g(\mathrm{\nu }{\mathsf{ϵ}}_i,{\mathsf{ϵ}}_i)\hfill \\ & +2g({\mathsf{ϵ}}_j,\mathrm{\nu }{\mathsf{ϵ}}_j)g(\mathrm{\nu }{\mathsf{ϵ}}_i,{\mathsf{ϵ}}_i)+g({\mathsf{ϵ}}_i,\mathrm{\nu }{\mathsf{ϵ}}_j)g({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i)\hfill \\ & +g({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_j)g(\mathrm{\nu }{\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i)-2g({\mathsf{ϵ}}_i,\mathrm{\nu }{\mathsf{ϵ}}_j)g(\mathrm{\nu }{\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i)]\hfill \\ & -\varrho \left({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_j\right)g\left({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_i\right)+\varrho \left({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_j\right)g\left({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i\right)\hfill \\ & -g\left({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_j\right)g\left(\kappa {\mathsf{ϵ}}_i,{\mathsf{ϵ}}_i\right)+g\left({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_j\right)g\left(\kappa {\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i\right)\hfill \\ & +g\left({\beta }^{\ast }\left({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_i\right),\beta \left({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_j\right)\right)-g\left(\beta \left({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i\right),{\beta }^{\ast }\left({\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i\right)\right).\hfill \end{array}$$

The CT corresponding to the dual connection, denoted as ${\mathsf{R}}^{\ast }({\mathsf{ϵ}}_i,{\mathsf{ϵ}}_j,{\mathsf{ϵ}}_j,{\mathsf{ϵ}}_i)$, can be derived from the above equation by simply replacing $\mathrm{\nu }$ with ${\mathrm{\nu }}^{\ast }$. By applying Definition (1) and the Gauss equations (13) to Equation (11), we obtain the following through straightforward calculations:

$$\begin{array}{cc}\hfill \top ={\displaystyle \frac{\tilde{c}}{3}}& [n^2-3n+{2\parallel \mathrm{\nu }\parallel }^2-2n\parallel \mathrm{\nu }\parallel ]-2\left(n-1\right)tr\varrho -{\top }_0\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\sum _{1\le i<j\le n}\left[{\beta }_{ii}^{\ast {\rm Y}}{\beta }_{jj}^{{\rm Y}}+{\beta }_{ii}^{{\rm Y}}{\beta }_{jj}^{\ast {\rm Y}}-2{\beta }_{ij}^{\ast {\rm Y}}{\beta }_{ij}^{{\rm Y}}\right],\hfill \end{array}$$

(24)

which can be rewritten as

$$\begin{array}{cc}\hfill \top ={\displaystyle \frac{\tilde{c}}{3}}& [n^2-3n+{2\parallel \mathrm{\nu }\parallel }^2-2n\parallel \mathrm{\nu }\parallel ]-2\left(n-1\right)tr\varrho \hfill \\ & -{\top }_0+2\sum _{{\rm Y}=n+1}^m\sum _{1\le i<j\le n}\left[{\beta }_{ii}^{0{\rm Y}}{\beta }_{jj}^{0{\rm Y}}-{\left({\beta }_{ij}^{0{\rm Y}}\right)}^2\right]\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\sum _{1\le i<j\le n}\left[\left\{{\beta }_{ii}^{{\rm Y}}{\beta }_{jj}^{{\rm Y}}-{\left({\beta }_{ij}^{{\rm Y}}\right)}^2\right\}+\left\{{\beta }_{ii}^{\ast {\rm Y}}{\beta }_{jj}^{\ast {\rm Y}}-{\left({\beta }_{ij}^{\ast {\rm Y}}\right)}^2\right\}\right].\hfill \end{array}$$

By using Equation (13) for the LCC, we have

$$\begin{array}{cc}\hfill \top ={\displaystyle \frac{\tilde{c}}{3}}& [n^2-3n+{2\parallel \mathrm{\nu }\parallel }^2-2n\parallel \mathrm{\nu }\parallel ]-2\left(n-1\right)tr\varrho -2{\widehat{\top }}_0\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\sum _{1\le i<j\le n}\left[\left\{{\beta }_{ii}^{{\rm Y}}{\beta }_{jj}^{{\rm Y}}-{\left({\beta }_{ij}^{{\rm Y}}\right)}^2\right\}+\left\{{\beta }_{ii}^{\ast {\rm Y}}{\beta }_{jj}^{\ast {\rm Y}}-{\left({\beta }_{ij}^{\ast {\rm Y}}\right)}^2\right\}\right],\hfill \end{array}$$

(25)

where ${\widehat{\top }}_0$ represents the scalar curvature of the primary SM. The sectional $\varkappa$-curvature $\varkappa (\pi )$ for the plane $\pi$ is defined by Equation (10), in which:

$$\varkappa (\pi )={\displaystyle \frac{1}{2}}\left[g\left(\mathsf{R}\left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2\right){\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right)+g\left({\mathsf{R}}^{\ast }\left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2\right){\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right)-2g\left({\mathsf{R}}^0\left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2\right){\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right)\right].$$

Thus, we obtain

$$\begin{array}{cc}\hfill \mathsf{R}({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1)& ={\displaystyle \frac{\tilde{c}}{3}}[g({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2)g({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1)-g({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2)g({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1)-g({\mathsf{ϵ}}_2,\mathrm{\nu }{\mathsf{ϵ}}_2)g({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1)\hfill \\ & -g({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2)g(\mathrm{\nu }{\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1)+2g({\mathsf{ϵ}}_2,\mathrm{\nu }{\mathsf{ϵ}}_2)g(\mathrm{\nu }{\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1)\hfill \\ & +g({\mathsf{ϵ}}_1,\mathrm{\nu }{\mathsf{ϵ}}_2)g({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1)+g({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2)g(\mathrm{\nu }{\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1)-2g({\mathsf{ϵ}}_1,\mathrm{\nu }{\mathsf{ϵ}}_2)g(\mathrm{\nu }{\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1)]\hfill \\ \hfill & -\varrho \left({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2\right)g\left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1\right)+\varrho \left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2\right)g\left({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right)\hfill \\ & -g\left({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2\right)g\left(\kappa {\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1\right)+g\left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2\right)g\left(\kappa {\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right)\hfill \\ \hfill & +g\left({\beta }^{\ast }\left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1\right),\beta \left({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2\right)\right)-g\left(\beta \left({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right),{\beta }^{\ast }\left({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right)\right).\hfill \end{array}$$

${\mathsf{R}}^{\ast }\left({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2,{\mathsf{ϵ}}_1\right)$ can be obtained from the above equation just by replacing $\mathrm{\nu }$ by ${\mathrm{\nu }}^{\ast }$. After doing some straightforward computations, we deduce

$$\begin{array}{cc}\hfill \varkappa (\pi )=& {\displaystyle \frac{\tilde{c}}{3}}\{1-tr\left({\mathrm{\nu }}_{\mid \pi }\right)\}+2\theta (\pi )-3tr\left({\beta }_{\mid \pi }\right)-{\varkappa }_0(\pi )\hfill \\ & +{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left[{\beta }_{11}^{{\rm Y}}{\beta }_{22}^{\ast {\rm Y}}+{\beta }_{11}^{\ast {\rm Y}}{\beta }_{22}^{{\rm Y}}-2{\beta }_{12}^{\ast {\rm Y}}{\beta }_{12}^{\ast {\rm Y}}\right],\hfill \end{array}$$

where $\theta (\pi )=\varrho ({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2)g({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2)$ and $tr{\beta }_{\mid \pi }=\varrho ({\mathsf{ϵ}}_1,{\mathsf{ϵ}}_1)=\varrho ({\mathsf{ϵ}}_2,{\mathsf{ϵ}}_2)$.

Using $\beta +{\beta }^{\ast }=2{\beta }^0$, we get

$$\begin{array}{cc}\hfill \varkappa (\pi )=& {\displaystyle \frac{\tilde{c}}{3}}\{1-tr\left({\mathrm{\nu }}_{\mid \pi }\right)\}+2\theta (\pi )-3tr\left({\beta }_{\mid \pi }\right)-{\varkappa }_0(\pi )+2\sum _{{\rm Y}=n+1}^m\left[{\beta }_{11}^{0{\rm Y}}{\beta }_{22}^{0{\rm Y}}-{\left({\beta }_{12}^{0{\rm Y}}\right)}^2\right]\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left\{\left[{\beta }_{11}^{{\rm Y}}{\beta }_{22}^{{\rm Y}}-{\left({\beta }_{12}^{{\rm Y}}\right)}^2\right]+\left[{\beta }_{11}^{\ast {\rm Y}}{\beta }_{22}^{\ast {\rm Y}}-{\left({\beta }_{12}^{\ast {\rm Y}}\right)}^2\right]\right\}.\hfill \end{array}$$

Using Equation (13) in the context of the LCC, we obtain

$$\begin{array}{cc}\hfill \varkappa (\pi )=& {\displaystyle \frac{\tilde{c}}{3}}\{1-tr\left({\mathrm{\nu }}_{\mid \pi }\right)\}+2\theta (\pi )-3tr\left({\beta }_{\mid \pi }\right)-2{\widehat{\varkappa }}_0(\pi )\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left[{\beta }_{11}^{{\rm Y}}{\beta }_{22}^{{\rm Y}}-{\left({\beta }_{12}^{{\rm Y}}\right)}^2\right]-{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left[{\beta }_{11}^{\ast {\rm Y}}{\beta }_{22}^{\ast {\rm Y}}-{\left({\beta }_{12}^{\ast {\rm Y}}\right)}^2\right],\hfill \end{array}$$

(26)

where ${\widehat{\varkappa }}_0$ is the SC with respect to the primary SM. Using Equations (25) and (26), we get

$$\begin{array}{cc}\hfill (\top -\varkappa (\pi ))-\left({\top }_0-{\varkappa }_0(\pi )\right)=& {\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n+{2\parallel \mathrm{\nu }\parallel }^2-2n\parallel \mathrm{\nu }\parallel \right]-2\left(n-1\right)tr\varrho \hfill \\ \hfill & -\left[{\displaystyle \frac{\tilde{c}}{3}}\left\{1-tr\left({\mathrm{\nu }}_{\mid \pi }\right)\right\}+2\theta (\pi )-3tr\left({\beta }_{\mid \pi }\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left[{\beta }_{ii}^{{\rm Y}}{\beta }_{jj}^{{\rm Y}}-{\left({\beta }_{ij}^{{\rm Y}}\right)}^2\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left[{\beta }_{ii}^{\ast {\rm Y}}{\beta }_{jj}^{\ast {\rm Y}}-{\left({\beta }_{ij}^{\ast {\rm Y}}\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left\{\left[{\beta }_{11}^{{\rm Y}}{\beta }_{22}^{{\rm Y}}-{\left({\beta }_{12}^{{\rm Y}}\right)}^2\right]\right.\hfill \\ \hfill & \left.+\left[{\beta }_{11}^{\ast {\rm Y}}{\beta }_{22}^{\ast {\rm Y}}-{\left({\beta }_{12}^{\ast {\rm Y}}\right)}^2\right]\right\}+2{\widehat{\varkappa }}_0(\pi )-2{\widehat{\top }}_0.\hfill \end{array}$$

(27)

Now, consider the terms of the second fundamental form. Let us define

$$A:=\sum _{{\rm Y}=n+1}^m\sum _{\begin{array}{c}1\le \mathsf{ı}<\mathsf{ȷ}\le n\end{array}}\left[{\beta }_{\mathsf{ı}\mathsf{ı}}^{{\rm Y}}{\beta }_{\mathsf{ȷ}\mathsf{ȷ}}^{{\rm Y}}-{\left({\beta }_{\mathsf{ı}\mathsf{ȷ}}^{{\rm Y}}\right)}^2\right],A^{\ast }:=\sum _{{\rm Y}=n+1}^m\sum _{\begin{array}{c}1\le \mathsf{ı}<\mathsf{ȷ}\le n\end{array}}\left[{\beta }_{\mathsf{ı}\mathsf{ı}}^{\ast {\rm Y}}{\beta }_{\mathsf{ȷ}\mathsf{ȷ}}^{\ast {\rm Y}}-{\left({\beta }_{\mathsf{ı}\mathsf{ȷ}}^{\ast {\rm Y}}\right)}^2\right].$$

According to Lemma 1, we have

$$A+A^{\ast }\le {\displaystyle \frac{n-2}{2(n-1)}}{\left(\sum _{{\rm Y}=n+1}^m\sum _{\mathsf{ı}=1}^n{\beta }_{\mathsf{ı}\mathsf{ı}}^{{\rm Y}}\right)}^2+{\displaystyle \frac{n-2}{2(n-1)}}{\left(\sum _{{\rm Y}=n+1}^m\sum _{\mathsf{ı}=1}^n{\beta }_{\mathsf{ı}\mathsf{ı}}^{\ast {\rm Y}}\right)}^2.$$

(28)

This can be written as

$$A+A^{\ast }\le {\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left({\parallel \mu \parallel }^2+{\parallel {\mu }^{\ast }\parallel }^2\right),$$

where $\mu$ and ${\mu }^{\ast }$ are the mean curvature vectors, defined as

$$\mu :={\displaystyle \frac{1}{n}}\sum _{\mathsf{ı}=1}^n{\beta }_{\mathsf{ı}\mathsf{ı}},{\mu }^{\ast }:={\displaystyle \frac{1}{n}}\sum _{\mathsf{ı}=1}^n{\beta }_{\mathsf{ı}\mathsf{ı}}^{\ast }.$$

Substituting this into (27) and applying Lemma (1), the equation reduces to Equation (23). This completes the proof. □

Remark 3. 

In the application of Theorem (1), we assume that the semi-symmetric metric connection ${▽}^{\ast }$ on the ambient manifold $(\tilde{M},g,\mathrm{\nu })$ satisfies the following compatibility conditions:

• ${▽}^{\ast }g=0$, i.e., ${▽}^{\ast }$ is metric-compatible;
• ${▽}^{\ast }\mathrm{\nu }=0$, i.e., the GS $\mathrm{\nu }$ is parallel with respect to ${▽}^{\ast }$;

These assumptions imply that the curvature operator ${\mathsf{R}}^{\ast }$ of ${▽}^{\ast }$ commutes with $\mathrm{\nu }$, namely,

$${\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2)\left(\nu {\alpha }_3\right)=\mathrm{\nu }\left({\mathsf{R}}^{\ast }({\alpha }_1,{\alpha }_2){\alpha }_3\right),\forall {\alpha }_1,{\alpha }_2,{\alpha }_3\in \mathsf{\Gamma }\left(T\tilde{M}\right).$$

This justifies the use of $\mathrm{\nu }$ in curvature expressions involving ${\mathsf{R}}^{\ast }$ or its restrictions. Furthermore, the Gauss equation employed is adapted to the semi-symmetric setting and preserves compatibility with the GS.

The following results follow immediately from the above theorem.

Corollary 1. 

Consider $M^n$ to be a totally real SSBM within an LDGLSM $\tilde{M}$ equipped with SSMC. Then, the following properties hold:

$$\begin{array}{cc}\hfill (\top -\varkappa (\pi ))-\left({\top }_0-{\varkappa }_0(\pi )\right)& \ge {\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n-1\right]-2\left(n-1\right)tr\varrho -2\theta (\pi )+3tr\left({\beta }_{\mid \pi }\right)\hfill \\ \hfill & -{\displaystyle \frac{n^2(n-2)}{4(n-+2\theta (\pi )-3tr\left({\beta }_{\mid \pi }\right)1)}}\left[{\parallel \mu \parallel }^2+{∥{\mu }^{\ast }∥}^2\right]\hfill \\ \hfill & +2{\widehat{\varkappa }}_0(\pi )-2{\widehat{\top }}_0.\hfill \end{array}$$

(29)

Proof. 

The proof follows directly from Theorem (2), since Equation (29) is an immediate consequence of the Equation (23) upon setting $\mathrm{\nu }$ = 0. □

Remark 4. 

Moreover, the equalities condition for Theorem (2) and Corollary (1) hold ∀${\rm Y}\in \{n+1,\dots ,m\}$ iff.

$$\begin{array}{c}{\beta }_{11}^{{\rm Y}}+{\beta }_{22}^{{\rm Y}}={\beta }_{33}^{{\rm Y}}=\cdots ={\beta }_{nn}^{{\rm Y}},\\ {\beta }_{11}^{\ast {\rm Y}}+{\beta }_{22}^{\ast {\rm Y}}={\beta }_{33}^{\ast {\rm Y}}=\cdots ={\beta }_{nn}^{\ast {\rm Y}},\\ {\beta }_{ij}^{{\rm Y}}={\beta }_{ij}^{\ast {\rm Y}}=0,\end{array}$$

$\forall i,j\in \{1,\dots ,n\}$ with $i\ne j$, excluding the pairs $(i,j)=(1,2)$ and $(2,1)$, and assuming $i<j$.

Remark 5. 

The equality conditions in Theorem (2) and Corollary (1) hold for all ${\rm Y}\in \{n+1,\dots ,m\}$ if and only if the following geometric properties are satisfied:

1. Totally umbilic submanifold: The second fundamental forms ${\beta }^{{\rm Y}}$ and ${\beta }^{\ast {\rm Y}}$ are umbilical, meaning the submanifold M curves equally in all normal directions. Specifically:

$${\beta }_{11}^{{\rm Y}}+{\beta }_{22}^{{\rm Y}}={\beta }_{33}^{{\rm Y}}=\cdots ={\beta }_{nn}^{{\rm Y}},{\beta }_{11}^{\ast {\rm Y}}+{\beta }_{22}^{\ast {\rm Y}}={\beta }_{33}^{\ast {\rm Y}}=\cdots ={\beta }_{nn}^{\ast {\rm Y}}.$$

This implies that M is intrinsically symmetric with respect to the planes ${\pi }_1$ and ${\pi }_2$, and the extrinsic curvatures are uniform in the remaining directions.

2. Vanishing mixed components: The mixed components of ${\beta }^{{\rm Y}}$ and ${\beta }^{\ast {\rm Y}}$ vanish, i.e.,

$${\beta }_{ij}^{{\rm Y}}={\beta }_{ij}^{\ast {\rm Y}}=0\mathit{for}\mathit{all}i\ne j,$$

except for the pairs $(i,j)=(1,2),(2,1),(3,4),(4,3)$. This condition ensures that the submanifold has no “twisting” in directions orthogonal to ${\pi }_1$ and ${\pi }_2$, further restricting M to a highly symmetric configuration.

Corollary 2. 

Consider $M^n$ to be a totally real SSBM of an LDGLSM $\tilde{M}$ equipped with SSMC. If there exists a point $p\in M$ and a plane $\pi \subset T_{p}M$ such that

$$\begin{array}{cc}\hfill \top -\varkappa (\pi )<& {\top }_0-{\varkappa }_0(\pi )+{\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n-1\right]-2\left(n-1\right)tr\varrho \hfill \\ \hfill & -2\theta (\pi )+3tr\left({\beta }_{\mid \pi }\right)+2\left({\widehat{\varkappa }}_0(\pi )-{\widehat{\top }}_0\right).\hfill \end{array}$$

(30)

Then M is non-minimal, i.e., $\mu \ne 0$ or ${\mu }^{\ast }\ne 0$.

Proof. 

We proceed by contradiction. Assume M is minimal at p, meaning the mean curvature vectors vanish:

$$\mu ={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\beta }_{ii}=0\mathrm{and}{\mu }^{\ast }={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\beta }_{ii}^{\ast }=0.$$

This implies:

$$\sum _{i=1}^n{\beta }_{ii}^{{\rm Y}}=0\mathrm{and}\sum _{i=1}^n{\beta }_{ii}^{\ast {\rm Y}}=0\forall {\rm Y}\in \{n+1,\dots ,m\}.$$

Consequently, the squared mean curvatures vanish:

$${\parallel \mu \parallel }^2=0\mathrm{and}{\parallel {\mu }^{\ast }\parallel }^2=0.$$

From Corollary (1), the inequality simplifies under minimality to:

$$\begin{array}{cc}\hfill \top -\varkappa \left(\pi \right)\ge & {\top }_0-{\varkappa }_0\left(\pi \right)+{\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n-1\right]-2(n-1)tr\varrho -2\theta \left(\pi \right)\hfill \\ & +3tr(\beta {|}_{\pi })+2\left({\varkappa }_0\left(\pi \right)-{\top }_0\right).\hfill \end{array}$$

This contradicts the given strict inequality in the corollary’s hypothesis. Therefore, our assumption of minimality must be false, and at least one of $\mu$ or ${\mu }^{\ast }$ must be non-zero at p. □

4. $\mathbf{\delta }(\mathbf{2},\mathbf{2})$ Chen’s Inequality

Definition 3 

(Chen’s Inequality). B.-Y. Chen [37,38] established sharp estimates of the squared mean curvature ${\parallel \mu \parallel }^2$ in terms of Chen invariants for submanifolds $M^n$ in Riemannian space forms ${\tilde{M}}^m\left(c\right)$, given by

$$\delta \left(n_1,\dots ,n_k\right)\le {\displaystyle \frac{n^2\left(n+k-{\sum }_{j=1}^k{n}_j-1\right)}{2\left(n+k-{\sum }_{j=1}^k{n}_j\right)}}{\parallel H\parallel }^2+{\displaystyle \frac{1}{2}}\left[n(n-1)-\sum _{j=1}^k{n}_j\left(n_j-1\right)\right]c.$$

These inequalities are known as Chen inequalities. After that, Chen inequalities for special classes of submanifolds in various space forms were obtained by several researchers.

Consider $p\in M$, and let ${\pi }_1,{\pi }_2\subset T_{p}M$ be two mutually perpendicular planes spanned by the sets $\{{\mathsf{ϵ}}_1,{\mathsf{ϵ}}_2\}$ and $\{{\mathsf{ϵ}}_3,{\mathsf{ϵ}}_4\}$, respectively. Additionally, let $T_{p}M$ and $T_p^{\perp }M$ be equipped with orthonormal bases $\{{\mathsf{ϵ}}_1,\dots ,{\mathsf{ϵ}}_n\}$ and $\{{\mathsf{ϵ}}_{n+1},\dots ,{\mathsf{ϵ}}_m\}$, respectively.

Through explicit evaluation of $\varkappa \left({\pi }_1\right)$ and $\varkappa \left({\pi }_2\right)$, together with the application of Lemma (2), we obtain the inequality representing the $\mathsf{\delta }(2,2)$ Chen inequality for a SSBM within a GLSM.

Theorem 3. 

Consider $M^n$ to be a SSBM of an LDGLSM $\tilde{M}$. Then, the following holds:

$$\begin{array}{cc}\hfill (\top -\varkappa ({\pi }_1)& -\varkappa \left({\pi }_2\right)-\left({\top }_0-{\varkappa }_0\left({\pi }_1\right)-{\varkappa }_0\left({\pi }_2\right)\right))\hfill \\ \hfill & \ge {\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n+{2\parallel \mathrm{\nu }\parallel }^2-2n\parallel \mathrm{\nu }\parallel \right]-2(n-1)tr\varrho \hfill \\ \hfill & -[{\displaystyle \frac{\tilde{c}}{3}}\left\{1-tr\left(\mathrm{\nu }{|}_{{\pi }_1}\right)+tr\left(\mathrm{\nu }{|}_{{\pi }_2}\right)\right\}\hfill \\ \hfill & +2\theta \left({\pi }_1\right)+\theta \left({\pi }_2\right)-3tr\left(\varrho {|}_{{\pi }_1}\right)+tr\left(\varrho {|}_{{\pi }_2}\right)]\hfill \\ \hfill & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \mu \parallel }^2+{\parallel {\mu }^{\ast }\parallel }^2\right]-2\left[{\widehat{\top }}_0-{\widehat{\varkappa }}_0\left({\pi }_1\right)-{\widehat{\varkappa }}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

(31)

The equalities hold for all ${\rm Y}$ from $n+1$ to m iff

$$\begin{array}{c}{\beta }_{11}^{{\rm Y}}+{\beta }_{22}^{{\rm Y}}={\beta }_{33}^{{\rm Y}}+{\beta }_{44}^{{\rm Y}}={\beta }_{55}^{{\rm Y}}\cdots ={\beta }_{nn}^{{\rm Y}},\\ {\beta }_{11}^{\ast {\rm Y}}+{\beta }_{22}^{\ast {\rm Y}}={\beta }_{33}^{\ast {\rm Y}}+{\beta }_{44}^{\ast {\rm Y}}={\beta }_{55}^{\ast {\rm Y}}\cdots ={\beta }_{nn}^{\ast {\rm Y}},\end{array}$$

for all $1\le i<j\le n$, the components satisfy ${\beta }_{ij}^{{\rm Y}}={\beta }_{ij}^{\ast {\rm Y}}=0$, except when $(i,j)=(1,2),(2,1),(3,4),$ or $(4,3)$.

Proof. 

The sectional curvatures for ${\pi }_1$ and ${\pi }_2$ are

$$\begin{array}{cc}\hfill \varkappa \left({\pi }_1\right)=& {\displaystyle \frac{\tilde{c}}{3}}\left\{1-\mathrm{tr}\left(\mathrm{\nu }{|}_{{\pi }_1}\right)\right\}+2\theta \left({\pi }_1\right)-3\mathrm{tr}\left(\varrho {|}_{{\pi }_1}\right)-2{\widehat{\varkappa }}_0\left({\pi }_1\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left[{\beta }_{11}^{{\rm Y}}{\beta }_{22}^{{\rm Y}}-{\left({\beta }_{12}^{{\rm Y}}\right)}^2+{\beta }_{11}^{\ast {\rm Y}}{\beta }_{22}^{\ast {\rm Y}}-{\left({\beta }_{12}^{\ast {\rm Y}}\right)}^2\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill \varkappa \left({\pi }_2\right)=& {\displaystyle \frac{\tilde{c}}{3}}\left\{1-\mathrm{tr}\left(\mathrm{\nu }{|}_{{\pi }_2}\right)\right\}+2\theta \left({\pi }_2\right)-3\mathrm{tr}\left(\varrho {|}_{{\pi }_2}\right)-2{\widehat{\varkappa }}_0\left({\pi }_2\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{{\rm Y}=n+1}^m\left[{\beta }_{33}^{{\rm Y}}{\beta }_{44}^{{\rm Y}}-{\left({\beta }_{34}^{{\rm Y}}\right)}^2+{\beta }_{33}^{\ast {\rm Y}}{\beta }_{44}^{\ast {\rm Y}}-{\left({\beta }_{34}^{\ast {\rm Y}}\right)}^2\right].\hfill \end{array}$$

Subtracting $\varkappa \left({\pi }_1\right)+\varkappa \left({\pi }_2\right)$ from Equation (24) and simplifying using Lemma (2), we obtain

$$\begin{array}{cc}\hfill (\top -\varkappa \left({\pi }_1\right)-& \varkappa \left({\pi }_2\right))-({\top }_0-{\varkappa }_0\left({\pi }_1\right)-{\varkappa }_0\left({\pi }_2\right))\ge \hfill \\ & {\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n+{2\parallel \mathrm{\nu }\parallel }^2-2n\parallel \mathrm{\nu }\parallel \right]-2(n-1)\mathrm{tr}\left(\varrho \right)\hfill \\ & -\left[{\displaystyle \frac{\tilde{c}}{3}}\left\{1-\mathrm{tr}\left(\mathrm{\nu }{|}_{{\pi }_1}\right)+\mathrm{tr}\left(\mathrm{\nu }{|}_{{\pi }_2}\right)\right\}+2\theta \left({\pi }_1\right)+2\theta \left({\pi }_2\right)\right.\hfill \\ & \left.-3\mathrm{tr}\left(\varrho {|}_{{\pi }_1}\right)-3\mathrm{tr}\left(\varrho {|}_{{\pi }_2}\right)\right]\hfill \\ & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \mu \parallel }^2+{\parallel {\mu }^{\ast }\parallel }^2\right]-2\left[{\widehat{\top }}_0-{\widehat{\varkappa }}_0\left({\pi }_1\right)-{\widehat{\varkappa }}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

This completes the proof. □

The next results are immediate consequences of the preceding theorem.

Corollary 3. 

Suppose $M^n$ is a totally real SSBM of an LDGLSM $\tilde{M}$. In that case, we obtain the following:

$$\begin{array}{cc}\hfill (\top -\varkappa ({\pi }_1)& -\varkappa \left({\pi }_2\right)-\left({\top }_0-{\varkappa }_0\left({\pi }_1\right)-{\varkappa }_0\left({\pi }_2\right)\right))\hfill \\ \hfill & \ge {\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n\right]-2(n-1)tr\varrho \hfill \\ \hfill & -\left[2\theta \left({\pi }_1\right)+\theta \left({\pi }_2\right)-3tr\left(\varrho {|}_{{\pi }_1}\right)+tr\left(\varrho {|}_{{\pi }_2}\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \mu \parallel }^2+{\parallel {\mu }^{\ast }\parallel }^2\right]-2\left[{\widehat{\top }}_0-{\widehat{\varkappa }}_0\left({\pi }_1\right)-{\widehat{\varkappa }}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

(32)

Proof. 

The proof follows directly from Theorem (3), since Equation (32) is an immediate consequence of the Equation (31) upon setting $\mathrm{\nu }$ = 0. □

Remark 6. 

Equality condition for Theorem (3) and Corollary (3) holds if and only if:

1. 

${\beta }^{{\rm Y}}$ and ${\beta }^{\ast {\rm Y}}$ satisfy:

$${\beta }_{11}^{{\rm Y}}+{\beta }_{22}^{{\rm Y}}={\beta }_{33}^{{\rm Y}}+{\beta }_{44}^{{\rm Y}}={\beta }_{55}^{{\rm Y}}=\cdots ={\beta }_{nn}^{{\rm Y}},$$

$${\beta }_{11}^{\ast {\rm Y}}+{\beta }_{22}^{\ast {\rm Y}}={\beta }_{33}^{\ast {\rm Y}}+{\beta }_{44}^{\ast {\rm Y}}={\beta }_{55}^{\ast {\rm Y}}=\cdots ={\beta }_{nn}^{\ast {\rm Y}}.$$

2. 

The mixed components vanish except for $(1,2),(2,1),(3,4),(4,3)$: ∀$1\le i<j\le n$, the components satisfy ${\beta }_{ij}^{{\rm Y}}={\beta }_{ij}^{\ast {\rm Y}}=0$, except when $(i,j)=(1,2),(2,1),(3,4),$ or $(4,3)$.

Corollary 4. 

Consider a totally real SSBM $M^n$ of an LDGLSM $\tilde{M}$. If ∃ a point $p\in M$ and orthogonal planes ${\pi }_1,{\pi }_2\subset T_{p}M$ satisfying

$$\begin{array}{cc}\hfill \top -\varkappa \left({\pi }_1\right)-\varkappa \left({\pi }_2\right)<& {\top }_0-{\varkappa }_0\left({\pi }_1\right)-{\varkappa }_0\left({\pi }_2\right)+{\displaystyle \frac{\tilde{c}}{3}}\left[n^2-3n\right]-2(n-1)tr\varrho \hfill \\ \hfill & -\left[2\theta \left({\pi }_1\right)+\theta \left({\pi }_2\right)-3tr\left(\varrho {|}_{{\pi }_1}\right)+tr\left(\varrho {|}_{{\pi }_2}\right)\right]\hfill \\ \hfill & -2\left[{\widehat{\top }}_0-{\widehat{\varkappa }}_0\left({\pi }_1\right)-{\widehat{\varkappa }}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

(33)

Consequently, M fails to be minimal, i.e., $\mu \ne 0$ or ${\mu }^{\ast }\ne 0$.

Proof. 

The proof of Corollary (4) follows analogously to that of Corollary (2) and is therefore omitted. □

5. Conclusions

In this study, we have systematically developed curvature inequalities for LDGLSM endowed with SSMC. By leveraging the interplay between the golden-like structure and the statistical framework, we derived new forms of Chen-type inequalities involving intrinsic and extrinsic invariants. These inequalities generalize the classical results known for Riemannian submanifolds and statistical manifolds, demonstrating how the presence of a golden-like tensor field modifies the curvature relations.

Furthermore, we established sufficient conditions under which the equalities in these inequalities are attained, thereby providing characterizations of special submanifolds in LDGLSM. The results presented herein contribute to the broader understanding of geometric inequalities in information geometry and related fields.

Applications and future directions: The derived curvature inequalities have potential applications in various fields:

• Information geometry: The inequalities can be employed to analyze the curvature of statistical models, particularly manifolds of probability distributions equipped with the Fisher information metric.
• Machine learning and optimization: Curvature bounds play a role in understanding the geometry of parameter spaces in deep learning and in analyzing convergence properties of optimization algorithms.
• Geometric data analysis: The golden-like tensor structure may be linked to divergence measures, offering new tools for dimensionality reduction and shape analysis.

Future research may focus on applying these inequalities to practical problems in information theory, statistical inference, and data-driven geometric modeling, as well as exploring extensions to metallic or complex statistical manifolds with semi-symmetric connections and quarter-symmetric connections.