Inequality Constraints on Statistical Submanifolds of Norden-Golden-like Statistical Manifold

Abstract

This paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants of submanifolds. The methodology involves deriving generalized Chen-type and $\delta (2,2)$ curvature inequalities using curvature tensor analysis and dual affine connections. A concrete example is provided to verify the theoretical framework. The novelty of this work lies in extending classical curvature inequalities to a newly introduced statistical structure, thereby opening new perspectives in the study of geometric inequalities in information geometry and related mathematical physics contexts.

1. Introduction

In many ancient societies, the golden mean, a family member of metallic means, served as the foundation for proportion when creating music, sculptures, paintings, temples, and palaces. Some of the metallic means family members, especially the silver and golden means, are thought to have an innate connection to the theoretical justification of quantum physics behavior. Physicists have examined the behavior of nonlinear dynamic systems as they transition from periodic to semi-periodic states, utilizing specific values related to the golden mean [1,2].

Using contemporary geometry, information geometry is a method for investigating the realm of information. Up until now, the majority of the methodologies used to study the concept of information have been algebraic, logical, analytical, and probabilistic. Geometry should give information science strong tools since it studies the relationships between variables like distance and curvature. Explorations of invariant geometric constructs in the context of statistical inference have given rise to information geometry. A Riemannian metric is exhibited on a manifold of probability distributions, accompanied by dually coupled affine connections. In addition to statistical inference, these structures are crucial in the more general domains of information science, encompassing domains like optimization, machine learning, neuroscience, and signal processing [3].

Amari used information geometry to develop the idea of statistical manifolds [3]. Dual connections, which are analogous to conjugate connections in affine geometry, are present in a statistical manifold [4]. It is difficult to define sectional curvature (SC) using classical Riemannian geometry because conjugate connections are not metric. Thus, B. Opozda presented a technique for defining the SC of a statistical manifold in [5]. Understanding the connections between extrinsic and intrinsic invariants is a fundamental aspect of exploring the differential geometric properties of a submanifold. Over the past few decades, significant progress has been made in uncovering many of these relationships. According to Euler’s inequality, the Gaussian curvature $\mathrm{K}$ of a surface is less than or equal to the square of the mean curvature ${\left|\varpi \right|}^2$, where both are intrinsic geometric properties. For a surface $\mathrm{M}$ in Euclidean 3-space, equality holds at umbilical points, where $\mathrm{M}$ locally resembles a plane or sphere. B.-Y. Chen extended this idea by formulating a similar inequality for submanifolds in real-space forms and later introduced the Chen–Ricci inequality, which precisely relates the Ricci curvature to the squared mean curvature of Riemannian submanifolds. For further details on Chen inequalities, see references [6,7,8,9,10,11,12,13,14].

Statistical manifolds have been the subject of much research lately. Takano defined a new class of statistical manifolds in [15] that includes almost-contact and almost-complex structures. G.E. and A.D. Vîlcu [16], in 2015, explored a statistical manifold within the quaternionic framework and outlined several open problems for future research. Responding to one of these challenges, M. Aquib [17] investigated the curvature behavior of submanifolds and derived several inequalities for totally real statistical submanifolds (TRSS) in quaternionic Kähler-like statistical space forms. B.-Y. Chen et al. [18] later formulated a Chen-type inequality for statistical submanifolds in a Hessian manifold with constant Hessian curvature. This was followed by the establishment of similar inequalities for submanifolds of Kähler-like statistical manifolds by H. Aytimur et al. [19]. In 2020, C. W. Lee and J. W. Lee introduced an inequality involving Casorati curvatures on a Sasakian statistical manifold. For additional insights into statistical submanifolds, refer to [15,17,20,21,22,23,24,25]. To date, statistical structures related to Norden-golden (NG) geometry remain uninvestigated.

Among the various classes of manifolds, the most prominent in differential geometry are characterized by constant curvature. In practice, these manifolds can serve as prototypes for a wide range of physical events. Furthermore, one can derive precise results regarding both the manifolds themselves and their submanifolds thanks to the unique shape of these manifolds concerning the curvature tensor fields. Complex manifolds, along with their specialized subclass, known as Kähler manifolds, represent the most prominent examples of manifolds equipped with an endomorphism that satisfies a specific condition. The SC of a Kähler manifold is zero if its curvature is a real constant (Proposition 4.3 in [26]). This naturally gives rise to the concept of holomorphic SC defined on holomorphic planes within a Kähler manifold. Complex space forms, which are Kähler manifolds with constant holomorphic SC, have been well-studied, and the expressions for their curvature tensor fields are explicitly known [27]. Similarly, if the holomorphic SC is constant, the curvature tensor fields associated with the manifold exhibit a specific form for almost-Kähler manifolds and locally conformal Kähler manifolds.

Crasmareanu and Hretcanu defined a golden manifold in [28], and other authors have examined their geometries. Şahin et. al. demonstrated in [29] that under specific circumstances, the SC in a locally decomposable golden manifold reduces to zero when the real and holomorphic-like SC is constant. The authors introduced a new concept of SC and also derived a representation for the curvature tensor fields when SC is constant.

In recent years, various statistical geometric structures have been explored, including Sasakian statistical manifolds [20,22,30], cosymplectic statistical manifolds [31], nearly-Kähler and nearly-Sasakian statistical manifolds [32], quaternion Kähler-like statistical manifolds [17,21], and Hessian statistical manifolds with constant Hessian curvature [18,24]. These studies primarily focus on curvature inequalities, submanifold geometry, and curvature pinching conditions. Notably, Chen-type inequalities have been extensively examined in Kähler-like and Hessian settings [18,19], reflecting the rich interplay between geometry and information structure. Taking motivation from [12,31,33] and given the geometric richness of NGLSM, their structure may prove useful in modeling dual connections and curvature phenomena in information geometry, quantum physics, and nonlinear dynamical systems, where golden ratios and affine structures naturally arise. Inspired by these studies, we introduce the concept of NGLSM and establish Chen-type inequalities on statistical submanifolds of NGLSM.

This paper is organized as follows. In Section 2, we revisit key background concepts, including the definitions of Norden golden manifolds and their statistical structure, and introduce the notion of NGLSM, along with a concrete example. Section 3 is devoted to deriving novel Chen-type inequalities on statistical submanifolds of NGLSM, focusing on sectional curvature estimates and their intrinsic–extrinsic relationships. In Section 4, we establish a $\delta (2,2)$-Chen inequality tailored to the NGLSM framework. These results extend classical inequalities from Kähler-like and Hessian statistical geometries to a new structural setting, opening avenues for further geometric investigations within information geometry.

2. Preliminaries

2.1. Norden Golden Manifold (NGM)

According to [28,34], we revisit the idea of an almost NGM. Given a manifold $\tilde{\mathrm{M}}$ and an endomorphism $\rho :T\tilde{\mathrm{M}}\to T\tilde{\mathrm{M}}$ defined as

$${\wp }^2=\wp -{\displaystyle \frac{3}{2}}I,$$

in which I is the identity mapping, in this setting, $(\tilde{\mathrm{M}},\wp )$ is termed an almost-complex golden manifold. Given a semi-Riemannian metric g on $\tilde{\mathrm{M}}$, for either vector field (${\cup }_1$ or ${\cup }_2$) on $\tilde{\mathrm{M}}$, we have

$$g\left(\wp {\cup }_1,{\cup }_2\right)=g\left({\cup }_1,\wp {\cup }_2\right).$$

(1)

Thus, $(\tilde{\mathrm{M}},\wp ,g)$ is termed an almost NGM. It can be observed that (1) is similar to

$$g\left(\wp {\cup }_1,\wp {\cup }_2\right)=g\left(\wp {\cup }_1,{\cup }_2\right)-{\displaystyle \frac{3}{2}}g\left({\cup }_1,{\cup }_2\right).$$

(2)

Moreover, the triple $(\tilde{\mathrm{M}},\wp ,g)$ is known as LDANGSRM if ℘ remains invariant under covariant differentiation along the vector field ${\cup }_1$ on $\tilde{\mathrm{M}}$, i.e.,

$${⊳}_{{\cup }_1}\wp =0.$$

The following result is also true:

Lemma 1

([33]). Let $(\tilde{\mathrm{M}},\wp ,g)$ be an LDANGSRM and $\mathrm{R}$ be its curvature tensor field. Then, the following is true:

$$\mathrm{R}\left(\wp {\cup }_1,\wp {\cup }_2\right)=\mathrm{R}\left({\cup }_1,\wp {\cup }_2\right)-{\displaystyle \frac{3}{2}}\mathrm{R}\left({\cup }_1,{\cup }_2\right).$$

(3)

$\tilde{\mathrm{M}}$ is said to be a space form (denoted as NGSF(c)), if its SC is constant c, independent of the plane $\mathbb{P}$, at every $x\in \tilde{\mathrm{M}}$.

Lemma 2

([33]). Let $\tilde{\mathrm{M}}\left(c\right)$ be an NGSF(c). Then, for any vector field (${\cup }_1$, ${\cup }_2$, or ${\cup }_3$) on $\tilde{M}$, we have

$$\begin{array}{cc}\hfill \mathrm{R}\left({\cup }_1,{\cup }_2\right){\cup }_3=& {\displaystyle \frac{c}{9}}\{2g\left({\cup }_2,\wp {\cup }_3\right){\cup }_1+2g\left({\cup }_2,{\cup }_3\right)\wp {\cup }_1+4g\left({\cup }_1,\wp {\cup }_3\right)\wp {\cup }_2\hfill \\ \hfill & -3g\left({\cup }_1,{\cup }_3\right){\cup }_2+3g\left({\cup }_2,{\cup }_3\right){\cup }_1-4g\left({\cup }_2,\wp {\cup }_3\right)\wp {\cup }_1\hfill \\ \hfill & -2g\left({\cup }_1,\wp {\cup }_3\right){\cup }_2-2g\left({\cup }_1,{\cup }_3\right)\wp {\cup }_2\}.\hfill \end{array}$$

(4)

Definition 1

([33]). Suppose that $\tilde{\mathrm{M}}\left(c\right)$ is an $\mathrm{NGSF}\left(c\right)$. The Ricci tensor field can be found through

$$r=S\left({\cup }_1,{\cup }_2\right)=\sum _{\mathsf{ı}=1}^{m}g\left(\mathrm{R}\left({\in }_{\mathsf{ı}},{\cup }_1\right){\cup }_2,{\in }_{\mathsf{ı}}\right),$$

(5)

for any vector field (${\cup }_1$ or ${\cup }_2$) on $\tilde{\mathrm{M}}$.

Using (5), (4), and (2), the scalar curvature is expressed as

$$S\left({\cup }_1,{\cup }_2\right)={\displaystyle \frac{c}{9}}\left\{4m\parallel \wp \parallel -{4\parallel \wp \parallel }^2+3m^2-9m\right\},$$

(6)

where $\left\{{\in }_1,{\in }_2,\dots ,{\in }_m\right\}$ is an orthonormal basis on $\tilde{\mathrm{M}}$, and ${\displaystyle \parallel \wp \parallel =\sum _{i=1}^{m}g\left(\wp {\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}}\right)}$.

2.2. Statistical Manifold and Statistical Submanifold

Now consider $({\tilde{\mathrm{M}}}^m,g)$, an m-dimensional semi-Riemannian manifold that admits two torsion-free affine connections, $\tilde{\nabla }$ and ${\tilde{\nabla }}^{*}$. Let $M^n$ be an n-dimensional Riemannian submanifold immersed in $({\tilde{\mathrm{M}}}^m,g)$, such that the following condition holds:

$${\cup }_3\tilde{g}({\cup }_1,{\cup }_2)=\tilde{g}\left({\widetilde{⊳}}_{{\cup }_3}{\cup }_1,{\cup }_2\right)+\tilde{g}\left({\cup }_1,{\widetilde{⊳}}_{{\cup }_3}^{*}{\cup }_2\right),$$

for any vector field (${\cup }_1$, ${\cup }_2$, or ${\cup }_3$) on ${\tilde{\mathrm{M}}}^m$. The connections $\widetilde{⊳}$ and ${\widetilde{⊳}}^{*}$ are referred to as conjugate connections [3,4,35], and they satisfy the duality relation

$${\left({\widetilde{⊳}}^{*}\right)}^{*}=\widetilde{⊳}.$$

The pair $(\widetilde{⊳},g)$ is referred to as a statistical structure.

The conjugate of a torsion-free affine connection $\widetilde{⊳}$ exists and it is explicitly defined as follows:

$$\widetilde{⊳}+{\widetilde{⊳}}^{*}=2{\widetilde{⊳}}^0,$$

(7)

Here, ${\widetilde{⊳}}^0$ represents the L.C.C. on ${\tilde{\mathrm{M}}}^m$. The curvature tensor fields equipped with conjugate connections $\widetilde{⊳}$ and ${\widetilde{⊳}}^{*}$ are represented by $\tilde{R}$ and ${\tilde{R}}^{*}$, respectively.

The Gauss formulas for the conjugate connections are given by [36]

$$\begin{array}{cc}\hfill & {\widetilde{⊳}}_{{\cup }_1}{\cup }_2={⊳}_{{\cup }_1}{\cup }_2+\perp ({\cup }_1,{\cup }_2),\hfill \\ \hfill & {\widetilde{⊳}}_{{\cup }_1}^{*}{\cup }_2={⊳}_{{\cup }_1}^{*}{\cup }_2+{\perp }^{*}({\cup }_1,{\cup }_2),\hfill \end{array}$$

(8)

for any ${\cup }_1,{\cup }_2\in \Gamma \left(T{\tilde{\mathrm{M}}}^m\right)$. The maps $\perp ,{\perp }^{*}:\Gamma \left(T{\mathrm{M}}^n\right)\times \Gamma \left(T{\mathrm{M}}^n\right)\to \Gamma \left(T^{\perp }{\mathrm{M}}^n\right)$ are bilinear and symmetric, and are referred to as the embedding curvature tensors of ${\mathrm{M}}^n$ in ${\tilde{\mathrm{M}}}^m$, corresponding to the connections $\widetilde{⊳}$ and ${\widetilde{⊳}}^{*}$, respectively. Moreover, the pairs $(⊳,g)$ and $({⊳}^{*},g)$ constitute dual statistical connections on ${\mathrm{M}}^n$.

Owing to the bilinearity of ⊥ and ${\perp }^{*}$, they define two linear operators, $A_{\zeta }$ and $A_{\zeta }^{*}$ on $T{\mathrm{M}}^n$, as

$$\begin{array}{cc}\hfill & g\left({\mathrm{A}}_{\zeta }{\cup }_1,{\cup }_2\right)=g(\perp ({\cup }_1,{\cup }_2),\zeta ).\hfill \\ \hfill & g\left({\mathrm{A}}_{\zeta }^{*}{\cup }_1,{\cup }_2\right)=g\left({\perp }^{*}({\cup }_1,{\cup }_2),\zeta \right),\hfill \end{array}$$

(9)

For any $\zeta$ in $T^{\perp }{\mathrm{M}}^n$ and ${\cup }_1$ and ${\cup }_2$ on $T{\mathrm{M}}^n$, the following holds. Moreover, the corresponding Weingarten formulae are given by [36]

$$\begin{array}{cc}\hfill {\widetilde{⊳}}_{{\cup }_1}\zeta & =-{\mathrm{A}}_{\zeta }^{*}{\cup }_1+{⊳}_{{\cup }_1}^{\perp }\zeta ,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\widetilde{⊳}}_{{\cup }_1}^{*}\zeta & =-{\mathrm{A}}_{\zeta }{\cup }_1+{⊳}_{{\cup }_1}^{*\perp }\zeta ,\hfill \end{array}$$

(11)

The connections ${⊳}^{\perp }$ and ${⊳}^{*\perp }$, given by (10) and (11), form a Riemannian conjugate connection concerning the inherited metric on $\Gamma \left(T^{\perp }{\mathrm{M}}^n\right)$.

Let $\{{\in }_1,\dots ,{\in }_n\}\subset T\mathrm{M}$ and $\{{\in }_{n+1},\dots ,{\in }_m\}\subset T^{\perp }\mathrm{M}$ form orthonormal frames for the tangent and normal distributions of $\mathrm{M}$, respectively. The mean curvature vector fields are then introduced as

$$\begin{array}{c}\varpi ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{\mathsf{ı}=1}^n\perp \left({\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}}\right)={\displaystyle \frac{1}{n}}\sum _{⋉=n+1}^m\left(\sum _{\mathsf{ı}=1}^n{\perp }_{\mathsf{ı}\mathsf{ı}}^{⋉}\right){\in }_{⋉},{\perp }_{\mathsf{ı}\mathsf{ȷ}}^{⋉}=g\left(\perp \left({\in }_{\mathsf{ı}},{\in }_{\mathsf{ȷ}}\right),{\in }_{⋉}\right),}\\ {\varpi }^{*}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{\mathsf{ı}=1}^n{\perp }^{*}\left({\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}}\right)={\displaystyle \frac{1}{n}}\sum _{⋉=n+1}^m\left(\sum _{\mathsf{ı}=1}^n{\perp }_{\mathsf{ı}\mathsf{ı}}^{*⋉}\right){\in }_{⋉},{\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*⋉}=g\left({\perp }^{*}\left({\in }_{\mathsf{ı}},{\in }_{\mathsf{ȷ}}\right),{\in }_{⋉}\right).}\end{array}$$

Vos [36] formulated the Gauss, Codazzi, and Ricci equations for statistical submanifolds in conjugate connection, covering the index ranges $\mathsf{ı},\mathsf{ȷ}\in \{1,\dots ,n\}$ and $⋉\in \{n+1,\dots ,m\}$.

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

Lemma 3

([18]). Let $t\in \mathbb{N}$ with $t\ge 3$, and consider real numbers $u_1,u_2,\dots ,u_t$. Then the following statement is valid:

$$\sum _{1\le \mathsf{ı}<\mathsf{ȷ}\le t}^t{u}_{\mathsf{ı}}{u}_{\mathsf{ȷ}}-u_1{u}_2\le {\displaystyle \frac{t-2}{2(t-1)}}{\left(\sum _{\mathsf{ı}=1}^t{u}_{\mathsf{ı}}\right)}^2.$$

Moreover, the equality holds iff $u_1+u_2=u_3=\dots =u_t$.

Lemma 4

([12]). Let $t\in \mathbb{N}$ with $t\ge 3$, and consider real numbers $u_1,u_2,\dots ,u_t$. Then the following statement is valid:

$$\sum _{1\le \mathsf{ı}<\mathsf{ȷ}\le t}^t{u}_{\mathsf{ı}}{u}_{\mathsf{ȷ}}-u_1{u}_2-u_3{u}_4\le {\displaystyle \frac{t-3}{2(t-2)}}{\left(\sum _{\mathsf{ı}=1}^t{u}_{\mathsf{ı}}\right)}^2.$$

Moreover, the equality holds iff $u_1+u_2=u_3+u_4=u_5=\dots =u_t$.

2.3. Norden Golden-like Statistical Manifold (NGLSM)

Definition 2.

Let $(\tilde{\mathrm{M}},g,\wp )$ denote an LDANGSRM endowed with a $(1,1)$-tensor field ${\wp }^{*}$ such that the subsequent condition holds:

$$g(\wp {\cup }_1,{\cup }_2)=g\left({\cup }_1,{\wp }^{*}{\cup }_2\right),$$

(12)

for any vector field (${\cup }_1$ or ${\cup }_2$). Using (12), we can readily deduce

$${\wp }^{*2}={\wp }^{*}-{\displaystyle \frac{3}{2}}I,$$

and

$$g\left(\wp {\cup }_1,{\wp }^{*}{\cup }_2\right)=g\left(\wp {\cup }_1,{\cup }_2\right)-{\displaystyle \frac{3}{2}}g\left({\cup }_1,{\cup }_2\right).$$

(13)

The structure $(\tilde{\mathrm{M}},g,\wp )$ defines an NGLSM.

2.4. Example

Example 1.

Let ${\mathbb{R}}_n^{2r+m}$ be an affine space with the standard system of coordinates

$$(x_1,\dots ,x_r,y_1,\dots ,y_r,z_1,\dots ,z_m),$$

where $r,m$ are positive integers.

Define the semi-Riemannian metric g as

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

where ${\delta }_{ij}$ is the Kronecker delta and $\kappa ={\displaystyle \frac{1+\sqrt{5}}{2}}$ is the golden ratio.

Define the $(1,1)$-tensor field ℘ as

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

The dual tensor field ${\wp }^{*}$ is given by

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

We now check that $({\mathbb{R}}_r^{2r+m},g,\wp )$ satisfies the conditions of an NGLSM.

• Verification of the Conditions:

Let ${\cup }_1=(x_1,\dots ,x_r,y_1,\dots ,y_r,z_1,\dots ,z_m)$ and ${\cup }_2=(u_1,\dots ,u_r,v_1,\dots ,v_r,w_1,\dots ,w_m)$ be the vector fields. We must show that

$$g(\wp {\cup }_1,{\cup }_2)=g({\cup }_1,{\wp }^{*}{\cup }_2).$$

First, we compute $\wp \left({\cup }_1\right)$ and ${\wp }^{*}\left({\cup }_2\right)$:

$$\begin{array}{cc}\hfill \wp \left({\cup }_1\right)& =\left({\displaystyle \frac{1}{2}}(x_i+(2\kappa -1)y_i),{\displaystyle \frac{1}{2}}(y_i+(2\kappa -1)x_i),{\displaystyle \frac{1-\kappa }{2}}{z}_j\right),\hfill \\ \hfill {\wp }^{*}\left({\cup }_2\right)& =\left({\displaystyle \frac{1}{2}}(u_i+(1-2\kappa )v_i),{\displaystyle \frac{1}{2}}(v_i+(1-2\kappa )u_i),{\displaystyle \frac{1-\kappa }{2}}{w}_j\right).\hfill \end{array}$$

We now have

$$\begin{array}{cc}\hfill g(\wp {\cup }_1,{\cup }_2)& =-\sum _{\mathsf{ı}=1}^r\left({\displaystyle \frac{1}{2}}(x_i+(2\kappa -1)y_i)\right)u_i+\sum _{\mathsf{ı}=1}^r\left({\displaystyle \frac{1}{2}}(y_i+(2\kappa -1)x_i)\right)v_i\hfill \\ \hfill & +(1-\kappa )\sum _{\mathsf{ȷ}=1}^m\left({\displaystyle \frac{1-\kappa }{2}}{z}_j\right)w_j\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\sum _{\mathsf{ı}=1}^r\left(-x_i{u}_i-(2\kappa -1)y_i{u}_i+y_i{v}_i+(2\kappa -1)x_i{v}_i\right)+{\displaystyle \frac{{(1-\kappa )}^2}{2}}\sum _{\mathsf{ȷ}=1}^m{z}_j{w}_j.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill g({\cup }_1,{\wp }^{*}{\cup }_2)& =-\sum _{\mathsf{ı}=1}^r{x}_i\left({\displaystyle \frac{1}{2}}(u_i+(1-2\kappa )v_i)\right)+\sum _{\mathsf{ı}=1}^r{y}_i\left({\displaystyle \frac{1}{2}}(vi+(1-2\kappa )u_i)\right)\hfill \\ \hfill & +(1-\kappa )\sum _{\mathsf{ȷ}=1}^m{z}_j\left({\displaystyle \frac{1-\kappa }{2}}{w}_j\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\sum _{\mathsf{ı}=1}^r\left(-x_i{u}_i-(1-2\kappa )x_i{v}_i+y_i{v}_i+(1-2\kappa )y_i{u}_i\right)+{\displaystyle \frac{{(1-\kappa )}^2}{2}}\sum _{\mathsf{ȷ}=1}^m{z}_j{w}_j.\hfill \end{array}$$

Thus,

$$g(\wp {\cup }_1,{\cup }_2)=g({\cup }_1,{\wp }^{*}{\cup }_2).$$

We check that

$${\wp }^{*2}={\wp }^{*}-{\displaystyle \frac{3}{2}}I.$$

With the block matrix representation:

$${\wp }^{*}={\displaystyle \frac{1}{2}}\left(\begin{array}{ccc}I& (1-2\kappa )I& 0\\ (1-2\kappa )I& I& 0\\ 0& 0& (1-\kappa )I\end{array}\right),⋉=1-2\kappa .$$

Focusing on the nontrivial block $A={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}I& ⋉I\\ ⋉I& I\end{array}\right)$,

$$\begin{array}{cc}\hfill A^2& ={\displaystyle \frac{1}{4}}\left(\begin{array}{cc}I+{⋉}^{2}I& 2⋉I\\ 2⋉I& I+{⋉}^{2}I\end{array}\right)\hfill \\ \hfill & ={\displaystyle \frac{1+{⋉}^2}{4}}{I}_{2r}+{\displaystyle \frac{⋉}{2}}\left(\begin{array}{cc}0& I\\ I& 0\end{array}\right).\hfill \end{array}$$

For $\kappa ={\displaystyle \frac{1+\sqrt{5}}{2}}$, we have:

$$\begin{array}{cc}\hfill {\kappa }^2& =\kappa +1.\hfill \\ \hfill {⋉}^2& ={(1-2\kappa )}^2=4{\kappa }^2-4\kappa +1=5.\hfill \\ \hfill \Rightarrow A^2& ={\displaystyle \frac{6}{4}}{I}_{2r}={\displaystyle \frac{3}{2}}{I}_{2r}.\hfill \end{array}$$

Thus, the full tensor satisfies

$${\wp }^{*2}={\wp }^{*}-\frac{3}{2}I.$$

Hence, (12) is satisfied and $({\mathbb{R}}_r^{2r+m},g,\wp )$ is an NGLSM.

3. Main Results

In this section, we derive fundamental curvature inequalities for statistical submanifolds of NGLSM. These results generalize classical Chen-type inequality within the new geometric context of NGLSM.

Theorem 1.

Consider $\tilde{\mathrm{M}}$ to be an NGLSM and $\mathrm{M}$ to be its statistical submanifold. Consequently, the following holds:

$$\begin{array}{cc}\hfill (⊺-\mathbb{k}\left(\pi \right))-\left({⊺}_0-{\mathbb{k}}_0\left(\pi \right)\right)\ge & {\displaystyle \frac{c}{9}}\left\{3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right\}\hfill \\ \hfill & -{\displaystyle \frac{c}{9}}\left\{6tr\left({\wp }_{\mid \pi }\right)-4{\rm Y}\left(\pi \right)-9\right\}\hfill \\ \hfill & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \varpi \parallel }^2+{∥{\varpi }^{*}∥}^2\right]+2{\widehat{\mathbb{k}}}_0\left(\pi \right)-2{\widehat{⊺}}_0.\hfill \end{array}$$

(14)

Furthermore, the equalities hold for all $\mathsf{\gamma }\in \{n+1,\dots ,m\}$ iff:

$$\begin{array}{c}{\perp }_{11}^{\mathsf{\gamma }}+{\perp }_{22}^{\mathsf{\gamma }}={\perp }_{33}^{\mathsf{\gamma }}=\dots ={\perp }_{nn}^{\mathsf{\gamma }},\\ {\perp }_{11}^{*\mathsf{\gamma }}+{\perp }_{22}^{*\mathsf{\gamma }}={\perp }_{33}^{*\mathsf{\gamma }}=\dots ={\perp }_{nn}^{*\mathsf{\gamma }},\\ \mathsf{ı}\ne \mathsf{ȷ},{\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}={\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}=0,(\mathsf{ı},\mathsf{ȷ})\ne (1,2)and(2,1),\mathsf{ı},\mathsf{ȷ}\in \{1,\dots ,n\},\mathsf{ı}<\mathsf{ȷ}.\end{array}$$

Proof. 

Consider orthonormal frames $\{{\in }_1,\dots ,{\in }_n\}\subset T\mathrm{M}$ and $\{{\in }_{n+1},\dots ,{\in }_m\}\subset T^{\perp }\mathrm{M}$. For $p\in \mathrm{M}$ and $\pi \subset T_p\mathrm{M}$, the quantity ${\rm Y}\left(\pi \right)$ is defined as:

$${\rm Y}\left(\pi \right)=g(\wp {\cup }_1,{\cup }_1)·g(\wp {\cup }_2,{\cup }_2),$$

where ${\cup }_1$ and ${\cup }_2$ are vectors spanning the plane section $\pi$.

The scalar curvature associated with the sectional K-curvature is

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

Thus, we have

$$\begin{array}{cc}\hfill \mathrm{R}({\in }_{\mathsf{ı}},{\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}})& ={\displaystyle \frac{c}{9}}\{-3g({\in }_{\mathsf{ı}},{\in }_{\mathsf{ȷ}})g({\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}})+3g({\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ȷ}})g({\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}})\hfill \\ \hfill & -4g({\in }_{\mathsf{ȷ}},\wp {\in }_{\mathsf{ȷ}})g(\wp {\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}})+2g({\in }_{\mathsf{ȷ}},\wp {\in }_{\mathsf{ȷ}})g({\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}})\hfill \\ \hfill & +2g({\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ȷ}})g(\wp {\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}})+4g({\in }_{\mathsf{ı}},\wp {\in }_{\mathsf{ȷ}})g(\wp {\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}})\hfill \\ \hfill & -2g({\in }_{\mathsf{ı}},\wp {\in }_{\mathsf{ȷ}})g({\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}})-2g({\in }_{\mathsf{ı}},{\in }_{\mathsf{ȷ}})g(\wp {\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}})\}\hfill \\ \hfill & +g\left({\perp }^{*}({\in }_{\mathsf{ı}},{\in }_{\mathsf{ı}}),\perp \left({\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ȷ}}\right)\right)-g\left(\perp \left({\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}}\right),{\perp }^{*}\left({\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}}\right)\right).\hfill \end{array}$$

To obtain the curvature tensor $R^{*}({\in }_{\mathsf{ı}},{\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ȷ}},{\in }_{\mathsf{ı}})$ corresponding to the dual connection, it suffices to replace ℘ with ${\wp }^{*}$ in the above equation. By applying (12) and (8), and a straightforward calculation, we obtain

$$\begin{array}{cc}\hfill ⊺=& {\displaystyle \frac{c}{9}}\left\{3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right\}-{⊺}_0\hfill \\ & +{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\sum _{\begin{array}{c}1\le \mathsf{ı}\le n\\ \mathsf{ı}<\mathsf{ȷ}\le n\end{array}}\left[{\perp }_{\mathsf{ı}\mathsf{ı}}^{*\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{\mathsf{\gamma }}+{\perp }_{\mathsf{ı}\mathsf{ı}}^{\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{*\mathsf{\gamma }}-2{\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}{\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}\right].\hfill \end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill ⊺=& {\displaystyle \frac{c}{9}}\left\{3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right\}-{⊺}_0+2\sum _{\mathsf{\gamma }=n+1}^m\sum _{\begin{array}{c}1\le \mathsf{ı}\le n\\ \mathsf{ı}<\mathsf{ȷ}\le n\end{array}}\left[{\perp }_{\mathsf{ı}\mathsf{ı}}^{0\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{0\mathsf{\gamma }}-{\left({\perp }_{\mathsf{ı}\mathsf{ȷ}}^{0\mathsf{\gamma }}\right)}^2\right]\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\sum _{\begin{array}{c}1\le \mathsf{ı}\le n\\ \mathsf{ı}<\mathsf{ȷ}\le n\end{array}}\left[\left\{{\perp }_{\mathsf{ı}\mathsf{ı}}^{\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{\mathsf{\gamma }}-{\left({\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}\right)}^2\right\}+\left\{{\perp }_{\mathsf{ı}\mathsf{ı}}^{*\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{*\mathsf{\gamma }}-{\left({\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}\right)}^2\right\}\right].\hfill \end{array}$$

Utilizing (8) with respect to the L.C.C., we derive

$$\begin{array}{cc}\hfill ⊺=& {\displaystyle \frac{c}{9}}\left\{3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right\}-2{\widehat{⊺}}_0\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\sum _{\begin{array}{c}1\le \mathsf{ı}\le n\\ \mathsf{ı}<\mathsf{ȷ}\le n\end{array}}\left[\left\{{\perp }_{\mathsf{ı}\mathsf{ı}}^{\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{\mathsf{\gamma }}-{\left({\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}\right)}^2\right\}+\left\{{\perp }_{\mathsf{ı}\mathsf{ı}}^{*\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{*\mathsf{\gamma }}-{\left({\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}\right)}^2\right\}\right].\hfill \end{array}$$

(15)

Here, ${\widehat{⊺}}_0$ denotes the scalar curvature of the principal statistical manifold. The sectional K-curvature $\mathbb{k}\left(\pi \right)$, associated with the plane $\pi$, is given by

$$\mathbb{k}\left(\pi \right)={\displaystyle \frac{1}{2}}\left[g\left(R\left({\in }_1,{\in }_2\right){\in }_2,{\in }_1\right)+g\left(R^{*}\left({\in }_1,{\in }_2\right){\in }_2,{\in }_1\right)-2g\left(R^0\left({\in }_1,{\in }_2\right){\in }_2,{\in }_1\right)\right].$$

Thus, we have

$$\begin{array}{cc}\hfill \mathrm{R}({\in }_1,{\in }_2,{\in }_2,{\in }_1)& ={\displaystyle \frac{c}{9}}\{-3g({\in }_1,{\in }_2)g({\in }_2,{\in }_1)+3g({\in }_2,{\in }_2)g({\in }_1,{\in }_1)\hfill \\ \hfill & -4g({\in }_2,\wp {\in }_2)g(\wp {\in }_1,{\in }_1)+2g({\in }_2,\wp {\in }_2)g({\in }_1,{\in }_1)\hfill \\ \hfill & +2g({\in }_2,{\in }_2)g(\wp {\in }_1,{\in }_1)+4g({\in }_1,\wp {\in }_2)g(\wp {\in }_2,{\in }_1)\hfill \\ \hfill & -2g({\in }_1,\wp {\in }_2)g({\in }_2,{\in }_1)-2g({\in }_1,{\in }_2)g(\wp {\in }_2,{\in }_1)\}\hfill \\ \hfill & +g\left({\perp }^{*}\left({\in }_1,{\in }_1\right),\perp \left({\in }_2,{\in }_2\right)\right)-g\left(\perp \left({\in }_2,{\in }_1\right),{\perp }^{*}\left({\in }_2,{\in }_1\right)\right),\hfill \end{array}$$

$R^{*}({\in }_1,{\in }_2,{\in }_2,{\in }_1)$ can be obtained from the above equation just by replacing ℘ with ${\wp }^{*}$. After performing some straightforward computations, we deduce

$$\begin{array}{cc}\hfill \mathbb{k}\left(\pi \right)=& {\displaystyle \frac{c}{9}}\left\{6tr\left({\wp }_{\mid \pi }\right)-4{\rm Y}\left(\pi \right)-9\right\}-{\mathbb{k}}_0\left(\pi \right)\hfill \\ & +{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{11}^{\mathsf{\gamma }}{\perp }_{22}^{*\mathsf{\gamma }}+{\perp }_{11}^{*\mathsf{\gamma }}{\perp }_{22}^{\mathsf{\gamma }}-2{\perp }_{12}^{*\mathsf{\gamma }}{\perp }_{12}^{*\mathsf{\gamma }}\right].\hfill \end{array}$$

Using $\perp +{\perp }^{*}=2{\perp }^0$, we get

$$\begin{array}{cc}\hfill \mathbb{k}\left(\pi \right)=& {\displaystyle \frac{c}{9}}\left\{6tr\left({\wp }_{\mid \pi }\right)-4{\rm Y}\left(\pi \right)-9\right\}-{\mathbb{k}}_0\left(\pi \right)+2\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{11}^{0\mathsf{\gamma }}{\perp }_{22}^{0\mathsf{\gamma }}-{\left({\perp }_{12}^{0\mathsf{\gamma }}\right)}^2\right]\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left\{\left[{\perp }_{11}^{\mathsf{\gamma }}{\perp }_{22}^{\mathsf{\gamma }}-{\left({\perp }_{12}^{\mathsf{\gamma }}\right)}^2\right]+\left[{\perp }_{11}^{*\mathsf{\gamma }}{\perp }_{22}^{*\mathsf{\gamma }}-{\left({\perp }_{12}^{*\mathsf{\gamma }}\right)}^2\right]\right\}.\hfill \end{array}$$

Using (8) in the context of the L.C.C., we obtain

$$\begin{array}{cc}\hfill \mathbb{k}\left(\pi \right)=& {\displaystyle \frac{c}{9}}\left\{6tr\left({\wp }_{\mid \pi }\right)-4{\rm Y}\left(\pi \right)-9\right\}-2{\widehat{\mathbb{k}}}_0\left(\pi \right)-{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{11}^{\mathsf{\gamma }}{\perp }_{22}^{\mathsf{\gamma }}-{\left({\perp }_{12}^{\mathsf{\gamma }}\right)}^2\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{11}^{*\mathsf{\gamma }}{\perp }_{22}^{*\mathsf{\gamma }}-{\left({\perp }_{12}^{*\mathsf{\gamma }}\right)}^2\right],\hfill \end{array}$$

(16)

where ${\widehat{\mathbb{k}}}_0$ is the SC concerning the statistical manifold. Using (15) and (16), we get

$$\begin{array}{cc}\hfill (⊺-\mathbb{k}\left(\pi \right))-\left({⊺}_0-{\mathbb{k}}_0\left(\pi \right)\right)=& {\displaystyle \frac{c}{9}}\left\{3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right\}\hfill \\ \hfill & -{\displaystyle \frac{c}{9}}\left\{6tr\left({\wp }_{\mid \pi }\right)-4{\rm Y}\left(\pi \right)-9\right\}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{\mathsf{ı}\mathsf{ı}}^{\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{\mathsf{\gamma }}-{\left({\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}\right)}^2\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{\mathsf{ı}\mathsf{ı}}^{*\mathsf{\gamma }}{\perp }_{\mathsf{ȷ}\mathsf{ȷ}}^{*\mathsf{\gamma }}-{\left({\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left\{\left[{\perp }_{11}^{\mathsf{\gamma }}{\perp }_{22}^{\mathsf{\gamma }}-{\left({\perp }_{12}^{\mathsf{\gamma }}\right)}^2\right]\right.\hfill \\ \hfill & \left.+\left[{\perp }_{11}^{*\mathsf{\gamma }}{\perp }_{22}^{*\mathsf{\gamma }}-{\left({\perp }_{12}^{*\mathsf{\gamma }}\right)}^2\right]\right\}+2{\widehat{\mathbb{k}}}_0\left(\pi \right)-2{\widehat{⊺}}_0.\hfill \end{array}$$

(17)

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

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

According to Lemma 3, we have

$$A+A^{*}\le {\displaystyle \frac{n-2}{2(n-1)}}{\left(\sum _{\mathsf{\gamma }=n+1}^m\sum _{\mathsf{ı}=1}^n{\perp }_{\mathsf{ı}\mathsf{ı}}^{\mathsf{\gamma }}\right)}^2+{\displaystyle \frac{n-2}{2(n-1)}}{\left(\sum _{\mathsf{\gamma }=n+1}^m\sum _{\mathsf{ı}=1}^n{\perp }_{\mathsf{ı}\mathsf{ı}}^{*\mathsf{\gamma }}\right)}^2.$$

(18)

This can be written as

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

where $\omega$ and ${\omega }^{*}$ are the mean curvature vectors, defined as

$$\omega :={\displaystyle \frac{1}{n}}\sum _{\mathsf{ı}=1}^n{\perp }_{\mathsf{ı}\mathsf{ı}},{\omega }^{*}:={\displaystyle \frac{1}{n}}\sum _{\mathsf{ı}=1}^n{\perp }_{\mathsf{ı}\mathsf{ı}}^{*}.$$

Substituting this into Equation (17), we obtain

$$\begin{array}{cc}\hfill (⊺-\mathbb{k}\left(\pi \right))-\left({⊺}_0-{\mathbb{k}}_0\left(\pi \right)\right)\ge & {\displaystyle \frac{c}{9}}\left\{3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right\}\hfill \\ & -{\displaystyle \frac{c}{9}}\left\{6\mathrm{tr}\left({\wp }_{\mid \pi }\right)-4{\rm Y}\left(\pi \right)-9\right\}\hfill \\ & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left({\parallel \omega \parallel }^2+{\parallel {\omega }^{*}\parallel }^2\right)+2{\widehat{\mathbb{k}}}_0\left(\pi \right)-2{\widehat{⊺}}_0.\hfill \end{array}$$

This completes the proof. □

From Theorem 1, we have the following consequences:

Corollary 1.

Let ${\mathrm{M}}^n$ be a TRSS within an NGLSM $\tilde{\mathrm{M}}$. Then, the following properties hold:

$$\begin{array}{cc}\hfill (⊺-\mathbb{k}(\pi \left)\right)-\left({⊺}_0-{\mathbb{k}}_0\left(\pi \right)\right)\ge & {\displaystyle \frac{nc}{3}}\left\{n-3\right\}+c\hfill \\ \hfill & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \varpi \parallel }^2+{∥{\varpi }^{*}∥}^2\right]+2{\widehat{\mathbb{k}}}_0\left(\pi \right)-2{\widehat{⊺}}_0.\hfill \end{array}$$

(19)

In addition, the equalities hold for all $\mathsf{\gamma }\in \{n+1,\dots ,m\}$ iff:

$$\begin{array}{c}{\perp }_{11}^{\mathsf{\gamma }}+{\perp }_{22}^{\mathsf{\gamma }}={\perp }_{33}^{\mathsf{\gamma }}=\dots ={\perp }_{nn}^{\mathsf{\gamma }},\\ {\perp }_{11}^{*\mathsf{\gamma }}+{\perp }_{22}^{*\mathsf{\gamma }}={\perp }_{33}^{*\mathsf{\gamma }}=\dots ={\perp }_{nn}^{*\mathsf{\gamma }},\\ {\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}={\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}=0,\mathsf{ı}\ne \mathsf{ȷ},(\mathsf{ı},\mathsf{ȷ})\ne (1,2),(2,1),\mathsf{ı},\mathsf{ȷ}\in \{1,\dots ,n\},\mathsf{ı}<\mathsf{ȷ}.\end{array}$$

Corollary 2.

Consider ${\mathrm{M}}^n$ to be a TRSS of an NGLSM $\tilde{\mathrm{M}}$. If there is a point $p\in \mathrm{M}$ and a plane $\pi \subset T_p\mathrm{M}$ such that

$$⊺-\mathbb{k}\left(\pi \right)<{⊺}_0-{\mathbb{k}}_0\left(\pi \right)+{\displaystyle \frac{nc}{3}}\left\{n-3\right\}+c+2\left({\widehat{\mathbb{k}}}_0\left(\pi \right)-{\widehat{⊺}}_0\right).$$

then $\mathrm{M}$ is non-minimal; more precisely, $\varpi \ne 0$ or ${\varpi }^{*}\ne 0$.

Remark 1.

• The proof of Corollary 1 follows directly from Theorem 1, since Equation (19) is an immediate consequence of Equation (14) upon setting $\wp =0$.
• The proof of Corollary 2 is obvious.
• The inequality in Theorem 1 generalizes corresponding Chen-type inequalities established for statistical submanifolds in Kähler-like, Sasakian-like, and Hessian statistical manifolds [30,31,37]. By incorporating the structure tensor ℘ of the Norden golden-like manifold, this result holds under broader geometric conditions, reducing to classical cases when ℘ assumes standard forms.

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

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

Through explicit evaluation of $\mathbb{k}\left({\pi }_1\right)$ and $\mathbb{k}\left({\pi }_2\right)$, together with the application of Lemma 4, we derive the inequality representing the $\delta (2,2)$ Chen inequality for a statistical submanifold within an NGLSM as follows:

Theorem 2.

Suppose ${\mathrm{M}}^n$ is a statistical submanifold of $\tilde{\mathrm{M}}$. Then, the following holds:

$$\begin{array}{cc}\hfill \left(⊺-\mathbb{k}\left({\pi }_1\right)\right.& -\mathbb{k}\left({\pi }_2\right)-\left({⊺}_0-{\mathbb{k}}_0\left({\pi }_1\right)-{\mathbb{k}}_0\left({\pi }_2\right)\right)\ge {\displaystyle \frac{c}{9}}\left\{3n^2+4n\parallel \wp \parallel \right.\hfill \\ & \left.-9n-{4\parallel \wp \parallel }^2\right\}\hfill \\ & -{\displaystyle \frac{c}{9}}\left\{6tr\left(\wp \mid {\pi }_1\right)+tr\left(\wp \mid {\pi }_2\right)-4{\rm Y}\left({\pi }_1\right)-9+{\rm Y}\left({\pi }_2\right)\right\}\hfill \\ & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \varpi \parallel }^2+{∥{\varpi }^{*}∥}^2\right]-2\left[{\widehat{⊺}}_0-{\widehat{\mathbb{k}}}_0\left({\pi }_1\right)-{\widehat{\mathbb{k}}}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

Moreover, the equalities hold ∀$\mathsf{\gamma }\in \{n+1,\dots ,m\}$ iff

$$\begin{array}{c}{\perp }_{11}^{\mathsf{\gamma }}+{\perp }_{22}^{\mathsf{\gamma }}={\perp }_{33}^{\mathsf{\gamma }}+{\perp }_{44}^{\mathsf{\gamma }}={\perp }_{55}^{\mathsf{\gamma }}\dots ={\perp }_{nn}^{\mathsf{\gamma }},\\ {\perp }_{11}^{*\mathsf{\gamma }}+{\perp }_{22}^{*\mathsf{\gamma }}={\perp }_{33}^{*\mathsf{\gamma }}+{\perp }_{44}^{*\mathsf{\gamma }}={\perp }_{55}^{*\mathsf{\gamma }}\dots ={\perp }_{nn}^{*\mathsf{\gamma }},\\ {\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}={\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}=0,\end{array}$$

where $\mathsf{ı}\ne \mathsf{ȷ}$, and both are integers such that $\mathsf{ı},\mathsf{ȷ}\in \{1,\dots ,n\},\mathsf{ı}<\mathsf{ȷ}$, excluding the pairs $(1,2)$, $(2,1)$, $(3,4)$, and $(4,3)$.

Proof. 

Consider a statistical submanifold ${\mathrm{M}}^n$ of $\tilde{\mathrm{M}}$. If ${\pi }_1=\mathrm{span}\{{\in }_1,{\in }_2\}$ and ${\pi }_2=\mathrm{span}\{{\in }_3,{\in }_4\}$ are two mutually orthogonal planes in $\mathrm{M}$, then its scalar curvature ⊺ can be expressed as

$$⊺={\displaystyle \frac{c}{9}}\left[3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right]-{⊺}_0+{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\sum _{1\le \mathsf{ı}<\mathsf{ȷ}\le n}\left[{\perp }_{ii}^{*\mathsf{\gamma }}{\perp }_{jj}^{\mathsf{\gamma }}+{\perp }_{ii}^{\mathsf{\gamma }}{\perp }_{jj}^{*\mathsf{\gamma }}-2{\perp }_{ij}^{*\mathsf{\gamma }}{\perp }_{ij}^{\mathsf{\gamma }}\right].$$

Using $\perp +{\perp }^{*}=2{\perp }^0$, this simplifies to:

$$\begin{array}{cc}\hfill ⊺=& {\displaystyle \frac{c}{9}}\left[3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right]-2{\widehat{⊺}}_0\hfill \\ & +2\sum _{\mathsf{\gamma }=n+1}^m\sum _{1\le \mathsf{ı}<\mathsf{ȷ}\le n}\left[{\perp }_{ii}^{0\mathsf{\gamma }}{\perp }_{jj}^{0\mathsf{\gamma }}-{\left({\perp }_{ij}^{0\mathsf{\gamma }}\right)}^2\right]\hfill \\ & -{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\sum _{1\le \mathsf{ı}<\mathsf{ȷ}\le n}\left[\left({\perp }_{ii}^{\mathsf{\gamma }}{\perp }_{jj}^{\mathsf{\gamma }}-{\left({\perp }_{ij}^{\mathsf{\gamma }}\right)}^2\right)+\left({\perp }_{ii}^{*\mathsf{\gamma }}{\perp }_{jj}^{*\mathsf{\gamma }}-{\left({\perp }_{ij}^{*\mathsf{\gamma }}\right)}^2\right)\right].\hfill \end{array}$$

Sectional curvatures $\mathbb{k}\left({\pi }_1\right)$ and $\mathbb{k}\left({\pi }_2\right)$ are given by:

$$\mathbb{k}({\pi }_1)={\displaystyle \frac{c}{9}}\left[6\mathrm{tr}(\wp {|}_{{\pi }_1})-4{\rm Y}\left({\pi }_1\right)-9\right]-{\mathbb{k}}_0({\pi }_1)+{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{11}^{\mathsf{\gamma }}{\perp }_{22}^{*\mathsf{\gamma }}+{\perp }_{11}^{*\mathsf{\gamma }}{\perp }_{22}^{\mathsf{\gamma }}-2{\perp }_{12}^{*\mathsf{\gamma }}{\perp }_{12}^{\mathsf{\gamma }}\right],$$

$$\mathbb{k}({\pi }_2)={\displaystyle \frac{c}{9}}\left[6\mathrm{tr}(\wp {|}_{{\pi }_2})-4{\rm Y}\left({\pi }_2\right)-9\right]-{\mathbb{k}}_0({\pi }_2)+{\displaystyle \frac{1}{2}}\sum _{\mathsf{\gamma }=n+1}^m\left[{\perp }_{33}^{\mathsf{\gamma }}{\perp }_{44}^{*\mathsf{\gamma }}+{\perp }_{33}^{*\mathsf{\gamma }}{\perp }_{44}^{\mathsf{\gamma }}-2{\perp }_{34}^{*\mathsf{\gamma }}{\perp }_{34}^{\mathsf{\gamma }}\right].$$

Subtracting $\mathbb{k}({\pi }_1)+\mathbb{k}({\pi }_2)$ from ⊺ and simplifying using Lemma 4, we obtain:

$$\begin{array}{cc}\hfill \left(⊺-\mathbb{k}({\pi }_1)-\mathbb{k}({\pi }_2)\right)-& \left({⊺}_0-{\mathbb{k}}_0({\pi }_1)-{\mathbb{k}}_0({\pi }_2)\right)\hfill \\ & \ge {\displaystyle \frac{c}{9}}\left[3n^2+4n\parallel \wp \parallel -9n-{4\parallel \wp \parallel }^2\right]\hfill \\ & -{\displaystyle \frac{c}{9}}\left[6\mathrm{tr}(\wp {|}_{{\pi }_1})+6\mathrm{tr}(\wp {|}_{{\pi }_2})-4{\rm Y}\left({\pi }_1\right)-4{\rm Y}\left({\pi }_2\right)-9\right]\hfill \\ & -{\displaystyle \frac{(n-2)n^2}{4(n-1)}}\left[{\parallel \varpi \parallel }^2+{\parallel {\varpi }^{*}\parallel }^2\right]\hfill \\ & -2\left[{\widehat{⊺}}_0-{\widehat{\mathbb{k}}}_0\left({\pi }_1\right)-{\widehat{\mathbb{k}}}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

Equality holds if and only if

• The second fundamental forms satisfy

  $${\perp }_{11}^{\mathsf{\gamma }}+{\perp }_{22}^{\mathsf{\gamma }}={\perp }_{33}^{\mathsf{\gamma }}+{\perp }_{44}^{\mathsf{\gamma }}={\perp }_{55}^{\mathsf{\gamma }}=\cdots ={\perp }_{nn}^{\mathsf{\gamma }},$$

  $${\perp }_{11}^{*\mathsf{\gamma }}+{\perp }_{22}^{*\mathsf{\gamma }}={\perp }_{33}^{*\mathsf{\gamma }}+{\perp }_{44}^{*\mathsf{\gamma }}={\perp }_{55}^{*\mathsf{\gamma }}=\cdots ={\perp }_{nn}^{*\mathsf{\gamma }}.$$
• The mixed components vanish, except for $(1,2),(2,1),(3,4),(4,3)$:

$${\perp }_{ij}^{\mathsf{\gamma }}={\perp }_{ij}^{*\mathsf{\gamma }}=0,(\mathsf{ı},\mathsf{ȷ})\notin \{(1,2),(2,1),(3,4),and(4,3)\},\mathsf{ı}\ne \mathsf{ȷ}.$$

This establishes the required inequality. □

From Theorem 2, we have the following consequences:

Corollary 3.

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

$$\begin{array}{cc}\hfill \left(⊺-\mathbb{k}\left({\pi }_1\right)\right.& -\mathbb{k}\left({\pi }_2\right)-\left({⊺}_0-{\mathbb{k}}_0\left({\pi }_1\right)-{\mathbb{k}}_0\left({\pi }_2\right)\right)\hfill \\ & \ge {\displaystyle \frac{nc}{3}}\left\{n-3\right\}+c\hfill \\ \hfill & -{\displaystyle \frac{n^2(n-2)}{4(n-1)}}\left[{\parallel \varpi \parallel }^2+{∥{\varpi }^{*}∥}^2\right]-2\left[{\widehat{⊺}}_0-{\widehat{\mathbb{k}}}_0\left({\pi }_1\right)-{\widehat{\mathbb{k}}}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

(20)

Moreover, the equalities hold for all $\mathsf{\gamma }\in \{n+1,\cdots ,m\}$ iff:

$$\begin{array}{c}{\perp }_{11}^{\mathsf{\gamma }}+{\perp }_{22}^{\mathsf{\gamma }}={\perp }_{33}^{\mathsf{\gamma }}+{\perp }_{44}^{\mathsf{\gamma }}={\perp }_{55}^{\mathsf{\gamma }}\cdots ={\perp }_{nn}^{\mathsf{\gamma }},\\ {\perp }_{11}^{*\mathsf{\gamma }}+{\perp }_{22}^{*\mathsf{\gamma }}={\perp }_{33}^{*\mathsf{\gamma }}+{\perp }_{44}^{*\mathsf{\gamma }}={\perp }_{55}^{*\mathsf{\gamma }}\cdots ={\perp }_{nn}^{*\mathsf{\gamma }},\\ {\perp }_{\mathsf{ı}\mathsf{ȷ}}^{\mathsf{\gamma }}={\perp }_{\mathsf{ı}\mathsf{ȷ}}^{*\mathsf{\gamma }}=0,\end{array}$$

where ı and ȷ are distinct integers such that $1\le \mathsf{ı}<\mathsf{ȷ}\le n$, excluding the pairs $(1,2)$, $(2,1)$, $(3,4)$, and $(4,3)$.

Corollary 4.

Consider ${\mathrm{M}}^n$ to be a TRSS of $\tilde{\mathrm{M}}$. If there is a point $p\in \mathrm{M}$ and mutually orthogonal planes ${\pi }_1,{\pi }_2\subset T_p\mathrm{M}$ such that

$$\begin{array}{cc}\hfill ⊺-\mathbb{k}\left({\pi }_1\right)-\mathbb{k}\left({\pi }_2\right)<& {⊺}_0-{\mathbb{k}}_0\left({\pi }_1\right)-{\mathbb{k}}_0\left({\pi }_2\right)+{\displaystyle \frac{nc}{3}}\left\{n-3\right\}+c\hfill \\ & -2\left[{\widehat{⊺}}_0-{\widehat{\mathbb{k}}}_0\left({\pi }_1\right)-{\widehat{\mathbb{k}}}_0\left({\pi }_2\right)\right].\hfill \end{array}$$

Then $\mathrm{M}$ is non-minimal, i.e., $\varpi \ne 0$ or ${\varpi }^{*}\ne 0$.

Remark 2.

• Corollary 3 is a direct consequence of Theorem 1, as Equation (20) follows immediately from Equation (14) by taking $\wp =0$.
• The proof of Corollary 4 is obvious.
• Theorem 2 provides a generalized $\delta (2,2)$ Chen-type inequality for statistical submanifolds in NGLSM, expanding upon the results in [12,18,19]. This highlights the role of golden structures and dual connections in shaping curvature bounds. The non-minimality condition in Corollary 4 expands upon prior findings in golden Riemannian settings [6,9].

5. Conclusions

In this work, we introduced and formalized the structure of NGLSM, a novel class that enriches the intersection of Norden geometry and information geometry. Through this framework, we derived generalized Chen-type and $\delta$(2,2) curvature inequalities for statistical submanifolds, offering new insights into the interaction between intrinsic and extrinsic invariants. Our results generalize several known inequalities from Kähler-like and Hessian settings, thereby highlighting the versatility of the NGLSM structure in supporting meaningful geometric constraints. Future developments may include the study of statistical submanifolds within more generalized Norden-type settings, such as quasi-Norden or para-Norden statistical manifolds. Additionally, exploring the behavior of these inequalities under semi-symmetric or quarter-symmetric metric connections could yield sharper geometric characterizations. Another promising direction is the application of Norden golden-like structures to geometric models arising in machine learning and information theory, particularly where dual connections and curvature constraints influence optimization dynamics.