Statistical Warped Product Immersions into Statistical Manifolds of (Quasi-)Constant Curvature

Abstract

Warped products provide an elegant and versatile framework for exploring and understanding a wide range of geometric structures. Their ability to combine two distinct manifolds through a warping function introduces a rich and diverse set of geometries, thus making them a powerful tool in various mathematical, physical, and computational applications. This article addresses the central query related to warped product submanifolds in the context of statistics. It focuses on deriving two new and distinct inequalities for a statistical warped product submanifold in a statistical manifold of a constant (quasi-constant) curvature. This article then finishes with some concluding remarks.

1. Introduction

Warped product manifolds are a fascinating class of mathematical structures that emerge from the field of differential geometry, offering a versatile framework for understanding and studying various geometric phenomena. These manifolds arise by combining two distinct manifolds using a special type of product known as the warped product. This construction introduces a notion of nontrivial warping or scaling along one of the manifold’s directions, resulting in intriguing and rich geometries.

At its core, a warped product manifold is formed by taking the Cartesian product of two different manifolds and then warping one of the manifolds using a smooth, positive function known as the warping function. This warping function acts as a scaling factor, and it varies along the directions of the other manifold, as well as determines how the two manifolds are interlinked. As a consequence, the geometry of the warped product manifold can significantly differ from the simple product manifold.

Warped product manifolds have found applications in different branches of mathematics and theoretical physics. They play a crucial role in general relativity where they are used to describe certain solutions to Einstein’s field equations, including the well-known Schwarzschild and Kerr metrics, which describe the geometry around non-rotating and rotating black holes, respectively. Beyond theoretical applications, warped product manifolds have also found utility in applied fields such as robotics, computer graphics, and computer-aided design. They offer a natural way to represent and manipulate complex shapes and curved structures, thus making them invaluable in generating realistic and visually appealing models.

Definition 1

([1]). Let $(\mathbb{B},G_1)$ and $(\mathbb{F},G_2)$ be two Riemannian manifolds, and let $\alpha :\mathbb{B}\to {\mathbb{R}}^{+}$ be a positive real-valued function defined on $\mathbb{B}$. Consider the canonical projections ${\pi }_1:\mathbb{B}\times \mathbb{F}\to \mathbb{B}$ and ${\pi }_2:\mathbb{B}\times \mathbb{F}\to \mathbb{F}$. Then, the warped product $\mathbb{M}=(\mathbb{B}{\times }_{\alpha }\mathbb{F},\mathbb{G})$ equipped with the Riemannian metric is such that

$$\begin{array}{c}\hfill \mathbb{G}(x,y)=G_1({\pi }_{1\ast }x,{\pi }_{1\ast }y)+{\alpha }^2{G}_2({\pi }_{2\ast }x,{\pi }_{2\ast }y)\end{array}$$

(1)

for all $x,y\in \mathsf{\Gamma }(T\mathbb{M})$, and ∗ represents the tangent maps. The function α is the “warping function” of the warped product.

Remark 1.

When the α of the warped product $\mathbb{M}=(\mathbb{B}{\times }_{\alpha }\mathbb{F},\mathbb{G})$ remains constant, then $\mathbb{M}$ is a trivial case. In particular, $\mathbb{M}$ essentially becomes a Riemannian product.

The terms “base” and “fiber” refer to the manifolds $\mathbb{B}$ and $\mathbb{F}$ of $\mathbb{M}$. We denote the collection of all vector fields on $\mathbb{B}\times \mathbb{F}$ that represent the horizontal lift of a vector field on $\mathbb{B}$ as $\mathcal{H}(\mathbb{B})$. Similarly, the collection representing the vertical lift of a vector field on $\mathbb{F}$ is denoted as $\mathcal{V}(\mathbb{F})$.

Remark 2.

In a warped product manifold $\mathbb{M}=\mathbb{B}{\times }_{\alpha }\mathbb{F}$, the base manifold $\mathbb{B}$ is characterized as completely geodesic in $\mathbb{M}$, while the fiber manifold $\mathbb{F}$ is identified as completely umbilical in $\mathbb{M}$.

When considering the unit vector fields denoted as x and z, which are tangent to $\mathbb{B}$ and $\mathbb{F}$, respectively, the following relationships hold:

$$\begin{array}{c}\hfill {\widehat{D}}_{x}z={\widehat{D}}_{z}x=(xln\alpha )z,\end{array}$$

(2)

where $\widehat{D}$ is the Levi-Civita connection of $\mathbb{G}$ on $\mathbb{M}$.

Since Bishop and O’Neill’s 1969 article, numerous studies on warped products have been conducted from an intrinsic perspective over the past fifty years. In contrast, the investigation of warped products from an extrinsic perspective began around the start of this century, and they were initiated by the very-well known geometer Bang-Yen Chen through a series of his articles (see [2]).

On the other hand, Amari [3] introduced an interesting manifold named as “statistical manifold” in the context of information geometry (see [4]). These manifolds are fascinating mathematical structures that have found extensive applications in the fields of statistics, machine learning, and information geometry. At their core, these concepts bring together the elegant framework of Riemannian geometry and statistical theory to provide a powerful and intuitive way of analyzing and understanding probability distributions. Furthermore, it can be regarded as the generalization of a Hessian manifold. Beyond expectations, statistical manifolds are familiar to geometers, and many interesting results have been obtained [5,6,7,8].

In traditional statistics, probability distributions are often represented by sets of parameters, such as the mean and variance for Gaussian distributions. However, this representation may not always capture the full complexity and interactions between variables in high-dimensional data. This is where statistical manifolds come into play. The study of statistical manifolds and their submanifolds goes beyond just representing probability distributions in a geometric framework. It enables the development of sophisticated statistical models, hypothesis testing, parameter estimation, and model comparison in a geometric context. Furthermore, the tools of information geometry offer a unique perspective on gradient-based optimization algorithms, thus making them efficient and robust for dealing with complex statistical models.

In [9], Todjihounde explored a statistical structure on a warped product of statistical manifolds. This article aims to make a meaningful contribution to the advancement of the current research on statistical warped product immersions, which was introduced in [10]. Several authors have worked on statistical warped products (for example, see [11,12,13,14,15,16,17,18]). In [19], Chen established a relationship between scalar curvature, the warping function, and the squared mean curvature for warped products that are isometrically immersed in real space forms. Furthermore, in [20], a distinct relationship was identified by Chen between the warping function and the extrinsic structures of warped products in the same ambient space. Chen and Blaga [21] presented an intriguing and comprehensive survey of the research conducted over the past two decades on warped product submanifolds, particularly in relation to this inequality. Indeed, Sular [22] extended this relationship to warped products in a Riemannian manifold of quasi-constant curvature. Drawing inspiration from the aforementioned research, and in the ongoing exploration of inequalities pertaining to a statistical warped product submanifold in a statistical manifold of constant (quasi-constant) curvature, we introduce two additional inequalities for this submanifold in the same ambient spaces. These inequalities involve the warping function denoted as $\alpha$, which are the intrinsic and extrinsic curvatures associated with the aforementioned submanifold. To build such relationships, we require the Lemma from [23]:

Lemma 1.

Let $A_1,\cdots ,A_n,\mathcal{B}\in \mathbb{R}$, $n\ge 2$, such that

$$\begin{array}{c}\hfill {\left({\displaystyle \sum _{l=1}^n}{A}_l\right)}^2=(n-1)\left({\displaystyle \sum _{l=1}^n}{A}_l^2+\mathcal{B}\right).\end{array}$$

Then, Inequality $2A_1{A}_2\ge \mathcal{B}$ is satisfied, and equality occurs if and only if $A_1+A_2=A_3=\cdots =A_n$.

2. Preliminaries

A Riemannian manifold $(\overline{B},G)$ equipped with an affine connection $\overline{D}$ is said to be a statistical manifold if

• $\overline{D}$ is symmetric (or torsion-free);
• $({\overline{D}}_{x}G)(y,z)=({\overline{D}}_{y}G)(x,z)$ for all $x,y,z\in \mathsf{\Gamma }(T\overline{B})$.

It is denoted by $(\overline{B},\overline{D},G)$. Also, the dual connection ${\overline{D}}^{\prime }$ of $\overline{D}$ is defined as

$$\begin{array}{c}\hfill xG(y,z)=G({\overline{D}}_{x}y,z)+G(y,{\overline{D}}_x^{\prime }z).\end{array}$$

(3)

Evidently, ${\left({\overline{D}}^{\prime }\right)}^{\prime }=\overline{D}$. If $(\overline{D},G)$ constitutes a statistical structure on $\overline{B}$, then $({\overline{D}}^{\prime },G)$ is also a statistical structure. It should be noted that the Levi-Civita connection ${\overline{D}}^0$ of G is defined as $2{\overline{D}}^0=\overline{D}+{\overline{D}}^{\prime }$.

Consider the curvature tensor fields $\overline{R}$ and ${\overline{R}}^{\prime }$ associated with $\overline{D}$ and ${\overline{D}}^{\prime }$, respectively. A statistical manifold $(\overline{B},\overline{D},G)$ is said to be of constant curvature $\epsilon \in \mathbb{R}$ if the following equation holds for all $x,y,z\in \mathsf{\Gamma }(T\overline{B})$, and it is denoted as $\overline{B}(\epsilon )$:

$$\begin{array}{c}\hfill \overline{R}(x,y)z=\epsilon \{G(y,z)x-G(x,z)y\}.\end{array}$$

(4)

Note that a statistical manifold $(\overline{B},\overline{D},G)$ with a constant curvature of 0 is a Hessian structure.

A statistical manifold $(\overline{B},\overline{D},G)$ is said to be of quasi-constant curvature if the following equation holds for all $x,y,z\in \mathsf{\Gamma }(T\overline{B})$ [7]:

$$\begin{array}{ccc}\hfill \overline{R}(x,y)z& =& f\{G(y,z)x-G(x,z)y\}\hfill \\ \hfill & \hfill & +g\{\mathsf{\Omega }(y)\mathsf{\Omega }(z)x-G(x,z)\mathsf{\Omega }(y)\mathcal{P}\hfill \\ \hfill & \hfill & +G(y,z)\mathsf{\Omega }(x)\mathcal{P}-\mathsf{\Omega }(x)\mathsf{\Omega }(z)y\},\hfill \end{array}$$

(5)

where f and g are scalar functions on $\overline{B}$, $\mathcal{P}$ is a unit vector field (called generator), and $\mathsf{\Omega }$ is a one-form dual to $\mathcal{P}$, where such a manifold is denoted as $\overline{B}(f,g,\mathcal{P})$. Note that $\overline{R}(x,y,z,w)=G(\overline{R}(x,y)w,z)$.

Consider a statistical manifold $(\overline{B},\overline{D},G)$, and let B represent a submanifold in $\overline{B}$. If $(B,D,G)$ also forms a statistical manifold, we refer to it as a statistical submanifold of $\overline{B}$. Here, D denotes the induced affine connection on B, and both the induced Riemannian metric for B and the metric for $\overline{B}$ are denoted by the same symbol G.

In the field of Riemannian submanifold geometry, as detailed in [24], the pivotal equations encompass the Gauss and Weingarten formulations along with the Gauss equation. In our specific context, when considering any $x,y\in \mathsf{\Gamma }(TB)$ and $v\in \mathsf{\Gamma }(T^{\perp }B)$, the Gauss and Weingarten formulas are elucidated as follows [25]:

$$\left.\begin{array}{c}\hfill {\overline{D}}_{x}y=D_{x}y+T(x,y),\\ \hfill {\overline{D}}_x^{\prime }y=D_x^{\prime }y+T^{\prime }(x,y),\\ \hfill {\overline{D}}_{x}v=-{\mathsf{\Omega }}_v(x)+D_x^{\perp }v,\\ \hfill {\overline{D}}_x^{\prime }v=-{\mathsf{\Omega }}_v^{\prime }(x)+D_x^{\prime \perp }v\end{array}\right\}$$

(6)

where T and $T^{\prime }$ are the imbedding curvature tensors of B in $\overline{B}$ for $\overline{D}$ and ${\overline{D}}^{\prime }$, respectively. Since T and $T^{\prime }$ are symmetric and bilinear, we have the linear transformations ${\mathsf{\Omega }}_v$ and ${\mathsf{\Omega }}_v^{\prime }$, which are defined by [25]

$$\begin{array}{c}\hfill G(T(x,y),v)=G({\mathsf{\Omega }}_v^{\prime }x,y),\mathrm{and}G(T^{\prime }(x,y),v)=G({\mathsf{\Omega }}_{v}x,y).\end{array}$$

(7)

Note that, from imbedding curvature tensor $\widehat{T}$ with respect to ${\overline{D}}^0$, we have $\widehat{T}=\frac{1}{2}(T+T^{\prime })$.

Furthermore, consider C and $C^{\prime }$ as the Casorati curvatures of M, where these curvatures are defined by the squared norms of T and $T^{\prime }$ over the dimension $m=r+s$, respectively.

The mean curvature vector fields of B of dimension m in $\overline{B}$ for ${\overline{D}}^{\prime }$ and ${\overline{D}}^{\prime }$ are, respectively, defined by

$$H=\frac{1}{m}trace(T)\mathrm{and}{H}^{\prime }=\frac{1}{m}trace(T^{\prime }).$$

The mean curvature vector field $\widehat{H}=\frac{1}{m}trace(\widehat{T})$ with respect to ${\overline{D}}^0$ can be written as $\widehat{H}=\frac{1}{2}(H+H^{\prime })$. The statistical submanifold B is a doubly auto-parallel (or doubly completely geodesic) when both $T=0$ and $T^{\prime }=0$ hold.

Consider the curvature tensor fields R and its dual $R^{\prime }$ associated with D and $D^{\prime }$, respectively. Then, the corresponding Gauss equation is expressed as [25]:

$$\begin{array}{ccc}\hfill \overline{R}(x,y,z,w)& =& R(x,y,z,w)+G(T(x,z),T^{\prime }(y,w))\hfill \\ \hfill & \hfill & -G(T^{\prime }(x,w),T(y,z))\hfill \end{array}$$

(8)

for all $x,y,z,w\in \mathsf{\Gamma }(TB)$.

In a similar manner, ${\overline{R}}^{\prime }$ and $R^{\prime }$ for ${\overline{D}}^{\prime }$ and $D^{\prime }$, respectively, can be represented using the same Equation (8), as explained in [25].

For an m-dimensional statistical submanifold $(B,D,G)$ in an n-dimensional statistical manifold $(\overline{B},\overline{D},G)$, we selected a local field of an orthonormal frame, which is denoted as $\{v_1,v_2,\cdots ,v_{m+1},\cdots ,v_n\}$ in $\overline{B}$. Here, $\{v_1,v_2,\cdots ,v_m\}$ was chosen to be tangent to the submanifold B, while $\{v_{m+1},v_{m+2},\cdots ,v_n\}$ was selected to be normal to B. The sectional curvature $S(v_i\wedge v_j)$ of the plane section is spanned by $v_i$ and $v_j$. Then, the scalar curvature $scal$ of B is given by

$$\begin{array}{ccc}\hfill scal& =& {\displaystyle \sum _{i<j}}S(v_i\wedge v_j)\hfill \\ \hfill & =& {\displaystyle \sum _{i<j}}G(R(v_i,v_j)v_j,v_i).\hfill \end{array}$$

(9)

3. Some Inequalities for Statistical Warped Product Immersions

Todjihounde, as presented in [9], introduced a technique for constructing a dualistic structure on the warped product manifold $B{\times }_{\alpha }F$. He demonstrated that, if both B and F are statistical manifolds, then the resulting warped product $M=B{\times }_{\alpha }F$ also becomes a statistical manifold and features a dualistic structure denoted as $(G,D,D^{\prime })$ on M. We fixed $dim(B)=r$ and $dim(F)=s$ so that $dim(M)=m=r+s$.

It is noteworthy that, when considering $x\in \mathcal{H}(B)$ and $z\in \mathcal{V}(F)$, the following equalities hold [9]:

$$\begin{array}{c}\hfill D_{x}z=D_{z}x=\frac{x\alpha }{\alpha }z,D_x^{\prime }z=D_z^{\prime }x=\frac{x\alpha }{\alpha }z.\end{array}$$

(10)

Now, let us consider $\tilde{M}$ as an n-dimensional Riemannian manifold, where D represents an affine connection and $\{v_1,v_2,\cdots ,v_n\}$ forms an orthonormal frame field. In this context, the Laplacian $\Delta \alpha$ of a function $\alpha$ with respect to D is defined as follows [12]:

$$\begin{array}{c}\hfill \Delta \alpha ={\displaystyle \sum _{i=1}^n}G(D_{v_i}grad\alpha ,v_i).\end{array}$$

(11)

Next, we consider the unit vector fields x and z that are tangent to B and F, respectively. Then, the sectional curvature S of $M=B{\times }_{\alpha }F$ can be expressed as follows:

$$\begin{array}{ccc}\hfill S(x\wedge z)& =& G(D_z{D}_{x}x,z)-G(D_x{D}_{z}x,z)\hfill \\ \hfill & =& \frac{1}{\alpha }\{(D_{x}x)\alpha -x^2\alpha \}.\hfill \end{array}$$

(12)

In cases where we choose a local orthonormal frame $\{v_1,\cdots ,v_r\}$ that is tangent to B and $\{v_{r+1},\cdots ,v_s\}$ tangent to F, then we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^r}S(v_i\wedge v_j)=\frac{\Delta \alpha }{\alpha }\end{array}$$

(13)

for every $j=r+1,\cdots ,s$. Here, $\Delta$ represents the Laplacian operator acting on B.

We established relationships between $\alpha$, the primary extrinsic properties, and the fundamental intrinsic characteristics of a statistical warped product submanifold in a statistical manifold of constant curvature.

Theorem 1.

Let $M=B{\times }_{\alpha }F$ be a statistical warped product submanifold in a statistical manifold $(\overline{B}(\epsilon ),\overline{D},G)$ of a constant curvature. Then, the scalar curvature of M satisfies

$$\begin{array}{ccc}\hfill scal& \le & 2\frac{\Delta \alpha }{r\alpha }+(\frac{(r+s+1)(r+s-2)-2}{2})\epsilon +\frac{(r+s)}{2}\widehat{C}\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2(r+s-2)}{(r+s-1)}\left|\right|\widehat{H}{\left|\right|}^2-\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(14)

Proof. 

Using (8) and summing over $1\le i,j\le m$, $(m=r+s)$, we find that

$$\begin{array}{ccc}\hfill (r+s)(r+s-1)\epsilon & =& 2scal-{(r+s)}^{2}G(H,H^{\prime })\hfill \\ \hfill & \hfill & +{\displaystyle \sum _{i,j=1}^{r+s}}G(T(v_i,v_j),T^{\prime }(v_i,v_j)).\hfill \end{array}$$

(15)

Further, we reduce (15) as

$$\begin{array}{ccc}\hfill (r+s)(r+s-1)\epsilon & =& 2scal-2{(r+s)}^2\left|\right|\widehat{H}{\left|\right|}^2+\frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +2\left|\right|\widehat{T}{\left|\right|}^2-\frac{1}{2}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(16)

Suppose that

$$\begin{array}{c}\hfill \delta =2scal-2\frac{{(r+s)}^2(r+s-2)}{r+s-1}\left|\right|\widehat{H}{\left|\right|}^2-(r+s+1)(r+s-2)\epsilon .\end{array}$$

(17)

Then, on combining (16) and (17), we arrive at

$$\begin{array}{ccc}\hfill 2{(r+s)}^2\left|\right|\widehat{H}{\left|\right|}^2& =& 2(r+s-1)\left|\right|\widehat{T}{\left|\right|}^2+(r+s-1)(\delta -2\epsilon )\hfill \\ \hfill & \hfill & +\frac{(r+s-1){(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{(r+s-1)}{2}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(18)

Consider two unit local vector fields, x and z, which are tangent to B and F, respectively. Select an orthonormal frame $\{v_1,\cdots ,v_n\}$, where $v_1$ aligns with x, $v_{r+s+1}$ aligns with z, and $v_{r+s+1}$ is parallel to $\widehat{H}$. Subsequently, Equation (18) can be rewritten as

$$\begin{array}{ccc}\hfill {({\displaystyle \sum _{i=1}^{r+s}}{\widehat{T}}_{ii}^{r+s+1})}^2& =& (r+s-1)\{{\displaystyle \sum _{i=1}^{r+s}}{({\widehat{T}}_{ii}^{r+s+1})}^2+{\displaystyle \sum _{i\ne j}}{({\widehat{T}}_{ij}^{r+s+1})}^2\hfill \\ \hfill & \hfill & +{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{i,j=1}^{r+s}}{({\widehat{T}}_{ij}^t)}^2\}+\frac{(r+s-1)}{2}(\delta -2\epsilon )\hfill \\ \hfill & \hfill & +\frac{(r+s-1){(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{(r+s-1)}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

At this stage, we utilized Lemma 1 on the last equation to derive that

$$\begin{array}{ccc}\hfill 2{\widehat{T}}_{11}^{r+s+1}{\widehat{T}}_{r+1r+1}^{r+s+1}& \ge & {\displaystyle \sum _{i\ne j}}{({\widehat{T}}_{ij}^{r+s+1})}^2+{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{i,j=1}^{r+s}}{({\widehat{T}}_{ij}^t)}^2\}+\frac{1}{2}\delta -\epsilon \hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{1}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(19)

From the equation of Gauss with respect to the Levi-Civita connection, we obtained

$$\begin{array}{c}\hfill \widehat{S}(v_1\wedge v_{r+1})=\epsilon -G({\widehat{T}}_{1r+1},{\widehat{T}}_{1r+1})+G({\widehat{T}}_{11},{\widehat{T}}_{r+1r+1}),\end{array}$$

where $\widehat{S}$ denotes the scalar curvature of M with respect to $\widehat{D}$.

For $\mathsf{\Gamma }=\{1,2,\cdots ,r+s\}\setminus \{1,r+1\}$, (19) provides

$$\begin{array}{ccc}\hfill \widehat{S}(v_1\wedge v_{r+1})& \ge & {\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{j\in \mathsf{\Gamma }}}\{{({\widehat{T}}_{1j}^t)}^2+{({\widehat{T}}_{r+1j}^t)}^2\}\hfill \\ \hfill & \hfill & +\frac{1}{2}\{{\displaystyle \sum _{i,j\in \mathsf{\Gamma };i\ne j}}{({\widehat{T}}_{ij}^{r+s+1})}^2+{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{i,j\in \mathsf{\Gamma }}}{({\widehat{T}}_{ij}^t)}^2\hfill \\ \hfill & \hfill & +{\displaystyle \sum _{t=r+s+2}^n}{({\widehat{T}}_{11}^t+{\widehat{T}}_{r+1r+1}^t)}^2+\frac{\delta }{4}+\frac{\epsilon }{2}\}\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{8}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{1}{8}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}\hfill \frac{\delta }{4}& \le & \widehat{S}(v_1\wedge v_{r+1})-\frac{\epsilon }{2}-\frac{{(r+s)}^2}{8}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +\frac{1}{8}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(20)

As we have a statistical warped product $B{\times }_{\alpha }F$, the gradients $D_{x}z$ and $D_{z}x$, respectively, are given by $(xln\alpha )z$. Consequently, we deduced that

$$\begin{array}{ccc}\hfill \widehat{S}(x\wedge z)& =& G({\widehat{D}}_z{\widehat{D}}_{x}x-{\widehat{D}}_x{\widehat{D}}_{z}x,z)\hfill \\ \hfill & =& ({\widehat{D}}_{x}x)ln\alpha -{(xln\alpha )}^2-x(xln\alpha )\hfill \\ \hfill & =& \frac{1}{\alpha }\{({\widehat{D}}_{x}x)\alpha -x^2\alpha \}.\hfill \end{array}$$

(21)

On putting Equations (17), (20), and (21) together, we arrived at

$$\begin{array}{ccc}\hfill scal& \le & \frac{2}{\alpha }\{({\widehat{D}}_{v_1}{v}_1)\alpha -v_1^2\alpha \}+\frac{(r+s+1)(r+s-2)-2}{2}\epsilon \hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2(r+s-2)}{r+s-1}\left|\right|\widehat{H}{\left|\right|}^2-\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +\frac{1}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2)\hfill \end{array}$$

(22)

Consider the Casorati curvatures C and $C^{\prime }$ of M; hence, (22) becomes

$$\begin{array}{ccc}\hfill scal& \le & \frac{2}{\alpha }\{({\widehat{D}}_{v_1}{v}_1)\alpha -v_1^2\alpha \}+\frac{(r+s+1)(r+s-2)-2}{2}\epsilon \hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2(r+s-2)}{r+s-1}\left|\right|\widehat{H}{\left|\right|}^2-\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +\frac{r+s}{4}(C+C^{\prime }).\hfill \end{array}$$

When denoting the Casorati curvature $\widehat{C}$ of M with respect to $\widehat{D}$, the last equation is

$$\begin{array}{ccc}\hfill scal& \le & \frac{2}{\alpha }\{({\widehat{D}}_{v_1}{v}_1)\alpha -v_1^2\alpha \}+\frac{(r+s+1)(r+s-2)-2}{2}\epsilon \hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2(r+s-2)}{r+s-1}\left|\right|\widehat{H}{\left|\right|}^2-\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +\frac{r+s}{2}\widehat{C},\hfill \end{array}$$

(23)

where $2\widehat{C}=C+C^{\prime }$.

Just like (23), we also possessed the following for $i=1,2,\cdots ,r$:

$$\begin{array}{ccc}\hfill scal& \le & \frac{2}{\alpha }\{({\widehat{D}}_{v_i}{v}_i)\alpha -v_i^2\alpha \}+\frac{(r+s+1)(r+s-2)-2}{2}\epsilon \hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2(r+s-2)}{r+s-1}\left|\right|\widehat{H}{\left|\right|}^2-\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +\frac{r+s}{2}\widehat{C}.\hfill \end{array}$$

(24)

Therefore, by summing i from 1 to r in (24), we acquired (14). □

In order to give a second result of this article, we assumed a new ${\delta }^{*}$ as

$$\begin{array}{c}\hfill {\delta }^{*}=2scal-{(r+s)}^2\left|\right|\widehat{H}{\left|\right|}^2-(r+s)(r+s-1)\epsilon .\end{array}$$

(25)

Then, by using (25), Equation (16) becomes

$$\begin{array}{ccc}\hfill {(r+s)}^2\left|\right|\widehat{H}{\left|\right|}^2& =& {\delta }^{*}+\frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +2{\displaystyle \sum _{i,j=1}^{r+s}}\left|\right|\widehat{T}{\left|\right|}^2-\frac{1}{2}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(26)

Again, on selecting an orthonormal frame comprising $v_{r+s+1},\cdots ,v_n$ for the normal bundle, such that $v_{r+s+1}$ aligns with $\widehat{H}$, Expression (26) transforms into

$$\begin{array}{ccc}\hfill {({\displaystyle \sum _{i=1}^{r+s}}{\widehat{T}}_{ii}^{r+s+1})}^2& =& 2\{\frac{{\delta }^{*}}{2}+{\displaystyle \sum _{i=1}^{r+s}}{({\widehat{T}}_{ii}^{r+s+1})}^2+{\displaystyle \sum _{i\ne j}}{({\widehat{T}}_{ij}^{r+s+1})}^2\hfill \\ \hfill & \hfill & +{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{i,j=1}^{r+s}}{({\widehat{T}}_{ij}^t)}^2\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{1}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2)\},\hfill \end{array}$$

which is equivalent to

$$\begin{array}{ccc}\hfill & \hfill & {\{{\widehat{T}}_{11}^{r+s+1}+({\widehat{T}}_{22}^{r+s+2}+\cdots +{\widehat{T}}_{rr}^{r+s+1})+({\widehat{T}}_{r+1r+1}^{r+s+1}+\cdots +{\widehat{T}}_{r+sr+s}^{r+s+1})\}}^2\hfill \\ \hfill & =& 2\{{({\widehat{T}}_{11}^{r+s+1})}^2+{({\widehat{T}}_{22}^{r+s+2}+\cdots +{\widehat{T}}_{rr}^{r+s+1})}^2+({\widehat{T}}_{r+1r+1}^{r+s+1}+\cdots \hfill \\ \hfill & \hfill & +{\widehat{T}}_{r+sr+s}^{r+s+1}{)}^2+2{\displaystyle \sum _{1\le i<j\le r+s}}{({\widehat{T}}_{ij}^{r+s+1})}^2+{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{i,j=1}^{r+s}}{({\widehat{T}}_{ij}^t)}^2\hfill \\ \hfill & \hfill & -2{\displaystyle \sum _{2\le j<k\le r}}{\widehat{T}}_{jj}^{r+s+1}{\widehat{T}}_{kk}^{r+s+1}-2{\displaystyle \sum _{r+1\le a<b\le r+s}}{\widehat{T}}_{aa}^{r+s+1}{\widehat{T}}_{bb}^{r+s+1}\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)-\frac{1}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2)+\frac{{\delta }^{*}}{2}\}.\hfill \end{array}$$

Utilizing Lemma 1 on the final equality leads us to

$$\begin{array}{ccc}\hfill & \hfill & 2{\displaystyle \sum _{2\le j<k\le r}}{\widehat{T}}_{jj}^{r+s+1}{\widehat{T}}_{kk}^{r+s+1}+2{\displaystyle \sum _{r+1\le a<b\le r+s}}{\widehat{T}}_{aa}^{r+s+1}{\widehat{T}}_{bb}^{r+s+1}\hfill \\ \hfill & \ge & 2{\displaystyle \sum _{1\le A<B\le r+s}}{({\widehat{T}}_{AB}^{r+s+1})}^2+{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{A,B=1}^{r+s}}{({\widehat{T}}_{AB}^t)}^2\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)-\frac{1}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2)+\frac{{\delta }^{*}}{2}.\hfill \end{array}$$

(27)

By utilizing (13) and the Gauss equation, we obtain

$$\begin{array}{ccc}\hfill \frac{s\Delta \alpha }{\alpha }& =& scal-{\displaystyle \sum _{1\le j<k\le r}}S(v_j\wedge v_k)-{\displaystyle \sum _{r+1\le a<b\le r+s}}S(v_j\wedge v_k)\hfill \\ \hfill & =& scal-\frac{1}{2}r(r-1)\epsilon -{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{1\le j<k\le r}}(T_{jj}^t{T}_{kk}^{\prime t}-T_{jk}^t{T}_{jk}^{\prime t})\hfill \\ \hfill & \hfill & -\frac{1}{2}s(s-1)\epsilon -{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}(T_{aa}^t{T}_{bb}^{\prime t}-T_{ab}^t{T}_{ab}^{\prime t}).\hfill \end{array}$$

Now, we can recreate it as

$$\begin{array}{ccc}\hfill \frac{s\Delta \alpha }{\alpha }& =& scal-\frac{1}{2}r(r-1)\epsilon -{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{1\le j<k\le r}}\{(2{\widehat{T}}_{jj}^t{\widehat{T}}_{kk}^t-\frac{1}{2}{T}_{jj}^t{T}_{kk}^t-\frac{1}{2}{T}_{jj}^{\prime t}{T}_{kk}^{\prime t})\hfill \\ \hfill & \hfill & -(2{({\widehat{T}}_{jk}^t)}^2-\frac{1}{2}{T}_{jk}^t{T}_{jk}^t-\frac{1}{2}{T}_{jk}^{\prime t}{T}_{jk}^{\prime t})\}\hfill \\ \hfill & \hfill & -\frac{1}{2}s(s-1)\epsilon -{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}\{(2{\widehat{T}}_{aa}^t{\widehat{T}}_{bb}^t-\frac{1}{2}{T}_{aa}^t{T}_{bb}^t-T_{aa}^{\prime t}{T}_{bb}^{\prime t})\hfill \\ \hfill & \hfill & -(2{({\widehat{T}}_{ab}^t)}^2-\frac{1}{2}{T}_{ab}^t{T}_{ab}^t-\frac{1}{2}{T}_{ab}^{\prime t}{T}_{ab}^{\prime t})\}.\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}\hfill \frac{s\Delta \alpha }{\alpha }& =& scal-\frac{1}{2}r(r-1)\epsilon -2{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{1\le j<k\le r}}({\widehat{T}}_{jj}^t{\widehat{T}}_{kk}^t-{({\widehat{T}}_{jk}^t)}^2)\hfill \\ \hfill & \hfill & -\frac{1}{2}s(s-1)\epsilon -2{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}({\widehat{T}}_{aa}^t{\widehat{T}}_{bb}^t-{({\widehat{T}}_{ab}^t)}^2)\hfill \\ \hfill & \hfill & +\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{1\le j<k\le r}}(T_{jj}^t{T}_{kk}^t+T_{jj}^{\prime t}{T}_{kk}^{\prime t})\hfill \\ \hfill & \hfill & -\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{1\le j<k\le r}}(T_{jk}^t{T}_{jk}^t+T_{jk}^{\prime t}{T}_{jk}^{\prime t})\hfill \\ \hfill & \hfill & +\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}(T_{aa}^t{T}_{bb}^t+T_{aa}^{\prime t}{T}_{bb}^{\prime t})\hfill \\ \hfill & \hfill & -\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}(T_{ab}^t{T}_{ab}^t+T_{ab}^{\prime t}{T}_{ab}^{\prime t}).\hfill \end{array}$$

From Inequality (27) and Equation (25), we can deduce that

$$\begin{array}{ccc}\hfill \frac{s\Delta \alpha }{\alpha }& \le & rs\epsilon +\frac{{(r+s)}^2}{2}\left|\right|\widehat{H}{\left|\right|}^2-2{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{1\le j<k\le r}}({\widehat{T}}_{jj}^t{\widehat{T}}_{kk}^t-{({\widehat{T}}_{jk}^t)}^2)\hfill \\ \hfill & \hfill & -2{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}({\widehat{T}}_{aa}^t{\widehat{T}}_{bb}^t-{({\widehat{T}}_{ab}^t)}^2)\hfill \\ \hfill & \hfill & -2{\displaystyle \sum _{1\le j<\le r;r+1\le b\le r+s}}{({\widehat{T}}_{jb}^{r+s+1})}^2-{\displaystyle \sum _{t=r+s+2}^n}{\displaystyle \sum _{A,B=1}^{r+s}}{({\widehat{T}}_{AB}^t)}^2\hfill \\ \hfill & \hfill & -\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)+\frac{1}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{1\le j<k\le r}}(T_{jj}^t{T}_{kk}^t+T_{jj}^{\prime t}{T}_{kk}^{\prime t})\hfill \\ \hfill & \hfill & -\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{1\le j<k\le r}}(T_{jk}^t{T}_{jk}^t+T_{jk}^{\prime t}{T}_{jk}^{\prime t})\hfill \\ \hfill & \hfill & +\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}(T_{aa}^t{T}_{bb}^t+T_{aa}^{\prime t}{T}_{bb}^{\prime t})\hfill \\ \hfill & \hfill & -\frac{1}{2}{\displaystyle \sum _{t=r+s+1}^n}{\displaystyle \sum _{r+1\le a<b\le r+s}}(T_{ab}^t{T}_{ab}^t+T_{ab}^{\prime t}{T}_{ab}^{\prime t}).\hfill \end{array}$$

And, we have

$$\begin{array}{ccc}\hfill \frac{s\Delta \alpha }{\alpha }& \le & rs\epsilon +\frac{{(r+s)}^2}{2}\left|\right|\widehat{H}{\left|\right|}^2+\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{1}{4}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(28)

Further, we used the following from [5]:

$$\begin{array}{ccc}\hfill {\left|\right|T\left|\right|}^2+\left|\right|T^{\prime }{\left|\right|}^2& \ge & \frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)-4{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}{\widehat{T}}_{ii}^t{\widehat{T}}_{jj}^t\hfill \\ \hfill & \hfill & +2{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}{T}_{ii}^t{T}_{jj}^{\prime t}+{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}\{{(T_{ij}^t)}^2+{(T_{ij}^{\prime t})}^2\},\hfill \end{array}$$

and then recreated it as

$$\begin{array}{ccc}\hfill {\left|\right|T\left|\right|}^2+\left|\right|T^{\prime }{\left|\right|}^2& \ge & \frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)-4{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}{\widehat{T}}_{ii}^t{\widehat{T}}_{jj}^t\hfill \\ \hfill & \hfill & +2{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}{T}_{ii}^t{T}_{jj}^{\prime t}+{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}\{{(T_{ij}^t+T_{ij}^{\prime t})}^2\hfill \\ \hfill & \hfill & -2T_{ij}^t{T}_{ij}^{\prime t}\}\hfill \\ \hfill & =& \frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)-4{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}{\widehat{T}}_{ii}^t{\widehat{T}}_{jj}^t\hfill \\ \hfill & \hfill & +2{\displaystyle \sum _{2\le i\ne j\le r+s}}R(v_i,v_j,v_i,v_j)-2(r+s-1)(r+s-2)\epsilon \hfill \\ \hfill & \hfill & +4{\displaystyle \sum _{t=1}^n}{\displaystyle \sum _{2\le i\ne j\le r+s}}{({\widehat{T}}_{ij})}^2\hfill \\ \hfill & =& \frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)+2{\displaystyle \sum _{2\le i\ne j\le r+s}}R(v_i,v_j,v_i,v_j)\hfill \\ \hfill & \hfill & -2(r+s-1)(r+s-2)\epsilon -4{\displaystyle \sum _{2\le i\ne j\le r+s}}\widehat{R}(v_i,v_j,v_i,v_j)\hfill \\ \hfill & \hfill & +4{\displaystyle \sum _{2\le i\ne j\le r+s}}\widehat{\overline{R}}(v_i,v_j,v_i,v_j),\hfill \end{array}$$

where $\widehat{\overline{R}}$ and $\widehat{R}$ are the notions used for the Riemannian curvature tensors of $\overline{B}$ and M for ${\overline{D}}^0$ and $\widehat{D}$, respectively.

Thus, we used the

$$\begin{array}{ccc}\hfill {\left|\right|T\left|\right|}^2+\left|\right|T^{\prime }{\left|\right|}^2& \ge & \frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)-2(r+s-1)(r+s-2)\epsilon \hfill \\ \hfill & \hfill & +4{\displaystyle \sum _{2\le i<j\le r+s}}\{S(v_i\wedge v_j)-2\widehat{S}(v_i\wedge v_j)+2\widehat{\overline{S}}(v_i\wedge v_j)\}\hfill \end{array}$$

in (28), and we thus obtained the following:

Theorem 2.

Let $B{\times }_{\alpha }F$ be a statistical warped product submanifold in a statistical manifold $(\overline{B}(\epsilon ),\overline{D},G)$ of a constant curvature. Then, we have

$$\begin{array}{ccc}\hfill \frac{\Delta \alpha }{\alpha }& \le & (r+\frac{(r+s-1)(r+s-2)}{2s})\epsilon +\frac{{(r+s)}^2}{2s}\left|\right|\widehat{H}{\left|\right|}^2\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{8s}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{1}{s}{\displaystyle \sum _{2\le i<j\le r+s}}\{S(v_i\wedge v_j)-2\widehat{S}(v_i\wedge v_j)+2\widehat{\overline{S}}(v_i\wedge v_j)\}.\hfill \end{array}$$

We observe that a Hessian manifold $(\overline{B}(c),\overline{D},G)$ of a constant Hessian curvature c is a statistical manifold of constant curvature 0, which is denoted by $\epsilon =0$, such that G is of a constant sectional curvature $-\frac{c}{4}$. Consequently, we expressed the following from Theorem 2:

Corollary 1.

Let $B{\times }_{\alpha }F$ be a statistical warped product submanifold in a Hessian manifold $(\overline{B}(c),\overline{D},G)$ of a constant Hessian curvature. Then, we have

$$\begin{array}{ccc}\hfill \frac{\Delta \alpha }{\alpha }& \le & \frac{(r+s-1)(r+s-2)c}{2s}+\frac{{(r+s)}^2}{2s}\left|\right|\widehat{H}{\left|\right|}^2\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{8s}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{1}{s}{\displaystyle \sum _{2\le i<j\le r+s}}\{S(v_i\wedge v_j)-2\widehat{S}(v_i\wedge v_j)\}.\hfill \end{array}$$

Using analogous reasoning to that employed in Theorems 1 and 2, one can easily give the following results for a statistical submanifold in a statistical manifold of a quasi-constant curvature $\overline{B}(f,g,\mathcal{P})$.

Theorem 3.

Let $B{\times }_{\alpha }F$ be a statistical warped product submanifold in a statistical manifold $(\overline{B}(f,g,\mathcal{P}),\overline{D},G)$ of a quasi-constant curvature. Then, the scalar curvature of M satisfies

$$\begin{array}{ccc}\hfill scal& \le & 2\frac{\Delta \alpha }{r\alpha }+(\frac{(r+s+1)(r+s-2)-2}{2})f+(r+s-3)g\hfill \\ \hfill & \hfill & +\frac{(r+s)}{2}\widehat{C}+\frac{{(r+s)}^2(r+s-2)}{(r+s-1)}\left|\right|\widehat{H}{\left|\right|}^2\hfill \\ \hfill & \hfill & -\frac{{(r+s)}^2}{4}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2).\hfill \end{array}$$

(29)

Remark 3.

To prove the theorem mentioned above, first we find [7,8]

$$\begin{array}{ccc}\hfill (r+s)(r+s-1)f+2(r+s-1)g& =& 2scal-2{(r+s)}^2\left|\right|\widehat{H}{\left|\right|}^2\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{2}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & +2\left|\right|\widehat{T}{\left|\right|}^2-\frac{1}{2}{(|\left|T\right||}^2+\left|\right|T^{\prime }{\left|\right|}^2),\hfill \end{array}$$

(30)

and we then set δ as

$$\begin{array}{ccc}\hfill \delta & =& 2scal-2\frac{{(r+s)}^2(r+s-2)}{r+s-1}\left|\right|\widehat{H}{\left|\right|}^2\hfill \\ \hfill & \hfill & -(r+s+1)(r+s-2)f-2(r+s-2)g.\hfill \end{array}$$

In the demonstration of the following theorem, we employed Expression (30) and considered

$$\begin{array}{c}\hfill {\delta }^{*}=2scal-{(r+s)}^2\left|\right|\widehat{H}{\left|\right|}^2-(r+s)(r+s-1)f-2(r+s-1)g.\end{array}$$

(31)

Theorem 4.

Let $B{\times }_{\alpha }F$ be a statistical warped product submanifold in a statistical manifold $(\overline{B}(f,g,\mathcal{P}),\overline{D},G)$ of a quasi-constant curvature. Then, we have

$$\begin{array}{ccc}\hfill \frac{\Delta \alpha }{\alpha }& \le & (r+\frac{(r+s-1)(r+s-2)}{2s})f-\frac{g}{s}{\displaystyle \sum _{i=1}^r}{\displaystyle \sum _{j=r+1}^{r+s}}\{F{(v_i)}^2+F{(v_j)}^2\}\hfill \\ \hfill & \hfill & +\frac{g}{s}\{(r+s-1)+(r+s-2)(1-F{(v_1)}^2)\}\hfill \\ \hfill & \hfill & +\frac{{(r+s)}^2}{2s}\left|\right|\widehat{H}{\left|\right|}^2+\frac{{(r+s)}^2}{8s}{(|\left|H\right||}^2+\left|\right|H^{\prime }{\left|\right|}^2)\hfill \\ \hfill & \hfill & -\frac{1}{s}{\displaystyle \sum _{2\le i<j\le r+s}}\{S(v_i\wedge v_j)-2\widehat{S}(v_i\wedge v_j)+2\widehat{\overline{S}}(v_i\wedge v_j)\}.\hfill \end{array}$$

Remark 4.

As we know, statistical manifolds with a quasi-constant curvature generalize statistical manifolds with a constant curvature. As such, by particularizing $f=\epsilon$ and $g=0$ in the main results of this paper, it is expected that the known results for statistical submanifolds in statistical manifolds with a constant curvature will be recovered.

4. Conclusions and Some Remarks

The statistical manifold is a powerful and abstract concept that originated from the practical needs of statistical physics and evolved through the integration of differential geometry and information theory. It provides a rich mathematical framework for understanding and solving problems in statistics and related fields. The terminology reflects the geometric nature of the construction, emphasizing the manifold structure and the statistical properties of the space being studied.

The construction of statistical manifolds provides a unified framework to study various aspects of statistical inference, estimation, and hypothesis testing using geometric methods. By abstracting the notion of probability distributions into a manifold structure, researchers can apply a wide range of mathematical tools from differential geometry, leading to broader and more general results. This geometric approach has applications beyond statistics, in fields such as machine learning, signal processing, and quantum mechanics, where understanding the geometry of probability distributions is crucial.

The idea of treating these probability distributions geometrically began to take shape with the work of scientists like Boltzmann and Gibbs. However, it was not until the 20th century that the formal mathematical framework for this concept started to develop. The key idea was to consider a family of probability distributions parameterized by a set of parameters, which can be viewed as points on a manifold. The first significant step toward formalizing the concept of a statistical manifold came from R.A. Fisher’s work on the Fisher information matrix. The Fisher information matrix provides a way to measure the amount of information that an observable random variable carries about an unknown parameter. Mathematically, it can be used to define a Riemannian metric on the space of probability distributions, thus turning it into a differentiable manifold. The field of information geometry, pioneered by Amari and others, helped with further developing these ideas. Information geometry studies the differential–geometric properties of families of probability distributions, and it uses concepts such as connections and curvature from differential geometry. It provides a framework to understand the geometry of statistical models and inference processes.

The application of differential geometry to statistics was further advanced by C.R. Rao in the 1940s. Rao introduced the concept of the statistical manifold explicitly by using the Fisher information metric to define a Riemannian structure on the parameter space of probability distributions. This geometric perspective allowed for the use of tools from differential geometry to study statistical problems. The notion of a statistical manifold has its roots in statistical physics, where the focus is on the macroscopic behavior of systems with a large number of particles. In statistical physics, different macrostates of a system can be described by probability distributions over the possible microstates. For example, the probability distribution of the velocities of particles in a gas can be described by a Maxwell–Boltzmann distribution.

One of the most fundamental problems in the theory of submanifolds is determining whether a Riemannian manifold can be immersed (or not immersed) in a Euclidean m-space ${\mathbb{E}}^m$ (or, more generally, in a real space form with a constant curvature c). According to a well-known theorem by Nash, every Riemannian manifold can be isometrically immersed in some Euclidean space with sufficiently high codimension. Nash’s theorem was developed with the expectation that if Riemannian manifolds can be consistently viewed as Riemannian submanifolds, then it would allow for the use of extrinsic help. Nash’s theorem implies that every warped product can always be regarded as a Riemannian submanifold in some Euclidean space.

Building on Nash’s theorem, the research goal of this article is to explore the control of extrinsic quantities in relation to the intrinsic quantities of Riemannian manifolds, as well as to investigate their applications.

Remark 5.

There are many examples of warped product immersions into Riemannian manifolds. Inspired by such studies, some authors have extended the notion of warped product immersion to statistical warped product immersions (for example, see [10,16]).

Remark 6.

There exist many minimal isometric immersions from some warped products $B{\times }_{\alpha }F$ with a harmonic warping function α into a real space form. Thus, in our setting, we take F as a minimal (with respect to the Levi-Civita connection) statistical submanifold of the unit-hypersphere ${\mathbb{S}}^m$ (see [11]) in a trivial statistical manifold ${\mathbb{E}}^{m+1}$ of a constant curvature 0 (called Euclidean space). This is centered at the origin, and then the minimal cone over F with vertex at the origin of ${\mathbb{E}}^{m+1}$ is the statistical warped product of type ${\mathbb{R}}_{+}{\times }_{\alpha }F$ with a warping function α as a harmonic function. Note that the function α is used as the coordinate function of the statistical manifold ${\mathbb{R}}_{+}$ (a positive real line).