Casorati Inequalities for Spacelike Submanifolds in Sasaki-like Statistical Manifolds with Semi-Symmetric Metric Connection

Abstract

In this paper, we establish some inequalities between the normalized $\delta$-Casorati curvatures and the scalar curvature (i.e., between extrinsic and intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Moreover, we study the submanifolds satisfying the equality cases of these inequalities. We also present an appropriate example.

1. Introduction

The research problem of assessing relationships between the main intrinsic and extrinsic invariants of submanifolds was identified in the late 1990s [1]. The curvature invariants are well known for their beauty and diversity, so many solutions of this problem have been developed over the years based on some geometric inequalities involving these invariants (see, e.g., [2,3,4,5,6,7,8,9,10,11]).

First, this type of inequality was investigated by B.-Y. Chen [12] in 1993, leading later to the theory of Chen inequalities. In the last decade, there has been renewed focus on Casorati inequalities. Since the (Chen) δ-invariants are intrinsic, the normalized δ-Casorati curvatures defined in [13,14] are extrinsic invariants. In fact, the concept called the Casorati curvature of a surface means the expression introduced by F. Casorati in 1890 as $C=\frac{k_1^2+k_2^2}{2}$, with $k_1$ and $k_2$ the principal curvatures of the surface in ${\mathbb{E}}^3$ [15]. This approach corresponds better than the traditional curvatures (Gauss and mean curvature) with the common intuition of curvature [15] because $C=0$ if and only if $k_1=k_2=0$, i.e., the Casorati curvature accurately measures the deviation of surface from being planar.

There is a growing interest in the contemporary research on the Casorati curvature. In computer science and mechanics, the Casorati curvature is known as the bending energy [16]. Applications of the Casorati curvature in the modeling of vision are revealed by L. Verstraelen in [17]. Moreover, a model of the isotropical Casorati curvature of production surfaces is developed in [18]. Very recently, L. Verstraelen proved that a submanifold is minimal and Casorati normally umbilical if and only if its polar hyperquadrics of Rouxel are hyperspheres centered on the submanifold in their first normal spaces [19].

Many recent studies established inequalities involving the extrinsic $\delta$-Casorati curvatures of submanifolds in a statistical manifold (see, e.g., [4,5,6,7,8,9,11]). Introduced initially by S. Amari in the context of information geometry [20], the concept of statistical manifold is also studied in machine learning, physics [21,22] etc.

The notion of Sasaki-like statistical manifold was defined by K. Takano in [23] as an almost contact metric manifold of certain kind with a statistical structure. Since the geometry of submanifolds in such manifolds is an important research topic, some geometric inequalities were obtained as solutions of above problem in this ambient space (see, e.g., [24,25,26]).

A central issue in the theory of Riemannian manifolds is based on the semi-symmetric metric connection, concept defined by H.A. Hayden in [27] and characterized later by K. Yano in [28], T. Imai in [29,30], and Z. Nakao in [31]. Although considerable research have recently turned to the geometric inequalities concerning intrinsic and extrinsic invariants of submanifolds in some manifolds endowed with a semi-symmetric metric connection (see, e.g., [32,33,34,35]), rather less attention has been paid to the statistical manifolds endowed with a semi-symmetric metric connection. The curvature properties of statistical manifolds with a semi-symmetric metric connection and their submanifolds are achieved in [36,37]. Very recently, some Casorati inequalities for submanifolds in Kenmotsu statistical manifolds of constant $\varphi$-sectional curvature with semi-symmetric metric connection are established in [38].

The aim of this paper is to prove some inequalities involving the extrinsic invariants $\delta$-Casorati curvatures and the intrinsic invariant scalar curvature of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Furthermore, the submanifolds satisfying the equality cases of these inequalities (i.e., the Casorati ideal submanifolds) are examined. In addition, an appropriate example is revealed.

2. Preliminaries

Let ($\overline{M}$, g) be a semi-Riemannian manifold, equipped with a pair of affine connections ($\overline{\nabla }$, ${\overline{\nabla }}^{*}$) which are dual connections with respect to the metric tensor g, i.e., $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$ satisfy the formula:

$$X_{1}g(X_2,X_3)=g({\overline{\nabla }}_{X_1}{X}_2,X_3)+g(X_2,{\overline{\nabla }}_{X_1}^{*}{X}_3),$$

for any $X_1$, $X_2$, $X_3$$\in \Gamma \left(T\overline{M}\right)$, where $\Gamma \left(T\overline{M}\right)$ is the set of smooth tangent vector fields on $\overline{M}$.

If the torsion tensor field of $\overline{\nabla }$ vanishes and $\overline{\nabla }g$ is symmetric [39], then ($\overline{\nabla }$, g) is called a statistical structure on $\overline{M}$. Moreover, ($\overline{M}$, g, $\overline{\nabla }$) is named a statistical manifold.

Obviously, $(\overline{M}$, g, ${\overline{\nabla }}^{*})$ is also a statistical manifold, because ${\overline{\nabla }}^{*}$ is a torsion-free affine connection, ${\overline{\nabla }}^{*}g$ is symmetric and ${\left({\overline{\nabla }}^{*}\right)}^{*}=\overline{\nabla }$.

Furthermore, the dual connections $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$ satisfy the relation $\overline{\nabla }+{\overline{\nabla }}^{*}=2{\overline{\nabla }}^0$, where ${\overline{\nabla }}^0$ is the Levi-Civita connection on $\overline{M}$ [40].

Consider M a submanifold of a statistical manifold $(\overline{M}$, g, $\overline{\nabla })$, with g the induced metric on M, and ∇ the induced connection on M. Clearly, $(M,g,\nabla )$ is also a statistical manifold.

Denote by h and $h^{*}$ the imbedding curvature tensor of M in $\overline{M}$ with respect to $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$, respectively. The formulas of Gauss [39] are given by:

$$\begin{array}{c}\hfill {\overline{\nabla }}_{X_1}{X}_2={\nabla }_{X_1}{X}_2+h(X_1,X_2),\\ \hfill {\overline{\nabla }}_{X_1}^{*}{X}_2={\nabla }_{X_1}^{*}{X}_2+h^{*}(X_1,X_2),\end{array}$$

for any $X_1,X_2\in \Gamma \left(TM\right)$.

Next, denote by R, $R^{*}$, $\overline{R}$ and ${\overline{R}}^{*}$ the $(0,4)$-curvature tensors for the connections ∇, ${\nabla }^{*}$, $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$, respectively. In this respect, the Gauss equations for the connections $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$, respectively, are represented by [40]:

$$\begin{array}{ccc}\hfill g(\overline{R}(X_1,X_2)X_3,X_4)& =& g(R(X_1,X_2)X_3,X_4)+g(h(X_1,X_3),h^{*}(X_2,X_4))\hfill \\ & -& g(h^{*}(X_1,X_4),h(X_2,X_3)),\hfill \end{array}$$

(1)

and

$$\begin{array}{ccc}\hfill g({\overline{R}}^{*}(X_1,X_2)X_3,X_4)& =& g(R^{*}(X_1,X_2)X_3,X_4)+g(h^{*}(X_1,X_3),h(X_2,X_4))\hfill \\ & -& g(h(X_1,X_4),h^{*}(X_2,X_3)),\hfill \end{array}$$

(2)

for any $X_1,X_2,X_3,X_4\in \Gamma \left(TM\right)$.

In addition, denote by S and $\overline{S}$ the statistical curvature tensor field [39] on M and $\overline{M}$, respectively, defined by:

$$S(X_1,X_2)X_3=\frac{1}{2}\{R(X_1,X_2)X_3+R^{*}(X_1,X_2)X_3\},$$

(3)

for any $X_1,X_2,X_3\in \Gamma \left(TM\right)$, and

$$\overline{S}(X_1,X_2)X_3=\frac{1}{2}\{\overline{R}(X_1,X_2)X_3+{\overline{R}}^{*}(X_1,X_2)X_3\},$$

(4)

for any $X_1,X_2,X_3\in \Gamma \left(T\overline{M}\right)$.

Let $\overline{K}$ be a $(1,2)$-tensor field on $\overline{M}$ defined as follows:

$${\overline{K}}_{X_1}{X}_2=\overline{K}(X_1,X_2)={\overline{\nabla }}_{X_1}{X}_2-{\overline{\nabla }}_{X_1}^0{X}_2.$$

(5)

Clearly, $\overline{K}$ also satisfies:

$$\overline{K}(X_1,X_2)={\overline{\nabla }}_{X_1}^0{X}_2-{\overline{\nabla }}_{X_1}^{*}{X}_2=\frac{1}{2}({\overline{\nabla }}_{X_1}{X}_2-{\overline{\nabla }}_{X_1}^{*}{X}_2),$$

$$\overline{K}(X_1,X_2)=\overline{K}(X_2,X_1),$$

$$g(\overline{K}(X_1,X_2),X_3)=g(X_2,\overline{K}(X_1,X_3)).$$

Next, we will explain the notion of Sasaki-like statistical manifold.

First, consider $\overline{M}$ a $(2m+1)$-dimensional manifold with $\varphi$ a $(1,1)$-tensor field, $\xi$ a vector field and $\eta$ a 1-form on $\overline{M}$. The manifold $\overline{M}$ is called an almost contact manifold if it has an almost contact structure$(\varphi ,\xi ,\eta )$, i.e., $\varphi$, $\xi$, $\eta$ satisfy the following conditions, for any vector field $X_1$ on $\overline{M}$:

$$\eta \left(\xi \right)=1,{\varphi }^2{X}_1=-X_1+\eta \left(X_1\right)\xi .$$

(6)

A semi-Riemannian manifold $(\overline{M},g)$ with an almost contact structure $(\varphi ,\xi ,\eta )$ is named almost contact metric manifold if:

$$g(\varphi X_1,\varphi X_2)=g(X_1,X_2)-\eta \left(X_1\right)\eta \left(X_2\right),\eta \left(X_1\right)=g(X_1,\xi ),$$

(7)

for any vector fields $X_1$, $X_2$ on $\overline{M}$.

Moreover, a semi-Riemannian manifold $(\overline{M},g)$ with an almost contact structure $(\varphi ,\xi ,\eta )$ is said to be an almost contact metric manifold of certain kind denoted by $(\overline{M},g,\varphi ,\xi ,\eta )$ if $\overline{M}$ has a tensor field ${\varphi }^{*}$ of type $(1,1)$ such that:

$$g(\varphi X_1,X_2)+g(X_1,{\varphi }^{*}{X}_2)=0,$$

(8)

for any vector fields $X_1$, $X_2$ on $\overline{M}$. Obviously, we notice that:

$${\left({\varphi }^{*}\right)}^2{X}_1=-X_1+\eta \left(X_1\right)\xi ,$$

(9)

$$g(\varphi X_1,{\varphi }^{*}{X}_2)=g(X_1,X_2)-\eta \left(X_1\right)\eta \left(X_2\right).$$

(10)

An almost contact metric manifold of certain kind $(\overline{M},g,\varphi ,\xi ,\eta )$ with a statistical structure $(\overline{\nabla },g)$ is called a Sasaki-like statistical manifold [23] if the following conditions hold:

$$\begin{array}{c}\hfill {\overline{\nabla }}_{X_1}\xi =-\varphi X_1,\end{array}$$

(11)

$$\begin{array}{c}\hfill \left({\overline{\nabla }}_{X_1}\varphi \right)X_2=g(X_1,X_2)\xi -\eta \left(X_2\right)X_1.\end{array}$$

(12)

On the Sasaki-like statistical manifold denoted by $(\overline{M},\overline{\nabla },g,\varphi ,\xi ,\eta )$, we consider that the curvature tensor $\overline{R}$ of the connection $\overline{\nabla }$ satisfies:

$$\begin{array}{ccc}\hfill \overline{R}(X_1,X_2)X_3& =& \frac{c+3}{4}\{g(X_2,X_3)X_1-g(X_1,X_3)X_2)\}\hfill \\ \hfill & +& \frac{c-1}{4}[\eta \left(X_1\right)\eta \left(X_3\right)X_2-\eta \left(X_2\right)\eta \left(X_3\right)X_1+g(X_1,X_3)\eta \left(X_2\right)\xi \hfill \\ \hfill & -& g(X_2,X_3)\eta \left(X_1\right)\xi -g(X_2,\varphi X_3)\varphi X_1+g(X_1,\varphi X_3)\varphi X_2\hfill \\ \hfill & +& \{g(X_1,\varphi X_2)-g(\varphi X_1,X_2)\}\varphi X_3],\hfill \end{array}$$

(13)

for all vector fields $X_1$, $X_2$, $X_3$ on $\overline{M}$, where c is a real constant.

Next, we will present the concept of semi-symmetric metric connection in the context of statistical manifolds. Let $\tilde{\nabla }$ be a linear connection on $\overline{M}$. If the torsion tensor $\tilde{T}$ of $\tilde{\nabla }$ defined by

$$\tilde{T}(X_1,X_2)={\tilde{\nabla }}_{X_1}{X}_2-{\tilde{\nabla }}_{X_2}{X}_1-[X_1,X_2]$$

satisfies the expression:

$$\tilde{T}(X_1,X_2)=\eta \left(X_2\right)X_1-\eta \left(X_1\right)X_2,$$

(14)

for $X_1,X_2\in \Gamma \left(T\overline{M}\right)$ and $\eta$ a 1-form, then $\tilde{\nabla }$ is said to be a semi-symmetric connection. If the connection $\tilde{\nabla }$ satisfies in addition $\tilde{\nabla }g=0$, then $\tilde{\nabla }$ is called a semi-symmetric metric connection on $\overline{M}$ [28].

Let $(\overline{M},g,\overline{\nabla })$ be a statistical manifold endowed with a semi-symmetric metric connection $\tilde{\nabla }$. Then, according to Balgeshir et al. [37], we have:

$${\tilde{\nabla }}_{X_1}{X}_2={\overline{\nabla }}_{X_1}{X}_2+\eta \left(X_2\right)X_1-g(X_1,X_2)U-{\overline{K}}_{X_1}{X}_2,$$

(15)

$${\tilde{\nabla }}_{X_1}{X}_2={\overline{\nabla }}_{X_1}^{*}{X}_2+\eta \left(X_2\right)X_1-g(X_1,X_2)U-{\overline{K}}_{X_1}{X}_2,$$

(16)

with U a vector field fulfilling $g(U,X_1)=\eta \left(X_1\right)$, $\overline{K}$ a tensor field defined in (5), and $X_1,X_2$ vector fields on $\overline{M}$.

Suppose M an n-dimensional submanifold of a statistical manifold $\overline{M}$ endowed with a semi-symmetric metric connection $\tilde{\nabla }$. Denote ${\nabla }^{\prime }$ the induced connection and $h^{\prime }$ the second fundamental form on M with respect to $\tilde{\nabla }$. Then, the Gauss formula with respect to $\tilde{\nabla }$ is represented by:

$${\tilde{\nabla }}_{X_1}{X}_2={\nabla }_{X_1}^{\prime }{X}_2+h^{\prime }(X_1,X_2).$$

(17)

The Gauss equation with respect to $\tilde{\nabla }$ is as follows [37]:

$$\begin{array}{c}\hfill g(\tilde{R}(X_1,X_2)X_3,X_4)=g(R^{\prime }(X_1,X_2)X_3,X_4)+g(h^{\prime }(X_1,X_3),h^{\prime }(X_2,X_4))\\ \hfill -g(h^{\prime }(X_2,X_3),h^{\prime }(X_1,X_4)),\end{array}$$

(18)

where $\tilde{R}$ and $R^{\prime }$ are the curvature tensor fields associated to the connections $\tilde{\nabla }$ and ${\nabla }^{\prime }$, respectively.

The curvature tensor $\tilde{R}$ performs the following two equations in relation with the curvature tensors $\overline{R}$ and ${\overline{R}}^{*}$ [36]:

$$\begin{array}{c}\hfill \tilde{R}(X_1,X_2)X_3=\overline{R}(X_1,X_2)X_3+\{\eta \left(X_1\right)U-\eta \left(U\right)X_1-{\overline{\nabla }}_{X_1}U+\overline{K}(X_1,U)\}g(X_2,X_3)\\ \hfill +\{\eta \left(X_2\right)U-\eta \left(U\right)X_2-{\overline{\nabla }}_{X_2}U+\overline{K}(X_2,U)\}g(X_1,X_3)\\ \hfill -g(\eta \left(X_1\right)U-{\overline{\nabla }}_{X_1}U+\overline{K}(X_1,U),X_3)X_2+g(\eta \left(X_2\right)U-{\overline{\nabla }}_{X_2}U+\overline{K}(X_2,U),X_3)X_1-\\ \hfill \left({\overline{\nabla }}_{X_1}\overline{K}\right)(X_2,X_3)+\left({\overline{\nabla }}_{X_2}\overline{K}\right)(X_1,X_3)+\overline{K}(X_1,\overline{K}(X_2,X_3))-\overline{K}(X_2,\overline{K}(X_1,X_3)),\end{array}$$

(19)

and

$$\begin{array}{c}\hfill \tilde{R}(X_1,X_2)X_3={\overline{R}}^{*}(X_1,X_2)X_3+\{\eta \left(X_1\right)U-\eta \left(U\right)X_1-{\overline{\nabla }}_{X_1}^{*}U-\overline{K}(X_1,U)\}g(X_2,X_3)\\ \hfill -\{\eta \left(X_2\right)U-\eta \left(U\right)X_2-{\overline{\nabla }}_{X_2}^{*}U-\overline{K}(X_2,U)\}g(X_1,X_3)\\ \hfill -g(\eta \left(X_1\right)U-{\overline{\nabla }}_{X_1}^{*}U-\overline{K}(X_1,U),X_3)X_2+g(\eta \left(X_2\right)U-{\overline{\nabla }}_{X_2}^{*}U-\overline{K}(X_2,U),X_3)X_1+\\ \hfill \left({\overline{\nabla }}_{X_1}^{*}\overline{K}\right)(X_2,X_3)-\left({\overline{\nabla }}_{X_2}\overline{K}\right)(X_1,X_3)+\overline{K}(X_1,\overline{K}(X_2,X_3))-\overline{K}(X_2,\overline{K}(X_1,X_3)),\end{array}$$

(20)

for any $X_1,X_2,X_3\in \Gamma \left(T\overline{M}\right)$.

Since $h^{\prime }$ coincides with the second fundamental form of the Levi–Civita connection, according to Balgeshir et al. [37], then $h^{\prime }$ can be written:

$$h^{\prime }(X_1,X_2)=\frac{1}{2}\{h(X_1,X_2)+h^{*}(X_1,X_2)\}.$$

(21)

Denote by K a $(1,2)$-tensor field on the submanifold M defined by $K=\frac{1}{2}(\nabla -{\nabla }^{*})$, where ∇ and ${\nabla }^{*}$ are the induced statistical connections of the dual connections $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$, respectively.

Moreover, the induced connection ${\nabla }^{\prime }$ of $\tilde{\nabla }$ is a semi-symmetric metric connection too [37]. In this respect, the Gauss Equation (18) leads to the following equation in terms of the curvature tensor R of the induced connection ∇ on M:

$$\begin{array}{c}\hfill g(\tilde{R}(X_1,X_2)X_3,X_4)=g(R(X_1,X_2)X_3,X_4)\\ \hfill +\{\eta \left(X_1\right)\eta \left(X_4\right)-\eta \left(U\right)g(X_1,X_4)-g({\nabla }_{X_1}U,X_4)+g(K_{X_1}U,X_4)\}g(X_2,X_3)\\ \hfill +\{\eta \left(X_2\right)\eta \left(X_4\right)-\eta \left(U\right)g(X_2,X_4)-g({\nabla }_{X_2}U,X_4)+g(K_{X_2}U,X_4)\}g(X_1,X_3)\\ \hfill -g(\eta \left(X_1\right)U-{\nabla }_{X_1}U+K(X_1,U),X_3)g(X_2,X_4)\\ \hfill +g(\eta \left(X_2\right)U-{\nabla }_{X_2}U+K(X_2,U),X_3)g(X_1,X_4)\\ \hfill -g(\left({\nabla }_{X_1}K\right)(X_2,X_3),X_4)+g(\left({\nabla }_{X_2}K\right)(X_1,X_3),X_4)\\ \hfill +g(K_{X_1}K(X_2,X_3),X_4)-g(K_{X_2}K(X_1,X_3),X_4)\\ \hfill -\frac{1}{4}g(h(X_1,X_4)+h^{*}(X_1,X_4),h(X_2,X_3)+h^{*}(X_2,X_3))\\ \hfill +\frac{1}{4}g(h(X_1,X_3)+h^{*}(X_1,X_3),h(X_2,X_4)+h^{*}(X_2,X_4)).\end{array}$$

(22)

In a similar manner, using the curvature tensor $R^{*}$ of the connection ${\nabla }^{*}$ on M in the Gauss Equation (18), we obtain:

$$\begin{array}{c}\hfill g(\tilde{R}(X_1,X_2)X_3,X_4)=g(R^{*}(X_1,X_2)X_3,X_4)\\ \hfill +\{\eta \left(X_1\right)\eta \left(X_4\right)-\eta \left(U\right)g(X_1,X_4)-g({\nabla }_{X_1}^{*}U,X_4)+g(K_{X_1}U,X_4)\}g(X_2,X_3)\\ \hfill +\{\eta \left(X_2\right)\eta \left(X_4\right)-\eta \left(U\right)g(X_2,X_4)-g({\nabla }_{X_2}^{*}U,X_4)+g(K_{X_2}U,X_4)\}g(X_1,X_3)\\ \hfill -g(\eta \left(X_1\right)U-{\nabla }_{X_1}^{*}U+K(X_1,U),X_3)g(X_2,X_4)\\ \hfill +g(\eta \left(X_2\right)U-{\nabla }_{X_2}^{*}U+K(X_2,U),X_3)g(X_1,X_4)\\ \hfill -g(\left({\nabla }_{X_1}^{*}K\right)(X_2,X_3),X_4)+g(\left({\nabla }_{X_2}^{*}K\right)(X_1,X_3),X_4)\\ \hfill +g(K_{X_1}K(X_2,X_3),X_4)-g(K_{X_2}K(X_1,X_3),X_4)\\ \hfill -\frac{1}{4}g(h(X_1,X_4)+h^{*}(X_1,X_4),h(X_2,X_3)+h^{*}(X_2,X_3))\\ \hfill +\frac{1}{4}g(h(X_1,X_3)+h^{*}(X_1,X_3),h(X_2,X_4)+h^{*}(X_2,X_4)).\end{array}$$

(23)

Forwards, we recall the definitions of the intrinsic and extrinsic curvature invariants on the submanifold involved in our research.

For a non-degenerate two-dimensional subspace $\pi$ of the tangent space $T_{x}M$, at a point $x\in M$, the sectional curvature $\sigma$ of $(M,\nabla ,g)$ [39] is given by:

$$\begin{array}{c}\hfill \sigma \left(\pi \right)=\sigma (X_1\wedge X_2)=\frac{g(S(X_1,X_2)X_2,X_1)}{g(X_1,X_1)g(X_2,X_2)-g^2(X_1,X_2)},\end{array}$$

(24)

where $\{X_1,X_2\}$ is a basis of $\pi$.

The scalar curvature $\tau$ of $(M,\nabla ,g)$ at a point $x\in M$ is defined by:

$$\begin{array}{c}\hfill \tau \left(x\right)=\sum _{1\le i<j\le n}\sigma (e_i\wedge e_j)=\sum _{1\le i<j\le n}g(S(e_i,e_j)e_j,e_i),\end{array}$$

(25)

where $\{e_1,\dots ,e_n\}$ is an orthonormal basis at x. On the other hand, the normalized scalar curvature $\rho$ of $(M,\nabla ,g)$ at a point $x\in M$ is given by

$$\rho \left(x\right)=\frac{2\tau \left(x\right)}{n(n-1)}.$$

(26)

The mean curvature vector fields of M are defined by, respectively:

$$H=\frac{1}{n}\sum _{i=1}^{n}h(e_i,e_i),H^{*}=\frac{1}{n}\sum _{i=1}^n{h}^{*}(e_i,e_i).$$

Then, we have $2h^0=h+h^{*}$ and $2H^0=H+H^{*}$, where $h^0$ and $H^0$ are the second fundamental form and the mean curvature field of M, respectively, with respect to the Levi–Civita connection ${\nabla }^0$ on M.

The squared mean curvatures of the submanifold M in $\overline{M}$ are given by:

$${\parallel H\parallel }^2=\frac{1}{n^2}\sum _{\alpha =n+1}^{2m+1}{\left(\sum _{i=1}^n{h}_{ii}^{\alpha }\right)}^2,{\parallel H^{*}\parallel }^2=\frac{1}{n^2}\sum _{\alpha =n+1}^{2m+1}{\left(\sum _{i=1}^n{h}_{ii}^{*\alpha }\right)}^2,$$

where $h_{ij}^{\alpha }=g(h(e_i,e_j),e_{\alpha })$ and $h_{ij}^{*\alpha }=g(h^{*}(e_i,e_j),e_{\alpha })$, for $i,j\in \{1,\dots ,n\}$, $\alpha \in \{n+1,\dots ,2m+1\}$.

The Casorati curvatures of the submanifold M in $\overline{M}$ are defined by the squared norms of h and $h^{*}$ over the dimension n, denoted by $\mathcal{C}$ and ${\mathcal{C}}^{*}$, respectively, as follows:

$$\mathcal{C}=\frac{1}{n}{\parallel h\parallel }^2=\frac{1}{n}\sum _{\alpha =n+1}^{2m+1}\sum _{i,j=1}^n{\left(h_{ij}^{\alpha }\right)}^2,$$

$${\mathcal{C}}^{*}=\frac{1}{n}{\parallel h^{*}\parallel }^2=\frac{1}{n}\sum _{\alpha =n+1}^{2m+1}\sum _{i,j=1}^n{\left(h_{ij}^{*\alpha }\right)}^2.$$

Let L be an s-dimensional subspace of $T_{x}M$, $s\ge 2$ and let $\{e_1,\dots ,e_s\}$ be an orthonormal basis of L. Then the Casorati curvatures $\mathcal{C}\left(L\right)$ and ${\mathcal{C}}^{*}\left(L\right)$ of L are given by:

$$\mathcal{C}\left(L\right)=\frac{1}{s}\sum _{\alpha =n+1}^{2m+1}\sum _{i,j=1}^s{\left(h_{ij}^{\alpha }\right)}^2,{\mathcal{C}}^{*}\left(L\right)=\frac{1}{s}\sum _{\alpha =n+1}^{2m+1}\sum _{i,j=1}^s{\left(h_{ij}^{*\alpha }\right)}^2.$$

The normalized δ-Casorati curvatures ${\delta }_{\mathcal{C}}(n-1)$ and ${\widehat{\delta }}_{\mathcal{C}}(n-1)$ of the submanifold M are given by:

$${\delta }_{\mathcal{C}}{(n-1)|}_x=\frac{1}{2}\mathcal{C}{\mid }_x+\frac{n+1}{2n}\inf \left\{\mathcal{C}\left(L\right)\right|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\}$$

and

$${\widehat{\delta }}_{\mathcal{C}}{(n-1)|}_x=2\mathcal{C}{\mid }_x-\frac{2n-1}{2n}\sup \left\{\mathcal{C}\left(L\right)\right|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\}.$$

Furthermore, the dual normalized ${\delta }^{*}$-Casorati curvatures ${\delta }_{\mathcal{C}}^{*}(n-1)$ and ${\widehat{\delta }}_{\mathcal{C}}^{*}(n-1)$ of the submanifold M in $\overline{M}$ are defined as follows:

$${\delta }_{\mathcal{C}}^{*}{(n-1)|}_x=\frac{1}{2}{\mathcal{C}}^{*}{\mid }_x+\frac{n+1}{2n}\inf \left\{{\mathcal{C}}^{*}\left(L\right)\right|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\}$$

and

$${\widehat{\delta }}_{\mathcal{C}}^{*}{(n-1)|}_x=2{\mathcal{C}}^{*}{\mid }_x-\frac{2n-1}{2n}\sup \left\{{\mathcal{C}}^{*}\left(L\right)\right|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\}.$$

The generalized normalized δ-Casorati curvatures ${\delta }_{\mathcal{C}}(r;n-1)$ and ${\widehat{\delta }}_{\mathcal{C}}(r;n-1)$ of M in $\overline{M}$ are defined in [14] by:

$${\delta }_{\mathcal{C}}{(r;n-1)|}_x=r\mathcal{C}{\mid }_x+a\left(r\right)\inf \{\mathcal{C}\left(L\right)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},$$

if $0<r<n(n-1)$, and

$${\widehat{\delta }}_{\mathcal{C}}{(r;n-1)|}_x=r\mathcal{C}{\mid }_x+a\left(r\right)\sup \{\mathcal{C}\left(L\right)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},$$

if $r>n(n-1)$, where $a\left(r\right)$ is set as

$$\begin{array}{c}\hfill a\left(r\right)=\frac{(n-1)(r+n)(n^2-n-r)}{nr},\end{array}$$

for any positive real number r, different from $n(n-1)$.

Moreover, the dual generalized normalized ${\delta }^{*}$-Casorati curvatures ${\delta }_{\mathcal{C}}^{*}(r;n-1)$ and ${\widehat{{\delta }^{*}}}_{\mathcal{C}}(r;n-1)$ of the submanifold M in $\overline{M}$ are given by:

$${\delta }_{\mathcal{C}}^{*}{(r;n-1)|}_x=r{\mathcal{C}}^{*}{\mid }_x+a\left(r\right)\inf \{{\mathcal{C}}^{*}\left(L\right)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},$$

if $0<r<n(n-1)$, and

$${\widehat{\delta }}_{\mathcal{C}}^{*}{(r;n-1)|}_x=r{\mathcal{C}}^{*}{\mid }_x+a\left(r\right)\sup \{{\mathcal{C}}^{*}\left(L\right)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},$$

if $r>n(n-1)$, where $a\left(r\right)$ is expressed above.

We remind that any vector field $X\in \Gamma \left(TM\right)$ can be decomposed uniquely into its tangent and normal components $PX$ and $PY$, respectively, as follows:

$$\varphi X=PX+FX.$$

Next, we suppose the following constrained extremum problem

$$\underset{x\in M}{min}f\left(x\right),$$

(27)

where M is a submanifold of a Riemannian manifold $(\overline{M},g)$, and $f:\overline{M}\to \mathbb{R}$ is a function of differentiability class $C^2$. We recall the following theorem, used later in our methodology.

Theorem 1

([41]). If the Riemannian submanifold M is complete and connected, $\left(gradf\right)\left(x_0\right)\in T_{x_0}^{\perp }M$ for a point $x_0\in M$, and the bilinear form $\mathcal{V}:T_{x_0}M\times T_{x_0}M\to \mathbb{R}$ defined by:

$$\mathcal{V}(X_1,X_2)=\mathrm{Hess}\left(f\right)(X_1,X_2)+g(\widehat{h}(X_1,X_2),\mathrm{grad}f),$$

(28)

is positive definite in $x_0$, then $x_0$ is the optimal solution of the problem (27), where $\widehat{h}$ is the second fundamental form of M.

Remark 1.

If the bilinear form $\mathcal{V}$ defined by (28) is positive semi-definite on the submanifold M, then the critical points of $f|M$ are global optimal solutions of the problem (27).

3. Main Inequalities

Theorem 2.

Let M be an n-dimensional ($n\ge 2$) spacelike statistical submanifold of a $(2m+1)$-dimensional Sasaki-like statistical manifold ($\overline{M},\overline{\nabla },g,\varphi ,\xi ,\eta$) such that $\overline{M}$ satisfies Equation (13), endowed with a semi-symmetric metric connection $\tilde{\nabla }$, with ξ a tangent vector field on M. Then the following inequalities hold:

(i) 

$$\begin{array}{ccc}\hfill {\delta }_C^0(r;n-1)& \ge & 2\tau -\frac{c+3}{4}n(n-1)+\frac{(c-1)(n-1)}{2}-\frac{c-1}{2}{\parallel P\parallel }^2\hfill \\ & +& \frac{c-1}{4}\{{\left[trace\left(P\right)\right]}^2+trace\left(P^2\right)\}+2(n-1)trace\left(\lambda \right)\hfill \end{array}$$

(29)

for any real number r such that $0<r<n(n-1)$, where ${\delta }_{\mathcal{C}}^0(r;n-1)$ is defined by $2{\delta }_{\mathcal{C}}^0(r;n-1)={\delta }_{\mathcal{C}}(r;n-1)+{\delta }_{\mathcal{C}}^{*}(r;n-1)$, and

(ii) 

$$\begin{array}{ccc}\hfill {\widehat{\delta }}_C^0(r;n-1)& \ge & 2\tau -\frac{c+3}{4}n(n-1)+\frac{(c-1)(n-1)}{2}-\frac{c-1}{2}{\parallel P\parallel }^2\hfill \\ & +& \frac{c-1}{4}\{{\left[trace\left(P\right)\right]}^2+trace\left(P^2\right)\}+2(n-1)trace\left(\lambda \right)\hfill \end{array}$$

(30)

for any real number r such that $r>n(n-1)$, where ${\widehat{\delta }}_{\mathcal{C}}^0(r;n-1)$ is defined by $2{\widehat{\delta }}_{\mathcal{C}}^0(r;n-1)={\widehat{\delta }}_{\mathcal{C}}(r;n-1)+{\widehat{\delta }}_{\mathcal{C}}^{*}(r;n-1).$

Moreover, the equality cases of the inequalities (29) and (30) are satisfied at all points $p\in M$ if and only if:

$$\begin{array}{c}\hfill h+h^{*}=0.\end{array}$$

Proof. 

Using the definitions (4), (5) in the Equations (19) and (20), the curvature tensor $\tilde{R}$ of $\tilde{\nabla }$ becomes:

$$\begin{array}{c}\hfill \tilde{R}(X_1,X_2)X_3=\overline{S}(X_1,X_2)X_3+\{\eta \left(X_1\right)U-\eta \left(U\right)X_1-{\overline{\nabla }}_{X_1}^{0}U\}g(X_2,X_3)\\ \hfill -\{\eta \left(X_2\right)U-\eta \left(U\right)X_2-{\overline{\nabla }}_{X_2}^{0}U\}g(X_1,X_3)\\ \hfill -g(\eta \left(X_1\right)U-{\overline{\nabla }}_{X_1}^{0}U,X_3)Y+g(\eta \left(X_2\right)U-{\overline{\nabla }}_{X_2}^{0}U,X_3)X_1.\\ \hfill \end{array}$$

(31)

From the expression (31), we obtain:

$$\begin{array}{c}\hfill g(\tilde{R}(X_1,X_2)X_3,X_4)=g(\overline{S}(X_1,X_2)X_3,X_4)-{\lambda }_1(X_2,X_3)g(X_1,X_4)\\ \hfill +{\lambda }_1(X_1,X_3)g(X_2,X_4)-{\lambda }_1(X_1,X_4)g(X_2,X_3)+{\lambda }_1(X_2,X_4)g(X_1,X_3),\end{array}$$

(32)

for any $X_1,X_2,X_3,X_4\in \Gamma \left(T\overline{M}\right)$, where ${\lambda }_1$ is given by:

$${\lambda }_1(X_1,X_2)=\left({\overline{\nabla }}_{X_1}^0\eta \right)X_2-\eta \left(X_1\right)\eta \left(X_2\right)+\frac{1}{2}\eta \left(U\right)g(X_1,X_2).$$

Let $\{e_1,\dots ,e_n=\xi \}$ and $\{e_{n+1},\dots ,e_{2m+1}\}$ be orthonormal bases of $T_{x}M$ and $T_x^{\perp }M$, respectively, with $x\in M$. Considering $X_1=X_4=e_i$ and $X_2=X_3=e_j$ ($i\ne j$; $i,j\in \{1,\dots ,n\}$) in the relation (32) such that the formula (13) is satisfied, we obtain:

$$\begin{array}{c}\hfill g(\tilde{R}(e_i,e_j)e_j,e_i)=\frac{c+3}{4}+\frac{c-1}{4}\{g^2(e_i,P{e}_j)+g^2(e_j,P{e}_i)\\ \hfill -{\eta }^2\left(e_i\right)-{\eta }^2\left(e_j\right)-g(e_i,P{e}_i)g(e_j,P{e}_j)-g(e_i,P{e}_j)g(e_j,P{e}_i)\}\\ \hfill -{\lambda }_1(e_i,e_i)-{\lambda }_1(e_j,e_j).\end{array}$$

(33)

Furthermore, the Gauss formulas (22) and (23) imply:

$$\begin{array}{c}\hfill g(\tilde{R}(X_1,X_2)X_3,X_4)=g(S(X_1,X_2)X_3,X_4)-{\lambda }_2(X_2,X_3)g(X_1,X_4)\\ \hfill +{\lambda }_2(X_1,X_3)g(X_2,X_4)-{\lambda }_2(X_1,X_4)g(X_2,X_3)+{\lambda }_2(X_2,X_4)g(X_1,X_3)\\ \hfill -\frac{1}{4}g(h(X_1,X_4)+h^{*}(X_1,X_4),h(X_2,X_3)+h^{*}(X_2,X_3))\\ \hfill +\frac{1}{4}g(h(X_1,X_3)+h^{*}(X_1,X_3),h(X_2,X_4)+h^{*}(X_2,X_4)),\end{array}$$

(34)

for any $X_1,X_2,X_3,X_4\in \Gamma \left(TM\right)$, where ${\lambda }_2$ is given by:

$${\lambda }_2(X_1,X_2)=\left({\nabla }_{X_1}^0\eta \right)X_2-\eta \left(X_1\right)\eta \left(X_2\right)+\frac{1}{2}\eta \left(U\right)g(X_1,X_2),$$

where ${\nabla }^0$ is the Levi–Civita connection on M. Denote by $\lambda (X_1,X_2)$ the following expression:

$$\lambda (X_1,X_2)={\lambda }_1(X_1,X_2)-{\lambda }_2(X_1,X_2)=\left({\overline{\nabla }}_{X_1}^0\eta \right)X_2-\left({\nabla }_{X_1}^0\eta \right)X_2,$$

for any $X_1,X_2\in \Gamma \left(TM\right)$.

On the other hand, supposing $X_1=X_4=e_i$ and $X_2=X_3=e_j$ in (34) we obtain:

$$\begin{array}{c}\hfill g(\tilde{R}(e_i,e_j)e_j,e_i)=g(S(e_i,e_j)e_j,e_i)-{\lambda }_2(e_i,e_i)-{\lambda }_2(e_j,e_j)\\ \hfill -\frac{1}{4}g(h(e_i,e_i)+h^{*}(e_i,e_i),h(e_j,e_j)+h^{*}(e_j,e_j))\\ \hfill +\frac{1}{4}g(h(e_i,e_j)+h^{*}(e_i,e_j),h(e_j,e_i)+h^{*}(e_j,e_i)).\end{array}$$

(35)

From (33) and (35), we obtain:

$$\begin{array}{c}\hfill \frac{c+3}{4}+\frac{c-1}{4}\{g^2(e_i,P{e}_j)+g^2(e_j,P{e}_i)\\ \hfill -{\eta }^2\left(e_i\right)-{\eta }^2\left(e_j\right)-g(e_i,P{e}_i)g(e_j,P{e}_j)-g(e_i,P{e}_j)g(e_j,P{e}_i)\}\\ \hfill -{\lambda }_1(e_i,e_i)-{\lambda }_1(e_j,e_j)+{\lambda }_2(e_i,e_i)+{\lambda }_2(e_j,e_j)=\\ \hfill g(S(e_i,e_j)e_j,e_i)-\frac{1}{4}g(h(e_i,e_i)+h^{*}(e_i,e_i),h(e_j,e_j)+h^{*}(e_j,e_j))\\ \hfill +\frac{1}{4}g(h(e_i,e_j)+h^{*}(e_i,e_j),h(e_j,e_i)+h^{*}(e_j,e_i)).\end{array}$$

(36)

By summing over $1\le i,j\le n$, it follows from (36) that:

$$\begin{array}{c}\hfill \frac{c+3}{4}n(n-1)+\frac{c-1}{4}{\{2\parallel P\parallel }^2-{\left(trace\left(P\right)\right)}^2-trace\left(P^2\right)-2n+2\}-2(n-1)trace\left(\lambda \right)=\\ \hfill 2\tau +n{\mathcal{C}}^0-n^2{\parallel H^0\parallel }^2,\end{array}$$

(37)

where ${\parallel P\parallel }^2$ is the squared norm of P defined by

$${\parallel P\parallel }^2=\sum _{1\le i,j\le n}{g}^2(P{e}_i,e_j).$$

Let $\mathcal{P}$ be a quadratic polynomial in the components of the second fundamental form given by:

$$\begin{array}{c}\hfill \mathcal{P}=r{\mathcal{C}}^0+a\left(r\right){\mathcal{C}}^0\left(L\right)+\frac{c+3}{4}n(n-1)+\frac{c-1}{2}{\parallel P\parallel }^2-\frac{(c-1)(n-1)}{2}\\ \hfill -\frac{c-1}{4}\{{\left(trace\left(P\right)\right)}^2+trace\left(P^2\right)\}-2(n-1)trace\left(\lambda \right)-2\tau .\end{array}$$

(38)

Without loss of generality, suppose that the hyperplane L is spanned by the tangent vectors $e_1,e_2,\dots ,e_{n-1}$. It follows that the latter equation implies:

$$\begin{array}{c}\hfill \mathcal{P}=(r+n){\mathcal{C}}^0+a\left(r\right){\mathcal{C}}^0\left(L\right)-n^2{\parallel H^0\parallel }^2.\end{array}$$

Furthermore, $\mathcal{P}$ becomes:

$$\begin{array}{c}\hfill \mathcal{P}=\sum _{\alpha =n+1}^{2m+1}\left[\frac{n+r}{n}\sum _{i,j=1}^n{\left(h_{ij}^{0\alpha }\right)}^2+\frac{a\left(r\right)}{n-1}\sum _{i,j=1}^{n-1}{\left(h_{ij}^{0\alpha }\right)}^2-{\left(\sum _{i=1}^n{h}_{ii}^{0\alpha }\right)}^2\right].\end{array}$$

(39)

The expression of $\mathcal{P}$ from (39) can also be represented by:

$$\begin{array}{ccc}\hfill \mathcal{P}& =& \sum _{\alpha =n+1}^{2m+1}\{\left[\frac{2(n+r)}{n}+\frac{2a\left(r\right)}{n-1}\right]\sum _{1\le i<j\le n-1}{\left(h_{ij}^{0\alpha }\right)}^2+\frac{2(n+r)}{n}\sum _{i=1}^{n-1}{\left(h_{in}^{0\alpha }\right)}^2\hfill \\ & & +\frac{r(n-1)+a\left(r\right)n}{n(n-1)}\sum _{i=1}^{n-1}{\left(h_{ii}^{0\alpha }\right)}^2-2\sum _{1\le i<j\le n}{h}_{ii}^{0\alpha }{h}_{jj}^{0\alpha }+\frac{r}{n}{\left(h_{nn}^{0\alpha }\right)}^2\}.\hfill \end{array}$$

Immediately, we obtain:

$$\begin{array}{ccc}\hfill \mathcal{P}& \ge & \sum _{\alpha =n+1}^{2m+1}\left[\frac{r(n-1)+a\left(r\right)n}{n(n-1)}\sum _{i=1}^{n-1}{\left(h_{ii}^{0\alpha }\right)}^2+\frac{r}{n}{\left(h_{nn}^{0\alpha }\right)}^2-2\sum _{1\le i<j\le n}{h}_{ii}^{0\alpha }{h}_{jj}^{0\alpha }\right].\hfill \end{array}$$

Next, we will use a method based on optimization theory.

First, we study the constrained extremum problem

$$min{f}_{\alpha },$$

where $f_{\alpha }$ is a quadratic form defined by $f_{\alpha }:{\mathbb{R}}^n\to \mathbb{R}$, for $\alpha \in \{n+1,\dots ,2m+1\}$:

$$\begin{array}{ccc}\hfill f_{\alpha }(h_{11}^{0\alpha },h_{22}^{0\alpha },\dots ,h_{nn}^{0\alpha })& =& \frac{r(n-1)+a\left(r\right)n}{n(n-1)}\sum _{i=1}^{n-1}{\left(h_{ii}^{0\alpha }\right)}^2\hfill \\ & & +\frac{r}{n}{\left(h_{nn}^{0\alpha }\right)}^2-2\sum _{1\le i<j\le n}{h}_{ii}^{0\alpha }{h}_{jj}^{0\alpha },\hfill \end{array}$$

with the constraint Q:

$$Q:h_{11}^{0\alpha }+h_{22}^{0\alpha }+\dots +h_{nn}^{0\alpha }=k^{\alpha },k^{\alpha }\in \mathbb{R}.$$

(40)

We solve the following first order partial derivatives system:

$$\left\{\begin{array}{c}\frac{\partial f_{\alpha }}{\partial h_{ii}^{0\alpha }}=\frac{2\left[r\right(n-1)+a(r)n+n(n-1\left)\right]}{n(n-1)}{h}_{ii}^{0\alpha }-2\sum _{k=1}^n{h}_{kk}^{0\alpha }=0\hfill \\ \frac{\partial f_{\alpha }}{\partial h_{nn}^{0\alpha }}=\frac{2r}{n}{h}_{nn}^{0\alpha }-2\sum _{k=1}^{n-1}{h}_{kk}^{0\alpha }=0,\hfill \end{array}\right.$$

for all $i\in \{1,\dots ,n-1\}$, $\alpha \in \{n+1,\dots ,2m+1\}$.

Then, we obtain the constrained critical point:

$$h_{ii}^{0\alpha }=\frac{n(n-1)k^{\alpha }}{a\left(r\right)n+(n-1)r+n(n-1)},$$

$$h_{nn}^{0\alpha }=\frac{n{k}^{\alpha }}{r+n},$$

for all $i\in \{1,\dots ,n-1\}$, $\alpha \in \{n+1,\dots ,2m+1\}$.

For $x\in$ Q, we define the two-form $\mathcal{V}:T_{x}Q\times T_{x}Q\to \mathbb{R}$ by:

$$\mathcal{V}(X,Y)=\mathrm{Hess}\left(f_{\alpha }\right)(X,Y)+\langle \widehat{h}(X,Y),\left(\mathrm{grad}{f}_{\alpha }\right)\left(x\right)\rangle ,$$

where $\widehat{h}$ is the second fundamental form of Q in ${\mathbb{R}}^n$ and $\langle ·,·\rangle$ is the standard inner product on ${\mathbb{R}}^n$.

The Hessian of $f_{\alpha }$ stands for:

$$\mathrm{Hess}\left(f_{\alpha }\right)=\left(\begin{array}{ccccc}a& -2& \cdots & -2& -2\\ -2& a& \cdots & -2& -2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -2& -2& \cdots & a& -2\\ -2& -2& \cdots & -2& \frac{2r}{n}\end{array}\right),$$

with a a real constant given by $a=\frac{2\left[\right(n-1)r+a(r\left)n\right]}{n(n-1)}$.

Let $X=(X_1,\dots ,X_n)$ be a tangent vector field to the hyperplane Q at x. Then, $\mathcal{V}(X,X)$ can be written:

$$\mathcal{V}(X,X)=a\sum _{i=1}^{n-1}{X}_i^2+\frac{2r}{n}{X}_n^2-4\sum _{1\le i<j\le n}{X}_i{X}_j.$$

(41)

Suppose that X satisfies the condition ${\sum }_{i=1}^n{X}_i=0$. Then, (41) becomes:

$$\mathcal{V}(X,X)=a\sum _{i=1}^{n-1}{X}_i^2+\frac{2r}{n}{X}_n^2+2\sum _{i=1}^n{X}_i^2\ge 0.$$

(42)

By using the Remark 1, we notice that the critical point $(h_{11}^{0\alpha },\dots ,h_{nn}^{0\alpha })$ is the global minimum point of the problem. Since $f_{\alpha }(h_{11}^{0\alpha },\dots ,h_{nn}^{0\alpha })=0$, it follows that $\mathcal{P}\ge 0$, i.e., the inequalities (29) and (30) hold in terms of normalized generalized $\delta$-Casorati curvatures.

In the end, we investigate the equality cases of the inequalities (29) and (30). In this respect, we solve the following system of equations:

$$\left\{\begin{array}{c}\frac{\partial \mathcal{P}}{\partial h_{ii}^{0\alpha }}=2\left[\frac{n+r}{n}+\frac{a\left(r\right)}{n-1}-1\right]h_{ii}^{0\alpha }-2\sum _{k\ne i,k=1}^n{h}_{kk}^{0\alpha }=0,\hfill \\ \frac{\partial \mathcal{P}}{\partial h_{nn}^{0\alpha }}=\frac{2r}{n}{h}_{nn}^{0\alpha }-2\sum _{k=1}^{n-1}{h}_{kk}^{0\alpha }=0,\hfill \\ \frac{\partial \mathcal{P}}{\partial h_{ij}^{0\alpha }}=4\left[\frac{n+r}{n}+\frac{a\left(r\right)}{n-1}\right]h_{ij}^{0\alpha }=0,i\ne j,\hfill \\ \frac{\partial \mathcal{P}}{\partial h_{in}^{0\alpha }}=\frac{4(n+r)}{n}{h}_{in}^{0\alpha }=0.\hfill \end{array}\right.$$

The solutions represent the critical points of $\mathcal{P}$ given by $h^c=h_{ij}^{0\alpha }=0$, for all $i,j\in \{1,\dots ,n\}$ and $\alpha \in \{n+1,\dots ,2m+1\}$. Furthermore, $h^c$ is a minimum point for $\mathcal{P}$ because $\mathcal{P}\ge 0$ and $\mathcal{P}\left(h^c\right)=0$. Consequently, the equality cases of the inequalities (29) and (30) hold if and only if $h_{ij}^{\alpha }=-h_{ij}^{*\alpha },$ for $i,j$$\in \{1,\dots ,n\}$, $\alpha \in \{n+1,\dots ,2m+1\}$. □

Similarly to Theorem 2, we establish the following inequalities in terms of the normalized $\delta$-Casorati curvatures ${\delta }_{\mathcal{C}}(n-1)$, ${\delta }_{\mathcal{C}}^{*}(n-1)$, ${\widehat{\delta }}_{\mathcal{C}}(n-1)$, ${\widehat{\delta }}_{\mathcal{C}}^{*}(n-1)$ (extrinsic invariants), and the normalized scalar curvature $\rho$ (intrinsic invariant) of the submanifold M. The proof is clear, since we obtain the inequalities (43) and (44) considering $r=\frac{n(n-1)}{2}$ in (29), and $r=2n(n-1)$ in (30), respectively.

Theorem 3.

Let M be an n-dimensional spacelike statistical submanifold of a $(2m+1)$-dimensional Sasaki-like statistical manifold (($\overline{M},\overline{\nabla },g,\varphi ,\xi ,\eta$) satisfying the Equation (13), endowed with a semi-symmetric metric connection $\tilde{\nabla }$, with ξ a tangent vector field on M. Then, the following inequalities hold:

(i) 

$$\begin{array}{c}\hfill {\delta }_C^0(n-1)\ge \rho -\frac{{(c-1)\parallel P\parallel }^2}{2n(n-1)}+\frac{c-1}{2n}-\frac{c+3}{4}\\ \hfill +\frac{c-1}{4n(n-1)}[{\left(trace\left(P\right)\right)}^2+trace\left(P^2\right)]+\frac{2}{n}trace\left(\lambda \right),\end{array}$$

(43)

where ${\delta }_{\mathcal{C}}^0(n-1)$ is expressed by $2{\delta }_{\mathcal{C}}^0(n-1)={\delta }_{\mathcal{C}}(n-1)+{\delta }_{\mathcal{C}}^{*}(n-1)$, and

(ii) 

$$\begin{array}{c}\hfill {\widehat{\delta }}_C^0(n-1)\ge \rho -\frac{{(c-1)\parallel P\parallel }^2}{2n(n-1)}+\frac{c-1}{2n}-\frac{c+3}{4}\\ \hfill +\frac{c-1}{4n(n-1)}[{\left(trace\left(P\right)\right)}^2+trace\left(P^2\right)]+\frac{2}{n}trace\left(\lambda \right),\end{array}$$

(44)

where ${\widehat{\delta }}_{\mathcal{C}}^0(n-1)$ is expressed by $2{\widehat{\delta }}_{\mathcal{C}}^0(n-1)={\widehat{\delta }}_{\mathcal{C}}(n-1)+{\widehat{\delta }}_{\mathcal{C}}^{*}(n-1)$.

Moreover, we have the equality cases of the inequalities (43) and (44) at all points $x\in M$ if and only if:

$$\begin{array}{c}\hfill h+h^{*}=0.\end{array}$$

Remark 2.

As mentioned above, the equality cases of the inequalities (29), (30), (43) and (44) hold if and only if $h+h^{*}=0$, meaning that the second fundamental form of the semi-symmetric metric connection $\tilde{\nabla }$ vanishes. This implies that the spacelike statistical submanifolds satisfying the equalities cases in Theorems 2 and 3 are totally geodesic with respect to the semi-symmetric metric connection, equivalently also with respect to the Levi–Civita connection.

4. Example

Consider $({\mathbb{R}}_m^{2m+1},\overline{\nabla },g,\varphi ,\xi ,\eta )$ a $(2m+1)$-dimensional Sasaki-like statistical manifold (example created in [23]). Suppose the standard coordinates $(x_1,\dots ,x_m,y_1,\dots ,y_m,z)$ and define a semi-Riemannian metric g on ${\mathbb{R}}_m^{2m+1}$ as follows:

$$\mathrm{g}=\left(\begin{array}{ccccc}2{\delta }_{ij}+y_i{y}_j& 0& & -y_i& \\ 0& -{\delta }_{ij}& & 0& \\ -y_j& 0& & 1\end{array}\right),$$

for $i,j\in \{1,\dots ,m\}$. We recall that $\varphi ,\xi ,\eta$ are defined by:

$$\varphi =\left(\begin{array}{ccccc}0& {\delta }_{ij}& & 0& \\ -{\delta }_{ij}& 0& & 0& \\ 0& y_j& & 0\end{array}\right),\xi =\frac{\partial }{\partial z}=\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\end{array}\right)$$

and $\eta =(-y_1,0,-y_2,0,\dots ,-y_m,0,1)$.

Moreover, ${\varphi }^{*}$ is given by:

$${\varphi }^{*}=\frac{1}{2}\left(\begin{array}{ccccc}0& -{\delta }_{ij}& & 0& \\ 4{\delta }_{ij}& 0& & 0& \\ 0& -y_j& & 0\end{array}\right).$$

Denote by $\overline{\nabla }$ and ${\overline{\nabla }}^{*}$ the dual connections on ${\mathbb{R}}_m^{2m+1}$. The affine connection $\overline{\nabla }$ is defined as follows:

$${\overline{\nabla }}_{\partial x_i}\partial x_j=-y_j\partial y_i-y_i\partial y_j,$$

$${\overline{\nabla }}_{\partial x_i}\partial y_j={\overline{\nabla }}_{\partial y_j}\partial x_i=y_i\partial x_j+(y_i{y}_j-2{\delta }_{ij})\partial z,$$

$${\overline{\nabla }}_{\partial x_i}\partial z={\overline{\nabla }}_{\partial z}\partial x_i=\partial y_i$$

$${\overline{\nabla }}_{\partial y_i}\partial y_j=0,$$

$${\overline{\nabla }}_{\partial y_i}\partial z={\overline{\nabla }}_{\partial z}\partial y_i=-\partial x_i-y_i\partial z,$$

$${\overline{\nabla }}_{\partial z}\partial z=0.$$

In addition, the dual connection ${\overline{\nabla }}^{*}$ is expressed by:

$${\overline{\nabla }}_{\partial x_i}^{*}\partial x_j=2y_j\partial y_i+2y_i\partial y_j,$$

$${\overline{\nabla }}_{\partial x_i}^{*}\partial y_j={\overline{\nabla }}_{\partial y_j}^{*}\partial x_i=-\frac{y_i}{2}\partial x_j-\frac{1}{2}(y_i{y}_j-2{\delta }_{ij})\partial z,$$

$${\overline{\nabla }}_{\partial x_i}^{*}\partial z={\overline{\nabla }}_{\partial z}^{*}\partial x_i=-2\partial y_i,$$

$${\overline{{\nabla }^{*}}}_{\partial y_i}\partial y_j=0,$$

$${\overline{\nabla }}_{\partial y_i}^{*}\partial z={\overline{\nabla }}_{\partial z}^{*}\partial y_i=\frac{1}{2}\partial x_i+\frac{1}{2}{y}_i\partial z,$$

$${\overline{\nabla }}_{\partial z}^{*}\partial z=0.$$

Then, as we already mentioned, $({\mathbb{R}}_m^{2m+1},\overline{\nabla },g,\varphi ,\xi ,\eta )$ is a Sasaki-like statistical manifold, where the curvature tensor of ${\mathbb{R}}_m^{2m+1}$ satisfies Equation (13) with $c=-3$ (see [23]).

Next, we study a semi-symmetric metric connection on the above Sasaki-like statistical manifold. First, we define an affine connection $\tilde{\nabla }$ by:

$${\tilde{\nabla }}_{\partial x_i}\partial x_j=-y_j\partial x_i+\frac{1}{2}{y}_j\partial y_i+\frac{1}{2}{y}_i\partial y_j-y_i{y}_j\partial z,$$

$${\tilde{\nabla }}_{\partial x_i}\partial y_j=\frac{1}{4}{y}_i\partial x_j+\frac{1}{4}(y_i{y}_j-2{\delta }_{ij})\partial z,$$

$${\tilde{\nabla }}_{\partial y_j}\partial x_i=\frac{1}{4}{y}_i\partial x_j-y_i\partial y_j+\frac{1}{4}(y_i{y}_j-2{\delta }_{ij})\partial z,$$

$${\tilde{\nabla }}_{\partial x_i}\partial z=\partial x_i-\frac{1}{2}\partial y_i+y_i\partial z,$$

$${\tilde{\nabla }}_{\partial z}\partial x_i=-\frac{1}{2}\partial y_i,$$

$${\tilde{\nabla }}_{\partial y_i}\partial y_j={\delta }_{ij}\partial z,$$

$${\tilde{\nabla }}_{\partial y_i}\partial z=-\frac{1}{4}(\partial x_i+y_i\partial z)+\partial y_i,$$

$${\tilde{\nabla }}_{\partial z}\partial y_i=-\frac{1}{4}(\partial x_i+y_i\partial z),$$

$${\tilde{\nabla }}_{\partial z}\partial z=0,$$

for all $i,j\in \{1,\dots ,m\}$.

Due to the fact that $\tilde{\nabla }$ satisfies (14) with $\eta \left(X\right)=g(X,\xi )$ and $\xi =\partial z$, it follows that $\tilde{\nabla }$ is a semi-symmetric connection. Moreover, we have $\tilde{\nabla }g=0$, implying that $\tilde{\nabla }$ is a semi-symmetric metric connection on the Sasaki-like statistical manifold $({\mathbb{R}}_m^{2m+1},\overline{\nabla },g,\varphi ,\xi ,\eta )$ where the relation (13) holds with $c=-3$. We mention that this manifold is not Sasaki with respect to the Levi–Civita connection.

Let M be an n-dimensional spacelike statistical submanifold of the above Sasaki-like statistical manifold endowed with the semi-symmetric metric connection $\tilde{\nabla }$. Then, the inequalities (29), (30), (43) and (44) are satisfied.

5. Conclusions

In this article, we demonstrated new inequalities in terms of the extrinsic invariants $\delta$-Casorati curvatures and the intrinsic invariant scalar curvature of spacelike statistical submanifolds in some Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. We also investigated the ideal Casorati submanifolds in this ambient space. Since there is little research on this interesting topic of statistical manifolds equipped with a semi-symmetric metric connection, there is the possibility to develop new other solutions for the above fundamental problem on varied classes of statistical (sub)manifolds and distinct curvature invariants.