Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

Abstract

In this paper, we prove some inequalities in terms of the normalized $\delta$-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented.

1. Introduction

The problem of discovering simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds is a basic problem in submanifold theory [1]. In this respect, beautiful results focus on certain types of geometric inequalities. Moreover, another basic problem in this field is to study the ideal submanifolds in a space form, namely to investigate the submanifolds which satisfy the equality case of such inequalities [2].

The method of looking for Chen invariants answers the problems posed above. First, Chen demonstrated in [3] an optimal inequality for a submanifold on a real space form between the intrinsically defined $\delta$-curvature and the extrinsically defined squared mean curvature. This approach initiated a new line of research and was extended to various types of submanifolds in several types of ambient spaces, e.g., submanifolds in complex space forms of constant holomorphic sectional curvature (see [4,5,6,7]). The submanifolds attaining the equality of these inequalities (called Chen ideal submanifolds) were also investigated. Recently, Chen et al. classified $\delta (2,n-2)$-ideal Lagrangian submanifolds in complex space forms in [8].

Moreover, new solutions to the above problems are given by the inequalities involving δ-Casorati curvatures, initiated in [9,10]. In the search for a true measure of curvature, Casorati in 1890 proposed the curvature which nowadays bears his name because it better corresponds with our common intuition of curvature than Gauss and mean curvature [11]. However, this notion of curvature was soon forgotten and was rediscovered by Koenderink working in the field of computer vision [12]. Verstraelen developed some geometrical models for early vision, presenting perception via the Casorati curvature of sensation [13]. A geometrical interpretation of this type of curvature for submanifolds in Riemannian spaces was given in [14]. In [15], the isotropical Casorati curvature of production surfaces was studied. The Casorati curvature was used to obtain optimal inequalities between intrinsic and extrinsic curvatures of submanifolds in real space forms in [9,10]. Later, this knowledge was extended (e.g., see [16,17,18,19,20,21]). Submanifolds which satisfy these equalities are named Casorati ideal submanifolds. Recently, Vîlcu established an optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature [22]. Aquib et al. obtained a classification of Casorati ideal Lagrangian submanifolds in complex space forms [23]. Very recently, Suceavă and Vajiac studied inequalities involving some Chen invariants, mean curvature, and Casorati curvature for strictly convex Euclidean hypersurfaces [24]. Brubaker and Suceavă investigated a geometric interpretation of Cauchy–Schwarz inequality in terms of Casorati curvature [25].

The concept of statistical manifold was defined by Amari in 1985, in the basic study on information geometry [26]. Currently, interest in the field of statistical manifolds is increasing, being focused on applications in differential geometry, information geometry, statistics, machine learning, etc. (see, e.g., [27,28,29]). Cuingnet et al. introduced a continuous framework to spatially regularize support vector machines (SVM) for brain image analysis, considering the images as elements of a statistical manifold, in order to classify patients with Alzheimer’s disease [30]. The study of curvature invariants of submanifolds in statistical manifolds gives other solutions to the above research problems. Aydin et al. established some inequalities (Chen–Ricci and Wintgen) for submanifolds in statistical manifolds of constant curvature in [31,32]. Lee et al. obtained inequalities on Sasakian statistical manifolds in terms of Casorati curvatures [33]. Aquib and Shahid [34] proved some inequalities involving Casorati curvatures on statistical submanifolds in quaternion Kähler-like statistical space forms. The quaternionic theory of statistical manifolds is investigated in [35]. Very recently, new results have been published. Aytimur et al. established some Chen inequalities for submanifolds in Kähler-like statistical manifolds [36]. Aquib et al. achieved generalized Wintgen-type inequalities for submanifolds in generalized space forms [37]. Chen et al. established a Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature [38]. Moreover, Siddiqui et al. studied a Chen inequality for statistical warped products statistically immersed in a statistical manifold of constant curvature [39].

Recently, Furuhata et al. [40] defined the notion of a holomorphic statistical manifold, which can be considered as a generalization of a special Kähler manifold. The authors establish the basics for statistical submanifolds in holomorphic statistical manifolds.

In order to find out new solutions for the problems under debate, we obtain inequalities for statistical submanifolds in holomorphic statistical manifolds. The invariants involved in such inequalities are the extrinsic normalized $\delta$-Casorati curvatures and the intrinsic scalar curvature. The method is focused on a constrained extremum problem. Moreover, the equality cases are investigated. This study revealed that the equality at all points characterizes submanifolds that are totally geodesic with respect to the Levi–Civita connection.

2. Preliminaries

Let ($\tilde{M}$, $\tilde{g}$) be a $2n$-dimensional manifold, $\tilde{\nabla }$ an affine connection on $\tilde{M}$, and $\tilde{g}$ a Riemannian metric on $\tilde{M}$. Consider $\tilde{T}\in \mathsf{\Gamma }(T{\tilde{M}}^{(1,2)})$ the torsion tensor field of $\tilde{\nabla }$.

A pair $(\tilde{\nabla },\tilde{g})$ is called a statistical structure on $\tilde{M}$ if the torsion tensor field $\tilde{T}$ vanishes and $\tilde{\nabla }\tilde{g}\in \mathsf{\Gamma }(T{\tilde{M}}^{(0,3)})$ is symmetric.

A Riemannian manifold $(\tilde{M}$, $\tilde{g})$ is called a statistical manifold if it is endowed with a pair of torsion-free affine connections $\tilde{\nabla }$ and ${\tilde{\nabla }}^{\ast }$ satisfying

$$Z\tilde{g}(X,Y)=\tilde{g}({\tilde{\nabla }}_{Z}X,Y)+\tilde{g}(X,{\tilde{\nabla }}_Z^{\ast }Y),$$

for any X, Y, Z$\in \mathsf{\Gamma }(T\tilde{M})$. Denote $(\tilde{M}$, $\tilde{g},\tilde{\nabla })$ as the statistical manifold. The connections $\tilde{\nabla }$ and ${\tilde{\nabla }}^{\ast }$ are named dual connections or conjugate connections.

Remark 1.

If $(\tilde{M}$, $\tilde{g}$, $\tilde{\nabla })$ is a statistical manifold, then we remark that

1. 

${({\tilde{\nabla }}^{\ast })}^{\ast }=\tilde{\nabla }$;

2. 

$(\tilde{M}$, $\tilde{g}$, ${\tilde{\nabla }}^{\ast })$ is also a statistical manifold;

3. 

$\tilde{\nabla }$ always has a dual connection ${\tilde{\nabla }}^{\ast }$ satisfying

$$\tilde{\nabla }+{\tilde{\nabla }}^{\ast }=2{\tilde{\nabla }}^0,$$

(1)

where ${\tilde{\nabla }}^0$ is the Levi–Civita connection on $\tilde{M}$.

Let M be an m-dimensional submanifold of a $2n$-dimensional statistical manifold $(\tilde{M},\tilde{g})$ and g the induced metric on M. The Gauss formulas are given by

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),\\ \hfill {\tilde{\nabla }}_X^{\ast }Y={\nabla }_X^{\ast }Y+h^{\ast }(X,Y),\end{array}$$

for any $X,Y\in \mathsf{\Gamma }(TM)$, where h and $h^{\ast }$ are symmetric and bilinear $(0,2)$-tensors, called the imbedding curvature tensor of M in $\tilde{M}$ for $\tilde{\nabla }$ and ${\tilde{\nabla }}^{\ast }$, respectively.

Denote the curvature tensor fields of ∇ and $\tilde{\nabla }$ by R and $\tilde{R}$, respectively. Then, the Gauss equation concerning the connection $\tilde{\nabla }$ is ([41])

$$\begin{array}{c}\hfill \tilde{g}(\tilde{R}(X,Y)Z,W)=g(R(X,Y)Z,W)+\tilde{g}(h(X,Z),h^{\ast }(Y,W))-\tilde{g}(h^{\ast }(X,W),h(Y,Z)),\end{array}$$

(2)

for any $X,Y,Z,W\in \mathsf{\Gamma }(TM)$.

In addition, denote the curvature tensor fields of the connections ${\nabla }^{\ast }$ and ${\tilde{\nabla }}^{\ast }$ by $R^{\ast }$ and ${\tilde{R}}^{\ast }$, respectively. Then the Gauss equation concerning the connection ${\tilde{\nabla }}^{\ast }$ is ([41])

$$\begin{array}{c}\hfill \tilde{g}({\tilde{R}}^{\ast }(X,Y)Z,W)=g(R^{\ast }(X,Y)Z,W)+\tilde{g}(h^{\ast }(X,Z),h(Y,W))-\tilde{g}(h(X,W),h^{\ast }(Y,Z)),\end{array}$$

(3)

for any $X,Y,Z,W\in \mathsf{\Gamma }(TM)$.

If M is a submanifold of a statistical manifold ($\tilde{M}$, $\tilde{g}$, $\tilde{\nabla })$, then $(M,g,\nabla )$ is also a statistical manifold with the induced metric g and the induced connection ∇.

Let S be the statistical curvature tensor field of a statistical manifold $(M,g,\nabla )$, where $S\in \mathsf{\Gamma }(T{M}^{(1,3)})$ is defined by [40]

$$S(X,Y)Z=\frac{1}{2}\{R(X,Y)Z+R^{\ast }(X,Y)Z\},$$

(4)

for $X,Y,Z\in \mathsf{\Gamma }(TM)$.

If $\pi =spa{n}_{\mathbb{R}}\{u_1,u_2\}$ is a 2-dimensional subspace of $T_{p}M$, for $p\in M$, then the sectional curvature of M is defined by [40]:

$$\begin{array}{c}\hfill \sigma (\pi )=\frac{g(S(u_1,u_2)u_2,u_1)}{g(u_1,u_1)g(u_2,u_2)-g^2(u_1,u_2)}.\end{array}$$

(5)

Let $\{e_1,\dots ,e_m\}$ be an orthonormal basis of the tangent space $T_{p}M$, for $p\in M$, and let $\{e_{m+1},\dots ,e_{2n}\}$ be an orthonormal basis of the normal space $T_p^{\perp }M$. The scalar curvature $\tau$ at p is given by

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

(6)

and the normalized scalar curvature $\rho$ of M is defined as

$$\rho =\frac{2\tau }{m(m-1)}.$$

(7)

The mean curvature vector fields of M, denoted by H and $H^{\ast }$, are given by

$$H=\frac{1}{m}{\displaystyle \sum }_{i=1}^{m}h(e_i,e_i),H^{\ast }=\frac{1}{m}{\displaystyle \sum }_{i=1}^m{h}^{\ast }(e_i,e_i).$$

From Equation (1), we get $2h^0=h+h^{\ast }$ and $2H^0=H+H^{\ast }$, 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 $\tilde{M}$ have the expressions

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

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

Denote by $\mathcal{C}$ and ${\mathcal{C}}^{\ast }$ the Casorati curvatures of the submanifold M, defined by the squared norms of h and $h^{\ast }$, respectively, over the dimension m, as follows:

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

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

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

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

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

$${\delta }_{\mathcal{C}}{(m-1)|}_p=\frac{1}{2}\mathcal{C}{\mid }_p+\frac{m+1}{2m}\inf \{\mathcal{C}(L)|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\}$$

and

$${\widehat{\delta }}_{\mathcal{C}}{(m-1)|}_p=2\mathcal{C}{\mid }_p-\frac{2m-1}{2m}\sup \{\mathcal{C}(L)|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\}.$$

Moreover, the dual normalized ${\delta }^{\ast }$-Casorati curvatures ${\delta }_{\mathcal{C}}^{\ast }(m-1)$ and ${\widehat{\delta }}_{\mathcal{C}}^{\ast }(m-1)$ of the submanifold M in $\tilde{M}$ are defined as

$${\delta }_{\mathcal{C}}^{\ast }{(m-1)|}_p=\frac{1}{2}{\mathcal{C}}^{\ast }{\mid }_p+\frac{m+1}{2m}\inf \{{\mathcal{C}}^{\ast }(L)|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\}$$

and

$${\widehat{\delta }}_{\mathcal{C}}^{\ast }{(m-1)|}_p=2{\mathcal{C}}^{\ast }{\mid }_p-\frac{2m-1}{2m}\sup \{{\mathcal{C}}^{\ast }(L)|L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\}.$$

Denote by ${\delta }_{\mathcal{C}}(r;m-1)$ and ${\widehat{\delta }}_{\mathcal{C}}(r;m-1)$, the generalized normalized δ-Casorati curvatures of M, defined in [10] as

$${\delta }_{\mathcal{C}}{(r;m-1)|}_p=r\mathcal{C}{\mid }_p+a(r)\inf \{\mathcal{C}(L)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\},$$

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

$${\widehat{\delta }}_{\mathcal{C}}{(r;m-1)|}_p=r\mathcal{C}{\mid }_p+a(r)\sup \{\mathcal{C}(L)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\},$$

if $r>m(m-1)$, for $a(r)$ set as

$$\begin{array}{c}\hfill a(r)=\frac{(m-1)(r+m)(m^2-m-r)}{mr},\end{array}$$

where $r\in {\mathbb{R}}_{+}$ and $r\ne m(m-1)$.

Furthermore, denote by ${\delta }_{\mathcal{C}}^{\ast }(r;m-1)$ and ${\widehat{{\delta }^{\ast }}}_{\mathcal{C}}(r;m-1)$ the dual generalized normalized ${\delta }^{\ast }$-Casorati curvatures of the submanifold M, defined as follows:

$${\delta }_{\mathcal{C}}^{\ast }{(r;m-1)|}_p=r{\mathcal{C}}^{\ast }{\mid }_p+a(r)\inf \{{\mathcal{C}}^{\ast }(L)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\},$$

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

$${\widehat{\delta }}_{\mathcal{C}}^{\ast }{(r;m-1)|}_p=r{\mathcal{C}}^{\ast }{\mid }_p+a(r)\sup \{{\mathcal{C}}^{\ast }(L)\mid L\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\},$$

if $r>m(m-1)$, for $a(r)$ set above.

A statistical submanifold $(M,g,\nabla )$ of $(\tilde{M},\tilde{g},\tilde{\nabla })$ is called totally geodesic with respect to the connection $\tilde{\nabla }$ if the second fundamental form h of M for $\tilde{\nabla }$ vanishes identically [40].

Let $\tilde{M}$ be an almost complex manifold with almost complex structure $J\in \mathsf{\Gamma }(T{\tilde{M}}^{(1,1)})$. A quadruplet $(\tilde{M},\tilde{\nabla },\tilde{g},J)$ is called a holomorphic statistical manifold if

• $(\tilde{\nabla },\tilde{g})$ is a statistical structure on $\tilde{M}$; and
• $\omega$ is a $\tilde{\nabla }$-parallel 2-form on $\tilde{M}$,

where $\omega$ is defined by $\omega (X,Y)=\tilde{g}(X,JY)$, for any $X,Y\in \mathsf{\Gamma }(T\tilde{M})$.

For a holomorphic statistical manifold, the following formula holds:

$$\tilde{g}(\tilde{S}(Z,W)JY,JX)=\tilde{g}(\tilde{S}(JZ,JW)Y,X)=\tilde{g}(\tilde{S}(Z,W)Y,X),$$

(8)

for any $X,Y,Z,W\in \mathsf{\Gamma }(T\tilde{M})$.

A holomorphic statistical manifold ($\tilde{M},\tilde{\nabla },\tilde{g},J$) is said to be of constant holomorphic sectional curvature $c\in \mathbb{R}$ if the following formula holds [42]:

$$\begin{array}{ccc}\hfill \tilde{S}(X,Y)Z& =& \frac{c}{4}\{\tilde{g}(Y,Z)X-\tilde{g}(X,Z)Y+\tilde{g}(JY,Z)JX-\tilde{g}(JX,Z)JY+2\tilde{g}(X,JY)JZ\},\hfill \end{array}$$

(9)

for any $X,Y,Z\in \mathsf{\Gamma }(T\tilde{M})$, where $\tilde{S}$ is the statistical curvature tensor field of $\tilde{M}$.

Remark 2

([43]). Let ($\tilde{M},\tilde{g},J$) be a Kähler manifold. If we define a connection $\tilde{\nabla }$ as $\tilde{\nabla }={\nabla }^{\tilde{g}}+K$, where $K\in \mathsf{\Gamma }(T{\tilde{M}}^{(1,2)})$ satisfying the conditions

$$\begin{array}{c}\hfill K(X,Y)=K(Y,X),\end{array}$$

(10)

$$\begin{array}{c}\hfill \tilde{g}(K(X,Y),Z)=\tilde{g}(Y,K(X,Z)),\end{array}$$

(11)

$$\begin{array}{c}\hfill K(X,JY)=-JK(X,Y),\end{array}$$

(12)

for any $X,Y,Z\in \mathsf{\Gamma }(T\tilde{M})$, then ($\tilde{M},\tilde{\nabla },\tilde{g},J$) is a holomorphic statistical manifold.

Let M be an m-dimensional statistical submanifold of a holomorphic statistical manifold $(\tilde{M},\tilde{\nabla },\tilde{g},J)$. For any vector field X tangent to M we can decompose

$$\begin{array}{c}\hfill JX=PX+FX,\end{array}$$

(13)

where $PX$ and $FX$ are the tangent component and the normal component, respectively, of $JX$. Given a local orthonormal frame $\{e_1,e_2,\cdots ,e_m\}$ of M, then the squared norm of P is expressed by

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

Next, we consider the constrained extremum problem

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

(14)

where M is a Riemannian submanifold of a Riemannian manifold $(\tilde{M},\tilde{g})$, and $f:\tilde{M}\to \mathbb{R}$ is a function of differentiability class $C^2$.

Theorem 1

([44]). If M is complete and connected, $(gradf)(p)\in T_p^{\perp }M$ for a point $p\in M$, and the bilinear form $\mathcal{A}:T_{p}M\times T_{p}M\to \mathbb{R}$ defined by

$$\mathcal{A}(X,Y)=\mathrm{Hess}(f)(X,Y)+\tilde{g}(h^0(X,Y),\mathrm{grad}f),$$

(15)

is positive definite in p, then p is the optimal solution of the Problem (14).

Remark 3

([44]). If the bilinear form $\mathcal{A}$ defined by Equation (15) is positive semi-definite on the submanifold M, then the critical points of $f|M$ are global optimal solutions of the Problem (14).

3. Main Inequalities

Theorem 2.

Let M be an m-dimensional statistical submanifold of a $2n$-dimensional holomorphic statistical manifold ($\tilde{M},\tilde{\nabla },\tilde{g},J$) of constant holomorphic sectional curvature c. Then we have

(i) 

$$\begin{array}{ccc}\hfill 2\tau & \le & {\delta }_{\mathcal{C}}^0(r;m-1)+m{\mathcal{C}}^0-2m^2{\parallel H^0\parallel }^2\hfill \\ & +& m^2\tilde{g}(H,H^{\ast })+\frac{3c}{4}{\parallel P\parallel }^2+\frac{c}{4}m(m-1),\hfill \end{array}$$

(16)

for any real number r such that $0<r<m(m-1)$, where ${\delta }_{\mathcal{C}}^0(r;m-1)=\frac{{\delta }_{\mathcal{C}}(r;m-1)+{\delta }_{\mathcal{C}}^{\ast }(r;m-1)}{2}$ and ${\mathcal{C}}^0=\frac{\mathcal{C}+{\mathcal{C}}^{\ast }}{2}$; and

(ii) 

$$\begin{array}{ccc}\hfill 2\tau & \le & {\widehat{\delta }}_{\mathcal{C}}^0(r;m-1)+m{C}^0-2m^2{\parallel H^0\parallel }^2\hfill \\ & +& m^2\tilde{g}(H,H^{\ast })+\frac{3c}{4}{\parallel P\parallel }^2+\frac{c}{4}m(m-1),\hfill \end{array}$$

(17)

for any real number r such that $r>m(m-1)$, where ${\widehat{\delta }}_{\mathcal{C}}^0(r;m-1)=\frac{{\widehat{\delta }}_{\mathcal{C}}(r;m-1)+{\widehat{\delta }}_{\mathcal{C}}^{\ast }(r;m-1)}{2}.$

Moreover, the equality cases of Inequalities (16) and (17) hold identically at all points $p\in M$ if and only if the following condition is satisfied:

$$h+h^{\ast }=0,$$

(18)

where h and $h^{\ast }$ are the imbedding curvature tensors of the submanifold associated to the dual connections $\tilde{\nabla }$ and ${\tilde{\nabla }}^{\ast }$, respectively.

Proof. 

The relations (Equations (2)–(4)) imply

$$\begin{array}{ccc}\hfill 2\tilde{g}(\tilde{S}(X,Y)Z,W)& =& 2g(S(X,Y)Z,W)-\tilde{g}(h(Y,Z),h^{\ast }(X,W))+\tilde{g}(h(X,Z),h^{\ast }(Y,W))\hfill \\ & & -\tilde{g}(h^{\ast }(Y,Z),h(X,W))+\tilde{g}(h^{\ast }(X,Z),h(Y,W)),\hfill \end{array}$$

(19)

where $X,Y,Z,W\in \mathsf{\Gamma }(TM)$.

For $p\in M$, we choose $\{e_1,\dots ,e_m\}$ and $\{e_{m+1},\dots ,e_{2n}\}$ orthonormal bases of $T_{p}M$ and $T_p^{\perp }M$, respectively. For $X=Z=e_i$ and $Y=W=e_j$ with $i,j\in \{1,\dots ,m\}$, from the Equation (19), it follows that

$$\begin{array}{ccc}\hfill 2\tau (p)& =& m^2\tilde{g}(H,H^{\ast })-\sum _{1\le i,j\le m}\tilde{g}(h^{\ast }(e_i,e_j),h(e_i,e_j))\hfill \\ & & +\frac{c}{4}(m^2-m+{3\parallel P\parallel }^2).\hfill \end{array}$$

(20)

Denoting $2H^0=H+H^{\ast }$ and $2{\mathcal{C}}^0=\mathcal{C}+{\mathcal{C}}^{\ast }$, Equation (20) becomes

$$\begin{array}{ccc}\hfill 2\tau (p)& =& 2m^2\parallel H^0{\parallel }^2-\frac{m^2}{2}{\parallel H\parallel }^2-\frac{m^2}{2}{\parallel H^{\ast }\parallel }^2\hfill \\ & & -2m{\mathcal{C}}^0+\frac{m}{2}(\mathcal{C}+{\mathcal{C}}^{\ast })+\frac{c}{4}(m^2-m+{3\parallel P\parallel }^2).\hfill \end{array}$$

(21)

Let $\mathcal{P}$ be the quadratic polynomial defined by

$$\begin{array}{ccc}\hfill \mathcal{P}& =& r{\mathcal{C}}^0+a(r){\mathcal{C}}^0(L)+\frac{m}{2}(\mathcal{C}+{\mathcal{C}}^{\ast })-\frac{m^2}{2}{(\parallel H\parallel }^2+\parallel H^{\ast }{\parallel }^2)\hfill \\ & & -2\tau (p)+\frac{c}{4}(m^2-m+{3\parallel P\parallel }^2),\hfill \end{array}$$

(22)

where L is a hyperplane of $T_{p}M$.

We consider that the hyperplane L is spanned by the tangent vectors $e_1,\dots ,e_{m-1}$, without loss of generality. Therefore, we get

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

(23)

Then, Equation (23) yields

$$\begin{array}{ccc}\hfill \mathcal{P}& =& {\displaystyle \sum }_{\alpha =m+1}^{2n}\{\left[\frac{2(2m+r)}{m}+\frac{2a(r)}{m-1}\right]\sum _{1\le i<j\le m-1}{(h_{ij}^{0\alpha })}^2+\left[\frac{2(2m+r)}{m}+\frac{2a(r)}{m-1}\right]{\displaystyle \sum }_{i=1}^{m-1}{(h_{im}^{0\alpha })}^2\hfill \\ & & +\left(\frac{2m+r}{m}+\frac{a(r)}{m-1}-2\right){\displaystyle \sum }_{i=1}^{m-1}{(h_{ii}^{0\alpha })}^2\hfill \\ & & -4{\displaystyle \sum }_{1\le i<j\le m}{h}_{ii}^{0\alpha }{h}_{jj}^{0\alpha }+\left(\frac{2m+r}{m}-2\right){(h_{mm}^{0\alpha })}^2\}\hfill \\ & \ge & {\displaystyle \sum }_{\alpha =m+1}^{2n}\left[\frac{r(m-1)+a(r)m}{m(m-1)}{\displaystyle \sum }_{i=1}^{m-1}{(h_{ii}^{0\alpha })}^2+\left(\frac{r}{m}\right){(h_{mm}^{0\alpha })}^2-4\sum _{1\le i<j\le m}{h}_{ii}^{0\alpha }{h}_{jj}^{0\alpha }\right].\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill f_{\alpha }(h_{11}^{0\alpha },h_{22}^{0\alpha },...,h_{mm}^{0\alpha })& =& {\displaystyle \sum }_{i=1}^{m-1}\frac{r(m-1)+a(r)m}{m(m-1)}{(h_{ii}^{0\alpha })}^2\hfill \\ & & +\frac{r}{m}{(h_{mm}^{0\alpha })}^2-4\sum _{1\le i<j\le m}{h}_{ii}^{0\alpha }{h}_{jj}^{0\alpha }.\hfill \end{array}$$

We investigate the constrained extremum problem

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

with the constraint

$$Q:h_{11}^{0\alpha }+h_{22}^{0\alpha }+\dots +h_{mm}^{0\alpha }=k^{\alpha },$$

where $k^{\alpha }$ is a real constant.

We obtain the system of first-order partial derivatives:

$$\left\{\begin{array}{c}\frac{\partial f_{\alpha }}{\partial h_{ii}^{0\alpha }}=2\frac{r(m-1)+a(r)m}{m(m-1)}{h}_{ii}^{0\alpha }-4\left({\displaystyle \sum }_{k=1}^m{h}_{kk}^{0\alpha }-h_{ii}^{0\alpha }\right)=0\hfill \\ \frac{\partial f_{\alpha }}{\partial h_{mm}^{0\alpha }}=\frac{2r}{m}{h}_{mm}^{0\alpha }-4{\displaystyle \sum }_{k=1}^{m-1}{h}_{kk}^{0\alpha }=0,\hfill \end{array}\right.$$

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

It follows that the constrained critical point is

$$h_{ii}^{0\alpha }=\frac{2m(m-1)}{(m-1)(2m+r)+ma(r)}{k}^{\alpha }$$

$$h_{mm}^{0\alpha }=\frac{2m}{2m+r}{k}^{\alpha },$$

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

For $p\in$ Q, let $\mathcal{A}$ be a 2-form, $\mathcal{A}:T_{p}Q\times T_{p}Q\to \mathbb{R}$ defined by

$$\mathcal{A}(X,Y)=\mathrm{Hess}(f_{\alpha })(X,Y)+\langle h^{{}^{\prime }}(X,Y),(\mathrm{grad}{f}_{\alpha })(p)\rangle ,$$

where $h^{\prime }$ is the second fundamental form of Q in ${\mathbb{R}}^{m+1}$ and $\langle ·,·\rangle$ is the standard inner product on ${\mathbb{R}}^m$.

The Hessian matrix of $f_{\alpha }$ is given by

$$\mathrm{Hess}(f_{\alpha })=\left(\begin{array}{ccccc}\lambda & -4& \cdots & -4& -4\\ -4& \lambda & \cdots & -4& -4\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -4& -4& \cdots & \lambda & -4\\ -4& -4& \cdots & -4& \frac{2r}{m}\end{array}\right),$$

where $\lambda =2\frac{(m-1)(r+2m)+ma(r)}{m(m-1)}$ is a real constant.

The condition ${\sum }_{i=1}^m{X}_i=0$ is satisfied, for a vector field $X\in T_{p}Q$, as the hyperplane Q is totally geodesic in ${\mathbb{R}}^m$. Then, we achieve

$$\begin{array}{cc}\hfill \mathcal{A}(X,X)& =\lambda {\displaystyle \sum }_{i=1}^{m-1}{X}_i^2+\frac{2r}{m}{X}_m^2-8{\displaystyle \sum }_{i,j=1(i\ne j)}^m{X}_i{X}_j\hfill \\ \hfill & =\lambda {\displaystyle \sum }_{i=1}^{m-1}{X}_i^2+\frac{2r}{m}{X}_m^2+4{\left({\displaystyle \sum }_{i=1}^m{X}_i\right)}^2-8{\displaystyle \sum }_{i,j=1(i\ne j)}^m{X}_i{X}_j\hfill \\ \hfill & =\lambda {\displaystyle \sum }_{i=1}^{m-1}{X}_i^2+\frac{2r}{m}{X}_m^2+4{\displaystyle \sum }_{i=1}^m{X}_i^2\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

Applying Remark 3, the critical point $(h_{11}^{0\alpha },\dots ,h_{mm}^{0\alpha })$ of $f_{\alpha }$ is the global minimum point of the problem. Since $f_{\alpha }(h_{11}^{0\alpha },\dots ,h_{mm}^{0\alpha })=0$, we get $\mathcal{P}\ge 0$.

We have then proved Inequalities (16) and (17), considering infimum and supremum, respectively, over all tangent hyperplanes L of $T_{p}M$.

In addition, we study the equality cases of Inequalities (16) and (17). First, we find out the critical points of $\mathcal{P}$

$$h^c=(h_{11}^{0m+1},h_{12}^{0m+1},\dots ,h_{mm}^{0m+1},\dots ,h_{11}^{02n},\dots ,h_{mm}^{02n})$$

as the solutions of following system of linear homogeneous equations:

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

The critical points satisfy $h_{ij}^{0\alpha }=0$, with $i,j\in \{1,\dots ,m\}$ and $\alpha \in \{m+1,\dots ,2n\}$. On the other hand, we know that $\mathcal{P}\ge 0$ and $\mathcal{P}(h^c)=0$, then the critical point $h^c$ is a minimum point of $\mathcal{P}$. Consequently, the cases of equality hold in both Inequalities (16) and (17) if and only if $h_{ij}^{\alpha }=-h_{ij}^{\ast \alpha },$ for $i,j$$\in \{1,\dots ,m\}$, $\alpha \in \{m+1,\dots ,2n\}$. □

Remark 4.

Under Equation (18), the submanifold M is totally geodesic with respect to the Levi–Civita connection ${\tilde{\nabla }}^0$. Then, the equality cases of Inequalities (16) and (17) hold for all unit tangent vectors at p if and only if p is a totally geodesic point with respect to the Levi–Civita connection.

By virtue of Theorem 2, the generalized normalized $\delta$-Casorati curvatures satisfy Inequalities (16) and (17). If the normalized $\delta$-Casorati curvatures ${\delta }_{\mathcal{C}}(m-1)$ and ${\delta }_{\mathcal{C}}^{\ast }(m-1)$, respectively, ${\widehat{\delta }}_{\mathcal{C}}(m-1)$ and ${\widehat{\delta }}_{\mathcal{C}}^{\ast }(m-1)$ are involved, then we can state the following result.

Corollary 1.

Let M be an m-dimensional statistical submanifold of a $2n$-dimensional holomorphic statistical manifold ($\tilde{M},\tilde{\nabla },\tilde{g},J$) of constant holomorphic sectional curvature c. Then, we have

(i) 

$$\begin{array}{ccc}\hfill \rho & \le & {\delta }_{\mathcal{C}}^0(m-1)+\frac{1}{m-1}{\mathcal{C}}^0-\frac{2m}{m-1}{\parallel H^0\parallel }^2\hfill \\ & +& \frac{m}{m-1}\tilde{g}(H,H^{\ast })+\frac{3c}{4m(m-1)}{\parallel P\parallel }^2+\frac{c}{4},\hfill \end{array}$$

(24)

where $2{\delta }_{\mathcal{C}}^0(m-1)={\delta }_{\mathcal{C}}(m-1)+{\delta }_{\mathcal{C}}^{\ast }(m-1)$ and $2{\mathcal{C}}^0=\mathcal{C}+{\mathcal{C}}^{\ast }$, and

(ii) 

$$\begin{array}{ccc}\hfill \rho & \le & {\widehat{\delta }}_{\mathcal{C}}^0(m-1)+\frac{1}{m-1}{\mathcal{C}}^0-\frac{2m}{m-1}{\parallel H^0\parallel }^2\hfill \\ & +& \frac{m}{m-1}\tilde{g}(H,H^{\ast })+\frac{3c}{4m(m-1)}{\parallel P\parallel }^2+\frac{c}{4},\hfill \end{array}$$

(25)

where $2{\widehat{\delta }}_{\mathcal{C}}^0(m-1)={\widehat{\delta }}_{\mathcal{C}}(m-1)+{\widehat{\delta }}_{\mathcal{C}}^{\ast }(m-1)$.

Moreover, the equality cases of Inequalities (24) and (25) hold identically at all points if and only if h and $h^{\ast }$ satisfy the condition in Equation (18), which implies that M is a totally geodesic submanifold with respect to the Levi–Civita connection.

4. An Example

Example 1.

Let $(x_1,x_2,y_1,y_2)$ be a standard system on ${\mathbb{R}}^4$, g the Euclidean metric. Define $t=(y_1^2+y_2^2)/2$ ($t\ge 0$) and the functions u, v on ${\mathbb{R}}^4$ as

$$u(x_1,x_2,y_1,y_2)=a(t),v(x_1,x_2,y_1,y_2)=b(t),$$

where a is a function $a:[0,\infty )\to (0,\infty )$, and $b(t)=-a(t)a^{\prime }(t){(2t{a}^{\prime }(t)-a(t))}^{-1}$, assuming that $a(t)+2tb(t)>0$ for $t\ge 0$.

Let G be a g-natural metric on ${\mathbb{R}}^4$ and J a complex structure defined by Oproiu ([45]) such that ${\mathbb{R}}^4$ is Kählerian, as follows:

$$\begin{array}{ccc}\hfill G& =& (u+v{y}_1^2)d{x}_{1}d{x}_1+2v{y}_1{y}_{2}d{x}_{1}d{x}_2+(u+v{y}_2^2)d{x}_{2}d{x}_2+\frac{u+v{y}_2^2}{u(u+2tv)}d{y}_{1}d{y}_1\hfill \\ & -& 2\frac{v{y}_1{y}_2}{u(u+2tv)}d{y}_{1}d{y}_2+\frac{u+v{y}_1^2}{u(u+2tv)}d{y}_{2}d{y}_2,\hfill \end{array}$$

(26)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}J\frac{\partial }{\partial x_1}=(u+v{y}_1^2)\frac{\partial }{\partial y_1}+v{y}_1{y}_2\frac{\partial }{\partial y_2},\hfill \\ J\frac{\partial }{\partial x_2}=v{y}_1{y}_2\frac{\partial }{\partial y_1}+(u+v{y}_2^2)\frac{\partial }{\partial y_2},\hfill \\ J\frac{\partial }{\partial y_1}=-\frac{u+v{y}_2^2}{u(u+2tv)}\frac{\partial }{\partial x_1}+\frac{v{y}_1{y}_2}{u(u+2tv)}\frac{\partial }{\partial x_2},\hfill \\ J\frac{\partial }{\partial y_2}=\frac{v{y}_1{y}_2}{u(u+2tv)}\frac{\partial }{\partial x_1}-\frac{u+v{y}_1^2}{u(u+2tv)}\frac{\partial }{\partial x_2}.\hfill \end{array}\right.\end{array}$$

(27)

Let the function u be defined as $u(x_1,x_2,y_1,y_2)=\frac{1+\sqrt{1+4t}}{2}$. Therefore, the function v becomes $v(x_1,x_2,y_1,y_2)=1$. Then, for the metric G and the complex structure J, there exists a tensor field K such that $({\mathbb{R}}^4,\tilde{\nabla }:={\nabla }^G+K,\tilde{g}:=G,J)$ is a special Kähler manifold [46]. Notice that a holomorphic statistical structure of holomorphic curvature 0 is nothing but a special Kähler manifold [43].

In this respect, define a $(1,2)$-tensor field K on ${\mathbb{R}}^4$:

$$K={\displaystyle \sum }_{i,j,l=1}^4{k}_{ij}^l\frac{\partial }{\partial x^l}\otimes d{x}^i\otimes d{x}^j.$$

(28)

Let ${\alpha }_1,\dots ,{\alpha }_7$ be functions on ${\mathbb{R}}^4$ and denote $p:=u+v{y}_1^2$, $q:=u+v{y}_2^2$, $r:=u+2tv$, $s:=v{y}_1{y}_2$. Suppose that ${\alpha }_2$ has the expression

$${\alpha }_2=\frac{1}{2}s(u_{y_1}+2y_1)+\frac{1}{2}q{u}_{y_2}.$$

(29)

Moreover, ${\alpha }_1$ and ${\alpha }_3$ satisfy the equation

$$\left({\alpha }_2\frac{q}{sur}-{\alpha }_3\frac{1}{ur}-{\alpha }_1\frac{q}{s}\right)\frac{pur}{q}+{\alpha }_1\frac{sur}{q}+{\alpha }_2\frac{s}{q}=\frac{1}{2}p(u_{y_1}+2y_1)+\frac{1}{2}s{u}_{y_2},$$

(30)

where $u_{y_1}:=\frac{\partial u}{\partial y_1}$ and $u_{y_2}:=\frac{\partial u}{\partial y_2}$.

If K performs the conditions in Equations (10)–(12) and also the conditions in Equations (29), (30), then we get $({\mathbb{R}}^4,\tilde{\nabla }:={\nabla }^G+K,\tilde{g}:=G,J)$ a special Kähler manifold [46] with K constructed as follows:

$$k_{14}^1=k_{41}^1=k_{13}^2=k_{31}^2=-k_{34}^3=-k_{43}^3={\alpha }_1,k_{11}^4=k_{12}^3=k_{21}^3={\alpha }_2,k_{12}^4=k_{21}^4=k_{22}^3={\alpha }_3,$$

$$k_{24}^1=k_{42}^1=k_{23}^2=k_{32}^2=-k_{34}^4=-k_{43}^4={\alpha }_4,k_{22}^2=-k_{24}^4=-k_{42}^4={\alpha }_5,$$

$$k_{11}^1=k_{12}^1=k_{21}^1=-k_{23}^3=-k_{32}^3=0,k_{12}^2=k_{21}^2=-k_{24}^3=-k_{42}^3={\alpha }_7\frac{q}{s},$$

$$k_{33}^1={\alpha }_6,k_{11}^2=k_{14}^3=k_{41}^3=0,k_{14}^2=k_{41}^2=-{\alpha }_2\frac{s}{urq}+{\alpha }_3\frac{p}{urq}-{\alpha }_1\frac{s}{q},$$

$$k_{23}^1=k_{32}^1={\alpha }_2\frac{q}{urp}-{\alpha }_4\frac{s}{p}-{\alpha }_3\frac{s}{urp},k_{11}^3={\alpha }_1\frac{2s^4-u^2{r}^2}{sq}+{\alpha }_2\frac{ur+2s^2}{sq}-{\alpha }_3\frac{p}{q},$$

$$k_{22}^4=-{\alpha }_2\frac{q}{p}-{\alpha }_4\frac{u^2{r}^2}{sp}+{\alpha }_3\frac{ur+2s^2}{sp},k_{13}^1=k_{31}^1={\alpha }_2\frac{q}{sur}-{\alpha }_3\frac{1}{ur}-{\alpha }_1\frac{q}{s},$$

$$k_{33}^1=-{\alpha }_2\frac{q}{urp}+{\alpha }_3\frac{s}{urp}+{\alpha }_4\frac{s}{p},k_{44}^1=-k_{24}^2=-k_{42}^2={\alpha }_2\frac{1}{ur}-{\alpha }_3\frac{p}{sur}+{\alpha }_4\frac{p}{s},$$

$$k_{44}^3={\alpha }_2\frac{s}{qur}-{\alpha }_3\frac{p}{qur}+{\alpha }_1\frac{s}{q},k_{33}^3=-{\alpha }_2\frac{q}{sur}+{\alpha }_3\frac{1}{ur}+{\alpha }_1\frac{q}{s},$$

$$k_{22}^1=-k_{23}^4=-k_{32}^4=-{\alpha }_5\frac{s}{p},$$

$$k_{34}^1=k_{43}^1=-{\alpha }_6\frac{s}{q}-{\alpha }_5\frac{s^2}{urpq},$$

$$k_{44}^1={\alpha }_6\frac{s^2}{q^2}+{\alpha }_5\frac{s(2s^2+ur)}{urp{q}^2},$$

$$k_{13}^3=k_{31}^3=k_{14}^4=k_{41}^4=0,$$

$$k_{13}^4=k_{31}^4=0,k_{33}^2=-{\alpha }_6\frac{p}{s},$$

$$k_{34}^2=k_{43}^2={\alpha }_6\frac{p}{q}+{\alpha }_5\frac{s}{urq},$$

$$k_{44}^2=-{\alpha }_5\frac{pq+s^2}{ur{q}^2}-{\alpha }_6\frac{sp}{q^2}.$$

Then, $\tilde{M}=({\mathbb{R}}^4,\tilde{\nabla }:={\nabla }^G+K,G,J)$ is a holomorphic statistical manifold of holomorphic curvature 0.

Next, let M be any m-dimensional submanifold ($m<4$) of $\tilde{M}$. Then, Inequalities (16) and (17) are satisfied. Moreover, the statistical submanifold M of $\tilde{M}$ attains equality in both these inequalities, provided that M is totally geodesic.

5. Conclusions

In this research study, we provided new solutions to the fundamental problem of finding simple relationships between various invariants (intrinsic and extrinsic) of the submanifolds. In this respect, we obtained inequalities involving the normalized $\delta$-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. In addition, we characterized the equality cases. These results may stimulate new research aimed at obtaining similar relationships in terms of various invariants, for statistical submanifolds in other ambient spaces.