Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature

Abstract

The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. De Smet et al. (Arch. Math. (Brno) 35:115–128, 1999) conjectured a generalized Wintgen inequality for submanifolds of arbitrary dimension and codimension in Riemannian space forms. This conjecture was proved by Lu (J. Funct. Anal. 261:1284–1308, 2011) and by Ge and Tang (Pac. J. Math. 237:87–95, 2008), independently. In the present paper we establish a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature.

1 Introduction

For surfaces $M^2$ of the Euclidean space $\mathbb {E}^3$, the Euler inequality $G\le \Vert H\Vert ^2$ is fulfilled, where G is the (intrinsic) Gauss curvature of $M^2$ and $\Vert H\Vert ^2$ is the (extrinsic) squared mean curvature of $M^2$.

Furthermore, $G=\Vert H\Vert ^2$ everywhere on $M^2$ if and only if $M^2$ is totally umbilical, or still, by a theorem of Meusnier, if and only if $M^2$ is (a part of) a plane $\mathbb {E}^2$ or, it is (a part of) a round sphere $S^2$ in $\mathbb {E}^3$.

In 1979, Wintgen [25] proved that the Gauss curvature G, the squared mean curvature $\left\| H\right\| ^2$ and the normal curvature $G^{\bot }$ of any surface $M^{2}$ in $\mathbb {E}^{4}$ always satisfy the inequality

$$\begin{aligned} G \le \left\| H\right\| ^2-|G^{\bot }|; \end{aligned}$$

the equality holds if and only if the ellipse of curvature of $M^{2}$ in $\mathbb {E}^{4}$ is a circle.

The Whitney 2-sphere satisfies the equality case of the Wintgen inequality identically.

A survey containing recent results on surfaces satisfying identically the equality case of Wintgen inequality can be read in [5].

Later, the Wintgen inequality was extended by Rouxel [20] and by Guadalupe and Rodriguez [10] independently, for surfaces $M^{2}$ of arbitrary codimension m in real space forms $\widetilde{M}^{2+m}(c)$; namely

$$\begin{aligned} G \le \left\| H\right\| ^2-|G^{\bot }|+c. \end{aligned}$$

The equality case was also investigated.

A corresponding inequality for totally real surfaces in n-dimensional complex space forms was obtained in [13]. The equality case was studied and a non-trivial example of a totally real surface satisfying the equality case identically was given.

In 1999, De Smet et al. [7] formulated the conjecture on Wintgen inequality for submanifolds of real space forms, which is also known as the DDVV conjecture.

This conjecture was proven by the authors for submanifolds $M^n$ of arbitrary dimension $n\ge 2$ and codimension 2 in real space forms $\tilde{M}^{n+2}(c)$ of constant sectional curvature c.

Recently, the DDVV conjecture was finally settled for the general case by Lu [12] and independently by Ge and Tang [9].

One of the present authors obtained generalized Wintgen inequalities for Lagrangian submanifolds in complex space forms [14] and Legendrian submanifolds in Sasakian space forms [15], respectively. Moreover, two of the present authors established in [3] a version of the Euler inequality and the Wintgen inequality for statistical surfaces in statistical manifolds of constant curvature.

In this paper, using the sectional curvature defined in [19], we derive a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature.

2 Statistical manifolds and their submanifolds

A statistical manifold is a Riemannian manifold $(\tilde{M}^{n+k},\tilde{g})$ of dimension $(n+k),$ endowed with a pair of torsion-free affine connections $\tilde{\nabla }$ and $\tilde{\nabla }^{*}$ satisfying

$$\begin{aligned} Z\tilde{g}(X,Y)=\tilde{g}(\tilde{\nabla }_{Z}X,Y)+ \tilde{g}(X,\tilde{\nabla }_{Z}^{*}Y), \end{aligned}$$

(2.1)

for any $X,Y,Z\in \Gamma (T\tilde{M}).$ The connections $\tilde{\nabla }$ and $\tilde{\nabla }^{*}$ are called dual connections (see [1, 17, 22]), and it is easily shown that $(\tilde{\nabla }^{*})^{*}=\tilde{\nabla }.$ The pair $(\tilde{\nabla },\tilde{g})$ is said to be a statistical structure. If $(\tilde{\nabla },\tilde{g})$ is a statistical structure on $\tilde{M}^{n+k},$ so is $(\tilde{\nabla }^{*},\tilde{g})$ [1, 24].

On the other hand, any torsion-free affine connection $\tilde{\nabla }$ always has a dual connection given by

$$\begin{aligned} \tilde{\nabla }+\tilde{\nabla }^{*}=2\tilde{\nabla }^{0}, \end{aligned}$$

(2.2)

where $\tilde{\nabla }^{0}$ is Levi-Civita connection on $\tilde{M}^{n+k}$.

Denote by $\tilde{R}$ and $\tilde{R}^{*}$ the curvature tensor fields of $\tilde{\nabla }$ and $\tilde{\nabla }^{*},$ respectively.

A statistical structure $(\tilde{\nabla },\tilde{g})$ is said to be of constant curvature $c\in \mathbb {R}$ if

$$\begin{aligned} \tilde{R}(X,Y)Z=c\{\tilde{g}(Y,Z)X-\tilde{g}(X,Z)Y\}. \end{aligned}$$

(2.3)

A statistical structure $(\tilde{\nabla },\tilde{g})$ of constant curvature 0 is called a Hessian structure.

The curvature tensor fields $\tilde{R}$ and $\tilde{R}^{*}$ of dual connections satisfy

$$\begin{aligned} \tilde{g}(\tilde{R}^{*}(X,Y)Z,W) =-\tilde{g} (Z,\tilde{R}(X,Y)W). \end{aligned}$$

(2.4)

From (2.4) it follows immediately that if $(\tilde{\nabla },\tilde{g})$ is a statistical structure of constant curvature c,  then $(\tilde{\nabla }^{*},\tilde{g})$ is also a statistical structure of constant curvature c. In particular, if $(\tilde{\nabla },\tilde{g})$ is Hessian, so is $(\tilde{\nabla }^{*},\tilde{g})$ [8].

On a Hessian manifold ($\tilde{M}^{n+k},\tilde{\nabla })$, let $\gamma =\tilde{\nabla }^0-\tilde{\nabla }$. The tensor field Q of type (1,3) defined by the covariant differential $Q=\tilde{\nabla }\gamma$ of $\gamma$ is said to be the Hessian curvature tensor for $\tilde{\nabla }$ (see [21]).

By using the Hessian curvature tensor Q, a Hessian sectional curvature can be defined on a Hessian manifold.

A Hessian manifold has constant Hessian sectional curvature $\tilde{c}$ if and only if (see [21])

$$\begin{aligned} Q(X,Y,Z,W)=\frac{\tilde{c}}{2}[g(X,Y)g(Z,W)+g(X,W)g(Y,Z)], \end{aligned}$$

for all vector fields on $\tilde{M}^{n+k}$.

If $(\tilde{M}^{n+k},\tilde{g})$ is a statistical manifold and $M^n$ a submanifold of dimension n of $\tilde{M}^{n+k},$ then $(M^{n},g)$ is also a statistical manifold with the induced connection by $\tilde{\nabla }$ and induced metric g. In the case that $(\tilde{M}^{n+k},\tilde{g})$ is a semi-Riemannian manifold, the induced metric g has to be non-degenerate. For details, see [23, 24].

In the geometry of Riemannian submanifolds (see [4]), the fundamental equations are the Gauss and Weingarten formulas and the equations of Gauss, Codazzi and Ricci.

Let denote the set of the sections of the normal bundle to $M^{n}$ by $\Gamma (TM^{n\perp }).$

In our case, for any $X,Y\in \Gamma (TM^{n}),$ according to [24], the corresponding Gauss formulas are

$$\begin{aligned}&\tilde{\nabla }_{X}Y =\nabla _{X}Y+h(X,Y),\end{aligned}$$

(2.5)

$$\begin{aligned}&\tilde{\nabla }_{X}^{*}Y =\nabla _{X}^{*}Y+h^{*}(X,Y), \end{aligned}$$

(2.6)

where h, $h^{*}:\Gamma (TM^n)\times \Gamma (TM^n)\rightarrow \Gamma ({TM^n}^\perp )$ are symmetric and bilinear, called the imbedding curvature tensor of $M^{n}$ in $\tilde{M}^{n+k}$ for $\tilde{\nabla }$ and the imbedding curvature tensor of $M^{n}$ in $\tilde{M}^{n+k}$ for $\tilde{\nabla }^{*},$ respectively.

In [24], it is also proved that $(\nabla ,g)$ and $(\nabla ^{*},g)$ are dual statistical structures on $M^{n}.$

Since h and $h^{*}$ are bilinear, we have the linear transformations $A_{\xi }$ and $A_{\xi }^{*}$ on $TM^n$ defined by

$$\begin{aligned}&g(A_{\xi }X,Y)=\tilde{g}(h(X,Y),\xi ),\end{aligned}$$

(2.7)

$$\begin{aligned}&g(A_{\xi }^{*}X,Y)=\tilde{g}(h^{*}(X,Y),\xi ), \end{aligned}$$

(2.8)

for any $\xi \in \Gamma (TM^{n\perp })$ and $X,Y\in \Gamma (TM^{n}).$ Further, see [24], the corresponding Weingarten formulas are

$$\begin{aligned}&\tilde{\nabla }_{X}\xi =-A_{\xi }^{*}X+\nabla _{X}^{\perp }\xi ,\end{aligned}$$

(2.9)

$$\begin{aligned}&\tilde{\nabla }_{X}^{*}\xi =-A_{\xi }X+\nabla _{X}^{*\perp }\xi , \end{aligned}$$

(2.10)

for any $\xi \in \Gamma (TM^{n\perp })$ and $X\in \Gamma (TM^{n}).$ The connections $\nabla _{X}^{\perp }$ and $\nabla _{X}^{*\perp }$ given by (2.9) and (2.10) are Riemannian dual connections with respect to induced metric on $\Gamma (TM^{n\perp }).$

Let $\{e_{1},\ldots ,e_{n}\}$ and $\{\xi _{1},\ldots ,\xi _{k}\}$ be orthonormal tangent and normal frames, respectively, on $M^{n}$. Then the mean curvature vector fields are defined by

$$\begin{aligned} H=\frac{1}{n}\sum _{i=1}^{n}h\left( e_{i},e_{i}\right) =\frac{1}{n} \sum _{\alpha =1}^{k}\left( \sum _{i=1}^{n}h_{ii}^{\alpha }\right) \xi _{\alpha },\quad h_{ij}^{\alpha }=\tilde{g}\left( h\left( e_{i},e_{j}\right) ,\xi _{\alpha }\right) , \end{aligned}$$

(2.11)

and

$$\begin{aligned} H^{*}=\frac{1}{n}\sum _{i=1}^{n}h^{*}\left( e_{i},e_{i}\right) =\frac{ 1}{n}\sum _{\alpha =1}^{k}\left( \sum _{i=1}^{n}h_{ii}^{*\alpha }\right) \xi _{\alpha },\quad h_{ij}^{*\alpha }=\tilde{g}\left( h^{*}\left( e_{i},e_{j}\right) ,\xi _{\alpha }\right) ,\qquad \end{aligned}$$

(2.12)

for $1\le i,j\le n$ and $1\le \alpha \le k$ (see also [6]).

The corresponding Gauss, Codazzi and Ricci equations are given by the following result.

Proposition 2.1

[24] Let $\tilde{\nabla }$ and $\tilde{\nabla }^*$ be dual connections on $\tilde{M}^{n+k}$ and $\nabla$ the induced connection by $\tilde{\nabla }$ on $M^{n}.$ Let $\tilde{R}$ and R be the Riemannian curvature tensors for $\tilde{\nabla }$ and $\nabla ,$ respectively. Then,

$$\begin{aligned} \tilde{g}(\tilde{R}(X,Y)Z,W)= & {} g(R(X,Y)Z,W)+\tilde{g}(h(X,Z),h^{*}(Y,W))\nonumber \\&-\tilde{g}(h^{*}(X,W),h(Y,Z)),\\ (\tilde{R}(X,Y)Z)^{\perp }= & {} \nabla _{X}^{\perp }h(Y,Z)-h(\nabla _{X}Y,Z)-h(Y,\nabla _{X}Z)\nonumber \\&-\{ \nabla _{Y}^{\perp }h(X,Z)-h(\nabla _{Y}X,Z) -h(X,\nabla _{Y}Z)\},\nonumber \end{aligned}$$

(2.13)

$$\begin{aligned} \tilde{g}(R^{\perp }(X,Y)\xi ,\eta )= & {} \tilde{g} (\tilde{R}(X,Y)\xi ,\eta )+g([A_{\xi }^{*},A_{\eta }]X,Y), \end{aligned}$$

(2.14)

where $R^{\perp }$ is the Riemannian curvature tensor of $\nabla ^\perp$ on $TM^{n\perp },\xi ,\eta \in \Gamma ( TM^{n\perp })$ and $[A_{\xi }^{*},A_{\eta }]=A_{\xi }^{*}A_{\eta }-A_{\eta }A_{\xi }^{*}.$

For the equations of Gauss, Codazzi and Ricci with respect to the connection $\tilde{\nabla }^{*}$ on $M^{n}$, we have

Proposition 2.2

[24]  Let $\tilde{\nabla }$ and $\tilde{\nabla }^*$ be dual connections on $\tilde{M}^{n+k}$ and $\nabla ^{*}$ the induced connection by $\tilde{\nabla }^*$ on $M^{n}.$ Let $\tilde{R}^{*}$ and $R^{*}$ be the Riemannian curvature tensors for $\tilde{\nabla }^{*}$ and $\nabla ^{*},$ respectively. Then, 

$$\begin{aligned} \tilde{g}(\tilde{R}^{*}(X,Y) Z,W)= & {} g(R^{*}(X,Y) Z,W)+\tilde{g}(h^{*}(X,Z),h(Y,W))\nonumber \\&-\tilde{g}(h(X,W),h^{*}(Y,Z)),\\ (\tilde{R}^{*}(X,Y)Z)^{\perp }= & {} \nabla _{X}^{*\perp }h^{*} (Y,Z)-h^{*}(\nabla _{X}^{*}Y,Z)-h^{*} (Y,\nabla _{X}^{*}Z)\nonumber \\&-\{\nabla _{Y}^{*\perp }h^{*}(X,Z)-h^{*}(\nabla _{Y}^{*}X,Z)-h^{*}(X,\nabla _{Y}^{*}Z) \},\nonumber \end{aligned}$$

(2.15)

$$\begin{aligned} \tilde{g}(R^{*\perp }(X,Y)\xi ,\eta )= & {} \tilde{g} (\tilde{R}^{*}(X,Y)\xi ,\eta )+g([A_{\xi },A_{\eta }^{*}]X,Y), \end{aligned}$$

(2.16)

where $R^{*\perp }$ is the Riemannian curvature tensor of $\nabla ^{\perp *}$ on $TM^{n\perp }, \xi ,\eta \in \Gamma ( TM^{n\perp })$ and $[A_{\xi },A_{\eta }^{*}]=A_{\xi }A_{\eta }^{*}-A_{\eta }^{*}A_{\xi }.$

Geometric inequalities for statistical submanifolds in statistical manifolds with constant curvature were obtained in [2].

3 Statistical surfaces in statistical manifolds of constant curvature

Let $(\tilde{M}^{3},\tilde{g})$ be a 3-dimensional statistical manifold of constant curvature c and $M^{2}$ a surface of $\tilde{M}.$ Denote the Gauss curvature, the mean curvature and the dual mean curvature of M,  by G,  H and $H^{*},$ respectively. In [3], a version of the Euler inequality for statistical surfaces was given.

Proposition 3.1

[3] Let $M^{2}$ be a surface in a 3-dimensional statistical manifold of constant curvature c. Then its Gauss curvature satisfies : 

$$\begin{aligned} G\le 2\Vert H\Vert \cdot \Vert H^{*}\Vert -c. \end{aligned}$$

(3.1)

Some examples of statistical surfaces satisfying the equality case of the above Euler inequality can be provided by the following.

Example 1

(A trivial example) Recall Lemma 5.3 of Furuhata [8].

Let $(\mathbb {H},\tilde{\nabla },\tilde{g})$ be a Hessian manifold of constant Hessian sectional curvature $\tilde{c}\ne 0,$ $( M,\nabla ,g)$ a trivial Hessian manifold and $f:M\longrightarrow \mathbb {H}$ a statistical immersion of codimension one. Then one has:

$$\begin{aligned} A^{*}=0,\quad h^{*}=0,\quad \Vert H^{*}\Vert =0. \end{aligned}$$

Thus, if dim $M=2$, the immersion f of codimension one satisfies the equality case of the statistical version of the Euler inequality given by Proposition 3.1.

Example 2

Let $(\mathbb {H}^{3},\tilde{g})$ be the upper half space of constant sectional curvature $-1$, i.e.,

$$\begin{aligned} \mathbb {H}^{3}=\{ y=(y^{1},y^{2},y^{3}) \in \mathbb {R}^{3}:y^{3}>0\},\quad \tilde{g}=(y^{3})^{-2}\sum _{k=1}^{3}dy^{k}dy^{k}. \end{aligned}$$

An affine connection $\tilde{\nabla }$ on $\mathbb {H}^3$ is given by

$$\begin{aligned} \tilde{\nabla }_{\frac{\partial }{\partial y^{3}}}\frac{\partial }{\partial y^{3}}=(y^{3})^{-1}\frac{\partial }{\partial y^{3}},\quad \tilde{\nabla }_{\frac{\partial }{\partial y^{i}}}\frac{\partial }{\partial y^{j}}=2\delta _{ij}(y^{3}) ^{-1}\frac{\partial }{\partial y^{3}} ,\quad \tilde{\nabla }_{\frac{\partial }{\partial y^{i}}}\frac{\partial }{ \partial y^{3}}=\tilde{\nabla }_{\frac{\partial }{\partial y^{3}}}\frac{ \partial }{\partial y^{j}}=0, \end{aligned}$$

where $i,j=1,2.$ The curvature tensor field $\tilde{R}$ of $\tilde{\nabla }$ is identically zero, i.e., $c=0$. Thus $( \mathbb {H}^{3},\tilde{\nabla }, \tilde{g})$ is a Hessian manifold of constant Hessian sectional curvature 4 (see [21]).

Now let consider a horosphere $M^{2}$ in $\mathbb {H}^{3}$ having null Gauss curvature, i.e., $G\equiv 0$ (for details, see [11]). If $f:M^{2}\longrightarrow \mathbb {H}^{3}$ is a statistical immersion of codimension one, then, by using Lemma 4.1 of [16], we deduce $A^{*}=0$, and then $H^{*}=0.$ This implies that the horosphere $M^{2}$ satisfies the equality case of the statistical version of the Euler inequality given by Proposition 3.1.

More generally, let consider a 4-dimensional statistical manifold of constant curvature c, i.e. $(\tilde{M}^{4},c),$ and a surface $M^{2}$ of $\tilde{M}^{4}.$ We respectively denote the Gauss curvature, the normal curvature and the Gauss curvature with respect to the Levi-Civita connection by G,  $G^{\perp }$ and $G^{0}.$ Similarly, we respectively denote the mean vector field, the dual mean curvature and the sectional curvature with respect to the Levi-Civita connection by H,  $H^{*}$ and $\tilde{K}^{0}.$ We have the following Wintgen inequalities.

Theorem 3.2

[3] Let $M^{2}$ be a statistical surface in a 4-dimensional statistical manifold $(\tilde{M}^{4},c)$ of constant curvature c. Then

$$\begin{aligned} G+\vert G^{\perp }\vert +2G^{0}\le \frac{1}{2}(\Vert H\Vert ^{2}+\Vert H^{*}\Vert ^{2})-c+2\tilde{K}^{0}(e_{1}\wedge e_{2}). \end{aligned}$$

In particular,  for $c=0$ we derive the following.

Corollary 3.3

[3] Let $M^{2}$ be a statistical surface of a Hessian 4-dimensional statistical manifold $\tilde{M}^{4}$ of Hessian curvature 0. Then : 

$$\begin{aligned} G+\vert G^{\perp }\vert +2G^{0}\le \frac{1}{2}(\Vert H\Vert ^{2}+\Vert H^{*}\Vert ^{2}). \end{aligned}$$

4 Wintgen inequality for statistical submanifolds

Let $M^{n}$ be an n-dimensional statistical submanifold of a $(n+m)$-dimensional statistical manifold $(\tilde{M}^{n+m},c)$ of constant curvature c.

The sectional curvature K on $M^{n}$ is defined by [3] (see also [18, 19])

$$\begin{aligned} K(X\wedge Y)=\frac{1}{2}[g(R( X,Y)X,Y)+g(R^{*}(X,Y)X,Y)], \end{aligned}$$

for any orthonormal vectors $X,Y\in T_{p}M^{n},$ $p\in M^{n}.$

In the case of the Levi-Civita connection, the above definition coincides (up to the sign) to the standard definition of the sectional curvature.

Let $p\in M^{n}$ and $\{e_{1},e_{2},\ldots ,e_{n}\}$ an orthonormal basis of $T_{p}M^{n}.$ Then the normalized scalar curvature $\rho$ is defined by (see [7]):

$$\begin{aligned} \rho= & {} \frac{2}{n\left( n-1\right) }\sum _{1\le i<j\le n}K\left( e_{i}\wedge e_{j}\right) \\= & {} \frac{1}{n\left( n-1\right) }\sum _{1\le i<j\le n}\left[ g\left( R\left( e_{i},e_{j}\right) e_{i},e_{j}\right) +g\left( R^{*}\left( e_{i},e_{j}\right) e_{i},e_{j}\right) \right] . \end{aligned}$$

By using the Gauss equations for the dual connections $\tilde{\nabla }$ and $\tilde{\nabla }^{*}$, respectively, we obtain

$$\begin{aligned} \rho= & {} \frac{1}{n\left( n-1\right) }\sum _{1\le i<j\le n}\left[ -c-g\left( h\left( e_{i},e_{i}\right) ,h^{*}\left( e_{j},e_{j}\right) \right) +g\left( h^{*}\left( e_{i},e_{j}\right) ,h\left( e_{i},e_{j}\right) \right) \right. \\&\left. -\,c-g\left( h^{*}\left( e_{i},e_{i}\right) ,h\left( e_{j},e_{j}\right) \right) +g\left( h\left( e_{i},e_{j}\right) ,h^{*}\left( e_{i},e_{j}\right) \right) \right] . \end{aligned}$$

Denoting as usual by

$$\begin{aligned} h_{ij}^{r}= & {} g\left( h\left( e_{i},e_{j}\right) ,\xi _{r}\right) ,\quad h_{ij}^{*r}=g\left( h^{*}\left( e_{i},e_{j}\right) ,\xi _{r}\right) , \qquad \\&\forall i,j=1,\ldots ,n\text { and }r=1,\ldots ,m, \end{aligned}$$

the above equation becomes

$$\begin{aligned} \rho =-c+\frac{1}{n\left( n-1\right) }\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( 2h_{ij}^{r}h_{ij}^{*r}-h_{ii}^{*r}h_{jj}^{r}-h_{ii}^{r}h_{jj}^{*r}\right) . \end{aligned}$$

(4.1)

On the other hand, the normalized normal scalar curvature $\rho ^{\perp }$ is defined by (see also [3]):

$$\begin{aligned} \rho ^{\perp }=\frac{1}{n\left( n-1\right) }\left\{ \sum _{1\le r<s\le m}\sum _{1\le i<j\le n}\left[ g\left( R^{\perp }\left( e_{i},e_{j}\right) \xi _{r},\xi _{s}\right) +g\left( R^{*\perp }\left( e_{i},e_{j}\right) \xi _{r},\xi _{s}\right) \right] ^{2}\right\} ^{\frac{1}{2}}. \end{aligned}$$

The Ricci equations for the dual connections $\tilde{\nabla },$ and $\tilde{ \nabla }^{*}$, respectively, imply

$$\begin{aligned} \rho ^{\perp }=\frac{1}{n\left( n-1\right) }\left\{ \sum _{1\le r<s\le m}\sum _{1\le i<j\le n}\left[ g\left( \left[ A_{\xi _{r}}^{*},A_{\xi _{s}}\right] e_{i},e_{j}\right) +g\left( \left[ A_{\xi _{r}},A_{\xi _{s}}^{*}\right] e_{i},e_{j}\right) \right] ^{2}\right\} ^{\frac{1}{2}} \end{aligned}$$

or equivalently,

$$\begin{aligned} \rho ^{\perp }=\frac{1}{n\left( n-1\right) }\left\{ \sum _{1\le r<s\le m}\sum _{1\le i<j\le n}\left[ \sum _{k=1}^{n}\left( h_{ik}^{s}h_{jk}^{*r}-h_{ik}^{*r}h_{jk}^{s}+h_{ik}^{*s}h_{jk}^{r}-h_{ik}^{r}h_{jk}^{*s}\right) \right] ^{2}\right\} ^{\frac{1 }{2}}. \end{aligned}$$

It follows that

$$\begin{aligned} \rho ^{\perp }= & {} \frac{1}{n\left( n-1\right) }\left\{ \sum _{1\le r<s\le m}\sum _{1\le i<j\le n}\left[ \sum _{k=1}^{n}\left( \left( h_{ik}^{s}+h_{ik}^{*s}\right) \left( h_{jk}^{r}+h_{jk}^{*r}\right) -h_{ik}^{s}h_{jk}^{r}-h_{ik}^{*s}h_{jk}^{*r}\right. \right. \right. \\&\left. \left. \left. -\left( h_{ik}^{r}+h_{ik}^{*r}\right) \left( h_{jk}^{s}+h_{jk}^{*s}\right) +h_{ik}^{r}h_{jk}^{s}+h_{ik}^{*r}h_{jk}^{*s}\right) \phantom {\sum _{1\le r<s\le m}\sum _{1\le i<j\le n}}\right] ^{2}\right\} ^{\frac{1}{2}}. \end{aligned}$$

It is known that the components of the second fundamental form $h^0$ of $M^n$ with respect to the Levi-Civita connection $\tilde{\nabla }^0$ are given by $2h_{ik}^{0r}=h_{ik}^{r}+h_{ik}^{*r},$ $\forall i,k=1,\ldots ,n,$ $r=1,\ldots ,m.$ Then we can write

$$\begin{aligned} \rho ^{\perp }= & {} \frac{1}{n\left( n-1\right) }\left\{ \sum _{1\le r<s\le m}\sum _{1\le i<j\le n}\left[ \sum _{k=1}^{n}\left( 4\left( h_{ik}^{0s}h_{jk}^{0r}-h_{ik}^{0r}h_{jk}^{0s}\right) +\left( h_{ik}^{r}h_{jk}^{s}-h_{ik}^{s}h_{jk}^{r}\right) \right. \right. \right. \nonumber \\&\left. \left. \left. ~~~~~~~~~~~~~~~~~+\left( h_{ik}^{*r}h_{jk}^{*s}-h_{ik}^{*s}h_{jk}^{*r}\right) \right) \right] ^{2}\right\} ^{\frac{1}{2}}. \end{aligned}$$

(4.2)

We shall use the algebraic inequality

$$\begin{aligned} \left( a+b+c\right) ^{2}\le 3(a^{2}+b^{2}+c^{2}),\quad \forall a,b,c\in \mathbb {R}. \end{aligned}$$

Therefore

$$\begin{aligned} \rho ^{\perp }\le & {} \frac{3}{n\left( n-1\right) }\left\{ \sum _{1\le r<s\le m}\sum _{1\le i<j\le n}\left( 16\left[ \sum _{k=1}^{n}\left( h_{ik}^{0s}h_{jk}^{0r}-h_{ik}^{0r}h_{jk}^{0s}\right) \right] ^{2}\right. \right. \nonumber \\&\left. \left. +\left[ \sum _{k=1}^{n}\left( h_{ik}^{r}h_{jk}^{s}-h_{ik}^{s}h_{jk}^{r}\right) \right] ^{2}+\left[ \sum _{k=1}^{n}\left( h_{ik}^{*r}h_{jk}^{*s}-h_{ik}^{*s}h_{jk}^{*r}\right) \right] ^{2}\right) \right\} ^{\frac{1}{2}}.\qquad \end{aligned}$$

(4.3)

Recall an inequality from [12] (see also [14])

$$\begin{aligned}&\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ii}^{r}-h_{jj}^{r}\right) ^{2}+2n\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ij}^{r}\right) ^{2}\\&\quad \ge 2n\left[ \sum _{1\le i<j\le n}\sum _{1\le r<s\le m}\left( \sum _{k=1}^n(h_{jk}^{r}h_{ik}^{s}-h_{ik}^{r}h_{jk}^{s}) \right) ^{2}\right] ^{\frac{1}{2}}. \end{aligned}$$

Similarly, we have

$$\begin{aligned}&\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ii}^{*r}-h_{jj}^{*r}\right) ^{2}+2n\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ij}^{*r}\right) ^{2}\\&\quad \ge 2n\left[ \sum _{1\le i<j\le n}\sum _{1\le r<s\le m}\left( \sum _{k=1}^n(h_{jk}^{*r}h_{ik}^{*s}-h_{ik}^{*r}h_{jk}^{*s})\right) ^{2} \right] ^{\frac{1}{2}} \end{aligned}$$

and

$$\begin{aligned}&\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ii}^{0r}-h_{jj}^{0r}\right) ^{2}+2n\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ij}^{0r}\right) ^{2}\\&\quad \ge 2n\left[ \sum _{1\le i<j\le n}\sum _{1\le r<s\le m}\left( \sum _{k=1}^n(h_{jk}^{0r}h_{ik}^{0s}-h_{ik}^{0r}h_{jk}^{0s})\right) ^{2}\right] ^{\frac{1 }{2}}. \end{aligned}$$

Summing up the above three inequalities, from (4.3) we obtain

$$\begin{aligned} \rho ^{\perp }\le & {} \frac{3}{2n^{2}\left( n-1\right) }\sum _{r=1}^{m}\sum _{1 \le i<j\le n}\left[ \left( h_{ii}^{r}-h_{jj}^{r}\right) ^{2}+\left( h_{ii}^{*r}-h_{jj}^{*r}\right) ^{2}+16\left( h_{ii}^{0r}-h_{jj}^{0r}\right) ^{2}\right] \nonumber \\&+\frac{3}{n\left( n-1\right) }\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left[ \left( h_{ij}^{r}\right) ^{2}+\left( h_{ij}^{*r}\right) ^{2}+16\left( h_{ij}^{0r}\right) ^{2}\right] . \end{aligned}$$

(4.4)

Also, we can write

$$\begin{aligned} n^{2}\left\| H\right\| ^{2}=\sum _{r=1}^{m}\left( \sum _{i=1}^{n}h_{ii}^{r}\right) ^{2}=\frac{1}{n-1}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ii}^{r}-h_{jj}^{r}\right) ^{2}+\frac{2n}{n-1} \sum _{r=1}^{m}\sum _{1\le i<j\le n}h_{ii}^{r}h_{jj}^{r} \end{aligned}$$

and similarly,

$$\begin{aligned} n^{2}\left\| H^{*}\right\| ^{2}=\frac{1}{n-1}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ii}^{*r}-h_{jj}^{*r}\right) ^{2}+\frac{2n}{n-1}\sum _{r=1}^{m}\sum _{1\le i<j\le n}h_{ii}^{*r}h_{jj}^{*r} \end{aligned}$$

and

$$\begin{aligned} n^{2}\left\| H^{0}\right\| ^{2}=\frac{1}{n-1}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ii}^{0r}-h_{jj}^{0r}\right) ^{2}+\frac{2n}{n-1} \sum _{r=1}^{m}\sum _{1\le i<j\le n}h_{ii}^{0r}h_{jj}^{0r}. \end{aligned}$$

Substituting in (4.4), we get

$$\begin{aligned} \rho ^{\perp }\le & {} \frac{3}{2}\left\| H\right\| ^{2}+\frac{3}{2}\left\| H^{*}\right\| ^{2}+24\left\| H^{0}\right\| ^{2} \\&-\frac{3}{n(n-1)}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left( h_{ii}^{r}h_{jj}^{r}+h_{ii}^{*r}h_{jj}^{*r}+16h_{ii}^{0r}h_{jj}^{0r}\right) \\&+\frac{3}{n(n-1)}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left[ \left( h_{ij}^{r}\right) ^{2}+\left( h_{ij}^{*r}\right) ^{2}+16\left( h_{ij}^{0r}\right) ^{2}\right] \\= & {} \frac{3}{2}\left\| H\right\| ^{2}+\frac{3}{2}\left\| H^{*}\right\| ^{2}+24\left\| H^{0}\right\| ^{2} \\&-\frac{3}{n(n-1)}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left[ \left( h_{ii}^{r}+h_{ii}^{*r}\right) \left( h_{jj}^{r}+h_{jj}^{*r}\right) -h_{ii}^{*r}h_{jj}^{r}-h_{ii}^{r}h_{jj}^{*r}+16h_{ii}^{0r}h_{jj}^{0r} \right] \\&+\frac{3}{n(n-1)}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left[ \left( h_{ij}^{r}+h_{ij}^{*r}\right) ^{2}-2h_{ij}^{r}h_{ij}^{*r}+16\left( h_{ij}^{0r}\right) ^{2}\right] . \end{aligned}$$

Using again $2h_{ij}^{0r}=h_{ij}^{r}+h_{ij}^{*r},$ $\forall i,j=1,\ldots ,n,$ $r=1,\ldots ,m,$ we obtain

$$\begin{aligned} \rho ^{\perp }\le & {} \frac{3}{2} \left\| H\right\| ^{2}+\frac{3}{2}\left\| H^{*}\right\| ^{2}+24\left\| H^{0}\right\| ^{2}\nonumber \\&-\frac{3}{n(n-1)}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left[ 20h_{ii}^{0r}h_{jj}^{0r}-h_{ii}^{*r}h_{jj}^{r}-h_{ii}^{r}h_{jj}^{*r}-20(h_{ij}^{0r})^{2}+2h_{ij}^{r}h_{ij}^{*r}\right] .\nonumber \\ \end{aligned}$$

(4.5)

Substituting (4.1) in (4.5), one leads to

$$\begin{aligned} \rho ^{\perp }\le & {} \frac{3}{2}\left\| H\right\| ^{2}+\frac{3}{2}\left\| H^{*}\right\| ^{2}+ 24\left\| H^{0}\right\| ^{2}-3\rho -3c\nonumber \\&-\,\frac{60}{n\left( n-1\right) }\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left[ h_{ii}^{0r}h_{jj}^{0r}-\left( h_{ij}^{0r}\right) ^{2}\right] . \end{aligned}$$

(4.6)

If we denote by

$$\begin{aligned} \tilde{\rho }^{0}=\frac{2}{n\left( n-1\right) }\sum _{1\le i<j\le n}\tilde{R} ^{0}\left( e_{i},e_{j},e_{i},e_{j}\right) , \end{aligned}$$

the Gauss equation for the Levi-Civita connection $\tilde{\nabla }^{0}$ gives

$$\begin{aligned} \tilde{\rho }^{0}=\rho ^0 -\frac{2}{n(n-1)}\sum _{r=1}^{m}\sum _{1\le i<j\le n}\left[ h_{ii}^{0r}h_{jj}^{0r}-\left( h_{ij}^{0r}\right) ^{2}\right] . \end{aligned}$$

(4.7)

From (4.6) and (4.7) we obtain

$$\begin{aligned} \rho ^{\perp }\le \frac{3}{2}\left\| H\right\| ^{2}+\frac{3}{2}\left\| H^{*}\right\| ^{2}+ 24\left\| H^{0}\right\| ^{2}-3\rho -3c+ 30\left( \tilde{\rho }^{0}-\rho ^{0}\right) . \end{aligned}$$

Summarizing, we proved the following generalized Wintgen inequality.

Theorem 4.1

Let $M^{n}$ be a submanifold in a statistical manifold $(\tilde{M}^{n+m},c)$ of constant curvature c. Then

$$\begin{aligned} \rho ^{\perp }+3\rho \le \frac{15}{2}\Vert H\Vert ^{2}+\frac{15}{2}\Vert H^{*}\Vert ^{2}+ 12g(H,H^{*})-3c+30(\tilde{\rho }^{0}-\rho ^{0}). \end{aligned}$$