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

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.


Introduction
For surfaces M 2 of the Euclidean space E 3 , the Euler inequality G ≤ H 2 is fulfilled, where G is the (intrinsic) Gauss curvature of M 2 and H 2 is the (extrinsic) squared mean curvature of M 2 .
Furthermore, G = H 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 E 2 or, it is (a part of) a round sphere S 2 in E 3 .
In 1979, Wintgen [25] proved that the Gauss curvature G, the squared mean curvature H 2 and the normal curvature G ⊥ of any surface M 2 in E 4 always satisfy the inequality the equality holds if and only if the ellipse of curvature of M 2 in 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 M 2+m (c); namely G ≤ H 2 − |G ⊥ | + c.
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 ≥ 2 and codimension 2 in real space formsM 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.

Statistical manifolds and their submanifolds
A statistical manifold is a Riemannian manifold (M n+k ,g) of dimension (n + k), endowed with a pair of torsion-free affine connections∇ and∇ * satisfying for any X, Y, Z ∈ (TM). The connections∇ and∇ * are called dual connections (see [1,17,22]), and it is easily shown that (∇ * ) * =∇. The pair (∇,g) is said to be a statistical structure. If (∇,g) is a statistical structure onM n+k , so is (∇ * ,g) [1,24]. On the other hand, any torsion-free affine connection∇ always has a dual connection given by∇ where∇ 0 is Levi-Civita connection onM n+k . Denote byR andR * the curvature tensor fields of∇ and∇ * , respectively.
A statistical structure (∇,g) of constant curvature 0 is called a Hessian structure. The curvature tensor fieldsR andR * of dual connections satisfỹ From (2.4) it follows immediately that if (∇,g) is a statistical structure of constant curvature c, then (∇ * ,g) is also a statistical structure of constant curvature c. In particular, if (∇,g) is Hessian, so is (∇ * ,g) [8].
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 curvaturec if and only if (see [21]) for all vector fields onM n+k . If (M n+k ,g) is a statistical manifold and M n a submanifold of dimension n of M n+k , then (M n , g) is also a statistical manifold with the induced connection by∇ and induced metric g. In the case that (M n+k ,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 (T M n⊥ ). In our case, for any X, Y ∈ (T M n ), according to [24], the corresponding Gauss formulas are∇ are symmetric and bilinear, called the imbedding curvature tensor of M n inM n+k for∇ and the imbedding curvature tensor of M n inM n+k for∇ * , respectively. In [24], it is also proved that (∇, g) and (∇ * , g) are dual statistical structures on M n .
Since h and h * are bilinear, we have the linear transformations A ξ and A * ξ on T M n defined by for any ξ ∈ (T M n⊥ ) and X, Y ∈ (T M n ). Further, see [24], the corresponding Weingarten formulas are∇ for any ξ ∈ (T M n⊥ ) and X ∈ (T M n ). The connections ∇ ⊥ X and ∇ * ⊥ X given by (2.9) and (2.10) are Riemannian dual connections with respect to induced metric on (T M n⊥ ). Let {e 1 , . . . , e n } and {ξ 1 , . . . , ξ k } be orthonormal tangent and normal frames, respectively, on M n . Then the mean curvature vector fields are defined by (2.11) and for 1 ≤ i, j ≤ n and 1 ≤ α ≤ k (see also [6]). The corresponding Gauss, Codazzi and Ricci equations are given by the following result. [24] Let∇ and∇ * be dual connections onM n+k and ∇ the induced connection by∇ on M n . LetR and R be the Riemannian curvature tensors for∇ and ∇, respectively. Then,

14)
where R ⊥ is the Riemannian curvature tensor of For the equations of Gauss, Codazzi and Ricci with respect to the connection∇ * on M n , we have Proposition 2.2 [24] Let∇ and∇ * be dual connections onM n+k and ∇ * the induced connection by∇ * on M n . LetR * and R * be the Riemannian curvature tensors for∇ * and ∇ * , respectively. Then,

16)
where R * ⊥ is the Riemannian curvature tensor of ∇ ⊥ * on T M n⊥ , ξ, η ∈ (T M n⊥ ) and Geometric inequalities for statistical submanifolds in statistical manifolds with constant curvature were obtained in [2].

Statistical surfaces in statistical manifolds of constant curvature
Let (M 3 ,g) be a 3-dimensional statistical manifold of constant curvature c and M 2 a surface ofM. 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. Some examples of statistical surfaces satisfying the equality case of the above Euler inequality can be provided by the following. [8].

Example 1 (A trivial example) Recall Lemma 5.3 of Furuhata
Let (H,∇,g) be a Hessian manifold of constant Hessian sectional curvaturec = 0, (M, ∇, g) a trivial Hessian manifold and f : M −→ H a statistical immersion of codimension one. Then one has: 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. (H 3 ,g) be the upper half space of constant sectional curvature −1, i.e.,

An affine connection∇ on H 3 is given bỹ
where i, j = 1, 2. The curvature tensor fieldR of∇ is identically zero, i.e., c = 0. Thus (H 3 ,∇,g) is a Hessian manifold of constant Hessian sectional curvature 4 (see [21]). Now let consider a horosphere M 2 in H 3 having null Gauss curvature, i.e., G ≡ 0 (for details, see [11]). If f : M 2 −→ 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. (M 4 , c), and a surface M 2 ofM 4 . We respectively denote the Gauss curvature, the normal curvature and the Gauss curvature with respect to the Levi-Civita connection by G, G ⊥ 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 * andK 0 . We have the following Wintgen inequalities.
In particular, for c = 0 we derive the following.

Wintgen inequality for statistical submanifolds
Let M n be an n-dimensional statistical submanifold of a (n+m)-dimensional statistical manifold (M n+m , c) of constant curvature c.
The sectional curvature K on M n is defined by [3] (see also [18,19]) for any orthonormal vectors X, Y ∈ T p M n , p ∈ 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 ∈ M n and {e 1 , e 2 , . . . , e n } an orthonormal basis of T p M n . Then the normalized scalar curvature ρ is defined by (see [7]): 1≤i< j≤n g R e i , e j e i , e j + g R * e i , e j e i , e j .
By using the Gauss equations for the dual connections∇ and∇ * , respectively, we obtain Denoting as usual by h r i j = g h e i , e j , ξ r , h * r i j = g h * e i , e j , ξ r , ∀i, j = 1, . . . , n and r = 1, . . . , m, the above equation becomes On the other hand, the normalized normal scalar curvature ρ ⊥ is defined by (see also [3]): The Ricci equations for the dual connections∇, and∇ * , respectively, imply or equivalently, It follows that .
It is known that the components of the second fundamental form h 0 of M n with respect to the Levi-Civita connection∇ 0 are given by 2h 0r ik = h r ik + h * r ik , ∀i, k = 1, . . . , n, r = 1, . . . , m. Then we can write We shall use the algebraic inequality Recall an inequality from [12] (see also [14]) (4.4)