A Note on Pseudo-Umbilical Submanifolds of Hessian Manifolds with Constant Hessian Sectional Curvature

The geometry of Hessian manifold, as a branch of statistics, physics, Kaehlerian, and affine differential geometry, is deeply fruitful and a new field for scientists. However, inspite of its importance submanifolds and curvature conditions of it have not been so well known yet. In this paper, we focus on the pseudo-umbilical submanifolds on Hessian manifold with constant Hessian sectional curvature and using sectional curvature conditions we obtain new results on it.


Introduction
A Riemannian metric on a flat manifold is called a Hessian metric if it is locally expressed by the Hessian of functions with respect to affine coordinate systems.The pair of D, g with flat connection D and Hessian metric g is called Hessian structure, and a manifold equipped with this structure is said to be a Hessian manifold.In 1, 2 , Hirohiko Shima introduced Hessian sectional curvature and its relations with Kaehlerian manifold.He also proved theorems and gave important remarks on the spaceform of Hessian manifolds.In the light of these studies Bektas ¸et al. obtained some curvature conditions, results, and integral inequalities on this type of manifolds, 3-5 .Let M n p be an n p -dimensional Hessian manifold of constant curvature c.Let M n be an n-dimensional Riemannian manifold immersed in M n p .Let h be the second fundamental form of the immersion, and ξ the mean curvature vector.Denote by g the scalar product of M n p .If there exists a function λ on M n such that g h X, Y , ξ λg X, Y * for any tangent vector X, Y on M n , then M n is called a pseudo-umbilical submanifold of M n p .It is clear that λ ≥ 0. If the mean curvature ξ 0 identically, then M n is called a minimal submanifold of M n p .
Every minimal submanifold of M n p is itself a pseudo umbilical submanifold of M n p .Cao 6 extended Bai's well-known theorem to the case in which M n p is pseudoumbilical.The aim of the present work is to obtain this theorem for compact pseudo-umbilical submanifold of a Hessian manifold and also give some results and examples of it.
Theorem A. Let M n be an n-dimensional compact pseudo-umbilical submanifold of n pdimensional Hessian manifold of constant Hessian sectional curvature c.Then where R 2 ijkl is the square length of the Riemannian curvature tensor, R 2 ij is the square length of the Ricci curvature tensor, R is the scalar curvature, and H is the mean curvature of M n .
We will use the same notation and terminologies as in 2 unless otherwise stated.Let M n p be a Hessian manifold with Hessian structure D, g .We express various geometric concepts for the Hessian structure D, g in terms of affine coordinate system {x 1 , . . ., x n p } with respect to D, that is, Ddx i 0.
i The Hessian metric, ii Let γ be a tensor field of type 1, 2 defined by where ∇ is the Riemannian connection for g.Then we have where Γ i jk are the Christoffel's symbols of ∇.
iii Define a tensor field S of type (see [2]).
Corollary 1.4.If a Hessian manifold M n p , D, g is a space of constant Hessian sectional curvature c, then the Riemannian manifold M, g is a space of constant sectional curvature −c/4, [2].
From now on, we shall construct, for each constant c, a Hessian manifold with constant Hessian sectional curvature c.We now recall the following result due to Shima and Yagi 7 .Let M n p , D, g be a simply connected Hessian manifold.If g is complete, then M n p , D, g is isomorphic to Ω, D, D 2 ϕ , where Ω is a convex domain in Ê n p , D is the canonical flat connection on Ê n p , and ϕ is a smooth convex function on Ω.
Let Ω be a domain in Ê n p given by where c is a positive constant, and let ϕ be a smooth function on Ω defined by Then Ω, D, g D 2 ϕ is a simply connected Hessian manifold of positive constant Hessian sectional curvature c.As Riemannian manifold Ω, g is isometric to the hyperbolic space H −c/4 , g of constant sectional curvature −c/4; 1.13 C Case c < 0 Theorem 1.6.Let ϕ be a smooth function on Ê n p defined by where c is a negative constant.Then Ê n p , D, g D 2 ϕ is a simply connected Hessian manifold of negative constant Hessian sectional curvature c.The Riemannian manifold Ê n p , g is isometric to a domain of the sphere For the proof of the theorems we refer to 1 .

Local Formulas
We choose a local field of orthonormal frames e 1 , . . ., e n p in M n p such that restricted to M n , e 1 , . . ., e n are tangent to M n .Let w 1 , . . ., w n p be its dual frame field.Then the structure equations of M n p are given by

2.1
We restrict these forms to M n , then we have where R ijkl are the components of the curvature tensor of M n .

2.3
We

2.7
Then we have

2.8
The Laplacian Δh α ij of h α ij is defined by Δh α ij h α ijkk .By a direct calculation we have 2.9

Proof of Theorem A
From * and 2.6 , we have

3.2
It is obvious that and, therefore,

3.4
On the other hand, from 2.2

3.5
From 2.3 , we have

3.15
Since M n is compact and we have 3.17 and we have Vol M n .

3.18
Corollary 3.1.Let M n be an n-dimensional compact pseudo-umbilical submanifold of Ê n p , D, g Proof.The Euclidean space Ê n p , D, g } is a simply connected Hessian manifold of constant Hessian sectional curvature 0. Taking into account of Theorem A, we conclude the corollary.

Corollary 3.2.
Let Ω be a domain in Ê n p given by where c is a positive constant, and let ϕ be a smooth function on Ω defined by

3.21
Let M n be an n-dimensional compact pseudo-umbilical submanifold of Ω, D, g D 2 ϕ .Then Theorem A holds.
Proof.It is obvious that Ω, D, g D 2 ϕ is a simply connected Hessian manifold of positive constant Hessian sectional curvature c.As Riemannian manifold Ω, g is isometric to the hyperbolic space H −c/4 , g of constant sectional curvature −c/4;

3.22
As a consequence of Theorem A, we conclude the proof.
On the other hand let us define ϕ as a smooth function on Ê n p as follows where c is a negative constant and M n be an n-dimensional compact pseudo-umbilical submanifold of 3.25

Applications in 3-Dimensional Spaces
Here we give some examples of the results indicated above.
Example 3.4.Let M 2 be a 2-dimensional compact pseudo-umbilical surface of Ê 3 , D, g also note that if the Ricci curvature tensor of the surface is given by R ij Kg ij , we may also compute the integral i terms of Gaussian curvature K.
Example 3.5.Let Ω be a domain in Ê 3 given by where c is a positive constant, and let ϕ be a smooth function on Ω defined by

1 e 1 ξ 2 A− 4 /Corollary 3 . 3 .
−cx A 1 , 3.23where c is a negative constant.Then Ê n p , D, g D 2 ϕ is a simply connected Hessian manifold of negative constant Hessian sectional curvature c.The Riemannian manifold Ê n p , g is isometric to a domain of the sphere n p 1 i c defined by ξ A > 0 for all A. Hence we acquire the following.Let ϕ be a smooth function on Ê n p defined by