ISRN GeometryVolume 2011 (2011), Article ID 374584, 12 pagesdoi:10.5402/2011/374584

Research Article

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

Münevver Yildirim Yilmaz and Mehmet Bektaş

Department of Mathematics, Faculty of Arts and Science, Firat University, 23119 Elazig, Turkey

Received 22 March 2011; Accepted 26 April 2011

Academic Editor: S. Sivasubramanian

Copyright © 2011 Münevver Yildirim Yilmaz and Mehmet Bektaş. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 with flat connection and Hessian metric 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 Bektaş et al. obtained some curvature conditions, results, and integral inequalities on this type of manifolds, [3–5].

Let be an -dimensional Hessian manifold of constant curvature . Let be an -dimensional Riemannian manifold immersed in . Let be the second fundamental form of the immersion, and the mean curvature vector. Denote by the scalar product of . If there exists a function on such that for any tangent vector on , then is called a pseudo-umbilical submanifold of . It is clear that . If the mean curvature identically, then is called a minimal submanifold of .

Every minimal submanifold of is itself a pseudo umbilical submanifold of . Cao [6] extended Bai’s well-known theorem to the case in which is pseudo-umbilical. 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 be an -dimensional compact pseudo-umbilical submanifold of -dimensional Hessian manifold of constant Hessian sectional curvature . Then where is the square length of the Riemannian curvature tensor, is the square length of the Ricci curvature tensor, is the scalar curvature, and is the mean curvature of .

We will use the same notation and terminologies as in [2] unless otherwise stated.

Let be a Hessian manifold with Hessian structure . We express various geometric concepts for the Hessian structure in terms of affine coordinate system with respect to , that is, .(i)The Hessian metric, (ii)Let be a tensor field of type () defined by where is the Riemannian connection for . Then we have where are the Christoffel’s symbols of .(iii)Define a tensor field of type ()by and call it the Hessian curvature tensor for . Then we have (iv)The Riemannian curvature tensor for , (see [2]).

Definition 1.1. Let be a Hessian curvature tensor on a Hessian manifolds . We define an endomorphism of the space of contravariant symmetric tensor fields of degree 2 by Then is a symmetric operator, [2].

Definition 1.2. For a nonzero contravariant symmetric tensor of degree 2 at , we set and call it the Hessian sectional curvature in the direction , [2].

Theorem 1.3. Let be a Hessian manifold of dimension ≥2. If the Hessian sectional curvature depends only on , then is of constant Hessian sectional curvature. is of constant Hessian sectional curvature if and only if (see [2]).

Corollary 1.4. If a Hessian manifold is a space of constant Hessian sectional curvature , then the Riemannian manifold is a space of constant sectional curvature , [2].

From now on, we shall construct, for each constant , a Hessian manifold with constant Hessian sectional curvature . We now recall the following result due to Shima and Yagi [7]. Let be a simply connected Hessian manifold. If is complete, then is isomorphic to , where is a convex domain in , is the canonical flat connection on , and is a smooth convex function on .(A)Case

It is obvious that the Euclidean space is a simply connected Hessian manifold of constant Hessian sectional curvature 0 [1].(B)Case

Theorem 1.5. Let be a domain in given by where is a positive constant, and let be a smooth function on defined by Then is a simply connected Hessian manifold of positive constant Hessian sectional curvature . As Riemannian manifold is isometric to the hyperbolic space of constant sectional curvature ;

(C)Case

Theorem 1.6. Let be a smooth function on defined by where is a negative constant. Then is a simply connected Hessian manifold of negative constant Hessian sectional curvature . The Riemannian manifold is isometric to a domain of the sphere defined by for all [1].

For the proof of the theorems we refer to [1].

2. Local Formulas

We choose a local field of orthonormal frames in such that restricted to are tangent to . Let be its dual frame field. Then the structure equations of are given by We restrict these forms to , then we have where are the components of the curvature tensor of .

We call the second fundamental form of the immersed manifold . Denote by the square length of , the mean curvature vector and the mean curvature of , respectively. Here is the trace of the matrix . Now let be parallel to . Then we have Let and denote the covariant derivative and the second covariant derivative of , respectively, defined by Then we have The Laplacian of is defined by By a direct calculation we have

3. Proof of Theorem A

From (*) and (2.6), we have therefore, It is obvious that and, therefore, On the other hand, from (2.2) From (2.3), we have while Let then we have Since it follows that that is Since we have From (2.9), (3.4)–(3.14), we have Since is compact and we have and we have

Corollary 3.1. Let be an -dimensional compact pseudo-umbilical submanifold of . Then

Proof. The Euclidean space 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 be a domain in given by where is a positive constant, and let be a smooth function on defined by Let be an -dimensional compact pseudo-umbilical submanifold of . Then Theorem A holds.

Proof. It is obvious that is a simply connected Hessian manifold of positive constant Hessian sectional curvature . As Riemannian manifold is isometric to the hyperbolic space of constant sectional curvature ; As a consequence of Theorem A, we conclude the proof.

On the other hand let us define as a smooth function on as follows where is a negative constant. Then is a simply connected Hessian manifold of negative constant Hessian sectional curvature . The Riemannian manifold is isometric to a domain of the sphere defined by for all . Hence we acquire the following.

Corollary 3.3. Let be a smooth function on defined by where is a negative constant and be an -dimensional compact pseudo-umbilical submanifold of . Then

3.1. Applications in 3-Dimensional Spaces

Here we give some examples of the results indicated above.

Example 3.4. Let be a 2-dimensional compact pseudo-umbilical surface of . Then also note that if the Ricci curvature tensor of the surface is given by , we may also compute the integral terms of Gaussian curvature .

Example 3.5. Let be a domain in given by where is a positive constant, and let be a smooth function on defined by Let be a 2-dimensional compact pseudo-umbilical surface of . Then

Example 3.6. Let be a smooth function on defined by where is a negative constant and be a 2-dimensional compact pseudo-umbilical surface of . Then and is isometric to a domain of the sphere .

References

H. Shima, The Geometry of Hessian Structures, World Scientific, Hackensack, NJ, USA, 2007.

H. Shima, “Hessian manifolds of constant Hessian sectional curvature,” Journal of the Mathematical Society of Japan, vol. 47, no. 4, pp. 735–753, 1995.

M. Bektaş and M. Yıldırım, “Integral inequalities for submanifolds of Hessian manifolds with constant Hessian sectional curvature,” Iranian Journal of Science and Technology. Transaction A, vol. 30, no. 2, pp. 235–239, 2006.

M. Y. Yilmaz and M. Bektaş, “A survey on curvatures of Hessian manifolds,” Chaos, Solitons and Fractals, vol. 38, no. 3, pp. 620–630, 2008.

M. Bektaş, M. Yildirim, and M. Külahci, “On hypersurfaces of Hessian manifolds with constant Hessian sectional curvature,” Journal of Mathematics and Statistics, vol. 1, no. 2, pp. 115–118, 2005.

X.-F. Cao, “Pseudo-umbilical submanifolds of constant curvature Riemannian manifolds,” Glasgow Mathematical Journal, vol. 43, no. 1, pp. 129–133, 2001.

H. Shima and K. Yagi, “Geometry of Hessian manifolds,” Differential Geometry and its Applications, vol. 7, no. 3, pp. 277–290, 1997.