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

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1 International Scolarly Researc Network ISRN Geometry Volume 011, Article ID , 1 pages doi:10.540/011/ Researc Article A Note on Pseudo-Umbilical Submanifolds of Hessian Manifolds wit Constant Hessian Sectional Curvature Münevver Yildirim Yilmaz and Memet Bektaş Department of Matematics, Faculty of Arts and Science, Firat University, 3119 Elazig, Turkey Correspondence sould be addressed to Münevver Yildirim Yilmaz, munyildirim@gmail.com Received Marc 011; Accepted 6 April 011 Academic Editor: S. Sivasubramanian Copyrigt q 011 M. Yildirim Yilmaz and M. Bektaş. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Te geometry of Hessian manifold, as a branc of statistics, pysics, Kaelerian, and affine differential geometry, is deeply fruitful and a new field for scientists. However, inspite of its importance submanifolds and curvature conditions of it ave not been so well known yet. In tis paper, we focus on te pseudo-umbilical submanifolds on Hessian manifold wit 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 te Hessian of functions wit respect to affine coordinate systems. Te pair of D, g wit flat connection D and Hessian metric g is called Hessian structure, and a manifold equipped wit tis structure is said to be a Hessian manifold. In 1,, Hiroiko Sima introduced Hessian sectional curvature and its relations wit Kaelerian manifold. He also proved teorems and gave important remarks on te spaceform of Hessian manifolds. In te ligt of tese studies Bektaş et al. obtained some curvature conditions, results, and integral inequalities on tis 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 be te second fundamental form of te immersion, and ξ te mean curvature vector. Denote by g te scalar product of M n p. If tere exists a function λ on M n suc tat g X, Y,ξ λg X, Y

2 ISRN Geometry for any tangent vector X, Y on M n,tenm n is called a pseudo-umbilical submanifold of M n p. It is clear tat λ 0. If te mean curvature ξ 0 identically, ten 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 teorem to te case in wic M n p is pseudoumbilical. Te aim of te present work is to obtain tis teorem for compact pseudo-umbilical submanifold of a Hessian manifold and also give some results and examples of it. Teorem A. Let M n be an n-dimensional compact pseudo-umbilical submanifold of n p dimensional Hessian manifold of constant Hessian sectional curvature c. Ten Mn { p R ijkl p R ij R n ( 3p n ) c 4 R n( 3p n ) H R n 1 n p c 4 H n 3 1 n c 4 H n 3 ph 4 n 4 H 4} n n 1 ( n p 1 ) c 16 Vol Mn, were R ijkl is te square lengt of te Riemannian curvature tensor, R ij is te square lengt of te Ricci curvature tensor, R is te scalar curvature, and H is te mean curvature of M n. We will use te same notation and terminologies as in unless oterwise stated. Let M n p be a Hessian manifold wit Hessian structure D, g. Weexpressvarious geometric concepts for te Hessian structure D, g in terms of affine coordinate system {x 1,...,x n p } wit respect to D, tatis,ddx i 0. i Te Hessian metric, g ij u x i x j. 1. ii Let γ be a tensor field of type 1, defined by γ X, Y X Y D X Y, 1.3 were is te Riemannian connection for g.tenweave γ i jk Γi jk 1 gir g rj x k, g ij γ ijk 1 x 1 3 u k x i x j x, k γ ijk γ jik γ kji, 1.4 were Γ i jk are te Cristoffel s symbols of.

3 ISRN Geometry 3 iii Define a tensor field S of type 1, 3 by S D γ 1.5 and call it te Hessian curvature tensor for D, g.tenweave S i jkl γ i jl x k, S ijkl 1 4 u 1 3 u x i x j x k x l grs x i x k x 3 u r x j x l x, s S ijkl S ilkj S kjil S jilk S klij. 1.6 iv Te Riemannian curvature tensor for, R i jkl γ i rk γ r jl γ i rl γ r jk, R ijkl 1 ( ) Sjikl S ijkl 1.7 see. Definition 1.1. Let S ik jl be a Hessian curvature tensor on a Hessian manifolds M, D, g. We define an endomorpism ς of te space of contravariant symmetric tensor fields of degree by ξ ξ ik S ik jl ξjl. 1.8 Ten ξ is a symmetric operator,. Definition 1.. For a nonzero contravariant symmetric tensor ξ x of degree at x, weset ξ x ξ ξ x,ξ x ξ x,ξ x 1.9 and call it te Hessian sectional curvature in te direction ξ x,. Teorem 1.3. Let M n p,d,g be a Hessian manifold of dimension. If te Hessian sectional curvature ξ x depends only on x, ten M, D, g is of constant Hessian sectional curvature. M, D, g is of constant Hessian sectional curvature c if and only if S ijkl c ( ) gij g kl g il g kj 1.10 (see []).

4 4 ISRN Geometry Corollary 1.4. If a Hessian manifold M n p,d,g is a space of constant Hessian sectional curvature c, ten te Riemannian manifold M, g is a space of constant sectional curvature c/4, []. From now on, we sall construct, for eac constant c, a Hessian manifold wit constant Hessian sectional curvature c. We now recall te following result due to Sima and Yagi 7. Let M n p,d,g be a simply connected Hessian manifold. If g is complete, ten M n p,d,g is isomorpic to Ω, D, D ϕ, wereω is a convex domain in R n p, D is te canonical flat connection on R n p,andϕ is a smoot convex function on Ω. A Case c 0 It is obvious tat te Euclidean space R n p, D, g 1/ D { n p xa } is a simply connected Hessian manifold of constant Hessian sectional curvature 0 1. B Case c>0 Teorem 1.5. Let Ω be a domain in R n p given by x n p > c ( x A), n p were c is a positive constant, and let ϕ be a smoot function on Ω defined by ϕ 1 c log { x n p c } (x A). 1.1 n p 1 Ten Ω, D, g D ϕ is a simply connected Hessian manifold of positive constant Hessian sectional curvature c. As Riemannian manifold Ω,g is isometric to te yperbolic space H c/4,g of constant sectional curvature c/4; H {( ξ 1,...,ξ n,ξ n p) } R n p ξ n p > 0, { n p 1 } 1 g (dξ A) 4 ξ n p c dξn p C Case c<0 Teorem 1.6. Let ϕ be a smoot function on R n p defined by ϕ 1 ( n p ) c log e cxa 1, 1.14 were c is a negative constant. Ten R n p, D, g D ϕ is a simply connected Hessian manifold of negative constant Hessian sectional curvature c. Te Riemannian manifold R n p,g is isometric to a domain of te spere n p 1 i 1 ξ A 4/c defined by ξ A > 0 for all A [1]. For te proof of te teorems we refer to 1.

5 ISRN Geometry 5. Local Formulas We coose a local field of ortonormal frames e 1,...,e n p in M n p suc tat restricted to M n,e 1,...,e n are tangent to M n.letw 1,...,w n p be its dual frame field. Ten te structure equations of M n p are given by dw A w AB w B, w AB w BA 0, dw AB w AC w CB 1 RABCD w C w D..1 We restrict tese forms to M n,tenweave w 0, w i ij w j, ij ji, dw ij w ik w kj 1 Rijkl w k w l, R ijkl c ( ) ( ) δik δ jl δ il δ jk ik 4 jl il jk,. were R ijkl are te components of te curvature tensor of M n. dw β w γ w γβ 1 Rβkl w k w l, R βkl ) ( ik β il β ik j il..3 We call ij w iw j e.4 te second fundamental form of te immersed manifold M n.denoteby S ( ij).5 te square lengt of, ξ 1/n tr H e te mean curvature vector and H 1/n tr H te mean curvature of M n, respectively. Here tr is te trace of te matrix H ij.nowlete n 1 be parallel to ξ. Tenweave tr H n 1 nh, tr H 0, / n 1..6

6 6 ISRN Geometry Let ijk and ijkl denote te covariant derivative and te second covariant derivative of ij, respectively, defined by ijk w k d ij ik w kj jk w ki β ij w β, ijkl w l d ijk ijl w lk ilk w lj ljk w li β ijk w β..7 Ten we ave ijk ikj 0, ijkl ijlk im R mjkl mj R mikl β ij R βkl..8 Te Laplacian Δ ij of ij is defined by Δ ij. By a direct calculation we ave ijkk 1 ΔS ) ( ijk ij Δ ij ( ijk ) ij kkij ij mk R mijk.9 ij mi R mkjk ij β ki R βjk. 3. Proof of Teorem A From and.6,weave g ( ( e i,e j ),Hen 1 ) H δ ij, 3.1 terefore, ( ) ( ijk ij kkij nhδh, n 1 iik ) n Δi H n H. 3. It is obvious tat 1 ΔH HΔH H 3.3 and, terefore, ( ) ijk ij kkij n H nhδh 1 nδh. 3.4

7 ISRN Geometry 7 On te oter and, from. ij mk R mijk 1 ( ) ij mk mj ik R mijk 1 { R imjk c ( ) } δij δ mk δ mj δ ik R mijk 4 1 R mijk c 4 R, ij mi R mkjk { 1 n c 4 δ mj nh δ mj R mj }R mj 3.5 R mj 1 n c 4 R nh R. From.3,weave ij β ki R βjk ij lj β ki β lk ij lk β ik β lj 3.6 wile ij lj β ki β lk { 1 n c 4 δ il nh δ il R il } n 1 n c 16 n 1 n c 4 H 1 n c 4 R nh R n 3 H 4 R il. 3.7 Let S i,j ( ij), 3.8 ten we ave S S. 3.9 Since ( ) S S S S β, <βs ( ) ( ) S S β p 1 S S β > 0 <βs <β 3.10 it follows tat ( ) p 1 S > S S β S <β S, 3.11

8 8 ISRN Geometry tat is S 1 p S, 3.1 Since ( ) ij lk i,j,l,k i,j,l,k β β ik β lj,β ij β ij lk β lk 3.13,β i,j ij β ij i,j ij ij S we ave ij lk β ik β lj ij lk β ik β lj 1 (,β,β i,j,l,k ij lk ) 1 i,j,l,k β β ik β lj S 1 { ( ) } ij lk ik lj S 1 { R iljk c ( ) } δij δ lk δ ik δ lj 4 S R iljk c c R n 1 n p S. From.9, , weave 1 ΔS 1 nδh R ijkl R ij 3 4 cnr 3nH R n 1 n c 16 n 1 n c 4 H n 3 H 4 1 p S Since M n is compact and S n 1 n c 4 n H R 3.16

9 ISRN Geometry 9 we ave M n { p R ijkl p R ij 3 4 cnpr 3npH R n 1 n c 16 p n 1 n c 4 H p n 3 H 4 p S } 1, S n 1 n c 16 n4 H 4 R n 3 1 n H c 4 n H R n 1 n c 4 R, { p R ijkl p R ij 3 4 ncrp 3nH Rp n 1 n c 16 p M n 3.17 n 1 n c 4 H p n 3 H 4 p n 1 n c 16 n4 H 4 R n 3 1 n H c 16 n H R n 1 n c } 4 R 1 and we ave Mn { p R ijkl p R ij R n ( 3p n ) c 4 R n( 3p n ) H R n 1 n p c 4 H n 3 1 n c 4 H n 3 ph 4 n 4 H 4} n 1 n ( n p 1 ) c 16 Vol Mn. Corollary 3.1. Let M n be an n-dimensional compact pseudo-umbilical submanifold of R n p, D, g 1/ D { x A }.Ten Mn { p R ijkl p R ij R n ( 3p n ) H R n 3 ph 4 n 4 H 4} Proof. Te Euclidean space R n p, D, g 1/ D { n p xa } is a simply connected Hessian manifold of constant Hessian sectional curvature 0. Taking into account of Teorem A, we conclude te corollary. Corollary 3.. Let Ω be a domain in R n p given by x n p > c ( x A), n p 1 3.0

10 10 ISRN Geometry were c is a positive constant, and let ϕ be a smoot function on Ω defined by ϕ 1 c log { x n p c } (x A). 3.1 n p 1 Let M n be an n-dimensional compact pseudo-umbilical submanifold of Ω, D, g Teorem A olds. D ϕ. Ten Proof. It is obvious tat Ω, D, g D ϕ is a simply connected Hessian manifold of positive constant Hessian sectional curvature c. As Riemannian manifold Ω,g is isometric to te yperbolic space H c/4,g of constant sectional curvature c/4; H {( ξ 1,...,ξ n,ξ n p) } R n p ξ n p > 0, { n p 1 } 1 g (dξ A) 4 ξ n p c dξn p. 3. As a consequence of Teorem A, we conclude te proof. On te oter and let us define ϕ asasmootfunctiononr n p as follows ϕ 1 ( n p ) c log e cxa 1, 3.3 were c is a negative constant. Ten R n p, D, g D ϕ is a simply connected Hessian manifold of negative constant Hessian sectional curvature c. Te Riemannian manifold R n p,g is isometric to a domain of te spere n p 1 i 1 ξ A 4/c defined by ξ A > 0forall A. Hence we acquire te following. Corollary 3.3. Let ϕ be a smoot function on R n p defined by ϕ 1 ( n p ) c log e cxa 1, 3.4 were c is a negative constant and M n be an n-dimensional compact pseudo-umbilical submanifold of R n p, D, g D ϕ.ten Mn { p R ijkl p R ij R n ( 3p n ) c 4 R n( 3p n ) H R n 1 n p c 4 H n 3 1 n c 4 H n 3 ph 4 n 4 H 4} n n 1 ( n p 1 ) c 16 Vol Mn.

11 ISRN Geometry Applications in 3-Dimensional Spaces Here we give some examples of te results indicated above. Example 3.4. Let M be a -dimensional compact pseudo-umbilical surface of R 3, D, g 1/ D { x A }.Ten M R ijkl R ij R H ( R 6H 4) also note tat if te Ricci curvature tensor of te surface is given by R ij Kg ij, we may also compute te integral i terms of Gaussian curvature K. Example 3.5. Let Ω be a domain in R 3 given by x 3 > c [ ( x 1) ( x ) ], 3.7 were c is a positive constant, and let ϕ beasmootfunctiononω defined by ϕ 1 { c log x 3 c [ ( x 1) ( x ) ]}. 3.8 Let M be a -dimensional compact pseudo-umbilical surface of Ω, D, g D ϕ.ten M R ijkl R ij R 5 cr H( R 3c 4H ) Example 3.6. Let ϕ beasmootfunctiononr 3 defined by c ϕ 1 ( ) 3 c log e cxa 1, 3.30 were c is a negative constant and M be a -dimensional compact pseudo-umbilical surface of R 3, D, g D ϕ.ten M R ijkl R ij R 3 cr H( R 3c 4H ) and R 3,g is isometric to a domain of te spere S 3 4/c.

12 1 ISRN Geometry References 1 H. Sima, Te Geometry of Hessian Structures, World Scientific, Hackensack, NJ, USA, 007. H. Sima, Hessian manifolds of constant Hessian sectional curvature, te Matematical Society of Japan, vol. 47, no. 4, pp , M. Bektaş andm.yıldırım, Integral inequalities for submanifolds of Hessian manifolds wit constant Hessian sectional curvature, Iranian Science and Tecnology. Transaction A, vol.30,no.,pp , M. Y. Yilmaz and M. Bektaş, A survey on curvatures of Hessian manifolds, Caos, Solitons and Fractals, vol. 38, no. 3, pp , M. Bektaş, M. Yildirim, and M. Külaci, On ypersurfaces of Hessian manifolds wit constant Hessian sectional curvature, Matematics and Statistics, vol. 1, no., pp , X.-F. Cao, Pseudo-umbilical submanifolds of constant curvature Riemannian manifolds, Glasgow Matematical Journal, vol. 43, no. 1, pp , H. Sima and K. Yagi, Geometry of Hessian manifolds, Differential Geometry and its Applications, vol. 7, no. 3, pp , 1997.

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