The Maximal Space-Like Submanifolds. in the Pseudo-Riemannian Space

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1 International Mathematical Forum, Vol. 6, 20, no. 60, The Maximal Sace-Like Submanifolds in the Pseudo-Riemannian Sace Yang Huizhang, Ding Yumin and Mu Feng Deartment of Mathematics, Honghe University Mengzi Yunnan 6600, P.R. China Abstract Let x : M n N n+ (r) be a seudo-riemannian manifold isometrically immersed into seudo-riemannian sace N n+ (r). In this aer, we generalize the Takahashi Theorem and obtain some results of maximal sace-like submanifolds in the seudo-riemannian sace. Keywords: Pseudo-Riemannian submanifolds, mean curvature, sace-like submanifolds Introduction Let x : M n S n+ be a n-dimensional Riemannian manifold isometrically immersed into a (n + )-dimensional shere S n+ (r), H is the mean curvature of M n. A famous Takahashi Theorem was obtained in [6] as Theorem (Takahashi Theorem)[6] Let x : M n S n+ be a n-dimensional Riemanian manifold isometrically immersed into a (n + )-dimensional shere S n+ (r). Then x(m) is a minimal submanifold if and only if Δx = λx, where Δ is Lalace oerator of M n, and λ = n r. 2 Now, we let x : M n N n+ be a n-dimensional Riemannian manifold isometrically immersed into a (n + )-dimensional seudo-riemannian sace N n+ (r), i.e., the Riemannian metric of M n is induced by the seudo-riemannian metric of N n+ (r). Then x is called sace-like immersion. H is the mean curvature of M n,ifh = 0, then M n is a maximal submanifold.

2 3006 Yang Huizhang, Ding Yumin and Mu Feng There are many results generalized by Takahashi Theorem, we can see [2]- [5]. In [7], we got generalized Theorems in seudo-riemannian manifolds as Theorem 2[7] Let x : M n S n+ (r) be a seudo-riemannian manifold isometrically immersed into the seudo-shere S n+ (r), then x is maximal if and only if Δx = n r x, where Δ is Lalace oerator of M n. 2 Theorem 3[7] Let x : M n S n+ (r) be a seudo-riemannian manifold isometrically immersed into the seudo-shere S n+ (r), where S n+ (r) ={x = (x,,x n++ ) R n++, x = r} with the sectional curvature c = r. if 2 Δ 2 x = n2 x, then x is a maximal immersion. r4 We denote S n+ (r) = {x R n++ x, x = r 2 } and H n+ (r) = {x R n++ x, x = r 2 } by the seudo-shere and the seudo-hyerbolic sace in the seudo-euclidean sace R n++ above results, we obtain resectively. For further study of the Theorem A Let x : M n N n+ (r) bean-dimensional Riemannian manifold isometrically immersed into a (n + )-dimensional seudo-riemannian sace N n+ (r). Δ is Lalace oerator of M n, and Δx = λx, then ()when λ = n r 2 S n+ (r); (2)when λ = n r 2 H n+ (r). > 0 if and only if x(m) is a maximal submanifold in < 0 if and only if x(m) is a maximal submanifold in Corollary Let x : M n N n+ (r)bean-dimensional Riemannian manifold isometrically immersed into a (n + )-dimensional seudo-riemannian sace N n+ (r). Δ is Lalace oerator of M n, and Δ 2 x = λx, then if λ = n2 r, x is a 4 maximal submanifold in S n+ (r) orh n+ (r). Theorem B Let x : M n S n+ (c) bean-dimensional comlete maximal sace-like submanifold isometrically immersed into a (n + )-dimensional seudo-shere sace S n+ (c), where c is the sectional curvature of M n.ifc 0, then M n is a geodesic sace-like submanifold. 2 Preliminaries For the convenience, we make the following convention on the ranges of indices: i,, k, n, A, B, C, n +, n + α, β, γ, n +.

3 Maximal sace-like submanifolds 3007 Suose that N n+ (r) is a seudo-riemannian manifold, M n is isometrically immersed into seudo-riemannian manifold N n+ (r). x =(x,,x n++ ) is the rectangular coordinate system of seudo-euclidean sace R n++ x, x = x 2 A + n++ A= A=+, and x 2 A. Let {e A} be a local orthonormal basis of R n++ with dual basis ω A. {e i }, i n are tangent vectors of M n and {e α },n+ α n++ are normal vectors of M n, e A,e B = ε A δ AB, ε A = ±, the basic formulas of R n++ are the following dx = A ω A e A, de A = B ω B A e B, where ε B ωa B + ε AωB A = 0, and ωb A is the connection form of Rn++ So the basic formulas of R n++ which is confined on M n are. dx = ω i e i, ω α =0 i de i = ω i e + ε α h α i ω e α (2.) α, de α = ε h α iω i e + ωαe β β i, β Where ω i is the connection form of M n and ω α =0,ω α i = h α i ω,h α i = hα i. (2.2) The structure equations of N n+ are given by dω A = ɛ B ω AB ω B, ω AB + ω BA =0 B dω AB = ɛ C ω AC ω CB R ABCD ω C ω D C 2 C,D K ABCD = ɛ A ɛ B c(δ AC δ BD δ AD δ BC ). From these formulas, we obtain the structure equations of M n : dω i = ω i ω, ω i + ω i =0 dω i = ω ik ω k R ikl ω k ω l k 2 k,l R ikl = K ikl (h α ik hα l hα il hα k ). α (2.3) (2.4) where R ikl,k ABCD are the comonents of the curvature tensor of M n and N n+ resectively.

4 3008 Yang Huizhang, Ding Yumin and Mu Feng Recall that h = α h α e α = h α i e α ω i ω α,i, is the second fundamental form of M n. The square length of the second fundamental form is defined by: S = α tr(h α ) 2 = α,i,(h α i )2 = h 2. Let h α ik and h α ikl denote the covariant derivative and the second covariant derivative of h α i. Then we have h α ik hα ik = K αik (2.5), h α ikl hα ilk = m h α im R mkl + m h α m R mikl + β h β ir αβkl, (2.6) where R αβkl are the comonents of the normal curvature tensor of M n, that is R αβkl = K αβkl + i (h α ik hβ il hα il hβ ik ). The Ricci curvature of M n is given by R i = k K ikk + αβ ɛ α ɛ k (h α i hα kk hα ik hα k ). (2.7) 3 Proof of the Theorems Proof of the Theorem A Let x(p )={x,,x n++ } be the osition vector of P M n in R n++, and {e A } be a fixed othtonormal basis of R n++, it s obvious that e i,e i >=, e α,e α =, e α,e i =0, e n++,e i =0. Denoting x =(x,x 2,,x n++ )= A x A e A = i ε i x A iie A, Since x A iω := dx A i x A ω i = e A,de i x A ω i = e A, ω i e + ε α h α i ω e α cε i ω i x α, x A ω i = ( ε αh α i e A,e α cε i x A δ i )ω. α

5 Maximal sace-like submanifolds 3009 Obviously, we have x A i = α ε αh α i e A,e α cε i x A δ i and Δx A = i ε i x A ii = e A,nH cx A. So Δx = nh cnx = λx. Then x is a maximal submanifold if and only if Δx = cnx = λx, i.e., λ = cn. However, from (2.3) and (2.4), we know if x(m) S n+ (r),c=,ifx(m) Hn+ r2, c =. Thus we finish the roof r2 of Theorem A. Using the same methods as [7] and from Theorem A, we can easily rove the Corollary. Proof of the Theorem B Suose that M n is the maximal sace-like submanifold of the seudo-riemannian manifold N n+, from (2.5) and (2.6), we obtain Δh α i = k h α ikk = k (K αkik + K αikk )+ (h α mk R mik + h α im R mkk)+ h β ik R αβk k,m k,β = (K αkik + K αikk )+ (h α mk K mik + h α im K mkk)+ h β ik K αβk k k,m k,β + (h β mk hα im hβ k + hβ mk hα mk hβ i 2h β mh α mk hβ ik + hβ ik hα m hβ mk ) k,m,β (2.8) Therefore, we have 2 ΔS = i,,k,α = (h α ik )2 + h α i Δhα i i,,α (h α ik )2 h α i (K αkik + K αikk )+ h α i (hα mk K mik + h α im K mkk) i,,k,α i,,k,m,α h α ih β ik K αβk + (h α ikh β k hβ ik hα k)(h α imh β m h β imh α m) i,,k,α + i,,k,α,β + i,,k,m,,β h α ih α mkh β mk hβ i i,,k,m,α,β Writing H α = (h α i ) n n and S αβ = i,,α,β (2.9) h α i hβ i, then (S αβ ) is a symmetric matrix, so we can choose a local orthonormal frame field {e i } of M n such that (S αβ ) is diagonalizable, i.e., S αβ = S αα δ αβ, and so S = S αα. For a matrix α A, we denote N(A) =tr(aa T ) and obviously have 2 ΔS = i,,k,α + α,β (h α ik )2 i,,k,α h α i (K αkik + K αikk )+ i,,k,m,α h α i (hα mk K mik + h α im K mkk) N(H α H β H β H α )+ α (S αα) 2 (2.0)

6 S2 (2.) 300 Yang Huizhang, Ding Yumin and Mu Feng Because of N(H α H β H β H α ) 0, α (S αα) 2 S2, and i,,k,m,α h α i (hα mk K mik+ h α im K mkk) ncs, where c is the sectional curvature of S n+ and c 0. Thus we have 2 ΔS = i,,k,α(h α ik )2 + α S β ) α<β(s 2 + ncs + N(H α H β H β H α )+ α,β S2 Then, if M is comact, from Hof Theorem, we obtain S =0,soM is totally geodesic sace-like submanifold. If M is not comact, since M is maximal and from (2.3) and (2.7), it is obvious that the Ricci curvature of M n is nonnegative, we denote u = S + δ, where δ is constant and δ>0, then we have infu = 0. From sus + δ (2.) we can obtain Δu u u 2 S2 u 2. Alying Omori-Yau s generalized maximum rincile to the function of u, then there exists oint range {x i } M such that 0 lim Δu(x t) t (sus)2 (infu) 2 0 Thus we have sus = 0, i.e., S = 0, therefore, M is totally geodesic. Acknowledgements This work was suorted by the Science Foundation of Yunnan Province Office of Education and the Science Foundation of Honghe University(0XJY2) of China. References [] Ishihara T.Maximal sace-like submanifolds of a seudo-riemannian sace of constant curvature.ann of Math.,04(976): [2] Montiel S., An integral inequality for comact sace-like hyersurfaces in desitter sace and alications to the case of constant mean curvature.indiana Univ Math J.,988,37(4):909. [3] Park J., Hyersurfaces satisfying the Equation Δx = Rx+b. Proc.Amer.Math.Soc. 20(994): [4] Qiu T.Z.,Ouyang C.Z.The sace-like hyoersurfaces with Δx = Rx in De sitter sace and Anti-de sitter sace.journal of Nanchang university.23(999):3 7

7 Maximal sace-like submanifolds 30 [5]Shen Yibin. On Maximal Submanifolds in Pseudo-Riemannian Manifolds. Journal of Hangzhou university.8(99): [6] Takahashi T.Minimal immersions of Riemannian manifolds. Jaan.J.Math. 8(896): [7] Huizhang Yang, Feng Mu. On Maximal Sace-Like Submanifolds in the Pseudo-Sheres. International Mathematical Forum,56( 5), Received: May, 20