Codimension Reduction for Real Submanifolds of a Complex Hyperbolic Space

Abstract

Wc study real submanifalds af a complex hyperbolic space and prove a cbdimension reductian thearem. O. INTRODUCTION. Recently Oknmura ([3]) defined itolaniorpitic first normal space for real submanifolds of a Kaeitler manifaid and praved a cadimettsiasi reductiasi titeorem far real submasiifalds of a complex projective space. Namely, ite sitawed following: Theorem. La M be a conneeted u-dimensional real submanifold of a real (u + p)-dirnensionul complez projective space ~p(n+v)/2 aud Jet No(x) be the orthogonul complement of first normal space itt 2’4(M). We prd Ho(x) = JNo(x)flNo(x) aud Jet H(x) be a J-invuriunt subspuce of Ho(x) where J is complez structure of CP(n+v)/2. If ihe orthogonul eornplement H 2(x) of H(z) itt 2’J(M) is invariant under purallel tratalution with respeci to ihe normal connection and {f ej is tite constunt 1991 Mathem.tics Subject Clasaification: 53H25, 53H30. Editorial Complutense. Madrid, 1994. http://dx.doi.org/10.5209/rev_REMA.1994.v7.n1.17784 120 Sbin-ichi K,w,mato di,nestsion of H2(x), then tites-e exists u real (u + q)-dimensional tota?ly geodesie complez projective subspace c£(n+Q)/2 in CP(tt+P)/ 2 such that M c ~p(n+q)/2 Tite purpose of titis paper is to prove titat tite similar result to tite aboye theorem Ls still itold in asubrnanifold of complex ityperbolic space. Tite autiter wonid like to express bis titanlcs to Prafessors M. Okuniura and M. Kimura for titeir valuable suggestiosis. 1. CODIMENSION REDUCTION POR SUBMANIFOLDS OF ANTI-DE SITTER SPACE. Let R~+l be a real vector space of (n+ 1) dlmension with a psendoRiemannian metric 4 of signature (u — 1, 2) given by n 4(x,y) = —xoyo—xiyi+>jx~y 1 (1.1) where x ~ (x ,xi,...,x,3, y ~ (Yo,Y1,...,Yn) E Rn+i. Let H’ = {x E R?+l 1 g(x,x) = —1}. Titen tite itypersurface Hf is a Lorentzian manifold witit tite isiduced Loresitzian metric 4 of constasit sectional curvature —1. We cail it n.dirnensional anti-De Sitter space. Let ffr+P be asi (u + p)-dimettsiosial anti-De Sitter space and let i: M —~ Hr+P be asi isametric imifiersion of a connected ii-dimensional Loretttzian manifaid witit tite Larentzian rnetric g into Hr+P. Titesi tite tangent bundle T(M) is identified witb a subitundie af T(Jfl+P) and tite normal busidle 2’ 1(M) is a subbusidle of 2’(JIr~~) cosisisting of ali elernent itt T(Rt+~) whicb are ortitagona] to 2’(M) witb respect to 4. We denote by ‘~ and sy tite Levi-Civita cannection of M and R?+P respectLvely and D tite induced normal cosinection fram y to 2”(M). Titen titey are related by tite followisig Ganss and Weingartesi formulae: VixiY = iSZxY+h(X,Y) (1.2)