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Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms

2006; Rocky Mountain Mathematics Consortium; Volume: 36; Issue: 5 Linguagem: Inglês

10.1216/rmjm/1181069385

1945-3795

Miguel Ortega, Juan de Dios Pérez, Florentino G. Santos,

Holomorphic and Operator Theory

We prove the nonexistence of real hypersurfaces in nonflat complex space forms whose Jacobi operator associated to the structure vector field is parallel.In order to prove this result we also obtain the nonexistence of several classes of non homogeneous real hypersurfaces in complex projective space. Introduction.Let CM m (c), m ≥ 2, c = 0, be a nonflat complex space form endowed with the metric g of constant holomorphic sectional curvature c.For the sake of simplicity, we will use c = 4ε, ε = 1 or ε = -1.When ε = 1 we will call it the complex projective space, CP m , and when ε = -1, the complex hyperbolic space, CH m .Let M be a connected real hypersurface in CM m (c) without boundary.Let J denote the complex structure of CM m (c) and N a locally defined unit normal vector field on M .Then -JN = ξ is a tangent vector field to M called the structure vector field on M .The study of real hypersurfaces in nonflat complex space forms is a classical topic in differential geometry.The classification of homogeneous real hypersurfaces in the case of complex projective space, CP m was obtained by Takagi, see [6, 11 13], and is given by the following list:A 1 : Geodesic hyperspheres.A 2 : Tubes over totally geodesic complex projective spaces CP k , 0 < k < m -1.B: Tubes over complex quadrics and RP m .C: Tubes over the Segre embedding of CP 1 xCP n , where 2n + 1 = m and m ≥ 5.