2010/12/13 by Pascual Lucas, Lucas, Pascual, H. Fabián Ramírez-Ospina +1
Mathematics · #53B25 #53B30 #53C50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #msc:53B25 #msc:53B30 #msc:53C50
paper · pdf · doi:10.48550/arxiv.1012.2778
28 pages. Final version submitted to Taiwanese Journal of Mathematics
openalex publication_date 2010/12/13 · arxiv created 2011/01/17 · arxiv updated 2011/01/18 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We study hypersurfaces either in the De Sitter space §1n+1⊂\R1n+2 or in the anti De Sitter space \H1n+1⊂\R2n+2 whose position vector ψ satisfies the condition Lkψ=Aψ+b, where Lk is the linearized operator of the (k+1)-th mean curvature of the hypersurface, for a fixed k=0,...,n-1, A is an (n+2)×(n+2) constant matrix and b is a constant vector in the corresponding pseudo-Euclidean space. For every k, we prove that when A is self-adjoint and b=0, the only hypersurfaces satisfying that condition are hypersurfaces with zero (k+1)-th mean curvature and constant k-th mean curvature, open pieces of standard pseudo-Riemannian products in §1n+1 (§1m(r)קn-m(√(1-r2)), \Hm(-r)קn-m(√(1+r2)), §1m(√(1-r2))קn-m(r), \Hm(-√(r2-1))קn-m(r)), open pieces of standard pseudo-Riemannian products in \H1n+1 (\H1m(-r)קn-m(√(r2-1)), \Hm(-√(1+r2))ק1n-m(r), §1m(√(r2-1))×\Hn-m(-r), \Hm(-√(1-r2))×\Hn-m(-r)) and open pieces of a quadratic hypersurface \x∈\mathbbMcn+1 | Rx,x=d\, where R is a self-adjoint constant matrix whose minimal polynomial is t2+at+b, a2-4b≤ 0, and \mathbbMcn+1 stands for §1n+1⊂\R1n+2 or \H1n+1⊂\R2n+2. When Hk is constant and b is a non-zero constant vector, we show that the hypersurface is totally umbilical, and then we also obtain a classification result (see Theorem 2).