2014/08/25 by Xingxiao Li, Li, Xingxiao
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Primary 53A15 #Secondary 53B25 #math.DG #msc:53A15 #msc:53B25
paper · pdf · doi:10.48550/arxiv.1408.5947
27 pages
arxiv created 2014/08/25 · openalex publication_date 2014/08/25 · arxiv updated 2014/08/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-one correspondence between the set of equivalence classes of connected, simply connected and nondegenerate equiaffine symmetric hypersurfaces with a given nonzero affine mean curvature and that of the equivalence classes of semi-simple real Jordan algebras. Then, via the existing classification theorem of the semi-simple real Jordan algebras with unity, a complete classification for the nondegenerate and locally equiaffine symmetric hypersurfaces with nonzero affine mean curvatures is established. As an direct application of the main theorems, we prove at the end of the paper a complete classification of nondegenerate hypersurfaces with parallel Fubini-Pick forms and nonzero affine mean curvatures.