Axioms of planes on real hypersurfaces in complex space forms

Abstract

A Riemannian manifold M satisfies the axiom of 2-planes if at each point $ p\in{M} $ and each tangent 2-plane V at the point, there exists a totally geodesic submanifold L with $ p\in{L} $ and $ T_pL=V $ . On the other hand, real hypersurfaces in complex space forms have nice families of tangent 2-planes. If we restrict the above definition to those planes, several definitions of axioms of 2-planes arise naturally. We classify real hypersurfaces in non-flat complex space forms satisfying these axioms.