Totally geodesic real hypersurfaces in complex quadric

Abstract

The Complex quadric is a complex hypersurface in complex projective space. It also can be regarded as a kind of real Grassmann manifold of compact type with rank 2. On the other hand Jacobi fields along geodesics of a given Riemannian manifold satisfy a well known differential equation, called Jacobi equation. This equation naturally inspires the so-called Jacobi operator. The purpose of this article is to give a non-existence result for any totally geodesic real hypersurfaces with recurrent normal Jacobi operator R_N and also with recurrent structure Jacobi operator R_ξ in Q^m. In the continuation, we obtain classifications of totally geodesic real hypersurfaces in Q^m with recurrent Ricci tensor Ric.