Submanifolds with splitting tangent sequence

Abstract

A theorem of Van de Ven states that a projective submanifold of complex projective space whose holomorphic tangent bundle sequence splits holomorphically is necessarily a linear subspace. Note that the sequence always splits differentiably but in general not holomorphically. We are interested in generalizations to the case, when the ambient space is a homogeneous manifold different from projective space: quadrics, Grassmannians or abelian manifolds, for example. Split submanifolds are closely related to totally geodesic submanifolds.