A splitting theorem for higher order parallel immersions

Abstract

We consider isometric immersions into space forms having the second fundamental form parallel at order k. We show that this class of immersions consists of local products, in a suitably defined sense, of parallel immersions and normally flat immersions of flat spaces.

A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds

“…Among others, for higher order parallel submanifolds in real space forms, see [134,174,175,173,177,178,179]; for higher order parallel surfaces in three-dimensional homogeneous spaces, see [136]; for higher order parallel surfaces in Bianchi-Cartan-Vranceanu spaces, see [132]; for higher order parallel surfaces in the Heisenberg group, see [130]; and for higher order parallel submanifolds of a complex space form, see [176]. For some further results on higher order parallel submanifolds, see Ü Lumiste's 2000 survey article [2].…”