A splitting theorem for higher order parallel immersions

TL;DR

This paper characterizes higher order parallel immersions into space forms, showing they decompose into products of simpler parallel and flat immersions, advancing understanding of their geometric structure.

Contribution

It introduces a splitting theorem for higher order parallel immersions, revealing their local product structure involving parallel and flat immersions.

Findings

Higher order parallel immersions are locally products of parallel and flat immersions.

The class of such immersions can be decomposed into simpler geometric components.

Provides a new structural understanding of 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.