A comprehensive survey on parallel submanifolds in Riemannian and pseudo-Riemannian manifolds

TL;DR

This paper provides a comprehensive survey of parallel submanifolds in Riemannian and pseudo-Riemannian manifolds, highlighting their geometric properties and significance.

Contribution

It offers an extensive overview of the theory, classification, and key properties of parallel submanifolds, consolidating existing research in the field.

Findings

Parallel submanifolds have constant extrinsic invariants.

They include totally geodesic submanifolds as a special case.

The survey covers both Riemannian and pseudo-Riemannian contexts.

Abstract

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper we provide a comprehensive survey on this important class of submanifolds.