Einstein submanifolds with flat normal bundle in space forms are holonomic

TL;DR

This paper extends a classical result by proving that Einstein submanifolds with flat normal bundle in space forms are holonomic, and explores their structure when the index of relative nullity is constant.

Contribution

It generalizes the holonomicity result from constant sectional curvature manifolds to Einstein manifolds, and characterizes their structure as generalized cylinders under certain conditions.

Findings

Einstein submanifolds with flat normal bundle are holonomic.

Submanifolds with constant index of relative nullity form generalized cylinders.

Extension of classical results to a broader class of manifolds.

Abstract

A well-known result asserts that any isometric immersion with flat normal bundle of a Riemannian manifold with constant sectional curvature into a space form is (at least locally) holonomic. In this note, we show that this conclusion remains valid for the larger class of Einstein manifolds. As an application, when assuming that the index of relative nullity of the immersion is a positive constant we conclude that the submanifold has the structure of a generalized cylinder over a submanifold with flat normal bundle.