On the nullity distribution of the second fundamental form of a submanifold of a space form

TL;DR

This paper investigates the nullity distribution of the second fundamental form in submanifolds of space forms, providing local descriptions and global non-integrability results for Euclidean and spherical cases, with counterexamples in hyperbolic space.

Contribution

It offers a new local description of submanifolds via their nullity distribution and proves a global non-integrability property for complete, irreducible submanifolds in Euclidean space and spheres.

Findings

Nullity distribution is locally descriptive of submanifolds.

Complete irreducible submanifolds in Euclidean space and spheres have non-integrable nullity distributions.

Counterexamples exist in hyperbolic space where the nullity distribution is not non-integrable.

Abstract

If M is a submanifold of a space form, the nullity distribution N of its second fundamental form is (when defined) the common kernel of its shape operators. In this paper we will give a local description of any submanifold of the Euclidean space by means of its nullity distribution. We will also show the following global result: if M is a complete, irreducible submanifold of the Euclidean space or the sphere then N is completely non integrable. This means that any two points in M can be joined by a curve everywhere perpendicular to N. We will finally show that this statement is false for a submanifold of the hyperbolic space.