Complete submanifolds with relative nullity in space forms

TL;DR

This paper investigates complete submanifolds with relative nullity in space forms, using splitting tensor techniques to derive new geometric results and recover known theorems across Euclidean, hyperbolic, and spherical spaces.

Contribution

It introduces a novel approach based on the splitting tensor to explicitly integrate the Codazzi equation, leading to new insights and results in the geometry of submanifolds with relative nullity.

Findings

No complete hyperbolic submanifold with high relative nullity has bounded away from zero extrinsic curvature.

Complete Euclidean submanifolds with integrable relative conullity are cylindrical.

Results relate to Milnor's conjecture on surfaces with bounded second fundamental form.

Abstract

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor's conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity.