Convexity, Rigidity, and Reduction of Codimension of Isometric Immersions into Space Forms

TL;DR

This paper proves that certain compact isometric immersions into space forms are convex, rigid, and of codimension one, extending classical theorems and settling a conjecture, with additional results on hypersurface existence and codimension reduction.

Contribution

It generalizes previous convexity and rigidity results for isometric immersions into space forms and confirms a longstanding conjecture, also providing new existence and codimension reduction theorems.

Findings

Immersions with semi-definite second fundamental form are convex and rigid.

The results extend Hadamard's theorem to space forms.

Existence of hypersurfaces with prescribed boundary and zero Gauss-Kronecker curvature.

Abstract

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by M. do Carmo and E. Lima, as well as by M. do Carmo and F. Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second fundamental form. We extend to the context of isometric immersions in space forms a classical theorem for Euclidean hypersurfaces due to Hadamard. In this same context, we prove an existence theorem of hypersurfaces with prescribed boundary and vanishing Gauss-Kronecker curvature. Finally, we show that isometric immersions into space forms which are regular outside the set of totally geodesic points admit a reduction of codimension to one.