Rigidity of Convex Surfaces in Homogeneous Spaces

TL;DR

This paper establishes that complete surfaces with the same positive extrinsic curvature in certain homogeneous 3-manifolds are uniquely determined up to isometry, extending rigidity results beyond classical space forms.

Contribution

It proves a new rigidity theorem for oriented isometric immersions of complete surfaces in homogeneous 3-manifolds with positive extrinsic curvature, generalizing classical results.

Findings

Rigidity holds for surfaces in E(k; τ) with positive extrinsic curvature.

Unique isometric immersions are determined by extrinsic curvature.

Extends classical rigidity results to non-space form homogeneous manifolds.

Abstract

We prove rigidity of oriented isometric immersions of complete surfaces in the homo- geneous 3-manifolds E(k; {\tau}) (different from the space forms) having the same positive extrinsic curvature.