On the Inner Curvature of the Second Fundamental Form of a Surface in the Hyperbolic Space

TL;DR

This paper investigates the geometric properties of compact surfaces in hyperbolic space with positive-definite second fundamental form, establishing conditions under which such surfaces are necessarily extrinsic spheres.

Contribution

It demonstrates that certain Gaussian curvature conditions on the second fundamental form uniquely characterize extrinsic spheres in hyperbolic space.

Findings

Conditions on Gaussian curvature imply the surface is an extrinsic sphere

Characterization of surfaces with positive-definite second fundamental form in hyperbolic space

Uniqueness of extrinsic spheres under specified curvature conditions

Abstract

The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.