The Gauss map and second fundamental form of surfaces in a Lie group

TL;DR

This paper establishes integrability conditions for surfaces in unimodular Lie groups with prescribed Gauss maps and positive extrinsic curvature, and explores their special cases in the sphere and Euclidean space.

Contribution

It provides new integrability criteria for isometric immersions with prescribed Gauss maps in Lie groups, linking harmonic Gauss maps to constant extrinsic curvature surfaces in spheres.

Findings

Characterization of surfaces in Lie groups with prescribed Gauss map and curvature.

Correspondence between surfaces in $ ext{S}^3$ and $ ext{R}^3$ with non-zero extrinsic curvature.

Harmonicity of the Gauss map characterizes constant extrinsic curvature surfaces in $ ext{S}^3$.

Abstract

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $\mathbb{S}^3$ and establish a correspondence between simply connected surfaces having extrinsic curvature $K$, $K$ different from 0 and -1, immersed in $\mathbb{S}^3$ with simply connected surfaces having non-vanishing extrinsic curvature immersed in the euclidean space $\mathbb{R}^3$. Moreover, we show that a surface isometrically immersed in $\mathbb{S}^3$ has positive constant extrinsic curvature if, and only if, its Gauss map is a harmonic map into the Riemann sphere.