Translation surfaces in Lie groups with constant Gaussian curvature

TL;DR

This paper proves that in certain Lie groups with bi-invariant metrics, surfaces of constant Gaussian curvature that are expressible as a product of two curves are necessarily flat, providing a local characterization in three dimensions.

Contribution

It establishes a classification of constant Gaussian curvature surfaces in Lie groups as products of curves, showing they are flat, and offers a local description in three dimensions.

Findings

Surfaces of constant Gaussian curvature as products of two curves are flat.

Complete characterization of such surfaces in 3D Lie groups.

Provides conditions under which these surfaces are flat.

Abstract

Let G be a n-dimensional Lie group (n>2) with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in G can be expressed as the product of two curves, then it must be flat. In particular, we can essentially characterize all such surfaces locally in 3-dimensional case.