The classification of totally umbilical surfaces in homogeneous 3-manifolds

TL;DR

This paper provides a comprehensive classification of totally umbilical surfaces within all simply-connected homogeneous 3-manifolds, covering both unimodular and non-unimodular Lie groups with arbitrary metrics.

Contribution

It completes the classification of totally umbilical surfaces in all homogeneous Riemannian 3-manifolds, extending previous partial results.

Findings

Complete classification of totally umbilical surfaces in all simply-connected homogeneous 3-manifolds.

Identification of all such surfaces in unimodular and non-unimodular Lie groups.

Advancement in understanding geometric structures of homogeneous 3-manifolds.

Abstract

We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of totally umbilical surfaces in homogeneous Riemannian 3-manifolds.