On the Gauss curvature of compact surfaces in homogeneous 3-manifolds

TL;DR

This paper classifies compact flat surfaces in certain homogeneous 3-manifolds and proves that compact surfaces with constant Gauss curvature do not exist in these spaces.

Contribution

It provides a complete classification of compact flat surfaces and establishes non-existence results for compact constant Gauss curvature surfaces in these manifolds.

Findings

Classification of compact flat surfaces in homogeneous 3-manifolds

Non-existence of compact constant Gauss curvature surfaces in these spaces

Results applicable to manifolds with 4-dimensional isometry groups

Abstract

Compact flat surfaces of homogeneous Riemannian 3-manifolds with isometry group of dimension 4 are classified. Non-existence results for compact constant Gauss curvature surfaces in these 3-manifolds are established.