The Bour's theorem for invariant surfaces in three-manifolds

TL;DR

This paper extends Bour's theorem to invariant surfaces within three-dimensional Riemannian manifolds using equivariant geometry techniques.

Contribution

It introduces a generalized version of Bour's theorem applicable to surfaces invariant under one-parameter isometry groups in 3D manifolds.

Findings

Proves a generalized Bour's theorem for invariant surfaces.

Demonstrates the applicability of equivariant geometry in this context.

Provides a framework for analyzing invariant surfaces in Riemannian manifolds.

Abstract

In this paper, we apply techniques from equivariant geometry to prove that a generalized Bour's theorem holds for surfaces that are invariant under the action of a one-parameter group of isometries of a three-dimensional Riemannian manifold.