Separation and symmetry on two dimensional manifolds

TL;DR

This paper introduces the concepts of separated solutions and simple symmetries on two-dimensional manifolds, proving conditions under which differential operators admit separated solutions based on symmetries.

Contribution

It defines new notions of separated solutions and simple symmetries, establishing a link between symmetries and solutions of differential operators on 2D manifolds.

Findings

Differential operators with certain symmetries have separated solutions.

Homogeneous simple symmetries of degree one influence the existence of solutions.

The paper provides conditions for separated solutions based on symmetries.

Abstract

We introduce notions of a separated solution and of a simple symmetry that generates a differential operator on a smooth manifold. We prove that a differential operator on a two dimensional manifold has a separated solution if it has a homogeneous simple symmetry of degree one which does not generate the operator.