Equivariant nonlinear partial differential operators on constant curvature spaces

TL;DR

This paper classifies nonlinear PDE operators on constant curvature spaces that are equivariant under isometry groups, using a novel graph-based space to analyze their relations and dependencies.

Contribution

It introduces a classifying space for polynomial nonlinear operators on constant curvature spaces, realized via equivalence classes of multigraphs, aiding in understanding their dependencies.

Findings

Classifying space constructed as vector space of multigraph equivalence classes

Identified non-trivial linear dependence relations among operators

Extended analysis to operators under isometry groups of sub-Riemannian spaces

Abstract

Motivated by PDE-learning, we give a classifying space for nonlinear operators on simply connected spaces with constant curvature which are also equivariant under the action of the isometry group. The nonlinear operators we are considering are those that can be written as a polynomial in linear operators. We show that the classifying space for such operators can be realized as the vector space spanned by equivalence-classes of multigraphs. We also illustrate how this realization can help us discover non-trivial linear dependence relations between nonlinear differential operators relative to the dimension of the manifold. We also give some comments on operators equivariant under the identity component of the isometry group and under isometry groups of sub-Riemannian model spaces.