Equivariant nonlinear partial differential operators on constant curvature spaces

Abstract

Motivated by PDE-learning, we give a classifying space for nonlinear partial differential operators equivariant under the action of isometries. This classifying space is for operators defined on Riemannian model spaces. 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. Finally, we give some comments on operators equivariant under the identity component of the isometry group and under isometry groups of sub-Riemannian model spaces.

30 pages