Equivariant nonlinear partial differential operators on constant curvature spaces

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.