  This work proposes a geometric insight into equivariant message passing on
Riemannian manifolds. As previously proposed, numerical features on Riemannian
manifolds are represented as coordinate-independent feature fields on the
manifold. To any coordinate-independent feature field on a manifold comes
attached an equivariant embedding of the principal bundle to the space of
numerical features. We argue that the metric this embedding induces on the
numerical feature space should optimally preserve the principal bundle's
original metric. This optimality criterion leads to the minimization of a
twisted form of the Polyakov action with respect to the graph of this
embedding, yielding an equivariant diffusion process on the associated vector
bundle. We obtain a message passing scheme on the manifold by discretizing the
diffusion equation flow for a fixed time step. We propose a higher-order
equivariant diffusion process equivalent to diffusion on the cartesian product
of the base manifold. The discretization of the higher-order diffusion process
on a graph yields a new general class of equivariant GNN, generalizing the ACE
and MACE formalism to data on Riemannian manifolds.
