Exact discretization of harmonic tensors

TL;DR

This paper extends the discretization techniques for harmonic functions on Riemannian manifolds to harmonic tensor fields, using holonomy restrictions and scalarization to facilitate the process.

Contribution

It introduces a method to discretize harmonic tensor fields by leveraging holonomy conditions and the scalarization technique, generalizing previous work on harmonic functions.

Findings

Discretization of harmonic tensor fields is possible under certain holonomy restrictions.

Lifted diffusion on the orthonormal frame bundle shares the recurrence property.

Scalarization reduces tensor discretization to scalar harmonic function discretization.

Abstract

Lyons and Sullivan have shown how to discretize harmonic functions on a Riemannian manifold $M$ whose Brownian motion satisfies a certain recurrence property called $\ast$-recurrence. We study analogues of this discretization for tensor fields which are harmonic in the sense of the covariant Laplacian. We show that, under certain restrictions on the holonomy of the connection, the lifted diffusion on the orthonormal frame bundle has the same $\ast$-recurrence property as the original Brownian motion. This observation permits us to reduce to the discretization of ordinary harmonic functions by a device called scalarization.