Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold

TL;DR

This paper introduces a uniformization method to compute the index of nonlocal elliptic operators induced by diffeomorphisms on manifolds, linking it to topological invariants via the Atiyah-Singer index theorem.

Contribution

It develops a novel uniformization approach that transforms nonlocal elliptic operators into pseudodifferential operators, enabling index calculation through topological invariants.

Findings

Established a method to assign pseudodifferential operators with the same index as nonlocal operators.

Connected the index of these operators to topological invariants of their symbols.

Applied the Atiyah-Singer index theorem to the uniformized operators.

Abstract

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.