  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.
