On the Index Formula for an Isometric Diffeomorphism

Abstract

We give an elementary solution to the problem of the index of elliptic operators associated with shift operator along the trajectories of an isometric diffeomorphism of a smooth closed manifold. This solution is based on index-preserving reduction of the operator under consideration to some elliptic pseudo-differential operator on a higher-dimension manifold and on the application of the Atiyah-Singer formula. The final formula of the index is given in terms of the symbol of the operator on the original manifold.

Equivariant Algebraic Index Theorem

“…The above theorem gives an algebraic version of the results of [24][25][26][27], without the requirement that acts by isometries. To recover the analytic version of the index theorem type results from [27] and [21] one can apply the methods of [19].…”

Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds

“…The second approach uses the idea of uniformization [49,55] (see also [46,47,59]) to reduce the index problem for a G-operator to a similar problem for a pseudodifferential operator on a manifold of a higher dimension. The index of the latter operator can be found using the celebrated Atiyah-Singer formula.…”