On the Index Formula for an Isometric Diffeomorphism

TL;DR

This paper presents an elementary method to compute the index of elliptic operators linked to isometric diffeomorphisms, reducing the problem to a higher-dimensional pseudodifferential operator and applying an Atiyah--Singer type formula.

Contribution

It introduces a simplified, index-preserving reduction technique and derives a new index formula based on the operator's symbol on the original manifold.

Findings

Provides an explicit index formula for elliptic operators associated with isometric diffeomorphisms.

Reduces the index problem to an elliptic pseudodifferential operator on a higher-dimensional manifold.

Uses an Atiyah--Singer type formula to compute the index in terms of the original operator's symbol.

Abstract

We give an elementary solution of the index problem for elliptic operators associated with the shift operator along the trajectories of an isometric diffeomorphism of a closed smooth manifold. This solution is based on a reduction (which preserves the index) of the operator to an elliptic pseudodifferential operator on a manifold of a higher dimension and an application of an Atiyah--Singer type formula. The final index formula is given in terms of the symbol of the operator on the original manifold.