Geodesic
On the index formula for an isometric diffeomorphism
Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 2, Tome 46 (2012), pp. 141-152

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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. 
@article{CMFD_2012_46_a7,
     author = {A. Yu. Savin and B. Yu. Sternin and E. Schrohe},
     title = {On the index formula for an isometric diffeomorphism},
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     pages = {141--152},
     publisher = {mathdoc},
     volume = {46},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2012_46_a7/}
}
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A. Yu. Savin; B. Yu. Sternin; E. Schrohe. On the index formula for an isometric diffeomorphism. Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 2, Tome 46 (2012), pp. 141-152. http://geodesic.mathdoc.fr/item/CMFD_2012_46_a7/