On the Index Formula for an Isometric Diffeomorphism
- 31 Downloads
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.
KeywordsManifold Elliptic Operator Noncommutative Geometry Topological Index Index Theorem
Unable to display preview. Download preview PDF.
- 1.A. B. Antonevich, Linear Functional Equations. Operator Approach [in Russian], Universitetskoe, Minsk (1988).Google Scholar
- 2.A. Antonevich, M. Belousov, and A. Lebedev, Functional Differential Equations II. C * -applications. Parts 1, 2, Longman, Harlow (1998).Google Scholar
- 3.A. Antonevich and A. Lebedev, Functional Differential Equations. I. C * -Theory, Longman, Harlow (1994).Google Scholar
- 4.M. F. Atiyah and I.M. Singer, “The index of elliptic operators III,” Ann. Math. (2), 87, 546–604 (1968).Google Scholar
- 8.A. Connes and H. Moscovici, “Type III and spectral triples,” in: Traces in number theory, geometry and quantum fields (Aspects of Mathematics), 38, 57–71, Vieweg+Teubner, Wiesbaden (2008).Google Scholar
- 14.A. Savin, E. Schrohe, and B. Sternin, “Uniformization and an index theorem for elliptic operators associated with diffeomorphisms of a manifold,” arXiv:1111.1525 (2011).Google Scholar
- 16.A. Savin and B. Sternin, “Index of elliptic operators for a diffeomorphism,” arXiv:1106.4195 (2011).Google Scholar