Journal of Mathematical Sciences

, Volume 201, Issue 6, pp 818–829 | Cite as

On the Index Formula for an Isometric Diffeomorphism

  • A. Yu Savin
  • B. Yu. Sternin
  • E. Schrohe


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.


Manifold Elliptic Operator Noncommutative Geometry Topological Index Index Theorem 
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  1. 1.
    A. B. Antonevich, Linear Functional Equations. Operator Approach [in Russian], Universitetskoe, Minsk (1988).Google Scholar
  2. 2.
    A. Antonevich, M. Belousov, and A. Lebedev, Functional Differential Equations II. C * -applications. Parts 1, 2, Longman, Harlow (1998).Google Scholar
  3. 3.
    A. Antonevich and A. Lebedev, Functional Differential Equations. I. C * -Theory, Longman, Harlow (1994).Google Scholar
  4. 4.
    M. F. Atiyah and I.M. Singer, “The index of elliptic operators III,” Ann. Math. (2), 87, 546–604 (1968).Google Scholar
  5. 5.
    A. Connes, “C * algèbres et géométrie différentielle,” C. R. Math. Acad. Sci. Paris, 290, No. 13, A599–A604 (1980).MathSciNetGoogle Scholar
  6. 6.
    A. Connes, Noncommutative Geometry, Academic Press, San Diego, CA (1994).zbMATHGoogle Scholar
  7. 7.
    A. Connes and N. Higson, “Déformations, morphismes asymptotiques et K-théorie bivariante,” C. R. Math. Acad. Sci. Paris, 311, No. 2, 101–106 (1990).zbMATHMathSciNetGoogle Scholar
  8. 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
  9. 9.
    V. M. Manuilov, “On asymptotical homomorphisms into Calkin algebras,” Funct. Anal. Appl., 35, No. 2, 81–84 (2001).CrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Moscovici, “Local index formula and twisted spectral triples,” Clay Math. Proc., 11, 465–500 (2010).MathSciNetGoogle Scholar
  11. 11.
    V. E. Nazaikinskii, A.Yu. Savin, and B.Yu. Sternin, “On homotopic classification of elliptic operators on stratified manifolds,” Izv. Ross. Akad. Nauk, Ser. Mat., 71, No. 6, 91–118 (2007).CrossRefMathSciNetGoogle Scholar
  12. 12.
    V. E. Nazaikinskii, A.Yu. Savin, and B.Yu. Sternin, Elliptic Theory and Noncommutative Geometry, Birkhäuser, Basel (2008).zbMATHGoogle Scholar
  13. 13.
    D. Perrot, “A Riemann-Roch theorem for one-dimensional complex groupoids,” Commun. Math. Phys., 218, No. 2, 373–391 (2001).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 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
  15. 15.
    A.Yu. Savin and B.Yu. Sternin, “Noncommutative elliptic theory. Examples,” Proc. Steklov Inst. Math., 271, 204–223 (2010).CrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Savin and B. Sternin, “Index of elliptic operators for a diffeomorphism,” arXiv:1106.4195 (2011).Google Scholar
  17. 17.
    A.Yu. Savin, B.Yu. Sternin, and E. Schrohe, “The problem of index of elliptic operators associated with a diffeomorphism of the maifold and uniformization,” Dokl. Ross. Akad. Nauk, 441, No. 5, 593–596 (2011).MathSciNetGoogle Scholar
  18. 18.
    L. B. Schweitzer, “Spectral invariance of dense subalgebras of operator algebras,” Int. J. Math., 4, No. 2, 289–317 (1993).CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Hannover Leibnitz-UniversitätHannoverGermany

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