Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Eta invariants of Dirac operators on foliated manifolds
HTML articles powered by AMS MathViewer

by Goran Perić PDF
Trans. Amer. Math. Soc. 334 (1992), 761-782 Request permission

Abstract:

We define the eta function of Dirac operators on foliated manifolds. We show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds. Under the hypothesis of invertibility of the operator in question, we prove the Atiyah-Singer relative index theorem for Dirac operators on foliated manifolds. Then we discuss the homotopy invariance of the index with respect to secondary data.
References
  • M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. MR 650828, DOI 10.1007/BF01425417
  • M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
  • J. M. Bismut and J. Cheeger, Families index for manifolds with boundary, superconnections and cones, Preprint.
  • Jean-Michel Bismut and Daniel S. Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles, Comm. Math. Phys. 106 (1986), no. 1, 159–176. MR 853982, DOI 10.1007/BF01210930
  • L. Boutet de Montvel, A course on pseudodifferential operators and their applications, Duke Univ. Math. Ser. II (to appear).
  • Jeff Cheeger and Mikhael Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 115–154. MR 780040
  • Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143 (French). MR 548112
  • —, A survey of foliations and operator algebras, Operator Algebras and Applications, Proc. Sympos. Pure Math., Vol. 38, Amer. Math. Soc., Providence, R.I., 1980.
  • A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183. MR 775126, DOI 10.2977/prims/1195180375
  • R. G. Douglas, S. Hurder, and J. Kaminker, Cyclic cocycles, renormalization and eta-invariants, Invent. Math. 103 (1991), no. 1, 101–179. MR 1079841, DOI 10.1007/BF01239510
  • M. A. Evgrafov, Analytic functions, W. B. Saunders Co., Philadelphia, Pa.-London, 1966. Translated from the Russian by Scripta Technica, Inc; Translation edited by Bernard R. Gelbaum. MR 0197686
  • Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
  • Peter B. Gilkey, The eta invariant and secondary characteristic classes of locally flat bundles, Algebraic and differential topology—global differential geometry, Teubner-Texte Math., vol. 70, Teubner, Leipzig, 1984, pp. 49–87. MR 792686
  • M. Gromov and B. Lawson, Positive scalar curvature and the Dirac operator on Riemannian complete manifolds, Publ. Math. Inst. Hautes Études Sci. 58 (1983), 295-408.
  • Calvin C. Moore and Claude Schochet, Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications, vol. 9, Springer-Verlag, New York, 1988. With appendices by S. Hurder, Moore, Schochet and Robert J. Zimmer. MR 918974, DOI 10.1007/978-1-4613-9592-8
  • H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
  • Calvin C. Moore and Claude Schochet, Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications, vol. 9, Springer-Verlag, New York, 1988. With appendices by S. Hurder, Moore, Schochet and Robert J. Zimmer. MR 918974, DOI 10.1007/978-1-4613-9592-8
  • Henri Moscovici and Robert J. Stanton, Eta invariants of Dirac operators on locally symmetric manifolds, Invent. Math. 95 (1989), no. 3, 629–666. MR 979370, DOI 10.1007/BF01393895
  • G. Perić, Ph.D. Thesis, The Ohio State University, 1989. —, Index theorem for foliated manifolds with boundary and cyclic cocycles, Preprint, 1990. —, Type III relative index theorem on foliated manifolds, in preparation, 1990.
  • John Roe, An index theorem on open manifolds. I, II, J. Differential Geom. 27 (1988), no. 1, 87–113, 115–136. MR 918459
  • David Ruelle and Dennis Sullivan, Currents, flows and diffeomorphisms, Topology 14 (1975), no. 4, 319–327. MR 415679, DOI 10.1016/0040-9383(75)90016-6
  • R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943
  • Patrick Shanahan, The Atiyah-Singer index theorem, Lecture Notes in Mathematics, vol. 638, Springer, Berlin, 1978. An introduction. MR 487910, DOI 10.1007/BFb0068264
  • V. Mathai, Positive scalar curvature and reduced eta invariants, Preprint. K. Yoshida, Functional analysis, Springer-Verlag, 1965.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G12, 57R30
  • Retrieve articles in all journals with MSC: 58G12, 57R30
Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 334 (1992), 761-782
  • MSC: Primary 58G12; Secondary 57R30
  • DOI: https://doi.org/10.1090/S0002-9947-1992-1068932-1
  • MathSciNet review: 1068932