×

The Atiyah-Singer theorems: A probabilistic approach. I: The index theorem. (English) Zbl 0538.58033

The author gives a proof of the Atiyah-Singer index theorem using heat equation methods. He uses a probabilistic construction of the heat equation kernel which permits a direct derivation of the index formula. This avoids use of the invariance theory of the reviewer. Stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem. This method also avoids the use of the complicated cancellation arguments of Patodi. The author will deal with the Lefschetz fixed point formulas in a subsequent paper using the same methods.
Reviewer: P.Gilkey

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
58J10 Differential complexes
58J65 Diffusion processes and stochastic analysis on manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atiyah, M. F., Classical groups and classical operators on manifolds, (Differential Operators on Manifolds (1975), CIME: CIME Cremonese, Rome) · Zbl 0614.57007
[2] Atiyah, M. F., Circular symmetry and stationary phase approximation, (Proceedings of the Conference in Honor of L. Schwartz (1984), Astérisque: Astérisque Paris), to appear · Zbl 0696.57002
[3] Atiyah, M. F.; Bott, R., A Lefschetz fixed point formula for elliptic complexes, II, Ann. of Math., 88, 451-491 (1968) · Zbl 0167.21703
[4] Atiyah, M. F.; Bott, R.; Patodi, V. K., On the heat equation and the index theorem, Invent. Math., 19, 279-330 (1973) · Zbl 0257.58008
[5] Atiyah, M. F.; Bott, R.; Shapiro, A., Clifford modules, Topology, 3, Supp. 1, 3-38 (1964) · Zbl 0146.19001
[6] Atiyah, M. F.; Singer, I. M., The index of elliptic operators, III, Ann. of Math., 87, 546-604 (1968) · Zbl 0164.24301
[7] Bismut, J. M., Mécanique Aléatoire, (Lecture Notes in Math, No. 866 (1981), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0528.60048
[8] Bismut, J. M., A generalized formula of Itô and some other properties of stochastic flows, Z. Wahrsch. Verw. Gebiete, 55, 331-350 (1981) · Zbl 0456.60063
[9] Bismut, J. M., An introduction to the stochastic calculus of variations, (Kohlmann, M.; Christopeit, N., Stochastic Differential Systems. Stochastic Differential Systems, Lecture Notes in Control Th. and Inf. Sc., No. 43 (1982), Springer-Verlag: Springer-Verlag Berlin), 33-72
[10] Bismut, J. M., Large deviations and the Malliavin calculus, (Progress in Math. (1984), Birkhaüser), to appear · Zbl 0537.35003
[11] Bismut, J. M., Calcul des variations stochastiques et grandes déviations, C. R. Acad. Sci. Paris Sér. I Math., 296, 1009-1012 (1983) · Zbl 0531.60071
[12] Bismut, J. M., Transformations différentiables du mouvement Brownien, (Proceedings of the Conference in Honor of L. Schwartz (1984), Astérique: Astérique Paris), to appear · Zbl 0572.60074
[13] Donelly, H.; Patodi, V. K., Spectrum and the fixed point set of isometries, Topology, 161, 1-11 (1977) · Zbl 0341.53023
[14] Duistermaat, J. J.; Heckman, G. J., Addendum, Invent. Math., 72, 153-158 (1983) · Zbl 0503.58016
[15] Eells, J.; Elworthy, K. D., Wiener integration on certain manifolds, (Some Problems in Nonlinear Analysis. Some Problems in Nonlinear Analysis, CIME IV (1971)), 69-94, Cremonese, Rome · Zbl 0226.58007
[16] Elworthy, K. D., Stochastic differential equations on manifolds, (London Math Soc. Lecture Notes, No. 70 (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0514.58001
[17] Garsia, A. M.; Rodemich, E.; Rumsey, H., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J., 20, 565-578 (1970) · Zbl 0252.60020
[18] Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents, Acta Math., 139, 96-153 (1977) · Zbl 0366.22010
[19] Gilkey, P., Curvature and the eigenvalues of the Laplacian, Advan. in Math., 10, 344-382 (1973) · Zbl 0259.58010
[20] Gilkey, P., Spectral geometry of a Riemannian manifold, J. Differential Geom., 10, 601-618 (1975) · Zbl 0316.53035
[21] Gilkey, P., Lefschetz fixed point formulas and the heat equation, (Byrnes, C., Partial Differential Equations and Geometry; Proceedings, Park City Conf.. Partial Differential Equations and Geometry; Proceedings, Park City Conf., 1977. Partial Differential Equations and Geometry; Proceedings, Park City Conf.. Partial Differential Equations and Geometry; Proceedings, Park City Conf., 1977, Lecture Notes Pure Appl. Math. No. 48 (1979), Dekker: Dekker New York), 91-147
[22] Hitchin, N., Harmonic spinors, Advan. in Math., 14, 1-55 (1974) · Zbl 0284.58016
[23] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Process (1981), North-Holland: North-Holland Amsterdam
[24] Kobayashi, S.; Nomizu, K., (Foundations of Differential Geometry, Vol. II (1969), Interscience: Interscience New York) · Zbl 0175.48504
[25] Lichnerowicz, A., Spineurs harmoniques, C. R. Acad. Sci. Paris Ser I Math., 257, 7-9 (1963) · Zbl 0136.18401
[26] Malliavin, P., Stochastic calculus of variations and hypoelliptic operators, (Itô, K., Proceedings of the Conference on Stochastic Differential Equations of Kyoto. Proceedings of the Conference on Stochastic Differential Equations of Kyoto, 1976 (1978), Kinokuniya: Kinokuniya Tokyo), 155-263, and Wiley, New York
[27] Malliavin, P., Géometrie différentielle stochastique (1978), Presses de l’Université de Montréal: Presses de l’Université de Montréal Montréal · Zbl 0393.60062
[28] Malliavin, P., Formule de la moyenne, calcul de perturbations et théorèmes d’annulation pour les formes harmoniques, J. Funct. Anal., 17, 274-291 (1974) · Zbl 0425.58022
[29] Mc Kean, H.; Singer, I. M., Curvature and the eigenvalues of the Laplacian, J. Differential Geom., 1, 43-69 (1967) · Zbl 0198.44301
[30] Meyer, P. A., (Un cours sur les intégrales stochastiques, Séminaire de Probabilités, No. X. Un cours sur les intégrales stochastiques, Séminaire de Probabilités, No. X, Lecture Notes in Math., No. 511 (1973), Springer-Verlag: Springer-Verlag Berlin), 245-400
[31] Milnor, J. W.; Stasshef, J. D., Characteristic Classes (1974), Princeton Univ. Press: Princeton Univ. Press Princeton, N. J
[32] Molchanov, S. A., Diffusion processes and Riemannian geometry, Russian Math. Surveys, 30, 1-53 (1975) · Zbl 0315.53026
[33] Patodi, V. K., Curvature and the eigenforms of the Laplace operator, J. Differential Geom., 5, 233-249 (1971) · Zbl 0211.53901
[34] Patodi, V. K., An analytic proof of the Riemann-Roch-Hirzebruch theorem for Kaehler manifolds, J. Differential Geom., 5, 251-283 (1971) · Zbl 0219.53054
[35] Patodi, V. K., Holomorphic Lefschetz fixed point formula, Bull. Amer. Math. Soc., 10, 288-307 (1971) · Zbl 0279.32015
[36] Seeley, R. T., Complex power of an elliptic operator, (Proc. Symp. Amer. Math. Soc., 10 (1967)), 288-307 · Zbl 0159.15504
[37] Spivak, M., (Differential Geometry, Vol. 5 (1975), Publish or Perish: Publish or Perish Boston)
[38] Varadhan, S. R.S, Diffusion processes on a small time interval, Commun. Pure and Appl. Math., 20, 659-685 (1967) · Zbl 0278.60051
[39] Ventcell, D.; Freidlin, M. I., On small random perturbations of dynamical systems, Russian Math. Survey, 25, 1-55 (1970) · Zbl 0297.34053
[40] Witten, E., Supersymmetry and Morse theory, J. Differential Geom., 17, 661-692 (1982) · Zbl 0499.53056
[42] Yor, M., Remarques sur une formule de P. Lévy, (Séminaire de Probabilités. Séminaire de Probabilités, Lecture Notes in Math., No. 784 (1980), Springer-Verlag: Springer-Verlag Berlin), 343-346, No. XIV
[43] Bismut, J. M., Le Théorème d’Atiyah-Singer pour les opérateurs elliptiques classiques: Une approche probabiliste, C. R. Acad. Sci. Paris Sér. I. Math., 297, 481-484 (1983) · Zbl 0539.58034
[44] Getzler, E., Pseudo-differential operators on supermanifolds and the Atiyah-Singer index theorem, Commun. Math. Phys., 92, 163-178 (1983) · Zbl 0543.58026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.