Abstract
The Gauss-Bonnet-Chern theorem for compact Riemannian manifold (without boundary) is discussed here to exhibit in a clear manner the role Riemannian Brownian motion plays in various probabilistic approaches to index theorems. The method with some modifications works also for the index theorem for the Dirac operator on the bundle of spinors, see Hsu.(7)
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REFERENCES
Azencott, R. (1986). Une approche probabiliste du théorème d'Atiyah-Singer, d'après J. M. Bismut, Astérisque 133–134,7. 18.
Bismut, J. M. (1984). The Atiyah-Singer theorems; a probabilistic approach, J. of Func. Anal. 57,I, 56–99: II, 329–348.
Cycon, H. L., Groese, R. G., Kirsch, W., and Simon, B. (1987). Schödinger Operators with Applications to Quantum Mechanics and Global Geometry, (Chap. 12.), Springer-Verlag, Berlin.
Elworthy, K. D. (1988). Geometric aspects of diffusions on manifolds, in École d'Été de Probabilités de Saint-Flour, Lecture Notes in Mathematics, Vol. 1362.
Getzler, E. (1986). A short proof of the local Atiyah-Singer index theorem, Topology 25, 111–117.
Gilkey, P. (1984). Invariant Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Inc., Wilmington, Delaware.
Hsu, Elton P. (1997). Brownian motion and Atiyah-Singer index theorem (to appear).
Hsu, Elton P. (1996). Estimates of derivatives of the heat kernel on compact manifolds (to appear).
Ikeda, N., and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, Second Edition, North-Holland and Kodansha, New York and Tokyo.
Ikeda, N., and Watanabe, S. (1987). Malliavin Calculus of Wiener functionals and its applications, in From Local Times to Global Geometry, Control and Physics, K. D. Elworthy, Pitman Research Notes in Math., Longman Scientific and Technical, Harlow, (150), pp. 132–178.
Léandre, R (1988). Sur le Théorème d'Atiyah-Singer, Prob. Th. Rel. Fields 80, 119–137.
MacKean, H. P., and Singer, I. M. (1967). Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1(1), 43–69.
Patodi, V. K. (1971). Curvature and the eigenforms of the Laplace operator, J. Differential Geometry 5, 233–249.
Pinsky, M. (1974). Multiplicative operator functionals and their asymptotic properties, in Advances in Probability, Marcel Dekker, New York, pp. 1–100.
Sheu, S.-Y. (1991). Some estimates of the transition density function of a nondegenerate diffusion Markov process, Ann. Prob. 19(2), 538–561.
Shigekawa, I., Ueki, N., and Watanabe, S. (1989). A probabilistic proof of the Gauss-Bonnet-Chern theorem for manifolds with boundary, Osaka J. Math. 26, 897–930.
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Hsu, E.P. Stochastic Local Gauss-Bonnet-Chern Theorem. Journal of Theoretical Probability 10, 819–834 (1997). https://doi.org/10.1023/A:1022691430720
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DOI: https://doi.org/10.1023/A:1022691430720