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

Bismut, Jean-Michel

J. Funct. Anal. 57, 56-99 (1984).

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

Keywords:

elliptic operator; de Rham complex; heat equation kernel; Lefschetz fixed point formulas

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