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Sur le théoreme d'atiyah-singer
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  • Published: December 1988

Sur le théoreme d'atiyah-singer

  • Rémi Léandre1 

Probability Theory and Related Fields volume 80, pages 119–137 (1988)Cite this article

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Résumé

On donne une version simplifiée de la démonstration probabiliste du théoreme de l'indice pour l'opérateur de Dirac, donnée par J.M. Bismut. Au lieu d'utiliser la construction du mouvement brownien au moyen du fibré des reperes, on utilise la construction de Schwartz en plongeant la variété ambiante dans un espace vectoriel de dimension plus grande. On évite aussi l'utilisation de la décomposition de l'espace de Wiener en deux utilisée par J.M. Bismut, le calcul des variations stochastiques étant notre outil principal. De ce fait, on perd la relation avec la cohomologie de l'espace des lacets.

Summary

We give a simplified version of Bismut's probabilistic proof of the index theorem for the Dirac operator. By using Schwartz's construction of a Brownian motion over a manifold, we expect to give a simpler approach to the computations of stochastic geometry. Our main tool is the calculus of stochastic variations, rather than the splitting of the Wiener space into two pieces. For that reason, we loose the relation with the cohomology of the loop space.

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  1. Faculté des Sciences, 16, Route de Gray, F-25030, Besancon Cedex, France

    Rémi Léandre

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  1. Rémi Léandre
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Léandre, R. Sur le théoreme d'atiyah-singer. Probab. Th. Rel. Fields 80, 119–137 (1988). https://doi.org/10.1007/BF00348755

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  • Received: 01 February 1987

  • Revised: 10 May 1988

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00348755

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