Chen Series and Atiyah-singer Theorem
نویسنده
چکیده
The goal of this paper is to give a new and short proof of the local Atiyah-Singer index theorem by using approximations of heat semigroups. The heat equation approach to index theorems is not new: It was suggested by Atiyah-Bott [1] and McKean-Singer [18], and first carried out by Patodi [21] and Gilkey [13]. Bismut in [8] introduces stochastic methods based on Feynman-Kac formula. For probabilistic approaches that are mainly based on Bismut’s ideas, we also refer to [15] and Chapter 7 of [14]. For a complete survey on (non probabilistic) heat equation methods for index theorems, we refer to the book [7].
منابع مشابه
v 1 [ m at h . D G ] 2 5 M ar 1 99 9 Real embeddings and the Atiyah - Patodi - Singer index theorem for Dirac operators ∗
We present the details of our embedding proof, which was announced in [DZ1], of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary [APS1]. Introduction The index theorem of Atiyah, Patodi and Singer [APS1, (4.3)] for Dirac operators on manifolds with boundary has played important roles in various problems in geometry, topology as well as mathematical physics. ...
متن کاملAn alternative stochastic proof for the Atiyah-Singer index theorem
According to Watanabe [6], by generalized Wiener functional we could give a simple stochastic proof of the index theorem for twisted Dirac operator (the Atiyah-Singer index theorem) on Clifford module. MR Subject Classification: 58J20 Chinese Library Classification: O186.12
متن کامل- Patodi - Singer Index Theorem ∗
In [Wu], the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case.
متن کاملApplications of Elliptic Operators and the Atiyah Singer Index Theorem
1. Review of Differential Geometry 2 2. Definition of an Elliptic Operator 5 3. Properties of Elliptic Operators 7 4. Example of an Elliptic Operator 9 5. Example: The Euler Characteristic 12 6. Example: The Signature Invariant 14 7. A Theorem of Atiyah, Frank and Mayer 18 8. Clifford Algebras 20 9. A Diversion: Constructing Vector Fields on Spheres using Clifford Algebras 23 10. Topological In...
متن کاملK-Theory and Elliptic Operators
This expository paper is an introductory text on topologicalK-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the material presented here is distilled from Atiyah’s classic “K-Theory” text, as well as his series of seminal papers “The Index of Elliptic Operators” with Singer. ...
متن کامل