Mathematical Topics S.

1 total work

Brownian Motion and Index Formulas for the de Rham Complex

by Kazuaki Taira

Published 25 November 1998

The purpose of this monograph is to give an analytic proof of an index formula for the relative de Rham cohomology groups which may be considered as a generalization of the celebrated Hodge-Kodaira theory for the absolute de Rham cohomology groups. More precisely, let X be a compact oriented smooth Riemannian manifold without boundary, and Y a submanifold of X. The purpose is to find an operator D such that ind D = X(X) - X(Y) where X(X) and X(Y) are the Euler-Poincare characteristics of X and Y, respectively. The crucial point is how to introduce spaces of currents on X and Y in which the index formula for D holds. In deriving this index formula, the theory of harmonic forms satisfying an interior boundary condition plays a fundamental role. The approach here has a great advantage of intuitive interpretation of the index formula in terms of Brownian motion from the point of view of probability theory, and the result may be stated as follows: Brownian motion describes the topology of a compact Riemannian manifold through its Euler-Poincare characteristic.