Brownian Motion and Index Formulas for the de Rham Complex
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Brownian Motion and Index Formulas for the de Rham Complex

Brownian Motion and Index Formulas for the de Rham Complex

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About the Book

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.

Table of Contents:
Elements of differential geometry, of functional analysis, of Markov processes, and of partial differential equations; index formulas for the de Rham complex; the Hodge-Kodaira decomposition theorem; the exterior derivative and the codifferential operator; the operator D; the long exact sequence and the operator D; proof of the main theorem.


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Product Details
  • ISBN-13: 9783527401390
  • Publisher: Wiley-VCH Verlag GmbH
  • Publisher Imprint: Wiley-VCH Verlag GmbH
  • Height: 240 mm
  • Returnable: N
  • Width: 168 mm
  • ISBN-10: 3527401393
  • Publisher Date: 25 Nov 1998
  • Binding: Paperback
  • Language: English
  • Weight: 440 gr


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