The Laplacian on a Riemannian manifold : an introduction to analysis on manifolds

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Bibliographic Information

The Laplacian on a Riemannian manifold : an introduction to analysis on manifolds

Steven Rosenberg

（London Mathematical Society student texts, 31）

Cambridge University Press, 1997

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  • : pbk

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Note

Reprinted 1998: x, 174 p., bibliography (p. 167-171)

Includes bibliographical references (p. 165-169) and index

Description and Table of Contents

Description

This text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The Atiyah-Singer index theorem and its applications are developed (without complete proofs) via the heat equation method. Zeta functions for Laplacians and analytic torsion are also treated, and the recently uncovered relation between index theory and analytic torsion is laid out. The text is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning. There are over 100 exercises with hints.

Table of Contents

  • Introduction
  • 1. The Laplacian on a Riemannian manifold
  • 2. Elements of differential geometry
  • 3. The construction of the heat kernel
  • 4. The heat equation approach to the Atiyah-Singer index theorem
  • 5. Zeta functions of Laplacians
  • Bibliography
  • Index.

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