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The Laplacian on a Riemannian Manifold

The Laplacian on a Riemannian Manifold
An Introduction to Analysis on Manifolds

Part of London Mathematical Society Student Texts

  • Date Published: March 1997
  • availability: Available
  • format: Paperback
  • isbn: 9780521468312

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  • 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.

    • Based on courses given in the US and UK
    • More modern treatment than competitors
    • There are over 100 exercises with hints
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    Product details

    • Date Published: March 1997
    • format: Paperback
    • isbn: 9780521468312
    • length: 188 pages
    • dimensions: 229 x 152 x 18 mm
    • weight: 0.286kg
    • contains: 50 exercises
    • availability: Available
  • 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.

  • Author

    Steven Rosenberg, Boston University

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