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

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Steven Rosenberg (Author)

9780521463003, Cambridge University Press

Hardback, published 9 January 1997

188 pages, 50 exercises
23.8 x 15.9 x 1.7 cm, 0.393 kg

"The book is well written.... This book provides a very readable introduction to heat kernal methods and it can be strongly recommended for graduate students of mathematics looking for a thorough introduction to the topic." Friedbert PrÜfer, Mathematical Reviews

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.

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.

Subject Areas: Differential & Riemannian geometry [PBMP]