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

by Steven Rosenberg

Overview

This text on analysis on 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 author develops the Atiyah-Singer index theorem and its applications (without complete proofs) via the heat equation method. Rosenberg also treats zeta functions for Laplacians and analytic torsion, and lays out the recently uncovered relation between index theory and analytic torsion. The text is aimed at students who have had a first course in differentiable manifolds, and the author develops the Riemannian geometry used from the beginning. There are over 100 exercises with hints.

Product Details

ISBN-13:	9780521468312
Publisher:	Cambridge University Press
Publication date:	01/09/1997
Series:	London Mathematical Society Student Texts , #31
Edition description:	New Edition
Pages:	188
Product dimensions:	5.98(w) x 9.02(h) x 0.71(d)

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.

Editorial Reviews

"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

From the Publisher