The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds (London Mathematical Society Student Texts, Series Number 31)

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. 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. Steven Rosenberg. Includes Index. Cover; Title; Copyright; Contents; Introduction; 1 The Laplacian on aRiemannian Manifold; 1.1 Basic Examples; 1.1.1 The Laplacian on S1 and R; 1.1.2 Heat Flow on S1 and R; 1.2 The Laplacian on a Riemannian Manifold; 1.2.1 Riemannian Metrics; 1.2.2 L2 Spaces of Functions and Forms; 1.2.3 The Laplacian on Functions; 1.3 Hodge Theory for Functions and Forms; 1.3.1 Analytic Preliminaries; 1.3.2 The Heat Equation Proof of the Hodge Theorem forFunctions; 1.3.3 The Hodge Theorem for Differential Forms; 1.3.4 Regularity Results; 1.4 De Rham Cohomology; 1.5 The Kernel of the Laplacian on Forms