This text examines the Atiyah-Singer theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The book presents a careful treatment of non-self-adjoint operators, asymptotics of the heat equation and variational formulas. It also introduces spectral geometry and provides a list of asymptotic formulas. The bibliography has been complied by Herbert...

This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z (r) Y is a Riemannian submersion.
After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure...

A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. Written by a leading pioneer in the field, it focuses on the functorial and special cases methods of computing asymptotic heat trace and heat content coefficients in the heat equation. It incorporates the work of many authors into the presentation, and includes a complete bibliography that serves as a roadmap to the literature on the subject. Geometers,...

Pseudo-Riemannian geometry is an active research field not only in differential geometry but also in mathematical physics where the higher signature geometries play a role in brane theory. An essential reference tool for research mathematicians and physicists, this book also serves as a useful introduction to students entering this active and rapidly growing field. The author presents a comprehensive treatment of several aspects of pseudo-Riemannian geometry, including the spectral geometry of the curvature tensor, curvature homogeneity, and Stanilov-Tsankov-Videv theory.