Mathematics and its Applications (Dordrecht). 534. Dordrecht: Kluwer Academic Publishers. ix, 445 p. EUR 145.00; $ 133.00; £ 91.00 (2001).
This text is designed to introduce spectral geometry to graduate students and non-specialists in the field. It contains an extensive bibliography. The first chapter contains an introduction to the theory of Riemannian manifolds -- tensor fields are defined, Riemannian structures are introduced, the Levi-Civita connection and the curvature are defined, and geodesics and the exponential map are presented. In the second chapter, several canonical differential operators are defined.
In the third chapter, the spectral properties of the Laplace-Beltrami operator are given -- the fundamental solution of the heat equation is studied, examples are presented of explicit spectra, and estimates of the eigenvalues through geometric data are discussed among several other topics. In the fourth chapter, isospectral closed Riemannian manifolds are discussed -- topics include asymptotic expansions for the trace of the heat kernel, isospectral flat tori, and Sunada’s theorem.
In Chapter 5, spectral properties for the de Rham complex are given. Chapter six deals with applications to geometry and topology with a discussion of the Hodge-de Rham theorem, Bochner type vanishing theorems, Lefschetz fixed point theorems, and the Chern-Gauss-Bonnet theorem. Chapter 7 gives an introduction to Witten-Helffer-Sjöstrand theory. The book concludes in Chapter 8 with open problems and some comments. The book also contains an appendix giving a review of linear algebra relevant to the task at hand.