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Spectral Theory in Riemannian Geometry
EMS Textbooks in Mathematics

Olivier Lablée (Université Joseph Fourier Grenoble 1, Saint Martin d’Hères, France)

Spectral Theory in Riemannian Geometry

ISBN print 978-3-03719-151-4, ISBN online 978-3-03719-651-9
DOI 10.4171/151
February 2015, 197 pages, hardcover, 16.5 x 23.5 cm.
38.00 Euro

Spectral theory is a diverse area of mathematics that derives its motivations, goals and impetus from several sources. In particular, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object in differential geometry. From a physical point a view, the Laplacian on a compact Riemannian manifold is a fundamental linear operator which describes numerous propagation phenomena: heat propagation, wave propagation, quantum dynamics, etc. Moreover, the spectrum of the Laplacian contains vast information about the geometry of the manifold.

This book gives a self-containded introduction to spectral geometry on compact Riemannian manifolds. Starting with an overview of spectral theory on Hilbert spaces, the book proceeds to a description of the basic notions in Riemannian geometry. Then its makes its way to topics of main interests in spectral geometry. The topics presented include direct and inverse problems. Direct problems are concerned with computing or finding properties on the eigenvalues while the main issue in inverse problems is “knowing the spectrum of the Laplacian, can we determine the geometry of the manifold?”

Addressed to students or young researchers, the present book is a first introduction in spectral theory applied to geometry. For readers interested in pursuing the subject further, this book will provide a basis for understanding principles, concepts and developments of spectral geometry.

Keywords: Spectral theory, linear operators, spectrum of operators, spectral geometry, eigenvalues, Laplacian, inverse problems, Riemannian geometry, analysis on manifolds

Further Information

MAA review

Zentralblatt Mathematik

Review in MR 3309800