# Spectral Theory in Riemannian Geometry

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*Olivier Lablée*

A publication of the European Mathematical Society

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-contained 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 to 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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Students and researchers interested in spectral geometry.