The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds (Q2785539)

This text studies how differential operators on a smooth manifold reveal deep relationships between the geometry and the topology of the manifold. The goal of the text is an introduction to central topics in analysis on manifolds through the study of Laplacian-type operators on manifolds. The main subjects covered are Hodge theory, heat operators for Laplacians on forms, and the Chern-Gauss Bonnet theorem in detail. Furthermore, Atiyah-Singer index theory and zeta functions for Laplacians are covered.NEWLINENEWLINENEWLINEThe book is organized in five Chapters: (1) The Laplacian on a Riemannian manifold; (2) Elements of differential geometry; (3) The construction of the heat kernel; (4) The heat equation approach to the Atiyah-Singer index theorem; (5) Zeta-functions of Laplacians.NEWLINENEWLINENEWLINEThe main technique used is the heat flow associated to a Laplacian.