Lectures on the Geometry of Manifolds

Description

The object of this book is to introduce the reader to some of the most important techniques of modern global geometry. It mainly deals with global questions and in particular the interdependence of geometry and topology, global and local. Algebraico-topological techniques are developed in the special context of smooth manifolds. The book discusses the DeRham cohomology and its ramifications: Poincare, duality, intersection theory, degree theory, Thom isomorphism, characteristic classes, Gauss-Bonnet etc. The authors seek to calculate the cohomology groups of as many as possible concrete examples without relying on the apparatus of homotopy theory (CW-complexes etc). Elliptic partial differential equations are also featured, requiring a familiarity with functional analysis. It describes the proofs of elliptic Lp and Holder estimates (assuming some deep results of harmonic analysis) for arbitrary elliptic operators with smooth coefficients. The book closes with alook at a class of elliptic operators, the Dirac operators. It discusses their algebraic structure in some detail, Weizenbock formulae and many concrete examples