Lectures on the Geometry of Manifolds

The goal of this book is to introduce the reader to some of the main techniques, ideas and concepts frequently used in modern geometry. It starts from scratch and it covers basic topics such as differential and integral calculus on manifolds, connections on vector bundles and their curvatures, basic Riemannian geometry, calculus of variations, DeRham cohomology, integral geometry (tube and Crofton formulas), characteristic classes, elliptic equations on manifolds and Dirac operators. The new edition contains a new chapter on spectral geometry presenting recent results which appear here for the first time in printed form.Contents: ManifoldsNatural Constructions on ManifoldsCalculus on ManifoldsRiemannian GeometryElements of the Calculus of VariationsThe Fundamental Group and Covering SpacesCohomologyCharacteristic ClassesClassical Integral GeometryElliptic Equations on ManifoldsSpectral GeometryDirac Operators
Readership: Graduate students and researchers in global analysis, differential geometry.Manifolds;Calculus on Manifolds;Vector Bundles and Connections;Curvature;Derham Cohomology;Characteristic Classes;Crofton Formula;Tube Formula;Elliptic Partial Differential Equations;Dirac Operators;Spectral Geometry0Key Features:The book discusses in great detail many concrete examples and computations that are very useful for the working mathematicianSome of these are recurring examples since they have many hidden facets that reveal connections with different subjects