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.
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