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The goal of this book is to introduce the reader to some of the most
frequently used techniques in modern global geometry. Suited to the beginning
graduate student willing to specialize in this very challenging field, the
necessary prerequisite is a good knowledge of several variables calculus,
linear algebra and point-set topology.
The book s guiding philosophy is, in the words of Newton, that in learning
the sciences examples are of more use than precepts . We support all the new
concepts by examples and, whenever possible, we tried to present several facets
of the same issue.
While we present most of the local aspects of classical differential
geometry, the book has a global and analytical bias . We develop many
algebraic-topological techniques in the special context of smooth manifolds
such as PoincarÃ© duality, Thom isomorphism, intersection theory,
characteristic classes and the Gauss Bonnet theorem.
We devoted quite a substantial part of the book to describing the analytic
techniques which have played an increasingly important role during the past
decades. Thus, the last part of the book discusses elliptic equations,
including elliptic Lp and HÃ¶lder estimates, Fredholm theory, spectral theory,
Hodge theory, and applications of these. The last chapter is an in-depth
investigation of a very special, but fundamental class of elliptic operators,
namely, the Dirac type operators.
The second edition has many new examples and exercises, and an entirely new
chapter on classical integral geometry where we describe some mathematical gems
which, undeservedly, seem to have disappeared from the contemporary