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 Poincaré 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 Lpand Hölder 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 mathematical limelight.
