This invaluable book is based on the notes of a graduate course on
differential geometry which the author gave at the Nankai Institute
of Mathematics. It consists of two parts: the first part contains
an introduction to the geometric theory of characteristic classes
due to Shiing-shen Chern and Andre Weil, as well as a proof of the
Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction
of Thom forms; the second part presents analytic proofs of the
Poincare-Hopf index formula, as well as the Morse inequalities
based on deformations introduced by Edward Witten.
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