by Weiping Zhang (Nankai Institute of Mathematics, Tianjin, P R China)

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 André 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 Poincaré–Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten.

• Chern–Weil Theory for Characteristic Classes
• Bott and Duistermaat–Heckman Formulas
• Gauss–Bonnet–Chern Theorem
• Poincaré–Hopf Index Formula: An Analytic Proof
• Morse Inequalities: An Analytic Proof
• Thom–Smale and Witten Complexes
• Atiyah Theorem on Kervaire Semi-characteristic