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.
Contents:
- 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
Readership: Graduate students and researchers in differential geometry, topology and mathematical physics.
“The last twenty years have seen widespread interest in the use of analytic techniques to calculate and refine invariants arising in algebraic and differential topology. Insights from mathematical physics have driven much of this activity. The book under review is an introduction to this field. This book is noteworthy for its combination of brevity, clarity, accessibility, and depth. This combination rests in part on the elegance of the writing and in part on the author's insight in using techniques due to J M Bismut and collaborators … A reader aware of the theorems discussed in the book but looking to understand their proofs through the eyes of an expert will find this book fascinating. This book is a valuable contribution to the literature.”
Mathematics Abstracts
“… the book's main strength is its clear presentation of analytic deformation techniques much simpler than those in the original work of Bismut and his collaborators (including the author), and Helffer Sjöstand … the book gives an excellent introduction to these analytic techniques …”
Mathematical Reviews
