By Weiping Zhang
This necessary publication relies at the notes of a graduate path on differential geometry which the writer gave on the Nankai Institute of arithmetic. It contains elements: the 1st half comprises an advent to the geometric idea of attribute sessions as a result of Shiing-shen Chern and Andre Weil, in addition to an explanation of the Gauss-Bonnet-Chern theorem in line with the Mathai-Quillen building of Thom types; the second one half provides analytic proofs of the Poincaré-Hopf index formulation, in addition to the Morse inequalities according to deformations brought by way of Edward Witten.
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Sample text
Since (x, and K is nowhere zero on M , 1 E R*(M) dK0 is well-defined. 1 follows directly from the Stokes formula. 5 39 References [A] M. F. Atiyah, Circular symmetry and stationary-phase approximation, Colloquium in honour of Laurent Schwartz, Vol. 2, AstLrique 131 (1985), 43-59. [AB] M. F. Atiyah and It. Bott, The moment map and equivariant cohomology. Topology 23 (1984), 1-28. [BC]P. Baum and J. Cheeger, Infinitesimal isometries and Pontryagin numbers. Topology 8 (1969), 173-193. [BGV] N. Berline, E.
Math. Phys. 121 (1989), 351-399. [YlY. Yu, Index Theorem and Heat Equation Method. (in Chinese) Shanghai Sci. & Tech. Press, 1996. Chapter 2 Bott and Duistermaat-Heckman Formulas In Chapter One we have defined characteristic classes and numbers in terms of curvatures of connections on vector bundles. A natural question is how to compute these characteristic numbers. In this chapter we will discuss a localization formula due to Bott [Bo] which shows that for a compact manifold admitting a compact Lie group action, the calculation of characteristic numbers on this manifold can be reduced to the fixed point set of the group action.
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