By Weiping Zhang

This valuable e-book 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 creation to the geometric concept of attribute periods as a result of Shiing-shen Chern and Andre Weil, in addition to an evidence of the Gauss-Bonnet-Chern theorem in accordance with the Mathai-Quillen development of Thom varieties; the second one half provides analytic proofs of the Poincaré-Hopf index formulation, in addition to the Morse inequalities in accordance with deformations brought through Edward Witten.

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**Extra resources for Lectures on Chern-Weil Theory and Witten Deformations**

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3 Let (V + 2,) A = 0. Proof. By Leibniz's rule, we have v (1x12) = -22,vx. 10) Mathai-Quillen's Thom Form 45 while by the Bianchi identity we have Vp*RE= 0. 10). 11) The following result shows that U is a Thom form* for E . 4 The form U is a closed n-form on E . Furthermore, one has the following formula for the fiberwise integration, (&)n/2 J,,, u = 1. 12) Prooj Since 2 A E @Oi(E,Ai(p*E)), i=O one gets n e W AE @Oi ( E , A i ( p * E ) ) . 11), one verifies easily that U is a closed n-form on E .

Math. Phys. 103 (1986)’ 127-166. [Bo] R. Bott, Vector fields and characteristic numbers. Michigan Math. J. 14 (1967)’ 231-244. [DH] J . J . Duistermaat and G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 (1982), 259-268. Addendum, 72 (1983), 153-158. Chapter 3 Gauss-Bonnet-Chern Theorem In this chapter we will present Mathai-Quillen's proof [MQ] of the GaussBonnet-Chern theorem [Cl], which expresses the Euler characteristic of a closed oriented Riemannian manifold as an integral of the Pfaffian of the curvature of the associated Levi-Civita connection.

5) follows. 0 / B (dx' A e l ) A . . A (dx" A en) Mathai-Quillen’s Thorn Form 43 Finally, let E be an oriented Euclidean vector bundle of rank n over a manifold M . 4). We still call it a Berezin integral. Let V E be a Euclidean connection on E (that is, V E preserves the metric on E ) , then it extends naturally to an action V on R * ( M , R * ( E ) ) . The following property is important for the applications in the next section. 2. For any a holds, E R * ( M , A * ( E ) ) ,the following identity dJ’Ba=J’BVa.