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9 There is also a heat kernel proof of the Gauss-Bonnet-Chern theorem due to Patodi [P], see the books [BGV] and [Y] for more details. 29) suggested by Witten [W]. We will present such a proof in the next chapter. 7 follows from the Poincare-Hopf index formula and also the Stokes formula. Here, instead of using the arguments in [C1], we adopt a simplified version due to Chern himself [C2]. We make the same assumptions and use the same notation as in previous sections. Recall that SM is the unit sphere bundle of the tangent bundle p : TM ----t M.
1. = V'x P [Y,Z]- V'y p [X,Z]- P [[X, YJ,Z] = p1. [Z, Xl] + [Z, [X, Y]]) = p1. ([X, [Y, Zll + [Y, [Z, Xll + [Z, [X, Y]]) _p1. [X, prY, Zll - p1. 26) and the Jacobi identity. 14. ,* denote the dual bundle of p1.. , VF-L) ... , VF-L) E r (A 2 (il+ .. * (M). 31) Since dimpl. 31) that when i 1 + ... 29) holds. 28). 12 is thus completed. 2 Adiabatic Limit and the Bott Connection One may argue that from the geometric point of view, the connection \IF-L is also a natural connection on pl.. In fact, by passing gTM to its adiabatic limit, one sees that the underlying limit of \I F-L and the Bott connection - F-L are ultimately related.
Bull. Amer. Math. Soc. 65 (1959), 276-281. [AH2] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces. Pmc. Symp. Pure Math. Vol. 3. pp. 7-38. Amer. Math. , 1961. [AS] M. F. Atiyah and 1. M. Singer, The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69 (1963), 422-433. [BD] P. Baum and R. G. Douglas, K-homology and index theory. in Pmc. Sympas. Pure and Appl. , Vol. 38, pp. 117-173, Amer. Math. Soc. Providence, 1982. [BGY] N. Berline, E. Getzler and M.