By Hans Grauert
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Additional resources for Complex Analysis and Algebraic Geometry: Proceedings of a Conference, Held in Gottingen, June 25 - July 2, 1985
Now, we trivially notice that (a) (c) and (b) * (c). 8. If <£ has the property Ch-W, then (30) and the following diagram commutes: ^T _JL^(Sec^(PY) A(P') PROOF. 9. The Chern-v/eil homomorphisms of pfb's are invariants of local isomorphisms having the property Ch-W. In the case of pfb's with connected structural Lie groups, the Chern-Weil homomorphisms are invariants of all local isomorphisms. D - 50 - CHAPTER 6 i A STRUCTURAL THEOREM Here we prove that any Lie algebroid A is uniquely determined (up to an isomorphism) by its Lie algebra bundle <£(A), a covariant derivative V in<£(A) and a 2-tensor SitSl* (M^(A)), fulfilling some conditions.
F 1 f . , . «* ^ n 4 . 1 . The elements S7 and &M fulfil the following as&tions (1°) RX y*-- C^M(X,Y),«5], X,Y£36(M), S"eSecq[, where R denotes the curvature tensor of V , ie ,O» *€Sec
PROOF, By corollary 7-3 , this correspondence is an injection. Let V be an arbitrary 21 -connection in or. Of course, T-V-V0 is a tensor where v is a £ -connection corresponding to an arbitrary but fixed connection V . Besides •f I We want to find a homomorphism c:TM such that ^vff" t( which will mean that vFirst, we notice that is a derivation of the Lie algebra (£ . Because of the fact that (f is semisimple, we see that the derivation T(v,») is inner 'x which means that there is an uniquely determined element c(v) such that T(v,O- [c(v),O.