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Extra info for [Article] Real Hypersurfaces Contained in Abelian Varieties

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Fulfils the Bianchi PROOF. It is easy to check that X for v,weT M is a derivation of the Lie algebra <£f , R being the curva ture tensor of V . Prom the assumption that

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. It remains to show that the mapping c(v), l is a C°°-vector bundle homoraorphism.

2921). The correspondence * h-> H* (24) sets up a bisection between connections in (6) and in P(M,G). PROOF. Let H be any connection in P(M,G). IX , x«M* By (23), we see that B I -A. '-**. :B IX IX IX —*• T XM is an isomorphism as a superposition ~~ B:. \J B|xCA(P) X is a vector subbundle. ,! and take a local cross-section n A tf:D —» P. etf, i *n, forms a basis of B on U, which proves that B is a vector subbundle* B defines H ~ H —1 a connection A :TM —*A(P) by ^ . » (t,vls,v) • The correspondence TT H >-* A I X is inverse to (24).

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