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Calabi-Yau manifolds have the nice property that $c_1(TM) = 0$ implies there is a Ricci flat metric: $\text{Ric}(\omega)$.

Is it possible to construct a similar theorem vor a Vector Bundle over a Calabi-Yau manifold? i.e. $c_1(V) = 0$ implies that there exists some flat connection on the bundle? Or something related?

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up vote 3 down vote accepted

There is a theorem by Uhlenbeck and Yau, according to wiki, which says that "they proved the existence and uniqueness of Hermitian–Einstein metrics (or equivalently Hermitian Yang–Mills connections) for stable bundles on any compact Kähler manifold". You can look at their paper entitled "On the Existence of Hermitian-Yang- Mills Connections in Stable Vector Bundles" for more details.

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Thanks, I believe this might do the trick :) –  FMN Jun 22 '12 at 14:27
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