# Extension of Yau's theorem to general bundles

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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