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As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion free connection $\Delta_g$, \nabla_g$, the Levi-Civita connection, that is compatible witht metric. I was wondering if one can reverse this situation: Given a manifold with$M$with connection$\Delta$, \nabla$, when does there exist a Riemannian metric $g$ for which $\Delta$ \nabla$is the Levi-Civita connection. If this were true for complex projective manifolds it would make me be very happy. 3 fixed title When can a Connection Induce a Riemannian Metric for which it is the Levi--CivitaLevi-Civita Connection? 2 added 12 characters in body As we all know, for a Riemannian manifold$(M,g)$, there exists a unique torsion free connection$\nabla_g$, he Levi--Civita \Delta_g$, the Levi-Civita connection, that is compatible witht metric.

I was wondering if one can reverse this situation: Given a manifold with $M$ with connection $\Delta$, when does there exist a Riemannian metric $g$ for which $\Nabla$ \Delta\$ is the Levi--Civita Levi-Civita connection.

If this were true for complex projective manifolds it would make me be very happy.

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