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Let $\Sigma$ be a surface endowed with a Riemannian metric $g.$ According to the fundamental theorem of Riemannian geometry, there exist a unique $\nabla$ symmetric connection ( i.e. torsionless) in the tangent bundle of $\Sigma$ that preserves the metric, that is $$ \nabla g \equiv 0.$$ Are there any examples of non symmetric connections that preserve a metrig $g$ on a surface $\Sigma?$ What if $g$ is flat?

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A slight extension of what you refer to as the "fundamental theorem of Riemannian geometry" says that mappsing linear connections on $TM$ which are compatible with a Riemannian metric $g$ to their torsion is an isomorphism between metric connections and smooth sections of the bundle $\Lambda^2T^*M\otimes TM$. So for any skew symmetric tensor field $T$, there is a unique metric connection with torsion $T$. It is not difficult to write out an explicit formula for the contorsion needed to modify the Levi-Civita connection to a metric connection with torsion $T$.

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  • $\begingroup$ @ Andreas Thank you Andreas. Can you provide a reference? $\endgroup$
    – Mike Cocos
    Jul 1 '16 at 15:18
  • $\begingroup$ I don't have a specific reference at hand. However, proofs that are based on the affine structure on the space of connections (i.e. which first take any linear connection, then change it to a metric connection and then remove the torsion) usually proceed by showing that mapping changes of metric linear connections to changes of torsion is injective in each point. But counting dimensions, this shows that it is also surjective and hence bijective in each point, so the stronger result is usually proved as well. $\endgroup$ Jul 4 '16 at 6:28

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