I have read and heard that there are several theories of gravity and Quantum Gravity which treat the metric (defining rods and clocks) and connection (defining free fall equation of geodesic) of the manifold as independent quantities. I know that the Levi-Civita connections are given as, $$ \Gamma_{\mu\nu}^{\alpha}=\frac{1}{2}g^{\alpha\delta}(g_{\mu\delta,\nu}+g_{\nu\delta,\mu}-g_{\mu\nu,\delta}) $$ But in this case, the connection is uniquely determined by the metric tensor. However, as I see it, the converse is not true (that is the metric cannot be uniquely determined by a given Levi-Civita connection).
So, Given that the Levi-Civita connection can be uniquely determined by the metric tensor, does this fact ascribe certain general properties to the manifold? Hence, by violating these properties of the manifold, one can find other mathematical forms of connection which can be uniquely determined by the metric tensor.