general relativity - Metric and Connection

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.

Answer

The Levi-Civita connection (or Christoffel symbols) are coming from the metric compatibility constraint :

$$\tag{1} \nabla_{\lambda} \, g_{\mu \nu} = 0,$$

plus the "desire" to get a symetrical connection : $\Gamma_{\mu \nu}^{\lambda} = \Gamma_{\nu \mu}^{\lambda}$. This last commandement is arbitrary, and is only justified to get the simplest theory possible (i.e classical General Relativity). If you remove the symetrical constraint but only imposes equ (1), you then get the Levi-Civita connection and a contorsion tensor :

$$\tag{2} \Gamma_{\mu \nu}^{\lambda} = \frac{1}{2} \, g^{\lambda \kappa} \, (\, \partial_{\mu} \, g_{\nu \kappa} + \partial_{\nu} \, g_{\mu \kappa} - \partial_{\kappa} \, g_{\mu \nu}) + K^{\lambda}_{\mu \nu}. $$ The antisymetrical part $T^{\lambda}_{\mu \nu} \equiv \Gamma_{\mu \nu}^{\lambda} - \Gamma_{\nu \mu}^{\lambda}$ is the torsion tensor, which is assumed to vanish in GR. This guy is not 0 in the Einstein-Cartan theory, which is the natural extension of GR (a bit more general than GR).

Some theories consider that the connection is an independant variable when they don't impose the symetric constraint, or even the compatibility (1) above.