I wonder how much information about metic tensor of Riemmanian manifold can be extracted if only the Levi-Civita connection is given. Conversly, if the metric on manifold is given there is formula for Christoffel symbols which define connection so there exists only one symetrical metric connection on Riemmanian manifold. I presume that connection should strongly limited the class of possible metrics but I don't know how. For example if Christoffel symbols in specific coordinate system vanish everywhere, the metric tensor in this base may be identity matrix times constant (are there any other possibilities??).