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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??).