Because a metric also defines the geodesics of a differential manifold, for some metric there is not only one connection defining the same geodesics (some examples can be found of a connection on formula_11 leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on formula_11, i.e. the usual directional derivative) , and given a metric, the only connection which defines the same geodesics (which leaves the metric unchanged by parallel transport) and which is torsion-free is the Levi-Civita connection (which is obtained from the metric by differentiation).