Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the connection can be determined by the metric from a relativity simple formula. BUT , doesn't whether the metric is symmetric or not, depend upon the choice of coordinates when specifying the metric. E.g- Schwarzschild metric given in spherical polar coordinates is diagonal, and so symmetric. However, the extended Schwarzschild metric in Eddington-Finkelstein coordinates is not diagonal, contains a dudr term, but not a drdu term. Does this mean that for the metric in Eddington-Finkelstein coordinates , the connection can no longer be simply computed from the metric? Thanks in advance.