It is pretty much as I had hoped You know a metric when you see it or at least Minkowski s metric You bring in Pythgarus but metrics go far beyong his insights A metric is an equation the describes the square of the separations between coordinates on a manifold In relativity metrics are Lorentzian with a temporal component having a the square of its separation given in a sign opposite that of the spatial separations Overall the metrics in Relativity give the proper time experienced in traveling a path which is why a geodesic in relativity maximizes the distance travelled since any deviation generated time dilation with respect to the inertial path instead of minimizing it as in a Riemannian manifold