: 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. 

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