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It is well known that a manifold is flat iff its Riemann tensor vanishes identically. However, the equation $R^{\mu}_{\nu\rho\sigma}=0$ is a differential equation for the metric tensor $g_{\mu \nu}$. Is there an analogous condition on the metric function $d(x,y)$ in the case of a flat manifold? For example, a translational invariant metric, $d(x,y)=d(x+z,y+z)$, with the proper scaling property $d(t x, t y) = |t| d (x,y)$ should be a sufficient condition for the flatness of the manifold. But, what are the necessary conditions?

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  • $\begingroup$ Just noting: Your proposed sufficient conditions seem to rely on an additive group structure and/or some multiplicative action of the real numbers, which in general a flat manifold does not possess. (A complete flat manifold is a quotient of Euclidean space by a discrete group of Euclidean motions, but these need not be translations, so Euclidean translations generally don't descend to isometries of the quotient.) $\endgroup$ – Andrew D. Hwang Feb 1 '15 at 13:42
  • $\begingroup$ So, what would you propose as a sufficient condition? $\endgroup$ – user54031 Feb 1 '15 at 14:46
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You can get such a condition by:

  1. Taking a neighborhood of a point into Euclidean space by the manifold property, and either 2a. using the translational properties of this Euclidean open set to define a flat metric (i.e. require d(x+z,y+z)=d(x,y)), or 2b. fixing a base point, and looking at the directional derivatives of the resulting distance function to recover the Riemannian metric, and use the condition on the Riemann tensor to get a condition on the distance function.
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