Conditions on the metric function in a flat manifold

Question

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?

Answer 1

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.