# Conditions on the metric function in a flat manifold

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?

• 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.) – Andrew D. Hwang Feb 1 '15 at 13:42
• So, what would you propose as a sufficient condition? – user54031 Feb 1 '15 at 14:46