"A proof" that Riemann tensor is zero

Question

Riemann tensor is zero in flat space, and well, it is tensor. Thus we have the tensor equation R=0 which means that Riemann tensor is zero in all the coordiantes systems, which is completely a lie. Where is my mistake here?

Answer 1

It's zero in all coordinate systems in that flat space, which is completely true. But if you're in a curved space, there is no coordinate system where $R=0$ to begin with.

Answer 2

The Riemann tensor transforms under a coordinate transformation as $R'^{\sigma}_{\rho \mu \nu}= (S^{−1})^{\sigma}_{\alpha} S^{\beta}_{\rho}S^{\gamma}_{\mu}S^{\delta}_{\nu}R^{\alpha}_{\beta \gamma \delta}$ which means that the components in the new coordinate system are linear combinations of its components in the old coordinate system.

So if the components are all zero in the old coordinate system, they'll also be zero in the new coordinate system. That's why you only need to prove they're zero in only one coordinate system to prove that the space is flat.