Spacetime in general relativity is supposed to be curved, despite this we assume that cartesian coordinates still make sense (locally?), don't we? So it should be possible to define cartesian coordinates locally on a curved manifold (Riemannian manifold?), shouldn't it?
So the question is: how are cartesian coordinates defined on a curved manifold (Riemannian manifold?)? An example with cartesian cooridnates on a sphere will be appreciated :)