I want to know two things:
1) If the riemann is zero the manifold is necessarily ${R^n}$ and if is true, how can I prove it?
2)Can we have 2 manifolds with the same Riemann tensor?
What I really want to know with these questions is if we can know our manifold only knowing the geodesic deviation.
The Riemann tensor only captures intrinsic curvature. Different manifolds embedded in an ambient space can have the same intrinsic but different extrinsic curvatures. For example, a plane and a cylinder both have zero intrinsic curvature (though different topologies) so their Riemann tensors are both identically zero. But there is obviously a sense in which the cylinder is "curved" while the plane isn't. This is captured by their extrinsic curvature (e.g. their mean curvature) but not by their Riemann tensors.