I remember encountering the following theorem many, many time ago:
if a connection $\nabla$ has $R = 0$ and $T = 0$ then...
The problem is that I cannot remember the conclusion. Was it that the manifold must be (locally?) a Euclidean space endowed with its standard connection? It was surely something about flatness, but I can't remember the exact details (it was about connections in the tangent bundle, not in general vector bundles).
Could anyone please provide the precise statement and just a sketch of the proof? Thank you.
(I adopt the less abstract point of view where I identify a connection with its associated covariant derivative, I can visualize this approach better.)