Let $(M, g)$ be a Riemannian manifold. Is it possible to construct two different affine (or metric) connections, say $\nabla$ and $\nabla'$, which induce the SAME curvature tensor, i.e. $R(X, Y)Z=R'(X, Y)Z$ for any $X, Y, Z\in\Gamma(TM)$?
Are there any references related to this question? Thank you very much.