More precisely, suppose we a given two metrics $g_0$ and $g_1$ on a manifold $M$. Let $\nabla_0$ and $\nabla_1$ be the corresponding Levi-Civita connections. Set $\nabla_t:=(1-t)\nabla_0+t\nabla_1$. Then $\nabla_t$ is a torsion free connection. Does there exist a continuous family of metrics $g_t$ such that $\nabla_t$ is the Levi-Civita connection of $g_t$?