TITLE: Is the space of Levi-Civita connections convex
QUESTION [12 upvotes]: 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 $g_t$?

REPLY [3 votes]: The answer of Professor Bryant shows, that the space of Levi-Civita connections is not convex. On the other hand, there are natural convex subspaces, at least in dimension 2, namely the space of Levi-Civita connections for metrics of a fixed conformal class. To prove this, let us assume we are in the oriented case, so we are dealing with Riemann surfaces. Let $K\to\Sigma$ denote the canonical bundle, i.e., its sections are the complex-valued complex linear 1-forms on $\Sigma$. A conformal metric on $\Sigma$ in its conformal class defines a unique connection $\nabla$ on $K$ whose $(0,1)$-part is just the exterior derivative, which is the natural holomorphic structure on $K$. If $g=e^{2\lambda}g_0$ their connection 1-forms  differ by  $$\nabla-\nabla^0=-2\partial\lambda.$$ For $\tilde g=e^{2\tilde\lambda}g_0$ we obtain $$\tilde\nabla-\nabla^0=-2\partial\tilde\lambda,$$ and $$t\tilde\nabla+(1-t)\nabla=\nabla^0-2(t\partial\tilde\lambda+(1-t)\partial\lambda)$$
which is the Levi-Civita connection for $e^{2(t\tilde\lambda+(1-t)\lambda)}g_0.$