Can I reconstruct a Riemannian metric out of its Levi-Civita connection? In other words: Given two Riemannian metrics $g$ and $h$ on a manifold $M$ with the same Levi-Civita connection, can I conclude that $g=h$ up to scalars?
If not, what can I say about the relationship between $g$ and $h$? How rigid is the Levi-Civita-Connection?
No. If $g$ is a metric, $2g$ is also a metric and they both have the same L-C connection.
More interestingly: In general, a connection is metric (that is, comes from a metric) if its holonomy at each point is contained in the orthogonal subgroup (this was discussed at https://mathoverflow.net/questions/54434/when-can-a-connection-induce-a-riemannian-metric-for-which-it-is-the-levi-civita), and any metric whose value at each point is preserved by the holonomy is one whose L-C connection is the one we started with. To get examples, pick any Riemannian manifold with trivial holonomy at each point: for example, a space form of curvature zero (the Euclidean plane or a flat torus, say,to keep things simple): there are many metrics which induce the same connection (and not multiples of each other), which can be constructed by picking any inner product in the tangent space at one point and transport it to the rest of the manifold parallelly w.r.t the connection of the flat metric we started with.