To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric.
Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics. (from Wikipedia)
Does this imply that for a given Riemannian manifold there is a Levi-Civita connection that is unique for that manifold?
I am still learning this stuff so I am often having trouble seeing whether something is true or not :)
Edit: To elaborate, from the fundamental theorem of riemannian geometry, there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.
Does this make this connection unique for that manifold? i.e. if I have a lot of manifolds can I identify them using the Levi-Civita connection?