I know that geodesics on a Riemannian manifold equipped with the Levi-Civita connection $\nabla$ have two equivalent characterizations:
- They are energy-extremizing curves (which is equivalent to being length-extremizing and being parametrized as to have constant velocity, of course w.r.t. the same background metric) ;
- They are self-parallel curves (i.e. the velocity vector of the curve is parallel transported along the curve) with respect to $\nabla$.
I'd like to know if this equivalence characterizes the Levi-Civita connection: given a Riemannian manifold equipped with a generic torsion-free connection $\nabla$, is it true that the equivalence of properties (1) and (2) forces $\nabla$ to be the Levi-Civita connection? (I guess the restriction to torsion-free connections is needed because only the symmetric part of the Christoffel symbol enters the geodesic equation.)
I am also curious about the possibility of weakening the hypotheses: is just one of the implications (e.g (2) $\Rightarrow$ (1)) enough to force $\nabla$ to be the Levi-Civita connection?
Finally, do these results carry over to metrics with a Lorentzian $(+---)$ signature or is a positive-definite metric essential? (I guess in this case we could restrict to 'timelike' curves, i.e. curves whose velocity vector $v$ satisfies $g(v,v) > 0$ everywhere, to ensure the length functional and its variation are real.)
If possible, I would appreciate proofs avoiding the explicit computation of the Christoffel symbols or the coordinate expression of the geodesic equation.