Geodesics are the same for two connections differing only by torsion. Seeking proof.

Question

I am seeking a proof of the fact that the geodesics of two different connections on a manifold will be identical if the two connections only differ in their respective torsions. This statement is mentioned in the Wikipedia article

https://en.wikipedia.org/wiki/Torsion_tensor

I would like the proof in the language of Riemannian geometry as used in the study of general relativity, if possible. I have found this point often cited in papers and articles regarding torsion but I have never seen it demonstrated and I am having trouble convincing myself that it is true.

Answer 1

The torsion is the antisymmetric part of the connection, so connections $\nabla,D$ "differing only by torsion" really means "having the same symmetric part", i.e. $$\nabla_X Y + \nabla_Y X = D_X Y + D_Y X$$ for all vector fields $X,Y$. Choosing $X=Y=\dot \gamma$ to be the velocity vector of a curve $\gamma$ we find $$\nabla_{\dot \gamma} \dot \gamma= D_{\dot \gamma} \dot \gamma,$$ so the geodesic equation $\nabla_{\dot \gamma} \dot \gamma = 0$ of $\nabla$ is the same as that of $D$.