Is the autoparallel equation same as the geodesic equation?

Question

My question may sound silly but i am self learning differential geometry using watching these lectures (Lecture 8) from 2015 by professor Frederic Schuller.

Can somebody please tell that the auto-parallel equation same as the geodesic equation?

Answer 1

You cannot talk about geodesics until you have a notion of distance, and curvature is insufficient for that. However, in lecture 10 he introduces the metric tensor, and he shows that the geodesic equation for a given metric takes the form of an autoparallel equation for one specific curvature / connection / covariant derivative.

Answer 2

Let there be given a pseudo-Riemannian manifold $(M,g)$ with a connection $\nabla$ that is compatible with the metric $g$ but not necessarily torsion-free. Let $\nabla^{LC}$ denote the Levi-Civita connection for $g$. Then the geodesic equation $\nabla^{LC}_{\dot{\gamma}}\dot{\gamma}=0$ and the auto-parallel equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$ are not necessarily the same. They are the same iff the torsion tensor is totally antisymmetric. See e.g. this Phys.SE post for details.