[Math] 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.