Intuition for basic concepts in differential geometry

Im studying a basic differential geometry course this semester. In the class, we defined the concept of covariant derivative (connection) as a function which takes 2 vectors fields into a vector field and satisfies some algebraic properties of derivative (linearity, additivity and product rule). From this definition we showed that the connection can be specified by the Christoffel symbols. The covariant derivative "should be" a way of differentiating vectors fields along vector fields. However, there are infinite covariant derivatives on any surface, so using a random one wont make sense. We proved the "fundamental theorem of Riemmanian geometry" - existence and uniqueness of Levi-Civita connection. Except for the nice properties of the Levi-Civita connection - why should I believe it is actually a good way to differentiate vector fields? An example from analytical mechanics could be great since Im familiar with the subject and didnt learn general relativity yet.

Another concern resulting from the first one is "Why (in what sense) is parallel transport parallel?". Parallel transport along a curve is defined using covariant derivative along the velocity field of the curve. In the euclidean space, the definition makes perfect sense - parallel transport is actually parallel. In other surfaces, the covariant derivative is 0, and I didn't find any geometric intuition about "what is actually parallel".

Thanks, Tsuf.

• Suppose c is a curve on the surface and X is a parallel vector field along c. Roll the surface on a flat ground along c, X will project to a (Euclidean) parallel vector field on the flat ground along the projected curve. – Xipan Xiao Dec 25 '14 at 18:23
• if you have a manifold embedded in $\mathbb{R}^n$, take the standard directional derivative, and project this onto the geometric tangent space, you get the Levi-Civita connection. – yoyo Dec 25 '14 at 18:57
• I'm pretty sure existence and uniqueness of the Levi-Civita connection is the fundamental theorem of Riemannian geometry. – Robin Goodfellow Dec 26 '14 at 3:14
• @RobinGoodfellow right, thanks. – tsufli Dec 26 '14 at 9:48
• @XipanXiao thanks. How can I show this (or where can i read a proof?) – tsufli Dec 26 '14 at 17:18