TITLE: What is the affine connection, and what is the intuition behind/for affine connection?
QUESTION [23 upvotes]: Here is the definition of affine connection, as appears in Milnor's book Morse Theory.

DEFINITION. An affine connection at a point $p \in \text{M}$ is a function which assigns to each tangent vector $\text{X}_p \in \text{TM}_p$ and to each vector field $\text{Y}$ a new tangent vector$$\text{X}_p \vdash \text{Y} \in \text{TM}_p$$called the covariant derivative of $\text{Y}$ in the direction $\text{X}_p$.
(Note that our $\text{X} \vdash \text{Y}$ coincides with Nomizu's $\nabla_\text{X} \text{Y}$. The notation is intended to suggest that the differential operator $\text{X}$ acts on the vector field $\text{Y}$.)
This is required to be bilinear as a function of $\text{X}_p$ and $\text{Y}$. Furthermore, if$$f: \text{M} \to \mathbb{R}$$is a real valued function, and if $f\text{Y}$ denotes the vector field$$(f\text{Y})_q = f(q)\text{Y}_q$$then $\vdash$ is required to satisfy the identity$$\text{X}_p \vdash (f\text{Y}) = (\text{X}_p f)\text{Y}_p + f(p) \text{X}_p \vdash \text{Y}.$$
(As usual, $\text{X}_p$ denotes the directional derivative of $f$ in the direction of $\text{X}_p$.)

I have two questions.

This definition of affine connection is quite terse here—I'm just seeing text on a page and not really understanding what is going on here. Is it possible somebody could help me parse through/explain what is really being said here with regards to affine connection?
Could somebody supply their intuitions behind/for affine connections?

Thanks.

REPLY [4 votes]: Intuition come from mechanics as usual in differential geometry. Assume you are in a car moving with a law $P(t)$. in your car, there is a compass which gives you the magnetic vector field say $ \vec M$, note that this vector field is globally defined on the earth, but what you see is  $ \vec M _{P(t)}$. Now, in your car you see the direction of the compass changing at every time, and you can compute ${d\over dt}  \vec M _{P(t)}$. It appears that this vector only depends on the speed ${\vec V}= {d\over dt} P(t)$ you have at the instant $t$. It is written either ${D\over dt}  \vec M _{P(t)}$ or $ \nabla _{\vec V}{\vec M}$. In order to prove this you can compute in coordinates, and check that this derivative is nothing else but the orthogonal projection of the usual derivative on the tangent plane.  Doing this carefully you will  "rediscover" Christoffel symbols, and find all properties of the affine connexion, which is nothing else but the operator which enable you to compute the derivate, called the "covariant"  derivative.