Manifold,affine connection,vector field

Question

An affine connection on $M$ is a differential operator, sending smooth vector fields $X$ and $Y$ to a smooth vector field $∇_X Y$ , which satisfies the some conditions.I would like to know the intuition behind the field $∇_X Y$,esp. what quantity it measures.

Answer 1

The intuition is that, if you ever compute it in the case that $M = \mathbb{R}^n$, a connection is the usual differentiation along the given vector field $X$.

So, a connection on a manifold $M$ should be some way to "differentiate" a vector field along another one. Or: A way to see what's the variation of a given vector field along the second one.

Thus, it can measure how a vector field varies along the trajectories of the other one (for an exact definition of this, look for affine conections along maps).

For example: If the manifold $M$ admits a Riemannian metric, we must want to "differentiate" the inner product of vector fields along a third one. From this, arises the concept of a metric connection: one of the (many) equivalent forms of stating that a connection is metric is that $\forall X,Y,Z \in \Gamma(TM)$ vector fields, we've that

$$ X\langle Y,Z\rangle = \langle \nabla_X Y, Z \rangle + \langle \nabla_X Z, Y \rangle $$

Where the inner product is the one in the right point of the Manifold. Note that this definition gives even more intuition: the usual derivative on $\mathbb{R}^n$ satisfies the given property, and thus an afine metric connection can also be seen as a way to measure how the angle of two vector fields varies along a third one.