Problem with the concept of connection

Question

I've been told that there is only a canonical way for doing the vertical subspace of the tangent bundle of a manifold and in order to do the horizontal subspace you need a connection. These are very elementary question, but I've just passed over them during my studies so I guess it is a good time to ask:

1 - I have no intuition whatsoever about a connection. What intuition is there to have?

2 - Why can't I simply take the orthogonal complement in each fiber to produce the "horizontal subspace"? (Answered).

Answer 1

I'll try to give an intuitive answer, and I'm aware that it might be imprecise in some points. For a more detailed answer, try and have a look at Choquet-Bruhat Y., DeWitt-Morette C., Analysis, manifolds and physics.

When you consider a tangent bundle, you are attaching "continuously" a vector space to every point in the manifold. This tells us that, at least locally, we can imagine the bundle as a copy of $\mathbb{R}^{2n}$ where the first coordinates (horizontal) are coordinates of the manifold, and the others are coordinates on the fibres ("vertical"). Hence, it is natural to think that a vector which is tangent $\textit{to the tangent bundle}$(!) can be decomposed in two different components, one along the fibre and one along the manifold. Locally everything is fine.

Now, the issue is the following: pick a point $x\in M$ and a vector $v$ in the fibre over $x$ (i.e. tangent to the manifold at $x$). Move some steps away from $x$, to a point $y$ on the manifold. There is NO canonical way (in general) to say what happened to your vector $v$ when you changed your point. In other words, there is no natural "parallel transportation" of your vector. The problem arises because moving from $x$ to $y$ you might change the basis of the fibre, and directional derivatives behave pretty badly with coordinate changes.

A connection is a tool designed to solve this problem: it "connects" the fibres to one another, and tells you how tangent vectors are transformed if you move along a path on the manifold. This might take some time to sink in, because it's pretty far away from the formal definition of a connection. However, it is exactly what you need to tell apart the two different directions of a tangent vector of the tangent bundle: vectors which are "horizontal" are sent to zero by the connection, which instead transforms vectors in the fibre, which are "vertical".

Answer 2

There are lots of good reasons for wanting to consider connections. Unfortunately these reasons are often historically motivated and aren't often discussed in intro courses. One reason is that a connection gives us a way to compare tangent vectors at different points on a manifold.

Even though each tangent space is identified with $\mathbb{R}^n$ on an n-dimensional manifold, this identification is not canonical and so we can't directly compare the tangent space at two different points.

A horizontal subspace gives us a way to do this. Given a vector $v \in T_pM$, we can slide that vector along the horizontal subspace to another point $q \in M$ and then project to $T_qM$ to get another tangent vector w. If we do this process infinitesimally along a path from p to q we call this parallel transport. The vectors v and w are in some sense "parallel" with respect to our given connection.

Another good reason is the differentiation of vector fields. We can differentiate vector fields in a local chart, but if we want to change from one chart to another things get difficult. Connections give us a way to globally differentiate vector fields by measuring the difference between the vector field and the chosen horizontal subspace.