Q:

How to construct the horizontal bundle?

I am learning the concept of connection. I am confused by the construction of the horizontal bundle. My question is:
For a fibre bundle $M\rightarrow B$, the vertical vector space $V$ can be easily determined from the projection map. Then how different horizontal vector spaces $H$ can be constructed so that $H_q\oplus V_q=T_q M$? 
Can I have a concrete example on how two different $H$s are chosen so that two different connections are constructed on the same fibre bundle? 
Thanks. 

A:

Consider $\mathbb{R}\hookrightarrow\mathbb{R}^3\to\mathbb{R}^2$ where the projection map is given by projection onto the first two coordinates.
I'm going to define connections in terms of one-forms. Since one-forms annihilate a two-dimensional subspace, the kernel of the one-form will give the horizontal subbundle of the tangent bundle of the total space.
The obvious connection is given by the one-form $dz$. This kills vectors of the form $a\partial_x + b\partial_y$ and so it gives what we might think of as the "usual" connection on this bundle.
A less obvious connection, but which shows up "in the wild," so to speak, is given by the one-form $dz - ydx$. The kernel of this form looks like this:

This is the "standard contact form," and it shows up in contact geometry.

