Why is the set of horizontal vector fields a vector space but not a Lie algebra?

Question

I can't think of a concrete counter-example that shows a horizontal vector field is not necessarily a Lie algebra. Is there any easy example for this?

Answer 1

I believe you are talking about the the horizontal vector fields defined by a connection on a principal bundle, it is a Lie algebra iff the curvature vanishes.

A vector $X$ is horizontal iff $\omega(X)=0$ where $\omega$ is the connection form, since this form is linear at each point, the set of horizontal vector fields is a vector space.

Answer 2

It is an immediate consequence of the definition of the curvature $\Omega$ of a principal bundle $G\rightarrow P\rightarrow M $ with $\mathfrak{g}$-valued connection $\omega$ that $\Omega$ vanishes if and only if the horizontal vector fields are closed under the Lie bracket (that is, form a Lie algebra). For reference, if you need it, check out José Figueroa-O'Farrill's notes: http://empg.maths.ed.ac.uk/Activities/GT/Lect2.pdf

Thus, to find an example where the horizontal fields are not a Lie algebra, it suffices to find a principal bundle with nonvanishing curvature. This is provided by the Hopf bundle, $S^1\rightarrow S^3\rightarrow S^2$. I will not provide details about this bundle, which are found amply in many references. Nor will I provide details of the curvature calculation, for which I heartily recommend anyone to Gregory Naber's book, "Geometry, Topology, & Gauge Fields, Vol. 1: Foundations."