Various definitions of Connections on bundles-2

Question

Starting with a principal $GL_n$ bundle, we could form the vector bundle associated to a representation $GL_n\to GL(V)$. Could someone please explain how we get a connection on this vector bundle.

Answer 1

Let $P \to M$ is a $G$-principal bundle with a connection $\omega$ on it and $R:G \to GL (V)$ is a representation, inducing a Lie algebra map $r: \mathfrak{g} \to \mathfrak{gl}(V)$. Assume that $P$ is trivial and pick a section $s$ of $P$. The section identifies sections of $P \times_G V$ with functions $M \to V$ and $s^* \omega$ is a $\mathfrak{g}$-valued $1$-form. The formula becomes (sections written as $V$-valued functions, $X$ a vector field)

$$ \nabla_X v:= (dv)(X) \pm r(s^* \omega)(X)\cdot v. $$

This is a connection on $V$ and it does not depend on the choice of $s$ when the sign is correctly chosen (exercise). The independence on $s$ shows that you can do these things locally, so that the procedure works for nontrivial bundles as well.

Answer 2

Here's another way to see the relation, complementary to the answer by Johannes Ebert.

Let $\pi: P \to M$ be a principal $G$-bundle, $\rho: G \to \mathrm{GL}(V)$ be a finite-dimensional representation and $P \times_\rho V \to M$ be the associated vector bundle.

Let us view connections on $P \to M$ in the sense of Ehresmann. That is, a connection on $P \to M$ is as a choice of a $G$-invariant horizontal distribution $\mathcal{H} \subset TP$, so that $TP = \mathcal{H} \oplus \mathcal{V}$ with $\mathcal{V} = \ker T\pi$. Let $h: TP \to \mathcal{H}$ be the projection along $\mathcal{V}$.

We can view sections of the associated bundle $P \times_\rho V \to M$ as $G$-equivariant functions $P \to V$; that is, functions $f: P \to V$ such that for all $p \in P$ and $g \in G$, $$f(p g) = \rho(g)^{-1} f(p)$$

Finally, let us view connections on $P \times_\rho V \to M$ in the sense of Koszul; that is, as a covariant derivative. Then we define for $f$ a section of $P\times_\rho V$, $$ \nabla f := h^* df,$$ where $df$ is the exterior derivative in $P$. In other words, if $X$ is a vector field on $P$, we define $$\nabla_X f = df(hX).$$

Answer 3

All you need to know is how to build a rank $n$ vector bundle from a principal $GL(n)$-bundle. The idea there is to view a local section of the principal bundle as a local frame of a vector bundle, and the transition functions for the principal bundle as change of frame maps for the vector bundle. In this case, it is easy to see that the connection on the principal bundle tell you how to "differentiate" each section of a local frame. This therefore defines a connection on the vector bundle.

Then given any principal $G$-bundle and a representation $G \rightarrow GL(V)$, there is a naturally defined principal $GL(V)$-bundle whose transition functions are defined by composing the transition functions of the original bundle with the representation. A connection on the original bundle defines a connection on the new bundle. Now use the construction above to construct the vector bundle and its connection from the principal $GL(V)$-bundle and its connection.