What is the relation between connections on principal bundles and connections on vector bundles?

Question

I'm reading Kobayashi / Nomizu 's vol. I. I am reading about connections in principal G-bundles. After that chapter there's a chapter on (linear) connections on Vector bundles. Since we can associate to every principal bundle a vector bundle (via the twisted product) and to every vector bundle a principal vector bundle, i was wondering if we can do this:

Given a connection on a principal bundle, define an associated connection in the associated vector bundle, and conversely given a (linear) connection in a vector bundle, define an associated connection in the associated principal bundle.

If anyone knows how to do this or can point me to a book that has this done, I'll be very grateful.

I tried to do it by myself but if there's an obvious way to do it I am missing it.

Answer 1

Let $P$ be a principal $G$-bundle, $\rho:G\to GL(V)$ a finite dimensional representation of $G$, $E = P \times_G V$ the associated vector bundle. To any principal connection $\Phi$ on $P$ is associated an induced linear connection $\bar \Phi$ on $E$. Conversely, any linear connection on a vector bundle $E$ is induced from a unique principal connection on the linear frame bundle $GL(\mathbb R^n, E)$ of $E$.

You can find all the details here: Topics in Differential Geometry - P. W. Michor.

Answer 2

Given a faithful representation $V$ of a group $G$, the associated bundle construction is an equivalence of categories between principal bundles on a locally ringed space $M$ with fiber $G$ and vector bundles on $M$ with fiber $V$. This is more or less in Steenrod's Topology of fiber bundles.

Given a principal bundle $P$ and a corresponding vector bundle $W$, there is a bijection between connections on $P$ and connections on $W$. One way of making this clear is to use Grothendieck's definition of connection: Let $\Delta: M \to M \times M$ be the diagonal on the base manifold, and let $I$ be the sheaf of ideals that defines $\Delta(M)$ as a submanifold of $M \times M$. The locally ringed space $M^{(1)}$defined by $I^2$ is the first-order neighborhood of the diagonal, and the inclusion into $M \times M$ induces maps $p_1, p_2: M^{(1)} \to M$. A connection on $P$ is an isomorphism $p_1^* P \to p_2^* P$, and a connection on $W$ is an isomorphism $p_1^* W \to p_2^* W$. Their correspondence follows from the equivalence in the first paragraph.

Possible references include Grothendieck's 1970 ICM lecture, available at the ICM proceedings site, and Berthelot's thesis (Springer Lecture Notes 407).