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.
Best Answer
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.