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

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.

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.

• I found this reference easier to follow. Thank You.
– Sak
Jun 28, 2011 at 5:52

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).

• ICM proceedings link is broken.. Sep 24, 2019 at 17:13
• Thanks. I think it is fixed now. Sep 25, 2019 at 0:58
• Thank you very much :) Sep 25, 2019 at 3:55