I searched on some standard references as Taubes's and Kobayashi's book but I did not find what I was searching.

Which is the explicit relation between the sectional curvature of the total space of a principal bundle and the curvature of the total space of some associated vector bundle? Does it provide some relation similar to the relation given by the O`neill's formulas for submersions? I mean, does the curvature of the associated vector bundle increases in contrast to the curvature of the principal bundle?

Where can I find some reference?

**EDIT**

If $G\hookrightarrow P\to B$ is a principal bundle and $F \hookrightarrow M \to B$ is an associated bundle, then there is a Riemannian submersion

$$\bar \pi : G\times P \to M.$$

More generally, if we start with a fiber bundle and let $G$ be the structure group of $F \hookrightarrow M \to B$, the same holds.

From this we can apply O'Neill formulae.

sectionalcurvature refers explicitly to a pseudo-Riemannian metric on a manifold and the associated Levi-Civita connection on the corresponding tangent bundle, so what do you mean by thesectionalcurvature in a principal bundle and on associated vector bundles? Unless you are actually referring to the curvature (Lie algebra-valued) 2-form of an equivariant connection on a principal bundle and the curvature of the induced (linear?) connection on an associated vector bundle. $\endgroup$ – Pedro Lauridsen Ribeiro May 23 '17 at 4:352more comments