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?
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.