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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.

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    $\begingroup$ The notion of sectional curvature 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 the sectional curvature 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 4:35
  • $\begingroup$ @PedroLauridsenRibeiro, sorry. I am searching for something: What is the relation between the sectional curvature of the total space of the principal bundle and the total space of the associated vector bundle? $\endgroup$ – L.F. Cavenaghi May 23 '17 at 14:59
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    $\begingroup$ I guess what he means is that he chooses an invariant metric on the structure group as well as a metric on the base space $B$ of the principal bundle $P$. Then having the additional information of a connection defines a metric on the total space of $P$ by transferring the invariant metric to the fibers via the action and lifting metric from the base to the horizontal subbundle of $TP$. $\endgroup$ – Matthias Ludewig May 23 '17 at 16:15
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    $\begingroup$ I think it should not be too hard to figure this out using O'Neil's formulas. Just calculate what these give for a principal bundle and then for a vector bundle; afterwards compare the two, remembering the formula that relates the curvature of the principal bundle to the curvature of the associated vector bundle. $\endgroup$ – Matthias Ludewig May 23 '17 at 16:21
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    $\begingroup$ It should also be relatively easy to do using moving frames and differential forms. $\endgroup$ – Deane Yang May 23 '17 at 20:09

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