Talk:Principal bundle

From Wikipedia, the free encyclopedia

[edit] Transitive action

Why did you suppress the fact that G acts transitively on the fiber. This is contained implicitly, but should be mentioned explicitly, since it is essential. Hottiger 17:39, 11 April 2006 (UTC)

[edit] Measure

For at least some principle bundles, there is a natural measure on he space of connections, is there not? I beleive this is the case when the fibers are compact; less clear of the situation when they're not compact. Would like to see a proper definition. May attempt to do this myself, if/when get around to it. linas 15:51, 20 July 2006 (UTC)

[edit] An Algebraic Characterization of Principal Bundles

Both transitivity and freeness of action on the fibres can be directly characterized by defining the quotient on the product bundle, along with the right multiplication by G, by the properties p\(pg) = g and p(p\q) = q, where p\q denotes the quotient. This has the advantage of making everything else that follows down the line more transparent. For instance, the connection form simply becomes p\dp, the differential of the quotient by the second argument. In general, the differential of the quotient serves as another equivalent way to define the connection. For a section S, the operator S dS\() + dS S\() is none other than the horizontal lift operator; and the connection relativized to a section is just S\dS. The local decomposition, through a section S, of the principal bundle becomes p = S(pG) S(pG)\p, where pG denotes the projection of p onto the base space. -- Mark, 11 October 2006