vertical differential form

Let $\pi : P \to X$ be a bundle in the category Diff of smooth manifolds.

The dg-algebra $\Omega^\bullet_{vert}(P)$ of *vertical differential forms* on $P$ is the quotient of the de Rham complex dg-algebra $\Omega^\bullet(P)$ of all forms on $P$, by the dg-ideal of all those forms that vanish when any one vector in their arguments is a vertical vector field in that it is in the kernel of the differential $d \pi : T P \to T X$.

For a trivial bundle $P = X \times F$ the underlying complex of $\Omega^\bullet_{vert}(P)$ is $\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F)$.

Revised on September 19, 2010 22:53:51
by Urs Schreiber
(188.20.66.18)