nLab vertical differential form

Definition

In differential geometry

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 horizontal differential forms, hence 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 $\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F)$.

Related concepts

• horizontal differential form

• Lie integration

References

• Philippe Bonnet, Alexandru Dimca, Relative differential forms and complex polynomials, Bulletin des Sciences Mathématiques 124 Issue 7 (2000) pp 557-571, doi:10.1016/s0007-4497(00)01055-1, (author pdf)