nLab vertical differential form

Definition

In differential geometry

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

The dg-algebra $\Omega^\bullet_{vert/X}(P)$ of vertical differential forms on $P$ is the quotient of the de Rham complex dg-algebra $\Omega^\bullet(P)$ of all differential forms on $P$, by the dg-ideal of horizontal differential forms:

$$\Omega^\bullet_{vert/X}\big( P \big) \simeq \Omega^\bullet\big(P\big) \big/ \big\langle \pi^\ast \Omega^1(X) \big\rangle$$

Examples

Example

For a trivial fiber bundle, $P \equiv X \times F \longrightarrow X$, the vertical differential forms are the smoothly $X$-parameterized differential forms on $F$:

$$\Omega^\bullet_{vert/X}\big( X \times F \big) \simeq C^\infty\big( X, \Omega^\bullet(F) \big) \mathrlap{\,.}$$

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)