nLab basic differential form

Definition

For general submersions

Given a submersion $p\colon E\to B$, one may ask: which differential forms on $E$ are pullbacks of differential forms on $B$?

If the fibers of $p$ are connected (otherwise the characterization given below is valid only locally in $E$), the answer is provided by the notion of a basic form: a differential form $\omega$ is basic if the following two conditions are met:

• The contraction of $\omega$ with any $p$-vertical vector field is zero.

• The Lie derivative of $\omega$ with respect to any $p$-vertical vector field is zero.

Using Cartan's magic formula, in the presence of the first condition, the second condition can be replaced by the following one:

• The contraction of $d\omega$ (where $d$ is the de Rham differential) with any $p$-vertical vector field is zero.

Via pullback to smooth principal bundles

Given a Lie group $G$ and a smooth $G$-principal bundle $P \overset{p}{\longrightarrow} X$ over a base smooth manifold $X$, a differential form $\omega \in \Omega^\bullet\big( P \big)$ on the total space $P$ is called basic if it is the pullback of differential forms along the bundle projection $p$ of a differential form $\beta \in \Omega^\bullet\big( X \big)$ of the base manifold

$$\omega \;=\; p^\ast(\beta) \,.$$

In terms of Cartan calculus

Equivalently, if $\mathfrak{g} \simeq T_e G$ denotes the Lie algebra of $G$, and for $v \in \mathfrak{g}$ we write

$$\hat v \;\colon\; P \overset{ (e,v), (-,0) }{\hookrightarrow} T G \times T P \simeq T ( G \times P ) \overset{ \;\;\; d \rho \;\;\; }{\longrightarrow} T P$$

for the vector field on $P$ which is the derivative of the $G$-action $G \times P \overset{\rho}{\to} P$ along $v$, then differential form $\omega$ is basic precisely if

1. it is annihilated by the contraction with $\hat v$

   $$\iota_{\hat v} \omega = 0$$

2. it is annihilated by the Lie derivative along $\hat v$:

   $$\mathcal{L}_{\hat v}\omega \;=\; [d_{dR}, \iota_{\hat v}] \omega \;=\; 0$$

   (where the first equality holds generally by Cartan's magic formula, we are displaying it just for emphasis)

for all $v \in \mathfrak{g}$.

In this form the definition of basic forms makes sense more generally whenever a Cartan calculus is given, not necessarily exhibited by smooth vector fields on actual manifolds. This more general concept of basic differential forms appears notably in the construction of the Weil model for equivariant de Rham cohomology.

Related concepts

• horizontal differential form