# nLab basic differential form

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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:

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

### 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.

## References

### Articles

• Peter W. Michor, Basic differential forms for actions of Lie groups, Proc. Amer. Math. Soc. 124 (1996), 1633–1642 doi

Last revised on April 2, 2024 at 12:51:53. See the history of this page for a list of all contributions to it.