# nLab Lie bracket of vector fields

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

Given a smooth manifold $X$, the Lie bracket of vector fields $u$ and $v$ can be defined in several ways.

### As commutator of derivations

Since derivations of smooth functions are vector fields, we can identify $u$ and $v$ with the corresponding derivations $C^\infty(X)\to C^\infty(X)$.

Taking the commutator $u v-v u$ of these derivations produces another derivation, which is denoted by $[u,v]$, and which can be identified with a vector field on $X$.

### As a Lie derivative

Alternatively, we can set

$[u,v]=\mathcal{L}_u v=-\mathcal{L}_v u,$

where $\mathcal{L}$ denotes the Lie derivative of a vector field.

## Properties

The real vector space of vector fields on $X$ equipped with the Lie bracket forms a Lie algebra.

Last revised on May 3, 2023 at 09:44:05. See the history of this page for a list of all contributions to it.