nLab
Lie derivative

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

Given a smooth manifold MM and a vector field XΓ(TM)X \in \Gamma(T M) on it, one defines a series of operators X\mathcal{L}_X on spaces of differential forms, of functions, of vector fields and multivector fields. For functions X(f)=X(f)\mathcal{L}_X(f) = X(f) (derivative of ff along an integral curve of XX); as multivector fields and forms can not be compared in different points, one pullbacks or pushforwards them to be able to take a derivative.

For vector fields XY=[X,Y]\mathcal{L}_X Y = [X,Y]. If ωΩ (M)\omega \in \Omega^\bullet(M) is a differential form on MM, the Lie derivative Xω\mathcal{L}_X \omega of ω\omega along XX is the linearization of the pullback of ω\omega along the flow exp(X):×MM\exp(X -) : \mathbb{R} \times M\to M induced by XX

Xω=ddt| t=0exp(tX) *(ω). \mathcal{L}_X \omega = \frac{d}{d t}|_{t = 0} \exp(t X)^*(\omega) \,.

Denote by ι X:Ω (M)Ω 1(M)\iota_X : \Omega^\bullet(M) \to \Omega^{\bullet -1}(M) be the graded derivation which is the contraction with a vector field XX. By Cartan's homotopy formula,

v=[d dR,ι v]=d dRι v+ι vd dR:Ω (X)Ω (X). \mathcal{L}_v = [d_{dR}, \iota_v] = d_{dR} \circ \iota_v + \iota_v \circ d_{dR} : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

References

An introduction in the context of synthetic differential geometry is in

  • Gonzalo Reyes, Lie derivatives, Lie brackets and vector fields over curves, pdf

A gentle elementary introduction for mathematical physicists

  • Bernard F. Schutz, Geometrical methods of mathematical physics (elementary intro) amazon, google

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Revised on September 1, 2011 21:00:48 by Zoran Škoda (161.53.130.104)