nLab
Lie derivative

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          \infty-Lie theory

          ∞-Lie theory (higher geometry)

          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

          .

          Last revised on March 23, 2017 at 09:29:00. See the history of this page for a list of all contributions to it.