nLab Killing vector field

Contents

Context

Riemannian geometry

Riemannian geometry

Applications

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

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

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

Idea

A Killing vector on a (pseudo-)Riemannian manifold is equivalently

Similarly a Killing spinor is a covariantly constant spinor.

Definition

For $(X,g)$ a Riemannian manifold (or pseudo-Riemannian manifold) a vector field $v \in \Gamma(T X)$ is called a Killing vector field if it generates isometries of the metric $g$. More precisely, if, equivalently

• the Lie derivative of $g$ along $v$ vanishes: $\mathcal{L}_v g = 0$;

• the flow $\exp(v) : X \times \mathbb{R} \to X$ is a flow by isometries.

Properties

The flows of Killing vectors are isometries of the Riemannian manifold onto itself.

Last revised on February 19, 2015 at 15:37:31. See the history of this page for a list of all contributions to it.