nLab
Killing vector field

Context

Riemannian geometry

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

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive Unknown characterUnknown character 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 }

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

  • the Lie derivative of gg along vv vanishes: vg=0\mathcal{L}_v g = 0;

  • the flow exp(v):X×X\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.

Revised on February 19, 2015 15:37:31 by David Corfield (129.12.18.141)