nLab Killing vector field

Contents

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \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)(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.

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