nLab
Killing tensor

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 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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

For (X,g)(X,g) a (pseudo-)Riemannian manifold a Killing tensor is a section of a symmetric power of the tangent bundle

KSym kΓ(TX) K \in Sym^k \Gamma(T X)

which is covariantly constant in that

(μK α 1,,α k)=0. \nabla_{(\mu} K_{\alpha_1, \cdots, \alpha_k)} = 0 \,.

For k=1k = 1 this reduces to the notion of Killing vector.

Properties

For every Killing tensor KK on (X,g)(X,g) the dynamics of the relativistic particle on XX has a further conserved quantity. In the canonical case K=gK = g this quantity is the Hamiltonian of the particle (in the case of a relativistic particle its four-velocity normalization).

The analog of this for spinning particles and superparticles are Killing-Yano tensors.

Revised on April 2, 2015 13:52:01 by Tim Porter (2.31.48.51)