nLab Killing tensor

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

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.

References

Named after Wilhelm Killing.

Last revised on April 24, 2018 at 13:45:57. See the history of this page for a list of all contributions to it.