nLab Killing tensor

Context

Riemannian geometry

Riemannian geometry

Applications

Differential geometry

differential geometry

synthetic differential geometry

Contents

Definition

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

$K\in {\mathrm{Sym}}^{k}\Gamma \left(TX\right)$K \in Sym^k \Gamma(T X)

which is covariantly constant in that

\nabla_{(\mu K_{\alpha_1, \cdots, \alpha_k}) = 0 \,. \nabla_{(\mu K_{\alpha_1, \cdots, \alpha_k}) = 0 \,.

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

Properties

For every Killing tensor $K$ on $\left(X,g\right)$ the dynamics of the relativistic particle on $X$ has a further conserved quantity. In the canonical case $K=g$ this quantity is the energy Hamiltonian of the particle.

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

Created on September 17, 2011 11:12:16 by Urs Schreiber (82.113.99.49)