nLab
Killing tensor

Definition

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

$$K\in {\mathrm{Sym}}^k\Gamma (TX)$$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 $(X,g)$ 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.

Related concepts

• Killing vector

• Killing spinor

• Killing-Yano tensor.