Conservation of quadratic geodesic integrals

Given: The metric-selected connection, a smooth rank-two Killing tensor $K$, and a smooth affinely parametrized geodesic $\gamma :I\to \mathcal{M}$.

Killing tensors and quadratic geodesic integrals defines a Killing tensor on a time-oriented Lorentzian four-manifold by symmetry and the cyclic equation ${\mathrm{\nabla }}_{(a}{K}_{bc)}=0$. In the Statement we explicitly use that same equation without a time-orientation assumption; symmetry and the equation are hypotheses, not an application of that definition outside its scope.

Geodesic of an affine connection: affine parameter means $D_{\lambda }v=0$ along $\gamma$.

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields supplies the unique metric-selected connection and the contraction product rule for every smooth nondegenerate symmetric metric on a smooth four-manifold, without time orientation; Covariant derivative along a curve defines its induced derivative along $\gamma$.

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant: a differentiable function on an interval whose derivative vanishes everywhere is constant.

The metric-selected connection and tangent-bundle convention use only countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).