Killing tensor (${\nabla }_{(a}{K}_{b_1\cdots b_r)}=0$)

A Killing tensor is a symmetric covariant tensor satisfying the displayed equation for the Levi-Civita connection. Its contraction with $r$ copies of a geodesic velocity is constant along every affinely parametrized geodesic. The converse follows because the resulting symmetric derivative is determined by its homogeneous polynomial on tangent vectors. Rank one recovers the Killing vector field after metric duality.

A symmetric two-tensor satisfying this equation gives a quadratic geodesic first integral. Examples include the Riemannian metric, symmetric products of metric-dual Killing vector fields, and the square of a Killing-Yano two-form.

A quadratic homogeneous polynomial in the fibre variables of the cotangent bundle is a first integral of the geodesic Hamiltonian exactly when its symmetric lowered coefficients form a rank-two Killing tensor. For $v=g^{-1}p$, the Poisson bracket is $\{K,H\}={\nabla }_{(a}{K}_{bc)}{v}^a{v}^b{v}^c$, so the equivalence follows by comparing cubic coefficients at every point.