= Killing tensor
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{title2=$\nabla_{(a}K_{b_1\cdots b_r)}=0$}
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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.
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