= Converse geodesic characterization of a Killing vector
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{title2=$\frac d{ds}(K_aV^a)=0\text{ for all timelike geodesics}\iff\nabla_{(a}K_{b)}=0$}
Along an affinely parametrized <geodesic>, differentiating the contraction of a <vector> field $K$ with its tangent $V$ gives $d(K_aV^a)/ds=V^aV^b\nabla_{(a}K_{b)}$. If this is zero along every <timelike geodesic>, arbitrary timelike initial tangents at each point make the symmetric <quadratic form> vanish on the whole <timelike cone>. That cone is open, so its <quadratic polynomial> must vanish identically. Thus $K$ is a <Killing vector>. The converse follows directly from the same identity. One selected orbit cannot establish the converse; the quantifier over all initial timelike directions is essential.
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