= Killing field is determined by its value and first covariant derivative
{c}
{title2=$K(p),\quad(\nabla K)_p$}
On a <connected> <Riemannian manifold>, these data determine a <Killing field> uniquely, without a completeness assumption. The <second covariant derivative of a Killing vector> makes its restriction to every <geodesic> a <Jacobi field>. The difference of two fields with identical data has zero Jacobi initial values and vanishes in a <convex normal neighborhood>. The set on which both its value and <covariant derivative> vanish is closed by continuity and open by the same local argument, so connectedness makes it the whole manifold.
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