= Killing transport
{c}
{title2=$\nabla_V\xi_a=L_{ab}V^b,\quad\nabla_VL_{ab}=-R_{abc}{}^dV^c\xi_d$}
= Killing transport equations
{c}
{synonym}
The <Killing equation> and the <second covariant derivative of a Killing vector> transport a Killing field and its antisymmetric derivative along any curve. Here $L_{ab}=\nabla_b\xi_a$ and the curvature convention is the one in <curvature-sign convention in Killing derivative identities>. In a parallel frame the equations form a homogeneous linear system of ordinary differential equations. Initial value zero has unique zero continuation. Thus on a connected <pseudo-Riemannian manifold>, not just on a Riemannian manifold, the field is determined by its value and first derivative at one point.
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