Potential symmetric 2-tensor
= Potential symmetric 2-tensor
{title2=$\beta=\mathcal L_Zg$}
For a vector field $Z$, define $\beta(v,w)=\langle\nabla_vZ,w\rangle+\langle\nabla_wZ,v\rangle$. Its diagonal evaluation is $2\langle\nabla_vZ,v\rangle$. Along a geodesic this equals $2\,d\langle Z,\dot\gamma\rangle/dt$, so it is invisible to closed-geodesic integrals. If a fibrewise linear function $u(x,v)=\eta_x(v)$ solves $Xu=\beta(v,v)$, then $Z=\eta^\sharp/2$ gives this convention.