Jacobi fields from conjugation at a central endpoint
= Jacobi fields from conjugation at a central endpoint
{c}
{title2=$J_X(t)=Xe^{tA}-e^{tA}X$}
For a <Lie group> with a <bi-invariant Riemannian metric>, vary $\gamma(t)=e^{tA}$ by conjugation: $F(s,t)=e^{sX}e^{tA}e^{-sX}$. Its <Jacobi field> is $J_X(t)=Xe^{tA}-e^{tA}X$. If $e^A$ is central, all these fields vanish at both endpoints. Their space has dimension $\dim\mathfrak g-\dim\mathfrak z(A)$, since the kernel is the <centralizer of an element of a Lie algebra>.