= Geodesic reparametrization under projective equivalence
{title2=$t''/t'=2V(\dot\gamma)$}
Let $\tau$ be an <affine parameter> for a <geodesic> of $\nabla$ and suppose $\bar\nabla_XY-\nabla_XY=V(X)Y+V(Y)X$. Then $\bar\nabla_{\dot\gamma}\dot\gamma=2V(\dot\gamma)\dot\gamma$. For the new velocity $d\gamma/dt=\dot\gamma/t'$, the acceleration is $(t')^{-2}[\bar\nabla_{\dot\gamma}\dot\gamma-(t''/t')\dot\gamma]$, which vanishes under the displayed <differential equation>. Its nonzero solution $t'=C\exp(2\int V(\dot\gamma)\,d\tau)$ changes the parameter while preserving the curve.
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