Geodesic reparametrization under projective equivalence ($t^{\prime \prime }/t^{\prime }=2V(\dot{\gamma })$)

Let $\tau$ be an affine parameter for a geodesic of $\nabla$ and suppose ${\bar{\nabla }}_{X}Y-{\nabla }_{X}Y=V(X)Y+V(Y)X$. Then ${\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }=2V(\dot{\gamma })\dot{\gamma }$. For the new velocity $d\gamma /dt=\dot{\gamma }/t^{\prime }$, the acceleration is $(t^{\prime }{)}^{-2}[{\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }-(t^{\prime \prime }/t^{\prime })\dot{\gamma }]$, which vanishes under the displayed differential equation. Its nonzero solution $t^{\prime }=C\exp (2\int V(\dot{\gamma })d\tau )$ changes the parameter while preserving the curve.