Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent field $\tilde X$ obeys $\nabla_{\tilde X} \tilde X = 0$ (affine parameterised).
Attempt: Let $t \mapsto \gamma(t)$ be such a geodesic with tangent vector field $X = \frac{d}{dt} g(\gamma(t))$ where $g$ is some smooth function defined over the manifold. Now reparamatrise, so that $t \rightarrow t(z) \mapsto \gamma(t(z))$ Then let $\tilde X$ be the tangent vector of this newly paramatrised curve. So $$\tilde X = \frac{dt}{dz} \frac{d}{dt} g(\gamma(t)) = \frac{dt}{dz} X \equiv X/h$$
Then $$\nabla_X X = fX \Rightarrow \nabla_{h \tilde X} (h \tilde X) = h^2 \nabla_{\tilde X} \tilde X + h \tilde X(h) \tilde X = fh \tilde X.$$ So provided $\tilde X(h) = f$ I am done. But I don't see why this is true or if there is a fault previously. Many thanks!