Let $(M,g)$ be a Riemannian manifold, and let $\lambda$ be a positive number. Then we know $(M, \lambda g)$ is also a Riemannian manifold.
Also, if we have a point $p$ in $M$ and a vector $V$ in $T_pM$ then there exists a unique geodesic $\gamma$ : $I$ → $M$ such that $\gamma(0) = p$, and $\dot\gamma(0) = V$.
My question is then, if we knew the form of the $g$-geodesic $\gamma$ would we then know the form of the $(\lambda g)$-geodesic? How will the constant scaling of the metric affect the geodesic?
