Given a Finsler metric $F$ on a compact boudaryless manifold $M$ and $\sigma : M \longrightarrow \mathrm{R}$ a $C^2$ strictly positive function, define a new Finsler metric by $\tilde{F}=\sigma F.$ Simple calculations shows that the fundamental tensor $\tilde{g}$ of the new metric is equal to $\sigma^2g,$ where $g$ is the fundamental tensor of $F.$

I've seen some bibliography about how the Cartan connection of the new metric behave depending on the old Cartan connection and on the derivative of $\sigma.$ Are there any bibliography about how the Chern-Rund behave?

Thank you!