Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$

Question

I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\mapsto\gamma(-t)$ is a geodesic of $\tilde{F}$. Let $\tilde{\gamma}(t):=\gamma(-t)$

The first part is done. But I have some difficulties doing the second part (i.e. $\tilde{\gamma}(t)$ is a geodesic of $\tilde{F}$). I first tried to prove that $\tilde{d}(\tilde{\gamma}(t_0),\tilde{\gamma}(t_1))=L(\tilde{\gamma}|_{[t_0,t_1]})$, but I got nowhere. Now I am thinking of proving that $\tilde\nabla^V=\nabla^{-V}$, where $\tilde\nabla^V$ is the connection of $\tilde{F}$. Is that make sense?

To prove this,first, It is not difficult to show that $\tilde{g}_{_V}=g_{_{-V}}$, where $\tilde{g}_{_V}$ and $g_{_V}$ are, resp., the fundamental tensors of $\tilde{F}$ and $F$. Then using this, one can show that $\tilde\nabla^V$ solves the same equation as $\nabla^{V}$.

Once we have that $\tilde\nabla^V$ is the connection of $\tilde{F}$, we can prove that $\tilde{\gamma}(t)$ is a geodesic of $\tilde{F}$ if and only if $\gamma(t)$ is a geodesic of $F$.

Is that right?

Answer 1

I solve this exercise using the properties of $\tilde{F}$. Indeed one can first prove that $\tilde{g}_v$ is an inner product. Next step prove that $\tilde{F}$ is a Finsler metric and so on. Then in is not difficult to prove that $\gamma$ is a $f$-geodesic iff $\tilde{\gamma}$ is a $\tilde{F}$ geodesic.