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?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.