Question about Finsler geometry

[Esmaeil]

Please read the attached file to answer my question. Thanks a lot.

 

[strangerep]

Please read the attached file to answer my question. Thanks a lot.

You should read the PF instructions on how to type mathematics.

As to your question, you mention $\beta$ in case (1), but it's not part of $F$. (Maybe you intended a Randers metric but omitted the $\beta$ part?)

In any case, since your 2nd $\phi(s)=1+rs$ is a 1st-order expansion of $s^r$ around $s=1$, the answer seems obvious. (Or were you thinking about something more sophisticated involving topologies on a continuum of metrics...??)