My question is linked to the foundations of Finsler metrics (with weak derivability assumptions).
Let $M$ be a manifold of dimension $n$, and $F$ is a function from the tangent bundle $TM$ to $\mathbb{R}^+$. The function is in fact given in a coordinates chart. That is, we have locally a function $f(x, v)$ (which represents $F$ in the coordinates chart) from $\mathbb{R}^n\times\mathbb{R}^n$ to $\mathbb{R}^+$.
We know that the function $f$ is homogeneous of the first degree relatively to the second variable. We are interested in the derivability conditions on $f$ that are preserved when we change the coordinates chart.
Alvarezpaiva, in the first answer below, gave an example that show that, if we suppose that $f$ is of class $\cal{C}^1$, only with regards to the first variable, then we cannot deduce that $\phi^*f$, representing $f$ in another coordinates chart, will have the same property.
The example show that, in order for $\phi^*f$ to be derivable, we need to assume that the change of coordinates $\phi$ is itself of class $\cal{C}^2$.
The question is now: is it always sufficient?
The correct derivability assumptions we have to make, in order for them to be preserved after a change of coordinates chart, seem to be:
1) That the change of coordinates is of class $\cal{C}^2$.
2) That the function $f$ is of class $\cal{C}^1$, with regards to BOTH variables.
The new function, after change of coordinates, is:
$\phi^*f=(f\circ \phi_*)$ with $\phi_*(x, v)=(\phi(x), D\phi_x(v))$.
Now $D\phi_*(x, v)(dx, dv)=(D\phi_x.dx, D^2\phi_x.v.dx+D\phi_x.dv)$. The chain rule gives us:
$D(\phi^*f).(x, v)(dx, dv)=Df_{\phi_*(x, v)}.D\phi_*(x, v)(dx, dv)$. And $f$ is still of class $\cal{C}^1$.
With Alvarezpaiva's example of a fixed norm $f(x, v)=\left\|v\right\|$, we get:\ $D(\phi^*f)(x, v)(dx, 0)=\frac{1}{\left\|D\phi_x.v\right\|}\left\langle D\phi_x(v)|D^2\phi_x.v.dx\right\rangle$.
Could you please confirm my reasoning and my notations (differential geometry is not my speciality)?
Thanks a lot.
