Let $M$ be a 2-dimensional manifold with local coordinate system $(x^i)$ and cotangent bundle $T^*\!M$ with induced coordinates $(x^i,y_i)$.
Define $F^*$ to be a (dual) Finsler metric on the cotangent bundle $T^*\!M{\backslash}\{y_i=0; y_2=0\}$: $$ F^* := \alpha\,\sqrt{\left(\dfrac{\alpha}{\beta}\right)^2-1} $$ where $\alpha^2$ is a Riemannian metric: $$ \alpha := \sqrt{a^{ij} y_i y_j} \,\,,\quad a^{ij} = f(x)\, \delta^{ij} $$ and $\beta$ is non-zero and dependent on one component of $y$ only: $$ \beta := b^{i} y_i \,\,,\quad b^{1} = 0 \,\,,\,\, b^{2} = f(x) $$ and where $f$ is a smooth, non-zero function.
Question: Can anyone comment on this metric and any properties of the Finsler space? Specifically:
- Is it appropriate to call it an $(\alpha,\beta)$ metric?
- Does this Finsler structure belong to any particular well-studied type?
Edit: Following up on a criticism by @TedShifrin that, unusually, the Finsler structure is not defined on the whole projective tangent bundle (it omits $y_2=0$), three comments:
- This Finsler structure describes a physical system in which $y_2 > 0$ always.
- If we allow $y_2=0$, the metric can take infinite values. Opinions seem to vary as to whether this is permissible. For example: Burago et al ("A course in metric geometry") are comfortable with it.
- Here is an example paper, albeit defining a Finsler metric on $TM$ rather than $T^*\!M$, in which their $\beta$ has $b^2 = 0$ and thus they have a similar issue if $y_1=0$:
Yajima, T., & Nagahama, H. (2015). Finsler geometry for nonlinear path of fluids flow through inhomogeneous media. Nonlinear Analysis: Real World Applications, 25, 1–8. http://doi.org/10.1016/j.nonrwa.2015.02.009