How to construct a Finsler metric on a product manifold and which are the short curves?

Question

Let $\mathcal{M}$ be a smooth manifold and let $f_{\mathcal{M}}^m$ be a Finsler structure on $\mathcal{M}$, i.e., let $f_{\mathcal{M}}^m : T_m\mathcal{M} \to \mathbb{R}$ be a metric that varies smoothly with base-point $m \in \mathcal{M}$. Then for all $m \in \mathcal{M}$, $f_{\mathcal{M}}^m$ is smooth on $T_m\mathcal{M} \setminus \{0\}$. Now, let $\mathcal{N}$ be another smooth manifold with Finsler structure $f_{\mathcal{N}}^n$ and consider the product manifold $\mathcal{O} := \mathcal{M} \times \mathcal{N}$. I would like to consider

$$f_{\mathcal{O}}^o : (X_\mathcal{M}, X_\mathcal{N}) \mapsto \sqrt{f_{\mathcal{M}}^m(X_{\mathcal{M}})^2 + f_{\mathcal{N}}^n(X_{\mathcal{N}})^2},$$

where $o = (m, n)$. Is this a valid Finsler structure? More specifically, does it pose a problem that $f_{\mathcal{O}}^o$ is not necessarily smooth on elements of the form $(X_\mathcal{M}, 0)$ and $(0, X_\mathcal{N})$?

Motivation/context: the question is motivated by curve length on product manifolds. In the Riemannian case, i.e., when the $f$s are induced by inner products on the corresponding tangent spaces (call them $g$ instead), the above construction is a well-known way to introduce a Riemannian metric on the prodct manifold (see, e.g, Example 13.2 in the book Introduction to smooth manifolds by J.M. Lee). A nice feature of this product metric is that $\mathcal{M}$ and $\mathcal{N}$ becomes orthogonal components of $\mathcal{O}$. More precisely, for each manifold the corresponding functional

$$L_{\mathcal{M}}(\gamma) = \int_0^1 g_{\mathcal{M}}^{\gamma(t)}(\dot{\gamma}(t)) dt$$

defines the curve length of the smooth curve $\gamma : [0,1] \to \mathcal{M}$, and the distance between two points $m_1, m_2 \in \mathcal{M}$ is defined as $\inf_{\gamma} L_{\mathcal{M}}(\gamma)$ where the $\inf$ is taken over all smooth curves such that $\gamma(0) = m_1$ and $\gamma(1) = m_2$. If $\gamma_{\mathcal{M}}$ is the minimizing curve between $m_1, m_2 \in \mathcal{M}$ and similarly for $\gamma_{\mathcal{N}}$, then a minimizing curve on $\mathcal{O}$ between $o_1 = (m_1, n_1)$ and $o_2 = (m_2, n_2)$ is given by $\gamma_\mathcal{O}(t) = (\gamma_\mathcal{M}(t), \gamma_\mathcal{N}(t))$ and the corresponding distance is $L_{\mathcal{O}}(\gamma_{\mathcal{O}}) = \sqrt{L_{\mathcal{M}}(\gamma_{\mathcal{M}})^2 + L_{\mathcal{N}}(\gamma_{\mathcal{N}})^2}$. Is the same true in the Finsler case?

Answer 1

I found the answer in the paper Minkowskian product of Finsler spaces and Berwald connection by T. Okada (www.doi.org/10.1215/kjm/1250521819). For potential benefit of others, I will post it here.

Yes, the $f^o_\mathcal{O}$ given in the question is a valid Finsler structure on the product manifold (by Okada termed "the Euclidean product of these Finsler spaces") and $\gamma_{\mathcal{O}}(t) = (\gamma_{\mathcal{M}}(t), \gamma_{\mathcal{N}}(t))$ is a corresponding length minimizing curve. However, there is no canonical Finsler strucutre on the product manifold. In fact, for any function $\Psi : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ such that

1. $\Psi(x_1, x_2) = 0$ if and only if $(x_1, x_2) = (0, 0)$,
2. $\Psi(\alpha x_1, \alpha x_2) = \alpha \Psi(x_1, x_2) $ for all $(x_1, x_2) \in \mathbb{R}_+ \times \mathbb{R}_+$ and all $\alpha \in \mathbb{R}_+$,
3. $\Psi$ is smooth on $\mathbb{R}_+ \times \mathbb{R}_+ \setminus \{ (0,0) \}$,
4. $\partial_{x_\ell} \Psi \neq 0$ on $\mathbb{R}_+ \times \mathbb{R}_+ \setminus \{ (0,0) \}$, for $\ell = 1, 2$,
5. $\partial_{x_1} \Psi \, \partial_{x_2} \Psi - 2 \Psi \partial_{x_1}\partial_{x_2} \Psi \neq 0$ on $\mathbb{R}_+ \times \mathbb{R}_+ \setminus \{ (0,0) \}$,

the function $$ f^o_{\mathcal{O}, \Psi} : (X_\mathcal{M}, X_\mathcal{N}) \mapsto \sqrt{\Psi\Big( f^m_{\mathcal{M}}(X_\mathcal{M})^2, f^n_{\mathcal{N}}(X_\mathcal{N})^2 \Big)} $$ is a Finsler structure on $\mathcal{O}$ with a length minimizing curve given by $\gamma_{\mathcal{O}}(t) = (\gamma_{\mathcal{M}}(t), \gamma_{\mathcal{N}}(t))$.