This question is inspired by Schrödinger operators with some "nice" $n \times n$ positive definite symmetric a.e. matrix potential defined on $\mathbb{R}^d$ acting on $\mathbb{R}^n$ valued functions defined on $\mathbb{R}^d$

Suppose we have some $n \times n$ positive definite symmetric a.e. matrix function $M$ defined on $\mathbb{R}^d$ (intuitively, some sort of "Agmon function" on $\mathbb{R}^d$ related to our matrix potential). If we are lucky and have $n = d$ then $M$ canonically induces a metric on $\mathbb{R}^d$. Namely, just define \begin{equation} d(x, y) = \inf_\gamma \int_0^1 \langle M(\gamma(t)) \gamma '(t), \gamma'(t) \rangle_{ \mathbb{R}^d} ^\frac12 \, dt \tag{1}\label{Metric}\end{equation} where the infimum is over all absolutely continuous curves $\gamma : [0, 1] \rightarrow \mathbb{R}^d$ where $\gamma(0) = x$ and $\gamma(1) = y$.

If $n \neq d$ then there seems no natural way to do this while "preserving" the matrix structure of $M$. That is, I don't want to look at something like $$d(x, y) = \inf_\gamma \int_0^1 \|M(\gamma(t))\| |\gamma '(t)|_{\mathbb{R}^d} \, dt. $$

Surely $M$ can induce a bundle metric on some rank $n$ vector bundle, but it seems impossible to do this canonically in a way that reduces to \eqref{Metric} when $n = d$.

Are there any papers in the Schrödinger operator literature that touch on this? I can't imagine an issue like this hasn't come up somewhere in the world of mathematics...

This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some matrix potential $V : \mathbb{R}^d \rightarrow \mathbb{M}^{n \times n}$ that is symmetric and positive definite a.e. on $\mathbb{R}^d$.

Suppose we have some matrix function $M : \mathbb{R}^d \rightarrow \mathbb{M}^{n \times n}$ that is symmetric and positive definite a.e. on $\mathbb{R}^d$ (intuitively, some sort of "Agmon function" on $\mathbb{R}^d$ related to our matrix potential). If we are lucky and have $n = d$ then $M$ canonically induces a metric on $\mathbb{R}^d$. Namely, just define \begin{equation} d(x, y) = \inf_\gamma \int_0^1 \langle M(\gamma(t)) \gamma '(t), \gamma'(t) \rangle_{ \mathbb{R}^d} ^\frac12 \, dt \tag{1}\label{Metric}\end{equation} where the infimum is over all absolutely continuous curves $\gamma : [0, 1] \rightarrow \mathbb{R}^d$ where $\gamma(0) = x$ and $\gamma(1) = y$.

If $n \neq d$ then there seems no natural way to do this while "preserving" the matrix structure of $M$. That is, I don't want to look at something like $$d(x, y) = \inf_\gamma \int_0^1 \|M(\gamma(t))\| |\gamma '(t)|_{\mathbb{R}^d} \, dt. $$

Surely $M$ can induce a bundle metric on some rank $n$ vector bundle, but it seems impossible to do this canonically in a way that reduces to \eqref{Metric} when $n = d$.

Are there any papers in the Schrödinger operator literature that touch on this? I can't imagine an issue like this hasn't come up somewhere before.