from now on assume that the cover $\{U_i\}$ is finite.

We say that ${\mathbb{E}}$ is a *$\theta$-Hölder vector bundle* if the coordinate transformations are locally $\theta$-Hölder. By compactness, we can reduce the coordinate neighborhoods so that the coordinate transformations become (uniformly) $\theta$-Hölder.

As mentioned in \[ss.intro\_extremal\], a *Finsler norm* is a continuous function $\| \mathord{\cdot} \|$ on ${\mathbb{E}}$ that restricts to a norm $\| \mathord{\cdot} \|_x$ on each fiber ${\mathbb{E}}_x$. A Finsler norm $\| \mathord{\cdot} \|$ is called *Riemannian* if each $\| \mathord{\cdot} \|_x$ is induced by an inner product $\langle \mathord{\cdot}, \mathord{\cdot} \rangle_x$. A Finsler norm $\| \mathord{\cdot} \|$ is called *$\theta$-Hölder* if for every $u \in {\mathbb{R}}^d$ and every coordinate neighborhood, the function $x \in U_i \mapsto \| h_i(x) u \| $ is $\theta$-Hölder. Every $\theta$-Hölder vector bundle ${\mathbb{E}}$ admits a $\theta$-Hölder Riemannian norm; the proof is straightforward using a $\theta$-Hölder partition of unity.

We will also need a way of “transporting” vectors from one fiber to another:

\[p.transport\] Let ${\mathbb{E}}$ be a $\theta$-Hölder vector bundle. There exists a family of linear maps $I_{y \gets x} \colon {\mathbb{E}}_x \to {\mathbb{E}}_y$ with the following properties:

1.  For every point $x \in X$, the linear map $I_{x \gets x}$ equals 