8

In fact there are very many ways to provide a Finsler manifold with a "canonical" volume. Personally I've gone from thinking that this is a nuisance and trying to pin down which one is really the best to thinking that this is part of the landscape and should be accepted.
There is a very good notion of volume that goes by the name of "Holmes-Thompson" ...

5

Without more restrictions on the Finsler structure, the answer is 'no, there need not be any finite $n$ such that the $n$-th power of the distance function is smooth'.
For example, consider the Finsler structure on the $xy$-plane whose norm is
$$
F(x,y;\dot x, \dot y) = \sqrt{\dot x^2+\dot y^2}+\sqrt{\dot x^2 + 2\dot y^2}.
$$
The distance from $p=(0,0)$ ...

answered May 22 '17 at 15:00

Robert Bryant

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3

I am one of the authors of the reference in question. Perhaps there is a confusion between "geodesics" and "minimizing geodesics". A minimizing geodesic between $p$ and $q$ is unique, as Martin Kell explained. However, a geodesic (which is just locally minimizing) may be non-unique.
For example, define $\|\cdot\|_M$ on $\{-1\le y\le 1\}$ by
$$
\|v\|_M=(1-\...

3

There is a whole theory of Lorentz-Finsler spaces, and their casuality has been considerably developed in recent years. This fact alone shows that it is not compelling to work with round light cones (i.e. cones that intersected by a hyperplane not passing through the origin of $T_xM$ give an ellipsoid. Notice that the notion of ellipsoid is affine, so the ...

2

Busemann's 1955 book The Geometry of Geodesics is a great presentation of his approach. It includes almost all of the content of his other two papers mentioned in the post.
By contrast, Papadopoulos's book is only partly about Busemann's approach to metric geometry.
Busemann also wrote a 1970 book called "Recent Synthetic Differential Geometry" with more ...

2

I don't follow quite the argument in the reference but maybe the following helps:
On the one hand assuming strict convexity of the inner metric there can be at most one geodesic (see (1) below). On the other hand, if you assume $T$-invariance then whenever $\gamma$ is geodesic connecting $p$ and $q$ then also $T(\gamma)$ is a geodesic connecting $p$ and $q$...

2

Here's a possible approach that goes backwards, where we define the tubular neighborhood first and the normal bundle second.
Let $M$ be a Finsler manifold and $S \subset M$ a smooth submanifold. Given any $x \in M$, we can define the distance $d(x,S)$ from $x$ to $S$ to be the shortest length of curves from $x$ to $S$. We'll call a curve segment $S$-...

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