Alexandrov geometry techniques for Finsler manifolds.

Question

Hi, first I would like to apologize for my English. It's not my native language and I feel my grasp of it is limited.

I've been reading Burago's book on metric geometry and I've that it mentions Finsler manifolds as an example of Alexandrov space. However it restricts to reversible Finsler metrics $F(x, \lambda y) = |\lambda|F(x,y).$

In general Finsler metrics only hold this:

$$F(x, \lambda y) = \lambda F(x,y) \quad \lambda > 0. $$

The problem is that since the goal of book is to study a metric induced like this:

$$d_{F}(x,y) = \inf_{\gamma}\int_{a}^{b}F(\gamma,\dot{\gamma})dt.$$

The resulting distance in the general case is not symmetric.My question is:

Does anyone know any approach that might be pursued to use similar techniques as those presented in the book to study Finsler manifolds in the general case?

Answer 1

As Anton mentioned in a comment, a non-Riemannian Finsler manifold cannot be an Alexandrov space. If you found an opposite statement in our book, I would appreciate the page number where it appears (I maintain a list of errors and misprints, there are way too many of them in the book).

Alexandrov spaces generalize Riemannian manifolds with lower or upper curvature bounds. I am not aware of any synthetic, metric space level definition that could work as a concept of a lower curvature bound (or even of nonnegative curvature) for Finsler metrics.

There is a generalization of nonpositive curvature, called "Busemann nonpositive curvature". It says that if $p$ and $q$ are midpoints of two segments $[xy]$ and $[xz]$, then $d(p,q)\le d(y,z)/2$. Equivalently, the distance between points on two geodesics is convex (as a function of two parameters). This is weaker than CAT(0). In the Riemannian case, this is equivalent to nonpositive sectional curvature, but, unlike Alexandrov's CAT(0) definition, this one has Finsler examples (e.g. any flat Finsler metric). The Busemann definition is not very popular for a number of reasons, but at least there is a globalization theorem (a la Cartan-Hadamard) and uniqueness of geodesics in homotopy classes.

As for non-reversible Finsler metrics, they are not very different from reversible ones. You just have to be careful with the order of arguments of a distance function, and then most things generalize in a straightforward way. Many geometers (myself included) often assume reversibility just because they are not comfortable with non-symmetric distances and think they are not interesting enough to worth the trouble.

Answer 2

There is a generalization of the Alexandrov curvature conditions in the dissertation `Construction of compatible Finsler structures on locally compact length spaces' of M.G. Knecht. He shows that a length space with this curvature condition has a differentiable structure and a Finsler function that generates the original lengths. The differentiable structure and the Finsler function are not $C^\infty$ though.