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For questions about Finsler geometry.

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4
votes
1answer
123 views

Are quadrics the cones of maximal symmetry?

A paper by Ehlers, Pirani, and Schild axiomatizes the geometry of general relativity in what seems like a nice way. However, Jacobson criticizes one aspect of the system as not natural: One deep ...
2
votes
0answers
81 views

Minkowski functional on infinite dimensional vector spaces

In finite dimensional Finsler geometry, we define Minkowski functional on tangent spaces that are finite dimensional vector spaces. The definition of Minkowski functional can be generalized to ...
7
votes
1answer
201 views

Reference request: Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach)

This question is a cross post from Math.SE. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this related question, ...
2
votes
2answers
94 views

Broken geodesic in Finsler polyhedral space

Here we assume that all norms has only one geodesic, i.e. locally minimizing, between any two points. Example : In $\mathbb{R}^2$, a line $y=kx,\ k>0$ divides $\mathbb{R}^2$ into two regions. We ...
1
vote
0answers
52 views

Second variation in saddle Finsler surface

Setting : Consider a two dimensional surface in $ (\mathbb{R}^n,\|\ \|)$. Here we define a function $f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ s.t. $L(v)(X)=\langle f(v),X\rangle$ where $\langle\ ,\...
3
votes
0answers
100 views

Existence of connections in a vector bundle whose parallel transport preserves a function on a total space

Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space ...
3
votes
0answers
71 views

How to find geodesics in a Randers spaces?

Consider a Randers space $(M,F)$ that is the solution of the zermelo's navigation problem associated to a wind $W$ which is homothety; $\mathcal{L}_Wh=\sigma h$, $\delta$ constant, on a Riemannian ...
1
vote
0answers
53 views

What happens if in Randers metric the norm of the wind is not less than 1

One way to define the Randers metric is using the data $(h,W)$ associated to the Zermelo problem. Here $h$ is the Riemannian metric and $W$ is the wind. In order to define the Randers metric we must ...
0
votes
0answers
96 views

Angle between two vectors in a Minkowski (Finsler) space!

Given a Minkowski (or Finsler) space $(V,F)$, I am wondering how to define the angle between two vectors $w$ and $v$. I first thought it must be as $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(...
6
votes
1answer
273 views

Volume form induced by a Finsler metric

I'm interested in knowing more about the volume form canonically induced by a Finsler metric. I've found some reasoning about it in this article http://www.ams.org/journals/bull/1950-56-01/S0002-9904-...
4
votes
0answers
40 views

Geodesics of non-smooth Finsler structure, or non-smooth Lagrange problem

I need to find the geodesics of a certain Finsler structure on $\mathbb R^n$. The structure is determined by quite nice $\ell^1$-like norms on tangent spaces, so that it is reversible. However the ...
2
votes
1answer
220 views

Existence of geometric Tubular Neighborhoods in Finsler spaces

I have not found any reference among the well-known books about the existence of a geometric tubular neighborhood in the Finsler spaces. I am wondering if there exists such a neighborhood for any ...
3
votes
1answer
81 views

Smoothness of some power of the geodesic distance in a Finsler geometry

I know that generally the geodesic distance $d_x$ from a point $x$ in a Finsler space is not smooth ($C^\infty$). According to Shen, the square of it is just $C^1$ at $x$. Now I am wondering if there ...
2
votes
0answers
126 views

Exponential Map for non-smooth Finsler manifolds

Context If I'm interested in studying reversible Finsler manifolds which do not have the strong convexity of the Hessian property (that is the Finsler function is a regular norm on every tangent ...