Questions tagged [finsler]

For questions about Finsler geometry.

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2
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1answer
189 views

Completeness on the tangent bundle

I was wondering if geodesics are defined for all time on compact Finsler manifolds, or more generally, for any spray on a compact manifold (where by geodesics, I simply mean the integral curves of the ...
8
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1answer
88 views

References on “not-quite” Finsler geometry?

In typical studies of Finsler geometry, the metric function $F: TM \to [0,\infty)$ is assumed to be smooth away from the zero section, and $F$ is assume to be sufficiently convex. Under these ...
3
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0answers
89 views

A non-Finsler metric on $\mathbb{R}^2$

I am looking for a inner metric on $\mathbb{R}^2$ (that induces the standard topology) which is not Finsler. By "Finsler" here I mean a metric that is obtained by the following construction: ...
4
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0answers
50 views

Divergence as infinitesimal volume change on a Finsler manifold

Let $M$ be a smooth manifold and $Z$ a smooth vector field on it. It generates a family of diffeomorphisms $\phi_t:M\to M$ by demanding that $\phi_0=\operatorname{id}$ and $\partial_t\phi_t(x)=Z(\...
4
votes
1answer
148 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
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0answers
91 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
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1answer
261 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
105 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
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0answers
53 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
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0answers
128 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
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0answers
77 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
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0answers
56 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
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0answers
123 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(...
8
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1answer
357 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
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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
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1answer
236 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
85 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
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0answers
161 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 ...
10
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1answer
509 views

The Finslerian version of the Nash embedding theorem

Is it true to say that every Finslerian manifold can be isometrically embedded in some $M_{n}(\mathbb{R})$ with operator norm? Note that every Riemannian manifold can be embedded in some matrix ...