Questions tagged [finsler]
For questions about Finsler geometry.
19
questions
2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
vote
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
votes
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
votes
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
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
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
votes
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
votes
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 ...
