# Questions tagged [finsler-geometry]

Finsler geometry is a metric generalization of Riemannian geometry, where the general definition of the length of a vector is not necessarily given in the form of the square root of a quadratic form as in the Riemannian case.

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### What is a measurable set in a Finsler manifold?

Let $(M,\phi)$ be a Finsler manifold. The following definition is taken from Wikipedia. Let $A\subseteq M$ be a measurable set, then the $n$-dimensional Holmes-Thompson volume is defined by \begin{...
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### What is convex geometry and what is the relation with Finsler geometry?

Would you be so kind as to give me a general idea of convex geometry? What is it about? And what is the relation with Finsler geometry (if any)? Many thanks.
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### Distintion between Finsler metrics

The main feature in Finsler manifolds is that the metric is not neccesary given by an inner product in the tangent apaces to every point. Are there theorems that tell you exactly when the Finsler ...
45 views

### Time-minimizing trajectories in the presence of a strong wind on a Riemannian manifold?

I want to find the solution to the Zermelo's navigation problem in a specific case where I have a strong wind $W$. I would like to apply the geometric formalism explained here or also here on arXiv. ...
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### Possible dual $(\alpha,\beta)$ metric on Finsler space?

Let $M$ be a 2-dimensional manifold with local coordinate system $(x^i)$ and cotangent bundle $T^*\!M$ with induced coordinates $(x^i,y_i)$. Define $F^*$ to be a (dual) Finsler metric on the ...
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### How to construct a Finsler metric on a product manifold and which are the short curves?

Let $\mathcal{M}$ be a smooth manifold and let $f_{\mathcal{M}}^m$ be a Finsler structure on $\mathcal{M}$, i.e., let $f_{\mathcal{M}}^m : T_m\mathcal{M} \to \mathbb{R}$ be a metric that varies ...
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### What is the difference/relation between Lorentzian and Finsler geometry?

I'm kind of lost among many similar concepts. What is the difference/relation among Lorentzian manifold, Finsler manifold, Minkowsky manifold (and I just came across such a Randers space, related to ...
184 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 MathOverflow. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I was reading about geometry in ...
181 views

### Geodesics and a general pregeodesic equation

Let $(M,g)$ be a Riemannian manifold, and let $\nabla$ denote the Levi-Civita connection. Then we say a smooth curve $\gamma:J\to M, t\mapsto\gamma(t)$ is a geodesic if $$D_t\gamma'=0.$$ We say a ...
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### Geodesic on Finsler surface

Consider $S$ which is a two-dimensional surface with an induced metric from a normed space $(\mathbb{R}^3,\|\ \|)$. When $c$ is a curve of unit speed $c : [0,l]\rightarrow S$, i.e. $\| c'(t)\|=1$, s.t....
Problem : $(\mathbb{R}^n,\|\ \|)$ has a smooth and strictly convex norm. When $f(x)=\|x\|$, then find a directional derivative of a function $f$, i.e. $\frac{d}{dt}f(x+tv)$ for $\|x\|=1$. Refer :  ...