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 ...
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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 ...
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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 ...
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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....
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Dual element of supporting function

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 : [1] ...
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Isometric embedding of $\mathbb{E}^3$ into Finsler space

Consider the following paper : Euclidean rank for Finsler Space If $(\mathbb{R}^3,\|\ \|_i)$ are normed space, then there is an isometric embedding of $\mathbb{E}^3$ into $(\mathbb{R}^3,\|\ \|_1)\...
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strongly convex normed space

A space is said to be convex if the unit ball is a convex set. This definition gives the intuition about the convex normed space. Now I am wondering what would be the similar definition for a ...
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Randers space and a Finsler space [closed]

What is the difference between Randers spaces and Finsler spaces?
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Most common name for function F in Finsler geometry

In Finsler geometry, a Finsler manifold $M$ is a differentiable manifold, together with a function $F$ defined on the tangent bundle of $M$, that is an asymmetric norm on each tangent space. I have ...
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Sasaki Metric for Finsler Manifolds

Question: is there a "generalized" Sasaki metric for Finsler manifolds? More directly, let $(M,F)$ be a Finsler manifold with the Cartan connection. Is there a Finsler manifold $(TM,\hat{F})$ such ...
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Equivalence of a set of norms depending continuously on a parameter

Background: In the paper "Lusternik-Schnirelman Theory on Banach Manifolds" by Richard Palais, the author defines a Finsler structure (Definition 2.1) in a general way, where roughly speaking, given a ...
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Is a differentiable manifold with a metric a Finsler space?

Let $M$ be a differentiable manifold and $d$ a metric on $M$ such that $d:M\times M\rightarrow \mathbb{R}$ is $C^\infty$. Is there some way $d$ will induce on $M$ a Finsler norm?
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Why aren't norms used in the definition of Finsler manifolds?

In the definition of Finsler manifolds, each tangent space is endowed with an asymmetric norm, which is more general than a norm. (I think Spivak calls them "Minkowski metrics" in Volume 2.) Question:...
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Is this a sensible/valid definition for "isotropic metric space"?

For a metric space, $(X,d)$, define the following: for every $x \in X$, the local similitude group, $Sim(X, x)$, is the set of all surjective similitudes $X \to X$ which fix $x$. for every $x \in X$...
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Conditions for a Chern Connection

If $(M,F)$ is an $n$-dimensional Finsler manifold, $\{ e_i \} $ a local orthonormal frame for $\pi^*TM$, $\{ \omega^i \}$ the dual coframe, and $$ \begin{align} C_{ijk}(x,y) &:= \frac{1}{4}e_k(e_j(...
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Minkowski norm basic properties

Background Given a manifold $M$ and a smooth curve ${C}$ between $p,q \in M$ given by $c: [a,b] \to M$, we want to construct the properties for a generic candidate function $F:TM \to \mathbb{R}_{+}$ ...
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Euclidean + Taxicab Minkowski Space is Finsler metric?

Define the function $F$ on the elements $(x,v)$ of the tangent bundle of $\mathbb{R}^D$ by $$ F(x,v) \triangleq \|x-v\|_2 + \|v\|_1, $$ where $\|\cdot\|_p$ is the $p$-norm on $\mathbb{R}^D$. Does $F$...
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Any diffeomorphism between the Minkowski indicatrix and the Euclidean sphere?

Consider the Finsler Minkowski space $(R^n,F)$ and the Euclidean space $(R^n,||.||)$. Consider the Finsler Minkowski indicatrix of radius $r$, that is $$\Sigma(r)=\{x\in R^n\ :\ F(x)=r\}$$ FYI. The ...
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Is the unit bundle of a Finsler vector bundle a sphere bundle?

Note: By now, I have asked this question also at mathoverflow. Let $E$ be a Finsler vector bundle* of rank $k$ over a manifold $M$. Does the unit "bundle" $UE$ admit a structure of a sphere ...