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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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Can two geodesics meet tangentially on a Finsler manifold?

On a Riemannian manifold, it is well-known that two distinct non-closed geodesics can not meet tangentially, that is, if they meet at time $t_0$ and $t_1$, then they will meet at different angles. The proof uses the symmetric property of the…
Sachchidanand Prasad
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Notion of `normal bundle' of a submanifold in a Finsler manifold

Given a submanifold $N$ in a Riemannian manifold $(M, g)$, we have the notion of normal bundle $$\nu(N) := \{(p,v) \in TM \; | \; p \in N, \, g(v, w) = 0 \, \forall w \in T_p N\}.$$ It is well-known that a distance minimimizing geodesic from $N$ has…
ChesterX
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Deriving Finsler geodesic equations from the energy functional

I'm struggling to derive the Finsler geodesic equations. The books I know either skip the computation or use the length functional directly. I want to use the energy. Let $(M,F)$ be a Finsler manifold and consider the energy functional $$E[\gamma] =…
Ivo Terek
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Coordinate-free proof that dual Minkowski norm is indeed a Minkowski norm on the dual space

Context: let's say that a Minkowski norm on a vector space $V$ is a map $F\colon V\to \Bbb R_{\geq 0}$ such that $F$ is smooth on $V\setminus \{0\}$, $F$ is positive-homogeneous of degree $1$, and for every $x\in V\setminus \{0\}$, the symmetric…
Ivo Terek
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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 metric spaces from different books, two of them are:…
Dante Grevino
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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 $(\mathbb{R}^3,\|\ \|_1)\times (\mathbb{R}^3,\|\ \|_2)$ where…
HK Lee
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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 $F$ define a Finsler function (in the sense that…
AB_IM
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Proving that Randers norm is a Minkowski norm

I am struggling to follow the proof that the Randers norm is a Minkowski norm from "Lectures on Finsler Geometry" by Zhongmin Shen. A Minkowski norm on finite dimensional vector space $V$ is a function $F:V\to[0,\infty)$ which has the following…
Mithrandir
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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 smooth curve $\hat{\gamma}:I\to M,…
Matt
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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 that if $(M,F)$ is actually Riemannian, then…
user3658307
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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?
user2846
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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 metric is actually given by a inner product in every…
Giovanny Soto
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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. However, I am not a mathematician (I'm a…
lorenzop
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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 topological space $\mathcal{B}$ and a vector…
ttb
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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(e_i(F^2(x,y)))), \\ g_{ij} (x,y) &:= \frac{1}{2} e_j…
Andrew Whelan
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