# Finsler geometry

Let $\mathcal{M}$ be an $n$-dimensional differential manifold and let $\varphi :T\mathcal{M}\to \mathbb{R}$ be a function $\varphi (x,\xi )$ defined for $x\in \mathcal{M}$ and $\xi \in {T}_{x}\mathcal{M}$ such that $\varphi (x,\cdot )$ is a possibly non symmetric norm on ${T}_{x}\mathcal{M}$. The couple $(\mathcal{M},\varphi )$ is called a Finsler space.

Let us define the $\varphi $-length of curves^{} in $\mathcal{M}$. If $\gamma :[a,b]\to \mathcal{M}$ is a differentiable^{} curve we define

$${\mathrm{\ell}}_{\varphi}(\gamma ):={\int}_{a}^{b}\varphi ({\gamma}^{\prime}(t))\mathit{d}t.$$ |

So a natural geodesic distance can be defined on $\mathcal{M}$ which turns the Finsler space into a quasi-metric space (if $\mathcal{M}$ is connected^{}):

$${d}_{\varphi}(x,y):=inf\{{\mathrm{\ell}}_{\varphi}(\gamma ):\gamma \text{is a differentiable curve}\gamma :[a,b]\to \mathcal{M}\text{such that}\gamma (a)=x\text{and}\gamma (b)=y\}.$$ |

Notice that every Riemann manifold $(\mathcal{M},g)$ is also a Finsler space, the norm $\varphi (x,\cdot )$ being the norm induced by the scalar product $g(x)$.

A finite dimensional Banach space is another simple example of Finsler space, where $\varphi (x,\xi ):=\parallel \xi \parallel $. Wulff Theorem is one of the most important theorems in this ambient space.

Title | Finsler geometry^{} |
---|---|

Canonical name | FinslerGeometry |

Date of creation | 2013-03-22 15:03:37 |

Last modified on | 2013-03-22 15:03:37 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 8 |

Author | paolini (1187) |

Entry type | Definition |

Classification | msc 58B20 |

Classification | msc 53B40 |

Classification | msc 53C60 |

Related topic | WulffTheorem |