Finsler manifold

In geometry , Finsler manifolds are a generalization of Riemannian manifolds .

They are named after Paul Finsler .

definition

A Finsler manifold is a differentiable manifold with a function that is smooth outside of the zero intersection, so that for all : ${\displaystyle M}$ ${\displaystyle F:TM\rightarrow \mathbb {R} }$ ${\displaystyle v,w\in T_{x}M,x\in M}$

• ${\displaystyle F(v)\geq 0}$ with equality only for ${\displaystyle v=0}$
• ${\displaystyle F(\lambda v)=\lambda F(v)}$ for all ${\displaystyle \lambda \geq 0}$
• ${\displaystyle F(v+w)\leq F(v)+F(w)}$.

Here the tangent space of the manifold in the point and the tangential bundle of, that is, the disjoint union of all tangent spaces. ${\displaystyle T_{x}M}$ ${\displaystyle M}$ ${\displaystyle x\in M}$ ${\displaystyle TM}$ ${\displaystyle M,}$

The Finsler manifold is called symmetric if holds for all . ${\displaystyle F(-v)=F(v)}$ ${\displaystyle v\in T_{x}M,x\in M}$

Examples

• Normalized vector spaces if the norm is smooth outside the zero vector.
• Riemannian manifolds : set . ${\displaystyle (M,g)}$ ${\displaystyle F(v)={\sqrt {g(v,v)}}}$
• Convex sets with the Hilbert metric : set for . ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle d_{\Omega }}$ ${\displaystyle F(v)={\frac {d}{dt}}\mid _{t=0}d_{\Omega }(x,x+tv)}$ ${\displaystyle v\in T_{x}\Omega ,x\in \Omega }$

Length and volume

The length of a rectifiable curve is defined by ${\displaystyle \gamma :\left[a,b\right]\rightarrow M}$

${\displaystyle L(\gamma )=\int _{a}^{b}F(\gamma ^{\prime }(t))dt}$.

The volume shape of a -dimensional Finsler manifold is defined as follows. Be , a base of , the dual base. Let be the Euclidean volume of . The volume shape is then given by ${\displaystyle n}$ ${\displaystyle x\in M}$ ${\displaystyle e_{1},\ldots ,e_{n}}$ ${\displaystyle T_{x}M}$ ${\displaystyle \eta _{1},\ldots ,\eta _{n}}$ ${\displaystyle V(x)}$ ${\displaystyle D(x)=\left\{y\in \mathbb {R} ^{n}:F(\sum _{i=1}^{n}y_{i}e_{i})\leq 1\right\}}$

${\displaystyle B_{F}(x)={\frac {C(n)}{V(x)}}\eta _{1}\wedge \ldots \wedge \eta _{n}}$,

where im denotes the Euclidean volume of the unit sphere . The Busemann volume of a measurable amount is defined by . ${\displaystyle C(n)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle A\subset M}$ ${\displaystyle \operatorname {vol} (A)=\int _{A}B_{F}(x)}$

literature

• D. Bao, S.-S. Chern, Z. Shen: An introduction to Riemann-Finsler geometry. (= Graduate Texts in Mathematics. 200). Springer-Verlag, New York 2000, ISBN 0-387-98948-X .
• Zhongmin Shen: Lectures on Finsler geometry. World Scientific Publishing, Singapore 2001, ISBN 981-02-4531-9 .