Geodetic metric space

The geodetic metric space is a term from mathematics. It describes spaces in which one can find a shortest connecting curve for every two points. The term generalizes the concept of complete Riemannian manifolds to general metric spaces. The terms length space or inner metric space can also be found in the literature .

Geodesics in metric spaces

Be a metric space . A path is a continuous map with a closed interval im . The length of the image curve is defined as ${\displaystyle (X,d)}$ ${\displaystyle \gamma :\left[a,b\right]\rightarrow X}$ ${\displaystyle \left[a,b\right]}$ ${\displaystyle \mathbb {R} ^{1}}$

${\displaystyle L(\gamma )=\sup \left\{\left.\sum _{i=1}^{r}d{\big (}\gamma (t_{i-1}),\gamma (t_{i}){\big )}\right|a=t_{0}<t_{1}<\cdots <t_{r}=b,r\in \mathbb {N} \right\}}$.

The inequality follows from the triangle inequality . The way is called minimizing geodesic if equality ${\displaystyle L(\gamma )\geq d(\gamma (a),\gamma (b))}$ ${\displaystyle \gamma :\left[a,b\right]\rightarrow X}$

${\displaystyle L(\gamma )=d(\gamma (a),\gamma (b))}$

applies.

definition

A metric space is geodesic if every two points a minimizing geodesic with ${\displaystyle (X,d)}$ ${\displaystyle x,y\in X}$ ${\displaystyle \gamma :\left[a,b\right]\rightarrow X}$

${\displaystyle \gamma (a)=x,\gamma (b)=y}$

gives.

Examples of non-geodetic metric spaces

Be

${\displaystyle X=\mathbb {C} ^{*}=\mathbb {C} -\left\{0\right\}}$

the dotted complex plane with the metric

${\displaystyle d(x,y)=|x-y|}$

for . This space is connected to the path , so two points can be connected by at least one curve. ${\displaystyle x,y\in X}$

Then for example or , in both cases the pairs of points cannot be connected by curves of length 2. ${\displaystyle d(1,-1)=2}$ ${\displaystyle d(i,-i)=2}$

More generally, it follows from Hopf-Rinow's theorem that a connected Riemannian manifold is a geodetic metric space if and only if all geodesics can be completely continued. ${\displaystyle \mathbb {R} }$

Hopf-Rinow's theorem

→ Main article : Hopf-Rinow theorem

A metric is defined for a Riemann manifold by ${\displaystyle (M,g)}$ ${\displaystyle d:M\times M\rightarrow \mathbb {R} }$

${\displaystyle d(x,y):=\inf\{L(\gamma )\mid \gamma \colon [0,1]\to M,\gamma (0)=x,\gamma (1)=y\}}$

for . It runs through all the piece-wise differentiable paths that connect and , and denotes the Riemann length of , according to ${\displaystyle x,y\in M}$ ${\displaystyle \gamma }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle L(\gamma )}$ ${\displaystyle \gamma }$

${\displaystyle L(\gamma )=\int _{0}^{1}\!{\sqrt {g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,\mathrm {d} t}$

is defined. The Riemann manifold thus becomes a metric space . ${\displaystyle (M,d)}$

From the Hopf-Rinow theorem it follows:

• ${\displaystyle (M,d)}$ is a geodetic metric space

if and only if one of the following equivalent conditions is met:

• the Riemannian manifold is geodetically complete , ${\displaystyle (M,g)}$
• there exists such that the exponential mapping is defined for all , ${\displaystyle p\in M,}$ ${\displaystyle \exp _{p}}$ ${\displaystyle v\in T_{p}M}$
• the metric space is complete as metric space . ${\displaystyle (M,d)}$

literature

• Bridson-Haefliger: Metric spaces of nonpositive curvature (PDF; 4 MB)