Inner metric

In mathematics , the inner metric or length metric measures the lengths of minimal connecting paths between points.

definition

It is a metric space . The inner metric (or length metric ) to be associated is defined as ${\displaystyle (X,d)}$ ${\displaystyle d}$ ${\displaystyle d_{i}}$

${\displaystyle d_{i}(x,y)=\inf L(\sigma )}$

for , the infimum over all rectifiable curves with is taken and represented by ${\displaystyle x,y\in X}$ ${\displaystyle \sigma \colon \left[0,1\right]\to X}$ ${\displaystyle \sigma (0)=x,\sigma (1)=y}$ ${\displaystyle L(\sigma )}$

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

defined length of the curve . ${\displaystyle \sigma }$

Geodetic metric spaces

→ Main article : Geodetic metric space

A metric space is called a geodetic metric space (also length space or inner metric space ) if is, i.e. if the inner metric matches the metric . ${\displaystyle d=d_{i}}$ ${\displaystyle d}$

Examples

• Let it be a Riemannian manifold and the through ${\displaystyle (M,g)}$ ${\displaystyle d}$

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

for defined metric. If geodetically complete , then is . (See Hopf-Rinow's theorem .) ${\displaystyle x,y\in M}$ ${\displaystyle (M,g)}$ ${\displaystyle d_{i}=d}$

• It be , for and . The restriction of to defines a metric space . The associated inner metric is ${\displaystyle X=\mathbb {R} ^{n}}$ ${\displaystyle d(x,y)=|x-y|}$ ${\displaystyle x,y\in \mathbb {R} ^{n}}$ ${\displaystyle S^{n-1}=\left\{x\in \mathbb {R} ^{n}\colon d(x,0)=1\right\}}$ ${\displaystyle d}$ ${\displaystyle S^{n-1}}$ ${\displaystyle (S^{n-1},d{\big |}_{S^{n-1}})}$ ${\displaystyle S^{n-1}}$

${\displaystyle d_{i}(x,y)=\arccos(\langle x,y\rangle )>d(x,y)\ \forall x\not =y}$.

literature

• Bridson, Martin R .; Haefliger, André: Metric spaces of non-positive curvature. Basic Teachings of Mathematical Sciences, 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9
• A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature , IRMA Lectures in Mathematics and Theoretical Physics 6, European Mathematical Society 2005, 2nd ed. 2014.
• Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume I, 908 p., Springer International Publishing, 2018.
• Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume II, 842 p., Springer International Publishing, 2018.

Web links

• Urs Lang: Length spaces .