Alexandrov room

Alexandrov spaces are metric spaces that are essential in differential geometry and topology . An Alexandrov space is a complete length space with a lower curvature bound and a finite Hausdorff dimension . They are named after Alexander Danilowitsch Alexandrov .

definition

A metric space is called length space, if the distance between two points in is given by the infimum of the lengths of all (continuous) curves that connect these points with each other. A shortest geodetic between two points is a curve parameterized according to the arc length from to , the length of which corresponds to the distance between these points. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle {\overline {xy}}}$ ${\displaystyle x,y\in X}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle |xy|}$

A triangle in a length space is determined by three points and three shortest geodetic tables . Referred to for a given real number , the symbol , the two-dimensional surface of constant curvature , as is understood by a comparison triangle for a triangle , a triangle in whose side lengths of the respective side lengths of the triangle match. Comparison triangles exist and are for or for and ${\displaystyle x,y,z}$ ${\displaystyle X}$ ${\displaystyle x,y,z\in X}$ ${\displaystyle {\overline {xy}},{\overline {xz}},{\overline {yz}}}$ ${\displaystyle \kappa }$ ${\displaystyle S_{\kappa }}$ ${\displaystyle \kappa }$ ${\displaystyle (\kappa -)}$ ${\displaystyle xyz\in X}$ ${\displaystyle {\tilde {x}}{\tilde {y}}{\tilde {z}}}$ ${\displaystyle S_{\kappa }}$ ${\displaystyle xyz}$ ${\displaystyle \kappa \leq 0}$ ${\displaystyle \kappa >0}$

${\displaystyle |xy|+|yz|+|xz|<{\frac {2\pi }{\sqrt {\kappa }}}}$

clearly determined except for congruence.

A length space is called a space with a lower curvature limit , or space with , if every point has a neighborhood , so that for every four points the comparison angles from in the corresponding comparison triangles into satisfy the following inequality: ${\displaystyle X}$ ${\displaystyle \kappa }$ ${\displaystyle K\geq \kappa }$ ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle a,b,c,d\in U_{x}}$ ${\displaystyle a}$ ${\displaystyle S_{\kappa }}$

${\displaystyle {\tilde {\measuredangle }}bac+{\tilde {\measuredangle }}cad+{\tilde {\measuredangle }}dab\leq 2\pi }$

If the length space is a one-dimensional manifold and , for reasons of consistency, it is additionally required that in this case the diameter does not exceed the value . In generalizing the theorems of Toponogov and Bonnet-Myers, the following applies : ${\displaystyle X}$ ${\displaystyle \kappa >0}$ ${\displaystyle {\frac {\pi }{\sqrt {\kappa }}}}$

The diameter of a complete room with is at most . ${\displaystyle K\geq \kappa >0}$ ${\displaystyle {\frac {\pi }{\sqrt {\kappa }}}}$

If one reverses the inequality sign in the above inequality, one obtains the definition of a space with an upper curvature bound . If a space is complete with and, then the above inequality applies globally, i.e. for any (different) points . ${\displaystyle K\leq \kappa }$ ${\displaystyle X}$ ${\displaystyle K\leq \kappa }$ ${\displaystyle a,b,c,d\in X}$

For locally compact spaces, the definition given above agrees with the usual distance comparison definition , according to which a locally compact length space is a space with a lower curvature limit , if every point has a neighborhood , so that for every triangle in and two points the distance equation ${\displaystyle K\leq \kappa }$ ${\displaystyle X}$ ${\displaystyle \kappa }$ ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle xyz}$ ${\displaystyle U_{x}}$ ${\displaystyle y_{0}\in {\overline {xy}},z_{0}\in {\overline {xz}}}$

${\displaystyle |y_{0}z_{0}|\leq |{\tilde {y_{0}}}{\tilde {z_{0}}}|}$

is fulfilled, where and denote the points and corresponding points in the comparison triangle corresponding to the triangle . ${\displaystyle {\tilde {y_{0}}}}$ ${\displaystyle {\tilde {z_{0}}}}$ ${\displaystyle y_{0}}$ ${\displaystyle z_{0}}$ ${\displaystyle xyz}$ ${\displaystyle \kappa }$

The first examples of spaces with are given by Riemannian manifolds with sectional curvature as well as quotients of spaces with generally metric and / or topological singularities on (?). ${\displaystyle K\leq \kappa }$ ${\displaystyle Sec\leq \kappa }$ ${\displaystyle K\leq \kappa }$

Rooms with a lower curvature barrier are often referred to synonymously as Alexandrov rooms. ${\displaystyle K\leq \kappa }$

(Definition quoted from, see also web link)

Special

Every point of an Alexandrov space has an open environment, which is homeomorphic to the tangential cone of this point. Furthermore: An Alexandrov space has a stratification into topological manifolds. The strata of the dimension consist of the points whose tangent cone is homeomorphic to the product of a cone with a Euclidean space of one dimension . ${\displaystyle l}$ ${\displaystyle \mathbb {R} ^{k}}$ ${\displaystyle k\leq l}$

literature

• Jonathan Alze: Hyperbolic Stretch Surgery. Diploma thesis 2002 ( http://www.mathematik.uni-muenchen.de/~jonathan/diplom/hdc_hyperref.pdf ( Memento from June 10, 2007 in the Internet Archive ) )
• Martin Weilandt: Isospectral Alexandrov Spaces. ( online )

Web links

• http://www.mis.mpg.de/preprints/ln/lecturenote-0900.pdf .

Individual evidence

1. ↑ Wilderich Tuschmann: Finiteness theorems and positive curvature Habilitation thesis Max Planck Institute for Mathematics, Leipzig 2000, pp. 18-19.