About: Alexandrov space

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In geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces. One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ k is either a non-negative integer or infinite. One can define a notion of "angle" and "tangent cone" in these spaces.

Attributes	Values
rdfs:label	Alexandrov-Raum (de) Alexandrov space (en) Александровская геометрия (ru) Александровська геометрія (uk)
rdfs:comment	Alexandrov-Räume sind metrische Räume, die in der Differentialgeometrie und in der Topologie von wesentlicher Bedeutung sind.Ein Alexandrov-Raum ist ein vollständiger Längenraum mit unterer Krümmungschranke und endlicher Hausdorff-Dimension. Sie sind nach Alexander Danilowitsch Alexandrow benannt. (de) Александровская геометрия — своеобразное развитие аксиоматического подхода в современной геометрии.Идея состоит в замене определённого равенства в аксиоматике евклидова пространства на неравенство. (ru) Александровська геометрія — своєрідний розвиток аксіоматичного підходу в сучасній геометрії.Ідея полягає в заміні певної рівності в аксіоматиці евклідового простору на нерівність. (uk) In geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces. One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ k is either a non-negative integer or infinite. One can define a notion of "angle" and "tangent cone" in these spaces. (en)
dcterms:subject	Differential geometry Riemannian manifolds Metric geometry
Wikipage page ID	14609741 (xsd:integer)
Wikipage revision ID	1067959814 (xsd:integer)
Link from a Wikipage to another Wikipage	Curvature of Riemannian manifolds Differential geometry Alexandrov topology Riemannian manifolds Geometry Complete metric space Hausdorff dimension Locally compact space Aleksandr Danilovich Aleksandrov Riemannian manifold Grigori Perelman Gromov's compactness theorem (geometry) Metric geometry Poincaré conjecture Mikhail Leonidovich Gromov Yuri Burago P. S. Alexandrov Gromov-Hausdorff metric Length space Geodesic triangle
sameAs	Alexandrov space Alexandrov space Alexandrov space Alexandrov space Alexandrov space Alexandrov space
dbp:wikiPageUsesTemplate	dbt:Reflist
has abstract	Alexandrov-Räume sind metrische Räume, die in der Differentialgeometrie und in der Topologie von wesentlicher Bedeutung sind.Ein Alexandrov-Raum ist ein vollständiger Längenraum mit unterer Krümmungschranke und endlicher Hausdorff-Dimension. Sie sind nach Alexander Danilowitsch Alexandrow benannt. (de) In geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces. One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ k is either a non-negative integer or infinite. One can define a notion of "angle" and "tangent cone" in these spaces. Alexandrov spaces with curvature ≥ k are important as they form the limits (in the Gromov-Hausdorff metric) of sequences of Riemannian manifolds with sectional curvature ≥ k, as described by Gromov's compactness theorem. Alexandrov spaces with curvature ≥ k were introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov in 1948 and should not be confused with Alexandrov-discrete spaces named after the Russian topologist Pavel Alexandrov. They were studied in detail by Burago, Gromov and Perelman in 1992 and were later used in Perelman's proof of the Poincaré conjecture. (en) Александровская геометрия — своеобразное развитие аксиоматического подхода в современной геометрии.Идея состоит в замене определённого равенства в аксиоматике евклидова пространства на неравенство. (ru) Александровська геометрія — своєрідний розвиток аксіоматичного підходу в сучасній геометрії.Ідея полягає в заміні певної рівності в аксіоматиці евклідового простору на нерівність. (uk)
prov:wasDerivedFrom	wikipedia-en:Alexandrov_space?oldid=1067959814&ns=0
page length (characters) of wiki page	2732 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Alexandrov_space
is Link from a Wikipage to another Wikipage of	Preissmann's theorem Alexandrov topology Tits metric Eugenio Calabi Glossary of Riemannian and metric geometry Grigori Perelman Collapsing manifold Metric space Mikhael Gromov (mathematician) Space of directions
is foaf:primaryTopic of	wikipedia-en:Alexandrov_space

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