About: CAT(k) space

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In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature . In a space, the curvature is bounded from above by . A notable special case is ; complete spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.

Attributes	Values
rdfs:label	CAT(k) space (en) Espace de Cartan-Alexandrov-Toponogov (fr) CAT(κ) 공간 (ko)
rdfs:comment	Les espaces de Cartan-Alexandrov-Toponogov ou espaces CAT(k) sont utilisés en géométrie. Ils permettent de définir dans le cadre des espaces métriques une notion de courbure qui relève traditionnellement de la géométrie riemannienne, par le truchement de relations de comparaison dans les triangles géodésiques. Le paramètre k est un réel qui permet de quantifier cette comparaison : on peut ainsi dire de certains espaces métriques qu'ils forment un espace CAT(k) pour un réel k donné. Les espaces CAT ont été dénommés ainsi par le géomètre Mikhail Gromov pour honorer les mathématiciens Élie Cartan, Alexandre Alexandrov et (en). (fr) 기하학에서 CAT(κ) 공간(-空間, 영어: CAT(κ) space)은 단면 곡률이 어디서나 이하인 거리 공간이다. (ko) In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature . In a space, the curvature is bounded from above by . A notable special case is ; complete spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard. (en)
foaf:depiction
dct:subject	Metric geometry
Wikipage page ID	12941912 (xsd:integer)
Wikipage revision ID	1117806206 (xsd:integer)
Link from a Wikipage to another Wikipage	Arc length Curve Victor Andreevich Toponogov Trivial group Complete space Mathematics Geodesic convexity Geometric group theory Neighbourhood (mathematics) François Bruhat Constant curvature Contractible space Convex function Sturm–Liouville theory Comparison triangle Élie Cartan Perimeter Unit sphere Mathematician Hadamard manifold Hadamard space Acronym Aleksandr Danilovich Aleksandrov Euclidean space France Simply connected space Riemannian manifold Interval (mathematics) Jacques Hadamard Jacques Tits Hyperbolic space Metric geometry Surface (mathematics) Homotopy group Triangle Diameter Punctured plane Inner product space Metric space Cartan–Hadamard theorem Real number Wronskian Non-positive curvature Normed vector space Bruhat–Tits building Simply connected Open ball Mikhail Gromov (mathematician) Geodesic metric space Geodesic triangle Homotopy type
Link from a Wikipage to an external page	http://www.math.uiuc.edu/~sba/the-defs-CBA.pdf http://www.math.upenn.edu/grad/dissertations/HindawiThesis.pdf
sameAs	CAT(k) space CAT(k) space CAT(k) space CAT(k) space CAT(k) space CAT(k) space
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thumbnail	wiki-commons:Special:FilePath/End_of_universe.jpg?width=300
has abstract	In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature . In a space, the curvature is bounded from above by . A notable special case is ; complete spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard. Originally, Aleksandrov called these spaces “ domain”.The terminology was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications). (en) Les espaces de Cartan-Alexandrov-Toponogov ou espaces CAT(k) sont utilisés en géométrie. Ils permettent de définir dans le cadre des espaces métriques une notion de courbure qui relève traditionnellement de la géométrie riemannienne, par le truchement de relations de comparaison dans les triangles géodésiques. Le paramètre k est un réel qui permet de quantifier cette comparaison : on peut ainsi dire de certains espaces métriques qu'ils forment un espace CAT(k) pour un réel k donné. Les espaces CAT ont été dénommés ainsi par le géomètre Mikhail Gromov pour honorer les mathématiciens Élie Cartan, Alexandre Alexandrov et (en). (fr) 기하학에서 CAT(κ) 공간(-空間, 영어: CAT(κ) space)은 단면 곡률이 어디서나 이하인 거리 공간이다. (ko)
prov:wasDerivedFrom	wikipedia-en:CAT(k)_space?oldid=1117806206&ns=0
page length (characters) of wiki page	13338 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:CAT(k)_space
is Link from a Wikipage to another Wikipage of	CAT(0) CAT(0) inequality CAT(0) space CAT(k) inequality CAT space Hyperbolic group Reshetnyak gluing theorem Curved space Victor Andreevich Toponogov Ptolemy's inequality Geodesic bicombing Graph of groups

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