Statements (47)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:characterizedBy |
convexity of distance function
triangle comparison property |
| gptkbp:defines |
A geodesic metric space where geodesic triangles are at least as thin as in Euclidean space
|
| gptkbp:example |
gptkb:Euclidean_space
gptkb:Hilbert_space gptkb:Hadamard_manifolds gptkb:finite-dimensional_real_hyperbolic_spaces gptkb:trees_(in_the_sense_of_metric_spaces) |
| gptkbp:field |
geometric group theory
metric geometry |
| https://www.w3.org/2000/01/rdf-schema#label |
CAT(0) spaces
|
| gptkbp:introducedIn |
1950s
|
| gptkbp:namedAfter |
gptkb:Élie_Cartan
gptkb:Alexandrov gptkb:Toponogov |
| gptkbp:property |
contractible
finite-dimensional CAT(0) spaces are locally geodesic finite-dimensional CAT(0) spaces are ANRs (absolute neighborhood retracts) Busemann convexity finite-dimensional CAT(0) spaces are locally uniquely geodesic contractible universal cover every closed ball is convex fixed point property for group actions by isometries finite-dimensional CAT(0) spaces are contractible finite-dimensional CAT(0) spaces are locally ANR midpoint map is 1-Lipschitz no local cut points finite-dimensional CAT(0) spaces are locally contractible simply connected if complete and locally CAT(0) non-positive curvature in the sense of comparison geometry unique geodesics between points uniqueness of geodesics between points finite-dimensional CAT(0) spaces are locally simply connected finite-dimensional CAT(0) spaces are geodesic metric spaces finite-dimensional CAT(0) spaces are uniquely geodesic finite-dimensional CAT(0) spaces are locally convex |
| gptkbp:relatedTo |
gptkb:Gromov_hyperbolic_spaces
gptkb:Hadamard_spaces non-positively curved spaces |
| gptkbp:usedIn |
gptkb:topology
fixed point theorems group actions rigidity theorems |
| gptkbp:bfsParent |
gptkb:Tits_metric
gptkb:CAT(k)_spaces |
| gptkbp:bfsLayer |
7
|