Ingeometry,Alexandrov spaces with curvature ≥k form a generalization ofRiemannian manifolds withsectional curvature ≥k, wherek is some real number. By definition, these spaces arelocally compactcompletelength spaces where the lower curvature bound is defined via comparison ofgeodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces.[1][2]
One can show that theHausdorff dimension of an Alexandrov space with curvature ≥k is either a non-negative integer or infinite.[1] One can define a notion of "angle" (seeComparison triangle#Alexandrov angles) and "tangent cone" in these spaces.
Alexandrov spaces with curvature ≥k are important as they form the limits (in theGromov–Hausdorff metric) of sequences of Riemannian manifolds with sectional curvature ≥k,[3] as described byGromov's compactness theorem.
Alexandrov spaces with curvature ≥k were introduced by the Russian mathematicianAleksandr Danilovich Aleksandrov in 1948[3] and should not be confused withAlexandrov-discrete spaces named after the Russian topologistPavel Alexandrov. They were studied in detail byBurago,Gromov andPerelman in 1992[4] and were later used in Perelman's proof of thePoincaré conjecture.
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