Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above

Alexander, Stephanie B.; Bishop, Richard L.

doi:10.1016/0926-2245(96)00008-3

Cited by 33 publications

(58 citation statements)

References 11 publications

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“…In CAT (K) spaces, curves with extrinsic curvature ≤ A have the globalization property, namely, they satisfy sharp global arc/chord bounds [AB1].…”

Section: ) (See Remark 32)mentioning

confidence: 99%

“…The base angle at p of a chord γ of ∂X at an endpoint p is the angle formed by the direction of γ at p and Σ p ∂X. The boundary ∂X has extrinsic curvature ≥ A > 0 in the base-angle sense at p if the base angles α at p of chords of length r from p satisfy (for further discussion, see [AB1]). Thus the following condition is equivalent to (4.1): the base angles of chords from p satisfy…”

Section: Curved Boundariesmentioning

confidence: 99%

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Extrinsic curvature of semiconvex subspaces in Alexandrov geometry

Alexander

Bishop

2009

Ann Glob Anal Geom

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Abstract. In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman's concavity theorem for distance from the boundary, deriving corollaries on sharp diameter bounds, contractibility, and rigidity.

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“…In CAT (K) spaces, curves with extrinsic curvature ≤ A have the globalization property, namely, they satisfy sharp global arc/chord bounds [AB1].…”

Section: ) (See Remark 32)mentioning

confidence: 99%

Section: Curved Boundariesmentioning

confidence: 99%

Extrinsic curvature of semiconvex subspaces in Alexandrov geometry

Alexander

Bishop

2009

Ann Glob Anal Geom

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“…The classical Jung Theorem [8,Theorem 2.6] states in essence that the diameter D of a compact set X in E n satisfies D ≥ R (2n + 2)/n, (1) where R is the circumradius of X. The theorem was extended recently in two directions.…”

Section: Introductionmentioning

confidence: 99%

The Jung Theorem in metric spaces of curvature bounded above

Dekster

1997

Proc. Amer. Math. Soc.

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Abstract. The classical Jung Theorem states in essence that the diameter D of a compact set X in E n satisfies D ≥ R[(2n + 2)/n] 1/2 where R is the circumradius of X. The theorem was extended recently to the hyperbolic and the spherical n-spaces. Here, the estimate above is extended to a class of metric spaces of curvature ≤ K introduced by A. D. Alexandrov. The class includes the Riemannian spaces. The extended estimate is of the form D ≥ f (R, K, n) where n is a positive integer suitably defined for the set X and its circumcenter. It can be that n is not unique or does not exist. In the latter case, no estimate is derived. In case of a Riemannian d-dimensional space, an integer n always exists and satisfies n ≤ d.

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“…In contrast, Haantjes curvature has received in recent literature far lesser attention, with a very few -but notable -exceptions, e.g. [1], where actualized, more elaborate versions of both Menger and Haantjes curvature are discussed in a contemporary framework. On the other hand, Haantjes curvature seems singularly well adapted for a multitude of applicative tasks -see [29], [30], [2], [28] (and perhaps even to applications in such fields as Geometric Group Theory).…”

Section: Introductionmentioning

confidence: 99%