×

zbMATH — the first resource for mathematics

A globalization for non-complete but geodesic spaces. (English) Zbl 1357.53090
The article shows that given a geodesic space \(X\) with the property that for any point \(x \in X\) there exists a neighborhood \(\Omega \ni x\) such that the \(\kappa\)-comparison holds for any quadruple of points in \(\Omega\), then the completion of \(X\) is an Alexandrov space with curvature \(\geq \kappa\). This answers a question asked by Viktor Schroeder around 2009.
Here an Alexandrov space with curvature \(\geq \kappa\) is a complete length space such that for any quadruple of points \((p; x^1; x^2; x^3)\) the \((1+3)\)-point comparison holds: \[ \measuredangle^\kappa(p^{x^1}_{x^2}) + \measuredangle^\kappa(p^{x^2}_{x^3}) + \measuredangle^\kappa(p^{x^3}_{x^1}) \leq 2 \pi. \]

MSC:
53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alexander, S., Bishop, R.: The Hadamard-Cartan theorem in locally convex spaces. l’Enseignement Math. 36, 309-320 (1990) · Zbl 0718.53055
[2] Alexander, S., Bishop, R.: Warped products admitting a curvature bound. arXiv:1509.00380 [math.DG] · Zbl 1352.53027
[3] Alexander, S., Kapovitch, V., Petrunin, A.: Alexandrov geometry. www.math.psu.edu/petrunin
[4] Burago, Y; Gromov, M; Perelman, G, A.D. Aleksandrov spaces with curvatures bounded below, Russ. Math. Surv., 47, 1-58, (1992) · Zbl 0822.20043
[5] Petrunin, A, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal., 8, 123-148, (1998) · Zbl 0903.53045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.