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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: \[ \measuredangle^\kappa(p^{x^1}_{x^2}) + \measuredangle^\kappa(p^{x^2}_{x^3}) + \measuredangle^\kappa(p^{x^3}_{x^1}) \leq 2 \pi. \]

53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
Full Text: DOI arXiv
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[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
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[5] Petrunin, A, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal., 8, 123-148, (1998) · Zbl 0903.53045
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