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

##### MSC:
 53C70 Direct methods ($$G$$-spaces of Busemann, etc.) 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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##### References:
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