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A finite quotient of join in Alexandrov geometry. (English) Zbl 1459.53048

Alexandrov geometry was introduced by Yu. Burago et al. [Russ. Math. Surv. 47, No. 2, 1 (1992; Zbl 0802.53018); translation from Usp. Mat. Nauk 47, No. 2(284), 3–51 (1992)], and extensive study has been done since then. An Alexandrov space with curvature greater than or equal to \(k\) is a locally complete length metric space such that every geodesic triangle looks fatter than a corresponding triangle in the simply connected 2-space form of constant curvature greater than or equal to \(k\). Several authors studied the Alexandrov geometry [T. Frankel, Pacific J. Math. 11, 165–174 (1961; Zbl 0107.39002); S. Y. Cheng, Math. Z. 143, No. 3, 289–297 (1975; Zbl 0329.53035); D. Gromoll and K. Grove, Ann. Sci. École Norm. Sup. (4) 20, No. 2, 227–239 (1987; Zbl 0626.53032); K. Grove and S. Markvorsen, J. Am. Math. Soc. 8, No. 1, 1–28 (1995; Zbl 0829.53033); K. Grove and K. Shiohama, Ann. of Math. (2) 106, No. 2, 201–211 (1977; Zbl 0341.53029); G. Perelman, J. Differ. Geom. 40, No. 1, 209–212 (1994; Zbl 0818.53056); A. Petrunin, Geom. Funct. Anal. 8, No. 1, 123–148 (1998; Zbl 0903.53045)].
The principal objective in this paper is to explore a rigidity of a finite quotient of join in Alexandrov geometry, which is a necessary step toward a classification for Alexandrov spaces of curvature greater than or equal to 1 and diameter \(\frac{\pi}{2}\).

MSC:

53C20 Global Riemannian geometry, including pinching
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C24 Rigidity results
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References:

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