Cohomogeneity one Alexandrov spaces in low dimensions
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Cohomogeneity one Alexandrov spaces in low dimensions Fernando Galaz‑García1,2 · Masoumeh Zarei1,3,4,5 Received: 15 January 2020 / Accepted: 5 May 2020 / Published online: 7 July 2020 © The Author(s) 2020
Abstract Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions 5, 6 and 7. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions at most 7. Keywords Cohomogeneity one · Alexandrov space · Orbifold Mathematics Subject Classification 53C23 · 57S10
F. Galaz-García and M. Zarei: Supported by the DFG Grant GA 2050/2-1 within the Priority Program SPP2026 “Geometry at Infinity.” F. Galaz-García: Supported by the RTG 2229 “Asymptotic Invariants and Limits of Groups and Spaces” at KIT/Universität Heidelberg. M. Zarei: Supported by the DFG Grant AM 342/4-1. * Masoumeh Zarei [email protected]‑augsburg.de Fernando Galaz‑García fernando.galaz‑[email protected] 1
Institut für Algebra und Geometrie, Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany
2
Present Address: Department of Mathematical Sciences, Durham University, Durham, UK
3
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
4
Beijing International Center for Mathematical Research, Peking University, Beijing, China
5
Present Address: Institut für Mathematik, Universität Augsburg, Augsburg, Germany
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Vol.:(0123456789)
110
Annals of Global Analysis and Geometry (2020) 58:109–146
1 Introduction Alexandrov spaces (with curvature bounded from below) are complete length spaces with a lower curvature bound in the triangle comparison sense; they generalize Riemannian manifolds with a uniform lower sectional curvature bound. Instances of Alexandrov spaces include Riemannian orbifolds (with a lower sectional curvature bound), orbit spaces of isometric actions of compact Lie groups on Riemannian manifolds with sectional curvature bounded below, or Gromov–Hausdorff limits of sequences of n-dimensional Riemannian manifolds with a uniform lower bound on the sectional curvature. The classification of spaces with compact Lie group actions is a central problem in the theory of transformation groups. In this context, a space with an effective action of a compact Lie group is of cohomogeneity one if its orbit space is one-dimensional. In the topological and smooth categories, the geometry and topology of cohomogeneityone manifolds have bee
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