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Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature

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Abstract

We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.

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Acknowledgments

The first-named author was supported in part through Karsten Grove’s grant from the US National Science Foundation. The second-named author was supported in part by CONACYT Project #SEP–106923. The authors are grateful to Stephanie Alexander, Christine Escher, Karsten Grove, Vitali Kapovitch, Alexander Lytchak, Ricardo Mendes, Anton Petrunin, and Conrad Plaut for helpful conversations, as well as to Fernando Galaz-García for initial conversations with C. Searle from which this paper evolved and for his later comments. Some of the work in this article forms a part of J. Harvey’s doctoral dissertation, completed under the supervision of Karsten Grove at the University of Notre Dame. C. Searle is grateful to the Mathematics Department there for its hospitality during a visit where a part of this research was carried out.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    John Harvey

  2. Department of Mathematics, Statistics, and Physics, Wichita State University, 1845 Fairmount Street, Wichita, KS, 67260, USA

    Catherine Searle

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Harvey, J., Searle, C. Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature. J Geom Anal 27, 1636–1666 (2017). https://doi.org/10.1007/s12220-016-9734-7

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