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Transactions of the American Mathematical Society

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A finite quotient of join in Alexandrov geometry


Authors: Xiaochun Rong and Yusheng Wang
Journal: Trans. Amer. Math. Soc. 374 (2021), 1095-1124
MSC (2020): Primary 53C20, 53C23, 53C24
DOI: https://doi.org/10.1090/tran/8194
Published electronically: November 18, 2020
MathSciNet review: 4196388
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Abstract: Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $\ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $\ge 1$, which contains $X_i$ as convex subsets such that their points are $\frac \pi 2$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called a quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $\ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $\frac \pi 2$ apart and $\dim (X_1)+\dim (X_2)=n-1$.


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Additional Information

Xiaochun Rong
Affiliation: Department of Mathematics, Capital Normal University, Beijing, People’s Republic of China; and Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
MR Author ID: 336377
Email: rong@math.rutgers.edu

Yusheng Wang
Affiliation: School of Mathematical Sciences (and Lab. math. Com. Sys.), Beijing Normal University, Beijing, 100875 People’s Republic of China
Email: wyusheng@bnu.edu.cn

Received by editor(s): November 17, 2018
Received by editor(s) in revised form: May 5, 2020
Published electronically: November 18, 2020
Additional Notes: The first author was supported in part by NSFC 11821101, BNSF Z19003, and a research fund from Capital Normal University
The second author was supported in part by NFSC 11971057 and BNSF Z190003.
Yusheng Wang is the corresponding author
Article copyright: © Copyright 2020 American Mathematical Society