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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 27, Issue 5


Small covers, infra-solvmanifolds and curvature

Shintarô Kuroki
  • Osaka City University Advanced Mathematical Institute (OCAMI), 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan; and Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada
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/ Mikiya Masuda / Li Yu
Published Online: 2014-01-10 | DOI: https://doi.org/10.1515/forum-2013-0084


It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structures of the underlying simple polytopes.

Keywords: Small cover; real moment-angle manifold; real Bott manifold; Ricci curvature; infra-solvmanifold; infra-nilmanifold; Coxeter group

MSC: 57N16; 57S17; 57S25; 53C25; 51H30

About the article

Received: 2013-05-22

Revised: 2013-10-12

Published Online: 2014-01-10

Published in Print: 2015-09-01

Funding Source: JSPS

Award identifier / Grant number: Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation

Funding Source: Grant-in-Aid for Scientific Research

Award identifier / Grant number: 22540094

Funding Source: JSPS

Award identifier / Grant number: P10018

Funding Source: Natural Science Foundation of China

Award identifier / Grant number: 11001120

Funding Source: PAPD (priority academic program development) of Jiangsu higher education institutions

Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 2981–3004, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2013-0084.

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