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Three-manifolds with many flat planes


Authors: Renato G. Bettiol and Benjamin Schmidt
Journal: Trans. Amer. Math. Soc. 370 (2018), 669-693
MSC (2010): Primary 53B21, 53C20, 53C21, 53C24, 58A07, 58J60
DOI: https://doi.org/10.1090/tran/6961
Published electronically: September 15, 2017
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Abstract: We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian $ 3$-manifold. We prove a rank rigidity theorem for complete $ 3$-manifolds, showing that having higher rank is equivalent to having reducible universal covering. We also study $ 3$-manifolds such that every tangent vector is contained in a flat plane, including examples with irreducible universal covering, and discuss the effect of finite volume and real-analyticity assumptions.


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Additional Information

Renato G. Bettiol
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: rbettiol@math.upenn.edu

Benjamin Schmidt
Affiliation: Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, Michigan 48824
Email: schmidt@math.msu.edu

DOI: https://doi.org/10.1090/tran/6961
Received by editor(s): November 24, 2015
Received by editor(s) in revised form: April 18, 2016, and April 21, 2016
Published electronically: September 15, 2017
Additional Notes: The first-named author was partially supported by the NSF grant DMS-1209387
The second-named author was partially supported by the NSF grant DMS-1207655
Article copyright: © Copyright 2017 American Mathematical Society

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