Three-manifolds with many flat planes
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- by Renato G. Bettiol and Benjamin Schmidt PDF
- Trans. Amer. Math. Soc. 370 (2018), 669-693 Request permission
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.References
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Additional Information
- Renato G. Bettiol
- Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
- MR Author ID: 903824
- ORCID: 0000-0003-0244-4484
- Email: rbettiol@math.upenn.edu
- Benjamin Schmidt
- Affiliation: Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, Michigan 48824
- MR Author ID: 803074
- Email: schmidt@math.msu.edu
- 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 - © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3717993