### 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.

Language | English (US) |
---|---|

Pages | 669-693 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2018 |

### Profile

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*370*(1), 669-693. DOI: 10.1090/tran/6961

**Three-manifolds with many flat planes.** / Bettiol, Renato G.; Schmidt, Benjamin.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 370, no. 1, pp. 669-693. DOI: 10.1090/tran/6961

}

TY - JOUR

T1 - Three-manifolds with many flat planes

AU - Bettiol,Renato G.

AU - Schmidt,Benjamin

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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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U2 - 10.1090/tran/6961

DO - 10.1090/tran/6961

M3 - Article

VL - 370

SP - 669

EP - 693

JO - Transactions of the American Mathematical Society

T2 - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -