### Abstract

We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form CD(0, 1). Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is CD(0, 1), we show that the fundamental group of M is the fundamental group of a compact manifold with non-negative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is CD(0, 1).

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

Pages | 6661-6681 |

Number of pages | 21 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 9 |

DOIs | |

State | Published - 2017 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**A warped product version of the cheeger-gromoll splitting theorem.** / Wylie, William.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 369, no. 9, pp. 6661-6681. https://doi.org/10.1090/tran/7003

}

TY - JOUR

T1 - A warped product version of the cheeger-gromoll splitting theorem

AU - Wylie, William

PY - 2017

Y1 - 2017

N2 - We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form CD(0, 1). Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is CD(0, 1), we show that the fundamental group of M is the fundamental group of a compact manifold with non-negative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is CD(0, 1).

AB - We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form CD(0, 1). Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is CD(0, 1), we show that the fundamental group of M is the fundamental group of a compact manifold with non-negative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is CD(0, 1).

UR - http://www.scopus.com/inward/record.url?scp=85020471589&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020471589&partnerID=8YFLogxK

U2 - 10.1090/tran/7003

DO - 10.1090/tran/7003

M3 - Article

VL - 369

SP - 6661

EP - 6681

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 - 9

ER -