### 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).

Original language | English (US) |
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Pages (from-to) | 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