### Abstract

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry–Émery–Ricci tensor. We extend the Hawking–Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions N≤1, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when N=1. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry–Émery–Ricci tensor for all N∈(−∞,2]∪(n,∞) and for the N=∞ case as well, with reduced rigidity if N=2. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-Émery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension.

Original language | English (US) |
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Pages (from-to) | 131-145 |

Number of pages | 15 |

Journal | Journal of Geometry and Physics |

Volume | 132 |

DOIs | |

State | Published - Oct 1 2018 |

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### Keywords

- Bakry–Emery–Ricci curvature
- Lorentzian geometry
- Singularity theorems
- Splitting theorems

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology

### Cite this

*Journal of Geometry and Physics*,

*132*, 131-145. https://doi.org/10.1016/j.geomphys.2018.06.001