### 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) |
---|---|

Pages (from-to) | 131-145 |

Number of pages | 15 |

Journal | Journal of Geometry and Physics |

Volume | 132 |

DOIs | |

State | Published - Oct 1 2018 |

### Fingerprint

### 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

**Curvature-dimension bounds for Lorentzian splitting theorems.** / Woolgar, Eric; Wylie, William.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 132, pp. 131-145. https://doi.org/10.1016/j.geomphys.2018.06.001

}

TY - JOUR

T1 - Curvature-dimension bounds for Lorentzian splitting theorems

AU - Woolgar, Eric

AU - Wylie, William

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - Bakry–Emery–Ricci curvature

KW - Lorentzian geometry

KW - Singularity theorems

KW - Splitting theorems

UR - http://www.scopus.com/inward/record.url?scp=85048752828&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048752828&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2018.06.001

DO - 10.1016/j.geomphys.2018.06.001

M3 - Article

VL - 132

SP - 131

EP - 145

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -