Curvature-dimension bounds for Lorentzian splitting theorems

Eric Woolgar, William Wylie

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)131-145
Number of pages15
JournalJournal of Geometry and Physics
Volume132
DOIs
StatePublished - Oct 1 2018

Fingerprint

theorems
Curvature
curvature
Theorem
rigidity
Rigidity
Null
Tensor
Lorentzian Geometry
tensors
Singularity
Warped Product
Space-time
Cover
products
geometry

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

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

In: Journal of Geometry and Physics, Vol. 132, 01.10.2018, p. 131-145.

Research output: Contribution to journalArticle

@article{913551566c5749ecb198403c46ceba08,
title = "Curvature-dimension bounds for Lorentzian splitting theorems",
abstract = "We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry–{\'E}mery–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–{\'E}mery–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-{\'E}mery 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.",
keywords = "Bakry–Emery–Ricci curvature, Lorentzian geometry, Singularity theorems, Splitting theorems",
author = "Eric Woolgar and William Wylie",
year = "2018",
month = "10",
day = "1",
doi = "10.1016/j.geomphys.2018.06.001",
language = "English (US)",
volume = "132",
pages = "131--145",
journal = "Journal of Geometry and Physics",
issn = "0393-0440",
publisher = "Elsevier",

}

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 -