Curvature-dimension bounds for Lorentzian splitting theorems

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 $N\le 1$, 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 $N\in (-\infty ,2]\cup (n,\infty )$ and for the $N=\infty$ 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.

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References in corpus (3)

• Cosmological singularities in Bakry-Émery spacetimes
• On the geometry of Riemannian manifolds with density
• Cosmological singularity theorems and splitting theorems for N-Bakry-Emery spacetimes

Cited by in corpus (5)

• Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
• Gravity from thermodynamics: optimal transport and negative effective dimensions
• Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem
• Comparison theorems on weighted Finsler manifolds and spacetimes with $\epsilon$-range
• Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case