Curvature-dimension bounds for Lorentzian splitting theorems

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.