Curvature-dimension bounds for Lorentzian splitting theorems

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Emery-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-Emery-Ricci tensor for all N is an element of (-infinity, 2] boolean OR (n, infinity) and for the N = infinity case as well, with reduced rigidity if N,= 2. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-Emery 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. (C) 2018 Elsevier B.V. All rights reserved.