Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor

We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature Ric(f)(m) timelike directions. We prove that they still hold when in is finite, and when m is infinite, they hold under the additional assumption that f is bounded from above. (C) 2009 Elsevier B.V. All rights reserved.