A proof of the splitting conjecture of S.-T. Yau

Limit curve theorems in Lorentzian geometry

“…14 Point ͑3͒ focuses on a very general case in which the limit points given in the beginning can be infinite but numerable. This case is very useful in applications, especially when it comes to prove the connectedness of spacetime through maximizing geodesics or similar results.…”

A Generalization of the Lorentzian Splitting Theorem

“…Much developments have been made to resolve this conjecture (cf. [4], [5]) and Newman [7] at last established a complete proof of the Lorentzian splitting theorem conjectured by Yau using the Lorentzian Busemann functions.…”

“…Thus the following theorem is a generalization of the Lorentzian splitting theorem in [7] (or in [9] …”

Null Geometry and the Einstein Equations

“…(Note that a complete Riemannian manifold with strictly positive Ricci curvature cannot contain any lines.) The standard Lorentzian splitting theorem [10,15,28], which is an exact Lorentzian analogue of the Cheeger-Gromoll splitting theorem, describes the rigidity of spacetimes obeying the strong energy condition, Ric (X, X) ≥ 0 for all timelike vectors X, which contain a timelike line. Yau [34] posed the problem of establishing a Lorentzian analogue of the Cheeger-Gromoll splitting theorem as an approach to establishing the rigidity of the Hawking-Penrose singularity theorems; see [3] for a more detailed discussion of these matters, as well as a nice presentation of the proof of the Lorentzian splitting theorem.…”