A compact manifold admits a Lorentzian structure iff its Euler characteristic vanishes. Therefore, every noncompact manifold admits a Lorentzian structure.
Marcus's Theorem
See also
Euler Characteristic, Lorentzian StructureExplore with Wolfram|Alpha
References
Dodson, C. T. J. and Parker, P. E. "Marcus's Theorem." §9.5 in A User's Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 289-291, 1997.Referenced on Wolfram|Alpha
Marcus's TheoremCite this as:
Weisstein, Eric W. "Marcus's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MarcussTheorem.html