Smooth Manifold admits Lorentzian Metric iff admits Rank-1 Tangent Distribution

Theorem
Let $M$ be a smooth manifold.

Then $M$ admits a Lorentzian metric $M$ admits a rank-$1$ tangent distribution.

That is, $M$ admits a Lorentzian metric $M$ admits a rank-$1$ subbundle of the tangent bundle $TM$.