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

Theorem

Let $M$ be a smooth manifold.

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

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

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics