In Riemannian geometry, we have
Proposition Any Manifold has a Riemannian metric.
However, we cannot place the proof on pseudo-Riemannian situation because we do not hold the signature on manifold. So my question is
Is there always a pseudo-Riemannian metric on manifold, which satisfies signature $(s,v)$?
If not, what condition on manifold keeps it having the pseudo-Riemannian metric?
Any advice is helpful. Thx.