General Relativity – Do All Metric Tensors Have Signature (1,3) or (3,1) in Spacetime?

Question

Do all metric tensors have signature (-,+,+,+) or (+,-,-,-) in the Pseudo-riemannian manifold describing spacetime in the Theory of General Relativity?

If yes:

In this answer by John Rennie, it is stated that:

Lorentzian manifolds are a special case of pseudo-Riemannian manifolds where the signature of the metric is (3,1) (or (1,3) depending on your sign convention).

Since Lorentzian manifolds ≡ signature (1,3), if the answer to my question is yes, it means that the General relativity spacetime is a 4D Lorentzian manifold.

Answer 1

Yes. In GR, spacetime is a 4D Lorentzian manifold