What's the difference between locally Lorentzian and locally Euclidean?

Question

What's the difference between locally Lorentzian and locally euclidean? Was the former (Lorentzian) the hyperbolic surface restriction of the latter (Euclidean)?

Answer 1

A pseudo-Riemannian manifold $(M,g)$ is locally Euclidean (Lorentzian) if the metric tensor $g$ has positive (Minkowski) signature, respectively.

NB: Concerning the use of the word Euclidean, see also my Phys.SE answer here.