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

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

• Is this from a reference? Which page? May 20 '20 at 18:39
• @Qmechanic I'm reading Gravitation chapter 13 the metric, where the concept of local Lorentz came up and a simple (necessary but not sufficient) criterion. I got curious, and started thinking, I mean, could the geometry resolve one of the sign change in the euclidean norm by itself? May 20 '20 at 18:49
• @Qmechanic The concept first came up at track 1 page 20 box 1.3. May 20 '20 at 19:01

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