What's the difference between locally Lorentzian and locally euclidean? Was the former (Lorentzian) the hyperbolic surface restriction of the latter (Euclidean)?
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asked May 20, 2020 at 18:33
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$\begingroup$ Is this from a reference? Which page? $\endgroup$– Qmechanic ♦Commented May 20, 2020 at 18:39
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$\begingroup$ @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? $\endgroup$– ShoutOutAndCalculateCommented May 20, 2020 at 18:49
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$\begingroup$ @Qmechanic The concept first came up at track 1 page 20 box 1.3. $\endgroup$– ShoutOutAndCalculateCommented May 20, 2020 at 19:01
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1 Answer
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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.
answered May 20, 2020 at 19:05