User talk:Svennik/sandbox

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So far in the references authors’ names, dates and pages are given but not the book title or isbn.

See Non-Euclidean geometry#Kinematic geometries for the spacetime situation, where the submanifold one moment into the future corresponds to the hyperboloid model. A new article CK geometry may not be justified as an extension of CK metric.

Generally, the presentation via the conic-absolute and projectivities is correct and quickly stated. Background is missing with over-reliance on certain links to significant topics. The comment is made “these examples are not exhaustive” with the M 1+1 thrown in, but rejected as non-metrizable. In projective geometry, elliptic, parabolic, and hyperbolic are exhaustive classes.

As the real projective plane is the platform, more development would be expected, such as the notion of quadric used for the absolute. Naming the circular points at infinity would be expected for a prepared approach to Euclidean geometry in the context of the real projective plane.Rgdboer (talk) 04:58, 31 December 2021 (UTC)[reply]

The objective is to replace the Cayley-Klein geometry article, because it's incomprehensible. I think you misunderstood me when you said "A new article CK geometry may not be justified as an extension of CK metric". The intention is to scrap the original article for being a poor summary and being incomprehensible. --Svennik (talk) 09:25, 31 December 2021 (UTC)[reply]

I'm in the middle of learning this topic, so there may be all sorts of mistakes. When you say "In projective geometry, elliptic, parabolic, and hyperbolic are exhaustive classes", this is intriguing because the absolute should correspond to the quadratic forms

\{x^{2}+y^{2}+z^{2},x^{2}+y^{2},x^{2}+y^{2}-z^{2},x^{2},x^{2}-z^{2}\}

(assuming I didn't miss one) and there are five of these. I think these are {Elliptic,Euclidean,Hyperbolic,Galilean,Minkowski} geometries respectively. Some lists of Cayley-Klein geometries also include co-Euclidean, co-Hyperbolic (sometimes called de Sitter space), co-Minkowski, and something called anti-de Sitter space. --Svennik (talk) 10:04, 31 December 2021 (UTC)[reply]