Talk:Lie sphere geometry

Wrong scalar products?

The scalar products are wrong, aren't they? If one has R^(n+1,2), then the signature suggests that there must be two minus signs in the scalar product. The same in the case of R(3,2).. — Preceding unsigned comment added by 130.149.14.27 (talk) 10:33, 16 June 2017 (UTC)

Sources

Cecil is a great source for curves and surfaces in Lie sphere geometry. Are there other important sources for the classical theory? If so, can someone add them, and cite them at suitable points? Geometry guy 23:06, 12 April 2008 (UTC)

THE classical source for sphere geometry is

W. Blaschke Differentialgeometrie 3 (in german). —Preceding unsigned comment added by 128.131.43.34 (talk) 11:12, 26 May 2008 (UTC)

Good point. I wish someone would translate this, as my German isn't good enough to be able to use it to provide citations. I've added it to the references anyway. Geometry guy 19:07, 27 May 2008 (UTC)

Wrong image

I have removed the photo of the wire model of "hyperboloid" because it actually shows a 4th degree surface. See Talk:ruled surface for details. --Jorge Stolfi (talk) 22:19, 21 February 2010 (UTC)

The article is claiming that PGL(2,D) is isomorphic to PGL(2,C) where D is the dual numbers and C is the complex numbers

> These groups also have a direct physical interpretation: As pointed out by Harry Bateman, the Lie sphere transformations are identical with the spherical wave transformations that leave the form of Maxwell's equations invariant. In addition, Élie Cartan, Henri Poincaré and Wilhelm Blaschke pointed out that the Laguerre group is simply isomorphic to the Lorentz group of special relativity (see Laguerre group isomorphic to Lorentz group). Eventually, there is also an isomorphism between the Möbius group and the Lorentz group (see Möbius group#Lorentz transformation).

This paragraph was introduced by User:D.H in June 2017. The claim is extremely strange. The group of Laguerre transformations is the group of linear fractional transformations over the dual numbers. The group of Mobius transformations is the group of linear fractional transformations over the complex numbers. The above paragraph is saying that these are isomorphic. On top of that, the Laguerre transformations have got two connected components, while the Mobius transformations have only got one. So such an isomorphism cannot respect the topological structure of either Lie group. This can only be an abstract group isomorphism. I seriously doubt that the claim is true. What do people think? --Svennik (talk) 21:56, 2 July 2021 (UTC)

The video series by Norman Wildberger here made the connection clearer. You can also approach this via Clifford algebra. The 2x2 dual number matrices form the Clifford algebra $Cl_{1,1,1}(\mathbb {R} )$, which on the one hand expresses the group of Laguerre transformations, but which also I believe according to projective geometric algebra is the Clifford algebra whose "rotors" express the Poincare group of special relativity. --Svennik (talk) 13:45, 16 June 2022 (UTC)