On 1/27/08, Eugene Salamin <gene_salamin@yahoo.com> wrote:
----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, January 27, 2008 5:25:36 AM Subject: Re: [math-fun] Reflections on Orientation
... The Lie group corresponding to Euclidean isometries has 4 connected components rather than 2. ...
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How can there be 4 connected component? I thought the general Euclidean isometry is (R,t) where R is a rotation, t is a translation, and the action is (R,t)x = Rx + t. Two connected components correspond to R being proper or improper. What are the other 2 components?
Are you confusing this with the case of Lorentz transformations, where space reflections and time reflections are independent? Or, is it me that is confused?
Gene
4 connected components ... ! I meant "analogous" rather than "corresponding" --- simply, the symmetry group defining the geometry. Yes, (physicist's) Lorentz group and "conformal" group respectively. WFL