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- Klein_quartic abstract "In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to PSL(2,7), the second-smallest non-abelian simple group. The quartic was first described in (Klein 1878b).Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's last theorem, and the Stark-Heegner theorem on imaginary quadratic number fields of class number one; see (Levy 1999) for a survey of properties.Originally, the "Klein quartic" referred specifically to the subset of the complex projective plane CP2 defined by the equation given in the As an algebraic curve section. This has a specific Riemannian metric (that makes it a minimal surface in CP2), under which its Gaussian curvature is not constant. But more commonly (as in this article) it is now thought of as any Riemann surface that is conformally equivalent to this algebraic curve, and especially the one that is a quotient of the hyperbolic plane H2 by a certain cocompact group G that acts freely on H2 by isometries. This gives the Klein quartic a Riemannian metric of constant negative curvature = −1 that it inherits from H2. This set of conformally equivalent Riemannian surfaces is precisely the same as all compact Riemannian surfaces of genus 3 whose conformal automorphism group is isomorphic to the unique simple group of order 168. This group is also known as PSL(2, Z/7Z), and also as the isomorphic group PSL(3, Z/2Z). By covering space theory, the group G mentioned above is isomorphic to the fundamental group of the compact surface of genus 3.".
- Klein_quartic thumbnail Uniform_tiling_73-t2.png?width=300.
- Klein_quartic wikiPageExternalLink KleinQuarticEq.html.
- Klein_quartic wikiPageExternalLink hypercolors.jpg.
- Klein_quartic wikiPageExternalLink mathieu.htm.
- Klein_quartic wikiPageExternalLink klein.html.
- Klein_quartic wikiPageExternalLink catalogue.asp?ISBN=9780521004190.
- Klein_quartic wikiPageExternalLink 9.html.
- Klein_quartic wikiPageExternalLink b44h2sin.pdf.
- Klein_quartic wikiPageExternalLink KleinQuartic.html.
- Klein_quartic wikiPageExternalLink bag44.htm.
- Klein_quartic wikiPageExternalLink bag442.
- Klein_quartic wikiPageExternalLink bagcon.htm.
- Klein_quartic wikiPageExternalLink eightfold.html.
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- Klein_quartic wikiPageExternalLink index.html.
- Klein_quartic wikiPageID "390976".
- Klein_quartic wikiPageRevisionID "575647321".
- Klein_quartic bot "H3llBot".
- Klein_quartic date "October 2010".
- Klein_quartic hasPhotoCollection Klein_quartic.
- Klein_quartic subject Category:Algebraic_curves.
- Klein_quartic subject Category:Differential_geometry_of_surfaces.
- Klein_quartic subject Category:Riemann_surfaces.
- Klein_quartic subject Category:Systolic_geometry.
- Klein_quartic type Artifact100021939.
- Klein_quartic type Object100002684.
- Klein_quartic type PhysicalEntity100001930.
- Klein_quartic type RiemannSurfaces.
- Klein_quartic type Surface104362025.
- Klein_quartic type Whole100003553.
- Klein_quartic comment "In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem.".
- Klein_quartic label "Klein quartic".
- Klein_quartic label "Quartique de Klein".
- Klein_quartic sameAs Quartique_de_Klein.
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- Klein_quartic sameAs Klein_quartic.
- Klein_quartic wasDerivedFrom Klein_quartic?oldid=575647321.
- Klein_quartic depiction Uniform_tiling_73-t2.png.
- Klein_quartic isPrimaryTopicOf Klein_quartic.
