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35(1): 2021/02/27(土)23:24 ID:f+hU2HEr(1/4) AAS
>>34
メモ
外部ﾘﾝｸ:en.wikipedia.org
Klein quartic
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).
Closed and open forms
It is important to distinguish two different forms of the quartic. The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space. The open or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are cusps. The open quartic may be obtained (topologically) from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The open and closed quartics have different metrics, though they are both hyperbolic and complete[1] – geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.
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36(1): 2021/02/27(土)23:25 ID:f+hU2HEr(2/4) AAS
>>35
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Considering the action of SL(2, R) on the upper half-plane model H2 of the hyperbolic plane by Möbius transformations, the affine Klein quartic can be realized as the quotient Γ(7)\H2. (Here Γ(7) is the congruence subgroup of SL(2, Z) consisting of matrices that are congruent to the identity matrix when all entries are taken modulo 7.)
Fundamental domain and pants decomposition
3-dimensional models
画像ﾘﾝｸ[gif]:upload.wikimedia.org
An animation by Greg Egan showing an embedding of Klein’s Quartic Curve in three dimensions, starting in a form that has the symmetries of a tetrahedron, and turning inside out to demonstrate a further symmetry.
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37(1): 2021/02/27(土)23:26 ID:f+hU2HEr(3/4) AAS
>>36
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外部ﾘﾝｸ:en.wikipedia.org
Klein quadric
In mathematics, the lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent each line in S lie on a quadric, Q known as the Klein quadric.
If the underlying vector space of S is the 4-dimensional vector space V, then T has as the underlying vector space the 6-dimensional exterior square Λ2V of V. The line coordinates obtained this way are known as Plücker coordinates.
以上
38: 2021/02/27(土)23:36 ID:f+hU2HEr(4/4) AAS
>>37
”Punctured spheres”
外部ﾘﾝｸ:en.wikipedia.org
Riemann surface
Contents
5 Classification of Riemann surfaces
5.1 Elliptic Riemann surfaces
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