Interval exchange maps and translation surfaces
Interval exchange maps and translation surfaces
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间隔交换图和平移面
DOI：
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发表时间：
2009
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通讯作者：
J. Yoccoz
中科院分区：
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作者：
J. Yoccoz
设T是二维环面，具有平坦的黎曼度量和与该度量平行的酉向量场。则存在唯一的格点Λ⊂R，使得T等距于R/Λ，T上的向量场对应于R/∂∂上的垂直向量场∂∂y。相应的“Teichmüler空间”(等同于恒等式的分类模同胚)是GL(2，R)，被看作是赋有基的格空间，“模空间”(分类模完全微分同胚群)是齐次空间GL(2，R)/GL(2，Z)，被看作是R中的格空间。
Let T be a 2-dimensional torus equipped with a flat Riemannian metric and a vector field which is unitary and parallel for that metric. Then there exists a unique lattice Λ ⊂ R such that T is isometric to R/Λ and the vector field on T corresponds to the vertical vector field ∂ ∂y on R/Λ. The corresponding “Teichmüller space” (classification modulo diffeomorphisms isotopic to the identity) is thus GL(2,R), viewed as the space of lattices equipped with a basis; the “moduli space” (classification modulo the full diffeomorphism group) is the homogeneous space GL(2,R)/GL(2,Z), viewed as the space of lattices in R.