解流形之间的保叶映射
Foliation-preserving Maps Between Solvmanifolds
摘要：
对i = 1，2，设Gamma_i是单连通可解李群G_i中的格，X_i是G_i的连通李群.双陪集Gamma_igX_i提供了齐性空间Gamma_i \G_i的一个叶状结构F_i。设f是从γ_1 \G_1到γ_2 \G_2的连续映射，其对F_1的每个叶的限制是到F_2的一个叶上的覆叠映射.如果我们假定F_1有一个稠密的叶，并对格Gamma_1和Gamma_2作某些技术上的假定，那么我们证明f一定是两种基本类型的映射的合成：一种是Gamma_1的把F_1的每一个叶都取到自身的同胚\M_1，另一种是把仿射映射通过同态扭成紧群而得到的映射。对于G_1和G_2既不是可解的也不是半单的情形，我们也证明了类似的结果。
英文摘要：
For i = 1,2, let Gamma_i be a lattice in a simply connected, solvable Lie group G_i, and let X_i be a connected Lie subgroup of G_i. The double cosets Gamma_igX_i provide a foliation F_i of the homogeneous space Gamma_i\G_i. Let f be a continuous map from Gamma_1\G_1 to Gamma_2\G_2 whose restriction to each leaf of F_1 is a covering map onto a leaf of F_2. If we assume that F_1 has a dense leaf, and make certain technical technical assumptions on the lattices Gamma_1 and Gamma_2, then we show that f must be a composition of maps of two basic types: a homeomorphism of Gamma_1\M_1 that takes each leaf of F_1 to itself, and a map that results from twisting an affine map by a homomorphism into a compact group. We also prove a similar result for many cases where G_1 and G_2 are neither solvable nor semisimple.
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