Bounded Isometries and Homogeneous Quotients

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• 作者：Joseph A. Wolf
• 关键词：Homogeneous space ; Isometry ; Constant displacement ; CW transformation ; Exponential solvable Lie group
• 刊名：The Journal of Geometric Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：27
• 期：1
• 页码：56-64
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Differential Geometry; Convex and Discrete Geometry; Fourier Analysis; Abstract Harmonic Analysis; Dynamical Systems and Ergodic Theory; Global Analysis and Analysis on Manifolds;
• 出版者：Springer US
• ISSN：1559-002X
• 卷排序：27

文摘

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author’s 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if \(\Gamma \backslash M\) is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold M, then \(\Gamma \backslash M\) is homogeneous if and only if each isometry \(\gamma \in \Gamma \) is of constant displacement. The description of bounded isometries in this paper gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford’s use of quaternions in non-Euclidean geometry, and we sketch that in a historical note.