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Mathematics > Differential Geometry

Title: Connection blocking in homogeneous spaces and nilmanifolds

Authors: Eugene Gutkin
Abstract: Let $G$ be a connected Lie group acting locally simply transitively on a manifold $M$. By connecting curves in $M$ we mean the orbits of one-parameter subgroups of $G$. To block a pair of points $m_1,m_2\in M$ is to find a finite set $B\subset M\setminus{m_1,m_2}$ such that every connecting curve joining $m_1$ and $m_2$ intersects $B$. The homogeneous space $M$ is blockable if every pair of points in $M$ can be blocked. Motivated by the geodesic security [4], we conjecture that the only blockable homogeneous spaces of finite volume are the tori. Here we establish the conjecture for nilmanifolds.
Comments: Minor editorial changes
Subjects: Differential Geometry (math.DG); Dynamical Systems (math.DS)
Cite as: arXiv:1211.7291 [math.DG]
  (or arXiv:1211.7291v2 [math.DG] for this version)

Submission history

From: Eugene Gutkin [view email]
[v1] Fri, 30 Nov 2012 15:43:38 GMT (18kb)
[v2] Fri, 11 Jan 2013 16:19:17 GMT (17kb)