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Let G be a locally compact group and \((H_1,H_2)\) a pair of closed subgroups of G. For the cases where G is a real linear reductive Lie group, T. Kobayashi [Math. Ann. ’89, J. Lie Theory ’96] established a criterion for properness of the \(H_1\) -action on the homogeneous space \(G/{H_2}\) in terms of Cartan’s KAK-decomposition of G. In this paper, we show that a similar theorem also holds if G is a locally compact group admitting a suitable isometric action on a metric space. This gives an alternative proof of Kobayashi’s criterion in terms of CAT(0) metric geometry on noncompact Riemannian symmetric spaces.

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A Proof of Kobayashi’s Properness Criterion from a Viewpoint of Metric Geometry

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  • 发表日期：

    2025年

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  • Kento Ogawa,
  • Takayuki Okuda

摘要

Let G be a locally compact group and \((H_1,H_2)\) a pair of closed subgroups of G. For the cases where G is a real linear reductive Lie group, T. Kobayashi [Math. Ann. ’89, J. Lie Theory ’96] established a criterion for properness of the \(H_1\) -action on the homogeneous space \(G/{H_2}\) in terms of Cartan’s KAK-decomposition of G. In this paper, we show that a similar theorem also holds if G is a locally compact group admitting a suitable isometric action on a metric space. This gives an alternative proof of Kobayashi’s criterion in terms of CAT(0) metric geometry on noncompact Riemannian symmetric spaces.