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Let $G$ be a locally-compact group and $(H,L)$ 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 ofthe $L$-action on the homogeneous space $G/H$ in terms of Cartan'sKAK-decomposition of $G$. In this paper, we show that a similar theorem alsoholds if $G$ is a locally-compact group admitting a suitable isometric actionon a metric space, and give a proof of Kobayashi's criterion in terms of CAT(0)metric geometry on non-compact Riemannian symmetric spaces.

A proof of Kobayashi's properness criterion from a viewpoint of metric geometry