  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 of
the $L$-action on the homogeneous space $G/H$ 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, and give a proof of Kobayashi's criterion in terms of CAT(0)
metric geometry on non-compact Riemannian symmetric spaces.
