Suppose $(M,g)$ is a complete Riemannian manifold on which a Lie group $H$ acts properly and isometrically. Then the induced Riemannian distance function $d:M\times M\rightarrow\mathbb{R}$ is a complete, proper $H$-invariant metric (in the sense of metric spaces).
The orbit space $M/H$ is a locally compact space that inherits a length metric $d'$ defined by letting $d'([x],[y])$ be the infimum of $d(w,z)$, where $w$ and $z$ range over elements in the orbits $[x]$ and $[y]$ respectively.
My question: is $d'$ a proper metric?
Thoughts: I think this would follow if $d'$ was a complete metric on $M/H$, but I'm not sure if it is.
Thanks!