Locally compact homogeneous spaces with inner metric

Abstract

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class $\Omega$ of all locally compact homogeneous spaces with inner metric is supplied with some metric $d_{BGH}$ such that 1) $(\Omega,d_{BGH})$ is a complete metric space; 2) a sequences in $(\Omega,d_{BGH})$ is converging if and only if it is converging in Gromov-Hausdorff sense; 3) the subclasses $\mathfrak{M}$ of homogeneous manifolds with inner metric and $\mathfrak{LG}$ of connected Lie groups with left-invariant Finslerian metric are everywhere dense in $(\Omega,d_{BGH}).$ It is given a metric characterization of Carnot groups with left-invariant sub-Finslerian metric. At the end are described homogeneous manifolds such that any invariant inner metric on any of them is Finslerian.