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 Ω of all locally compact homogeneous spaces with inner metric is supplied with some metric dBGH such that 1) (Ω, dBGH) is a complete metric space; 2) a sequences in (Ω, dBGH) is converging if and only if it is converging in Gromov-Hausdor sense; 3) the subclasses M of homogeneous manifolds with inner metric and L G of connected Lie groups with leftinvariant Finslerian metric are everywhere dense in (Ω, dBGH): 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.
Highlights
Introduction is geodesicThe last statement means that any two points of the space can be joined by a segment or shortest arc, i.e., a curve of lengthOne can observe in last decades an intensive development of non- which is equal to the distance between these points.holonomic metric geometry and its applications to geometric group theory, analysis, CR-manifolds, the theory of hypo-elliptic differential equations, non-holonomic mechanics, mathematical physics, thermodynamics, neurophysiology of vision etc
The last statement of theorem 9 implies that the search of geodesics and shortest arcs of invariant Finslerian or Riemannian metric on homogeneous manifolds reduces in many respects to the case of Lie groups with left-invariant Finslerian or Riemannian metric
Let us note that using Pontryagin maximum principle (PMP), the author found in paper [27] all geodesics and shortest arcs of arbitrary left-invariant sub-Finslerian metric on three-dimensional Heisenberg group
Summary
The following statements for locally compact homogeneous space with inner metric (M, ρ ) are equivalent:. (4) (M, ρ) is isometric to (G/H, d); where G is a connected Lie group, H is a compact Lie subgroup of G, and d is some inner metric on G/H; invariant relative to the canonical left action of G on G/H. Theorem 3 implies that G is a connected Lie group, H is a compact Lie subgroup of G; which proves (4). Let M=G/ H be the quotient space of a connected Lie group G by its compact Lie subgroup H, (D, F) is a pair with conditions (a1) (a5) [4,5,7]
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