Homogeneous manifolds with intrinsic metric. II
[For part I, cf. Sib. Math. J. 29, No.6, 887-897 (1988); translation from Sib. Mat. Zh. 29, No.6(172), 17-29 (1988; Zbl 0671.53036).] This paper completes the investigation of the author in the domain of the metric structure of homogeneous manifolds with intrinsic metric. The main result is: the homogeneous spaces with intrinsic metric are exactly factor-spaces G/H of connected Lie groups G by their compact subgroups H equipped with an (invariant with respect to the canonical action G on G/H) Carnot-\(Caratheodory\)-\(Finsler\) metric. Such metric \(d_ c\) is given by a ``completely nonholonomic'' G-invariant distribution \(\Delta\) on \(M=G/H\) and by the G-invariant norm \(F=F(p,\cdot)\), \(p\in G/H\) on the linear subspace \(\Delta\) (p) of the tangent space \(M_ p\) to M in the point p. A necessary and sufficient condition that the G-invariant intrinsic metric on G/H is of Finsler type is found and is given in terms of the corresponding Lie algebras. Homogeneous Riemannian and Finsler manifolds are characterized and described.
Carnot- \(Caratheodory\)-\(Finsler\) metric
scientific article; zbMATH DE number 4115459