Homogeneous manifolds with intrinsic metric. II. (English. Russian original) Zbl 0681.53029
Sib. Math. J. 30, No. 2, 180-191 (1989); translation from Sib. Mat. Zh. 30, No. 2(174), 14-28 (1989).
[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.
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.
Reviewer: A.Fleischer
MSC:
| 53C30 | Differential geometry of homogeneous manifolds |
| 53C70 | Direct methods (\(G\)-spaces of Busemann, etc.) |
| 53C20 | Global Riemannian geometry, including pinching |
Keywords:
homogeneous manifolds; intrinsic metric; Lie groups; Carnot- \(Caratheodory\)-\(Finsler\) metric; invariant distributionCitations:
Zbl 0671.53036References:
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