TY - JOUR

T1 - Curvatures of homogeneous sub-Riemannian manifolds

AU - Berestovskii, Valerii N.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In the case of a trivial compact subgroup, the invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by being in one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In the present paper, in connection with this, the author discusses briefly the operator definition of lower bound for Ricci curvature by Baudoin–Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail old definitions of curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev. To calculate the Solov’ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant bracket generating distributions on manifolds. As a justification, we find a foliation on the cotangent bundle [InlineEquation not available: see fulltext.] over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin–Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the coadjoint representation of the Lie group G. We also use the canonical symplectic form on [InlineEquation not available: see fulltext.] and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. In Sects. 5 and 6, some examples are presented.

AB - The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In the case of a trivial compact subgroup, the invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by being in one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In the present paper, in connection with this, the author discusses briefly the operator definition of lower bound for Ricci curvature by Baudoin–Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail old definitions of curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev. To calculate the Solov’ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant bracket generating distributions on manifolds. As a justification, we find a foliation on the cotangent bundle [InlineEquation not available: see fulltext.] over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin–Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the coadjoint representation of the Lie group G. We also use the canonical symplectic form on [InlineEquation not available: see fulltext.] and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. In Sects. 5 and 6, some examples are presented.

KW - Coadjoint representation

KW - Contact form

KW - Cotangent bundle

KW - Hamiltonian vector field

KW - Homogeneous sub-Riemannian manifold

KW - Left invariant sub-Riemannian metric

KW - Lie algebra

KW - Lie group

KW - Pontryagin–Hamilton function

KW - Sub-Riemannian curvature

KW - Submetry

KW - Symplectic form

KW - METRIC-MEASURE-SPACES

KW - SPHERES

KW - PROJECTIVE SPACES

KW - GAUSS SEMIGROUPS

KW - GEODESICS

KW - RICCI CURVATURE

KW - LIE GROUP

KW - Pontryagin-Hamilton function

KW - GEOMETRY

UR - http://www.scopus.com/inward/record.url?scp=85037355257&partnerID=8YFLogxK

U2 - 10.1007/s40879-017-0171-3

DO - 10.1007/s40879-017-0171-3

M3 - Review article

AN - SCOPUS:85037355257

VL - 3

SP - 788

EP - 807

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 4

ER -