Research output: Contribution to journal › Review article › peer-review
Curvatures of homogeneous sub-Riemannian manifolds. / Berestovskii, Valerii N.
In: European Journal of Mathematics, Vol. 3, No. 4, 01.12.2017, p. 788-807.Research output: Contribution to journal › Review article › peer-review
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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 -
ID: 9429237