Curvatures of homogeneous sub-Riemannian manifolds

Abstract

Abstract. 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 a case of trivial compact subgroup, invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by their 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 connection with this, the author discusses briefly the operator definition of Ricci lower bounds for sub-Riemannian manifolds by Baudoin-Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail the old definitions of the 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 completely nonholonomic distributions on manifolds. As a justification, we find a foliation on the cotangent bundle T * G 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 co-adjoint representation of the Lie group G. Also we use the canonical symplectic form on T * G 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. At the end, some examples are presented.

Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups with three-dimensional generating distributions

“…is a solution of equation (10) with the initial condition ψ 4 (0) = ϕ 4 = 0. Obviously, it is possible to find u(t), ψ(t) by the above formulas for all t ∈ R. Now Proposition 8 follows from (11) and the first equation in (5).…”

“…Proof. We can consider the covector function ψ(t) ∈ T * g(t) G from PMP as a leftinvariant 1-form on (G, •) and so naturally identify the latter with a covector function ψ(t) ∈ g * = T * e G. In [5], [6] for an extremal g(t) ∈ G, are proved the following relations satisfying for almost all t in the domain:…”

“…All main results of the paper are obtained by using results of [1], [2], [5], [6], [7], [8], [9], [10]. In the last of the listed papers we considered a similar problem for two-dimensional bracket generating subspaces and gave some detailed commentary on the four-dimensional Lie algebras.…”

Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups

“…Although not strictly related to the present work, for the sake of completeness we warn the reader that there are other lines of research in the Lagrangian direction for the definition of curvature in the sub-Riemannian context besides the ALZ approach. We refer the interested reader to [48,49,88] for generalizations of the notion of connection and to [50] for the so-called Solov'ev method.…”