Sub-Riemannian homogeneous spaces in dimensions 3 and 4

Abstract

Let (M, 79, g) be a sub-Riemannian manifold (i.e. M is a smooth manifold, 79 is a smooth distribution on M and g is a smooth metric defined on 79) such that the dimension of M is either 3 or 4 and 79 is a contact or odd-contact distribution, respectively. We construct an adapted connection V on M and use it to study the equivalence pl:oblem. Furthermore, we classify the 3-dimensional sub-Riemannian manifolds which are sub-homogeneous and show the relation to Cartan's list of homogeneous CR manifolds. Finally, we classify the 4-dimensional sub-Riemannian manifolds which are sub-symmetric. (1991): Primary, 53C15, 53C30; Secondary, 32C16. Mathematics Subject Classifications O. IntroductionSub-Riemannian geometry is concerned with the study of a smooth manifold M equipped with a metric defined only on a sub-bundle D of the tangent bundle TM, henceforth a sub-Riemannian manifold, and of the related geometric structures in analogy with Riemannian geometry. When 7) = TM we recover Riemannian geometry. Despite the similarities between the two geometries, there are new interesting phenomena occurring in sub-Riemannian geometry; see [11] for a survey and references.It is worth noting that this subject is of more than only formal interest since the several applications and connections range from control theory and mechanics with non-holonomic constraints, sub-Laplacians and hypoelliptic differential equations, to contact geometry and Cauchy-Riemann structures. Now we come to the subject of this paper. A sub-Riemannian homogeneous space is a sub-Riemannian manifold which admits a transitive group of subRiemannian isometries. A sub-Riemannian symmetric space ([13]) is a homogeneous sub-Riemannian manifold for which there is an involutive isometry hich is a central symmetry when restricted to the distribution.This work is divided into two parts. In the first part, we study 3-dimensional sub-Riemannian homogeneous spaces. We use a connection adapted to the subRiemannian structure (the pseudo-Hermitian connection of Webster [ 15] which was

Small sub-Riemannian balls onR 3

“…In case (i), we show also pictures of the wave fronts of small radius, and their intersection with the conjugate locus (Figs 1,3,4,5,6).…”

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

“…10.4], [25], that a simply connected, homogeneous, contact sub-Riemannian 3-manifold can be described in terms of some geometric and algebraic invariants, see [26]. For a Sasakian sub-Riemannian 3-manifold the most important invariant is the Webster scalar curvature K .…”

“…Moreover, the function K determines locally a Sasakian sub-Riemannian 3-manifold. In particular, if two such manifolds have the same constant Webster scalar curvature then they are locally isometric [26,Theorem 1.2]. All this might suggest that K plays for a Sasakian 3-manifold the same role as the Gaussian curvature for a Riemannian surface.…”

“…There are some reasons for which the spaces M(κ) may be understood as sub-Riemannian counterparts to the Riemannian 2-space forms. For example, it is known that any simply connected, homogeneous, contact sub-Riemannian 3-manifold has an isometry group of dimension 3 or 4, see [33], [23] and [26]. From this perspective, the spaces M(κ) are the unique ones with isometry group of maximal dimension.…”

Delaunay-type surfaces in the 2×2 real unimodular group

“…It is interesting to notice that from the point of view of sub-Riemannian geometry, H 3 , S 3 and SL(2, R) exactly represent those spaces of constant sub-Riemannian curvature in dimension 3[3].…”