Related papers: SubRiemannian geometry on the sphere $\mathbb{S}^3…
The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding…
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a…
Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian…
In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean…
The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through…
This paper is the third in a series dedicated to the fundamentals of sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", math.MG/0210189, available at…
We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it.…
Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division…
We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two…
Utilizing the framework of quaternionic contact geometry, we define a sequence of Riemannian metrics $\{g_L\}$ on the quaternionic Heisenberg group $\mathfrak{H}_{\mathbb{H}}$ by rescaling the vertical directions. By analyzing the limit of…
We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the…
In this article we study the sub-Riemannian geometry of the spheres $S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure defined by the Hopf map and the principal $S^3-$bundle structure given by the quaternionic Hopf…
This is a collection of notes on the properties of left-invariant metrics on the eight-dimensional compact Lie group SU(3). Among other topics we investigate the existence of invariant pseudo-Riemannian Einstein metrics on this manifold. We…
In order to detect salient lines in spherical images, we consider the problem of minimizing the functional $\int \limits_0^l C(\gamma(s)) \sqrt{\xi^2 + k_g^2(s)} \, {\rm d}s$ for a curve $\gamma$ on a sphere with fixed boundary points and…
We consider the sub-Riemannian $3$-sphere $(\mathbb{S}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that…
Following our approach to metric Lie algebras developed in math.DG/0312243 we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated…
In this article we continue the study of couplings of subelliptic Brownian motions on the subRiemannian manifolds SU (2) and SL(2, R). Similar to the case of the Heisenberg group, this subelliptic Brownian motion can be considered as a…
The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint…
In this paper we discuss how to associate a suitable non-transitive version of a Cartan connection to sub-Riemannian manifolds of corank 1 (including contact and quasi-contact sub-Riemannian manifolds) with non-necessarily constant…
In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost $3$-contact metric structure which allows us to define the metric connection…