Related papers: Sub-Riemannian structures on 3D Lie groups

The aim of the paper is to determine left-invariant,anti-self-dual, non conformally flat, Riemannian metrics on four-dimensional Lie groups.

We study left-invariant foliations $\mathcal{F}$ on Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

We find geodesics, shortest arcs, cut loci, first conjugate loci, distances between arbitrary elements for some left-invariant sub-Riemannian metrics on the Lie groups $SU(2)\times\mathbb{R}$ and $SO(3)\times\mathbb{R}$.

This paper is about non-Euclidean analysis on Lie groups endowed with left invariant distributions, seen as sub-Riemannian manifolds. This is a an updated version, which will be modified according to the contributions of the other…

We classify left invariant metrics with nonnegative curvature on SO(3) and U(2).

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket;…

The geodesics for a sub-Riemannian metric on a three-dimensional contact manifold $M$ form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on $M$, locally equivalent to the solutions…

We study the geometry of a family of Lie groups, which contained the classical affine Lie groups, endowed with an exact left invariant symplectic form. We show that this family is closed by symplectic reduction and symplectic double…

The main goal is to classify 4-dimensional real Lie algebras $\g$ which admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore…

We consider the completeness problem for left-invariant Lorentzian metrics on 3-dimensional non-unimodular Lie groups, all of which have Lie algebra of the form $\mathbb{R} \ltimes_A \mathbb{R}^2$, where $A$ is a real $2 \times 2$ matrix…

The subject of this paper is six-dimensional nearly (para-)K\"ahler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly K\"ahler manifold and prove…

We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension $\leq$ 6.

We provide the first known family of examples of integrable homogeneous sub-Riemannian structures admitting strictly abnormal geodesics. These examples were obtained through the analysis of the equivalence problem for sub-Riemannian Engel…

Let G be a 3-dimensional connected Lie group with a left-invariant nondegenerate CR structure. We show that all the chains on G are closed if and only if G is CR equivalent to one of the left-invariant spherical CR structures on SU(2),…

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative…

Semi-Riemannian manifolds that satisfy (homogeneous) linear differential conditions of arbitrary order on the curvature are analyzed. They include, in particular, the spaces with (higher-order) recurrent curvature, (higher-order) symmetric…

This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate…

We consider 5-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we give a partial classification of left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local…