Sub-Riemannian structures on 3D Lie groups (original) (raw)
SUB-RIEMANNIAN STRUCTURES ON GROUPS OF DIFFEOMORPHISMS
Journal of the Institute of Mathematics of Jussieu, 2015
In this paper, we define and study strong right-invariant sub-Riemannian structures on the group of diffeomorphisms of a manifold with bounded geometry. We establish some approximate and exact reachability properties, and we derive the Hamiltonian geodesic equations for such structures. We provide examples of normal and of abnormal geodesics in that infinitedimensional context.
The Geometry of Sub-Riemannian Three-Manifolds
2003
The local equivalence problem for sub-Riemannian structures on threemanifolds is solved. In the course of the solution, it is shown how to attach a canonical Riemannian metric and connection to the given sub-Riemannian metric and it is shown how all of the differential invariants of the sub-Riemannian structure can be calculated. The relation between the completeness of the sub-Riemannian metric, the associated Riemannian metric, and geodesic completeness is investigated, and an example is given of a manifold that is complete in the sub-Riemannian metric but not complete in the canonical associated Riemannian metric. It is shown that the Jacobi equations for subRiemannian geodesics can be interpreted as a scalar, fourth-order, self-adjoint linear operator along each geodesic. The influence of the differential invariants of the subRiemannian structure on the conjugate points is investigated, and the results are used to prove a Bonnet-Myers-type theorem for complete sub-Riemannian 3-m...
SubRiemannian geometry on the sphere mathbbS3\mathbb{S}^3mathbbS3
2008
The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on mathbbS3\mathbb{S}^3mathbbS3 a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than those obtained in the case of the Heisenberg group, which is a noncompact Lie group. Like in the Heisenberg group case, we introduce a nonintegrable distribution on the sphere and a metric on it using two of the noncommutative left invariant vector fields. This way mathbbS3\mathbb{S}^3mathbbS3 becomes a subRiemannian manifold. It is known that the group SU(2)SU(2) SU(2) is isomorphic with the sphere mathbbS3\mathbb{S}^3mathbbS3 and represents an example of subRiemannian manifold where the elements are matrices. The main issue here is to study the connectivity by horizontal curves and its geodesics on this manifold. In this paper, we are using Lagrangian method to study the connectivity theorem on ${\mathbb ...
HAL (Le Centre pour la Communication Scientifique Directe), 2021
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 recover the known examples (Killing metric and Jensen metric) in the Riemannian case (signature (8, 0)), as well as a Gibbons et al example of signature (6, 2), and we describe a new example, which is Lorentzian (i.e., of signature (7, 1). In the latter case the associated metric is left-invariant, with isometry group SU(3) × U(1), and has positive Einstein constant. It seems to be the first example of a Lorentzian homogeneous Einstein metric on this compact manifold. These notes are arranged into a paper that deals with various other subjects unrelated with the quest for Einstein metrics but that may be of independent interest: among other topics we describe the various groups that may arise as isometry groups of left-invariant metrics on SU(3), provide parametrizations for these metrics, give several explicit results about the curvatures of the corresponding Levi-Civita connections, discuss modified Casimir operators (quadratic, but also cubic) and Laplace-Beltrami operators. In particular we discuss the spectrum of the Laplacian for metrics that are invariant under SU(3) × U(2), a subject that may be of interest in particle physics.
Riemannian geometry of tangent Lie groups using two left invariant Riemannian metrics
arXiv (Cornell University), 2023
In this paper, we consider a Lie group G equipped with two left-invariant Riemannian metrics g 1 and g 2. Using these two left-invariant Riemannian metrics we define a left-invariant Riemannian metricg on the tangent Lie group T G. The Levi-Civita connection, tensor curvature, and sectional curvature of (T G,g) in terms of g 1 and g 2 are given. Also, we give a sufficient condition forg to be bi-invariant. Finally, motivated by the recent work of D. N. Pham, using symplectic forms ω1 and ω2 on G we define a symplectic formω on T G.
SubRiemannian geometry on the sphere S^3
2008
The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on S^3 a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than those obtained in the case of the Heisenberg group, which is a noncompact Lie group. Like in the Heisenberg group case, we introduce a nonintegrable distribution on the sphere and a metric on it using two of the noncommutative left invariant vector fields. This way S^3 becomes a subRiemannian manifold. It is known that the group SU(2) is isomorphic with the sphere S^3 and represents an example of subRiemannian manifold where the elements are matrices. The main issue here is to study the connectivity by horizontal curves and its geodesics on this manifold. In this paper, we are using Lagrangian method to study the connectivity theorem on S^3 by horizontal curves with minimal arc-leng...
Riemannian Geometry of Two Families of Tangent Lie Groups
Bulletin of the Iranian Mathematical Society, 2018
Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considered by J. Milnor and the second one is the class of Lie groups with one-dimensional commutator groups. The Levi-Civita connection, sectional and Ricci curvatures have been investigated.
ON THE SYMMETRIES OF THE Sol 3 LIE GROUP
2020
In this work we consider the Sol 3 Lie group, equipped with the left-invariant metric, Lorentzian or Riemannian. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations.
Left invariant degenerate metrics on Lie groups
Journal of Geometry, 2016
We consider left invariant degenerate metrics on the group SO(3). We prove that the isometry group of such a metric is SO(3) itself unless the metric is transversally Riemannian in which case the isometry group has infinite dimension.
Left invariant geometry of Lie groups
Cubo, 2004
Section 3 Poisson manifolds (3.1) Definition of a Poisson manifold (3.2) Reformulation of the Jacobi identity (3.3) Examples of Poisson manifolds (brief summary) Example 1 Symplectic manifolds Example 2 Dual space H* of a Lie algebra H Example 3 Lie algebra H with an inner ...
Isometry groups of three-dimensional Riemannian metrics
Journal of Mathematical Physics, 1992
The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group Gr of isometries acting on s-dimensional orbits are given. This provides the list of (abstract) groups that can act isometrically and maximally on such metrics. The conditions are expressed in terms of the eigenvalues and eigenvectors of the Ricci tensor. In any case, the order of differentiability of these data necessary to determine the isometry group is less than 4.
SOME RESULTS ON SUB-RIEMANNIAN GEOMETRY
Sub-Riemannian structures naturally occur in different branches of Mathematics in the study of constrained systems in classical mechanics, in optimal control, geometric measure theory and differential geometry. In this paper, we show that Sub-Riemannian structures on three manifolds locally depend on two functions 1 and K of three variables and we investigate how these differential invariants influence the geometry.
Left invariant semi Riemannian metrics on quadratic Lie groups
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannian metric. We give a useful necessary and sufficient condition that guaranties the flatness of a left invariant semi Riemannian metric defined on a quadratic Lie group. All these semi Riemannian metrics are complete. We show that there are no Riemannian or Lorentzian flat left invariant metrics on non Abelian quadratic Lie groups, and that every quadratic 3 step nilpotent Lie group admits a flat left invariant semi Riemannian metric. The case of quadratic 2 step nilpotent Lie groups is also addressed.
Homogeneous geodesics of left invariant Randers metrics on a three-dimensional Lie group
In this paper we study homogeneous geodesics in a three-dimensional connected Lie group G equipped with a left invariant Randers metric and investigates the set of all homogeneous geodesics. We show that there is a three-dimensional unimodular Lie group with a left invariant non-Berwaldian Randers metric which admits exactly one homogeneous geodesic through the identity element.
The sub-Riemannian geometry of screw motions with constant pitch
arXiv (Cornell University), 2023
We consider a family of Riemannian manifolds M such that for each unit speed geodesic γ of M there exists a distinguished bijective correspondence L between infinitesimal translations along γ and infinitesimal rotations around it. The simplest examples are R 3 , S 3 and hyperbolic 3-space, with L defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M. Given λ ∈ R, we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch λ. We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on R 7 ⋊ SO (7) involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of "small rotations".
Notes on a Three-Dimensional Riemannian Manifold with an Additional Structure
2013
We consider a three-dimensional Riemannian manifold equipped with two circulant structures-a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.
The Fine Structure of Transitive Riemannian Isometry Groups. I
Transactions of the American Mathematical Society, 1985
Let M be a connected homogeneous Riemannian manifold, G the identity component of the full isometry group of M and H a transitive connected subgroup of G. G = HL. where L is the isotropy group at some point of M. M is naturally identified with the homogeneous space H/H n L endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of M as a Riemannian homogeneous space of a connected Lie group H, describe the structure of the full connected isometry group G in terms of H. This problem has already been studied in case H is compact, semisimple of noncompact type, or solvable. We use the fact that every Lie group is a product of subgroups of these three types in order to study the general case.
Invariant totally geodesic unit vector fields on three-dimensional Lie groups
Journal of Mathematical Physics, Analysis, Geometry
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic unit vector field. From geometrical viewpoint, the field is either parallel or characteristic vector field of a natural almost contact structure on the group.