Left invariant contact structures on lie groups pdf

On invariant semisimple cr structures of maximal rank on. Invariant poissonnijenhuis structures on lie groups and. Left invariant connections ron g are the same as bilinear. Generalizing results of conti, we prove that for large classes of solvable lie groups g these. Request pdf left invariant contact structures on lie groups a result from gromov ensures the existence of a contact structure on any connected noncompact odd dimensional lie group. Invariant structures on lie groups are of the most interest objective classify all left invariant structures on 3d lie groups characterise equivalence classes in terms of scalar invariants dennis i. Leftinvariant connections ron g are the same as bilinear. We determine certain classes that a fivedimensional nilpotent lie group can not be equipped with.

Leftinvariant hypercontact structures on threedimensional. Almost contact metric structures on 5dimensional nilpotent lie. We study also the particular case of bi invariant riemannian metrics. Left invariant contact structures on lie groups request pdf. Complete leftinvariant affine structures on nilpotent lie groups. We construct explicit left invariant quaternionic contact structures on lie groups with zero and nonzero torsion, and with nonvanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere.

I am reading these lines from a text which shows why the bracket of two left invariant vector fields is also a left invariant vector field. Lie gyrovector spaces kasparian, azniv and ungar, abraham a. We prove that any threedimensional riemannian lie group. In addition, leftinvariant normal almost contact metric structures on three dimensional nonunimodularlie groups are classi.

Among all the riemannian metrics on a lie groups, those for which the left translations or the right translations are isometries are of particular interest because they take the group structure of g into account. We consider leftinvariant almost contact metric structures on threedimensional lie groups, satisfying a quite natural and mild condition. The problem of finding which lie groups admit a left invariant contact structure contact lie groups, is then still open. Next we will discuss some generalities about lie algebras. Pdf left invariant contact structures on lie groups andre. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra. We investigate contact lie groups having a left invariant riemannian or pseudoriemannian metric with specific properties such as being bi invariant, flat, negatively curved, einstein, etc. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We study leftinvariant almost paracontact metric structures on arbitrary threedimensional lo rentzian lie groups. In this paper, we pursue our study initiated in 7 of invariant semisimple cr structures on compact lie groups. Invariant structures on lie groups are of the most interest objective classify all leftinvariant systems on 3d lie groups restrict to unimodular groups dennis i. Amongst other results, we perform a contactization method to construct, in every odd dimension, many contact lie groups with a discrete center, unlike the usual classical contactization which only produces lie groups with a nondiscrete center. Lie groups relies on the same ideas which, supported by additional machinery from homotopy theory, give structure theorems for pcompact groups.

But in general such structures are not invariant under left translations of the lie group. For other leftinvariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. Multiplicative and ane poisson structures on lie groups. We study rightinvariant respectively, leftinvariant poissonnijenhuis structures pn on a lie group g and introduce their infinitesimal counterpart, the socalled rn structures on the corresponding lie algebra we show that rn structures can. Classification of 3dimensional leftinvariant almost paracontact. Left invariant connections on a lie group mathoverflow. We introduce the twisted cartesian product of two special kahler lie algebras according with two linear representations by infinitesimal kahler transformations. From the relationship quoted above between left invariant objects on lie groups and their lie algebra counterpart, we will only need to work locally, i. They form an interesting class of poisson structures on its own. It includes poisson lie group structures, left or right invariant poisson structures and their linear combinations. Riemannian geometry on contact lie groups springerlink.

Thepcompact groups seem to be the best available homotopical analogues of compact lie groups 10, 11, 12, but analytical objects like lie algebras are not available for them. Homogeneous paracontact metric threemanifolds calvaruso, g. In the second part of the paper, we present a method to compute explicitly the kernel of the hypoelliptic heat equation on a wide class of leftinvariant subriemannian structures on lie groups. Then, in 3, left invariant kcontact structures on five. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Geodesics of left invariant metrics on matrix lie groups. It is well known that left invariant structures on a lie group can be described in terms of. A left invariant distribution is uniquely determined by a two dimensional subspace of the lie algebra of the group. We give the classification of all symplectic structures on nilpotent lie algebras up the dimension 6. The topology on homeom induced by any of these metrics is separable a countable subbasis. How do i tell if a vector field on a lie group is leftinvariant.

The maximal altas is also called a differentiable structure on m. In this paper, leftinvariant almost contact metric structures on threedimensional nonunimodular lie groups are investigated. If g is a lie group, we denote by g the vector space of left invariant vector fields on. While there are few known obstruction for a closed manifold. This is a special case of lemma 3 in oneill 44 chapter 11. I am reading these lines from a text which shows why the bracket of two leftinvariant vector fields is also a leftinvariant vector field. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups.

If g is a lie group, an affine structure is called leftinvariant if for each g. A remark on left invariant metrics on compact lie groups lorenz j. We construct left invariant quaternionic contact qc structures on lie groups with zero and. We will discuss the classi cation of semisimple lie algebras, root systems, the weyl group, and dynkin diagrams. We can compute the left invariant vector elds on h. Conference on ordered structures in geometry and analysis. Thus, the geodesic exponential map of that connection starting from the identity is surjective. In recent years, substantial progress in this direction. The metric is complete, so any two points can be joined by a geodesic hopfrinow. A result from gromov ensures the existence of a contact structure on any connected noncompact odd dimensional lie group. Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at the end we will have formulas ready for applications. But in general such structuresare not invariant under left translations of the lie group. The topology on homeom induced by any of these metrics is separable.

Curvature of left invariant riemannian metrics on lie. Almost contact metric structures on 5dimensional nilpotent. We study almost contact metric structures on 5dimensional nilpotent lie algebras and investigate the class of left invariant almost contact metric structures on corresponding lie groups. Curvatures of left invariant metrics on lie groups john milnor institute for advanced study, princeton, new jersey 08540 this article outlines what is known to the author about the riemannian geometry of a lie group which has been provided with a riemannian metric invariant under left translation. Abstract amongst other results, we perform a contactization method to construct, in every odd dimension, many contact lie groups with a discrete center, unlike the usual classical contactization which only produces lie groups with a nondiscrete center. The beginning of this study goes back to auslander 3 and milnor 64. We recall that all compact and all nilpotent lie groups are unimodular. Pdf left invariant contact structures on lie groups. Left invariant structures on lie groups are the basic toy models of subriemannian manifolds and the study of. We investigate contact lie groups having a left invariant riemannian or pseudoriemannian metric with specific properties such as being biinvariant, flat, negatively curved, einstein, etc. We approach with geometrical tools the contactization and symplectization of filiform structures and define hamiltonian structures and momentum mappings on lie groups. The intrinsic hypoelliptic laplacian and its heat kernel. Scalar curvatures of leftinvariant metrics on some.

Chapter 17 metrics, connections, and curvature on lie groups. We consider left invariant almost contact metric structures on threedimensional lie groups, satisfying a quite natural and mild condition. The existence of a metric invariant under both left and right multiplication is a nontrivial question. Canonical fstructures and the generalized hermitian geometry. Furthermore, we classify sasakian lie algebras of dimension 5 and determine which of them carry a sasakian. A leftinvariant distribution is uniquely determined by a two dimensional subspace of the lie algebra of the group. In addition, a classification of fivedimensional sasakian lie algebras were obtained.

Unlike the symplectic case there exist contact lie groups with no left invariant af. Invariant structures on lie groups are of the most interest objective classify all left invariant systems on 3d lie groups restrict to unimodular groups dennis i. Section 4 geometry of lie groups with a left invariant metric. We construct left invariant special kahler structures on the cotangent bundle of a flat pseudoriemannian lie group. Just take the levicivita connection of any left invariant riemannian metric on the lie group.

Since a lie group g is a smooth manifold, we can endow g with a riemannian metric. In this paper, we seek to fill this gap, by proposing a new contactization that produces contact lie groups with a discrete center, from lie groups having left invariant exact symplectic structures also called frobenius lie groups. Sorry, we are unable to provide the full text but you may find it at the following locations. The distribution is bracket generating and contact if and only if the subspace is not a lie subalgebra. A remark on left invariant metrics on compact lie groups. Two definitions of leftinvariant vector fields of a lie. The geometry of leftinvariant structures on nilpotent lie groups. In the third section, we study riemannian lie groups with. It is proved that for every riemannian lie group, there is one of these structures. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex lie groups admitting irreducible left invariant flat complex projective structures.

On lie groups with leftinvariant symplectic or kahlerian structures. The problem offinding which lie groups admit a left invariant contact structure contact liegroups, is then still open. Complex, symplectic, and contact structures on low. We show the correspondence between left invariant flat projective structures on lie groups and certain prehomogeneous vector spaces. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. Hamiltonian structures for projectable dynamics on symplectic fiber bundles.

In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis. We classify some of such contact lie groups and derive some obstruction results to the existence of left invariant contact structures on lie groups. Curvatures of left invariant metrics on lie groups john. We denote this subspace by g and call it the lie algebra of g, for reasons that will. Unlike lie groups with left invariant symplectic structures which are widely studied by a great number of authors 5, 6. The main result of this paper gives a characterization of leftinvariant almost. Invariant nonholonomic riemannian structures on three. Invariant structures on lie groups are of the most interest objective classify all leftinvariant structures on 3d lie groups characterise equivalence classes in terms of scalar invariants dennis i. It is well known that on any lie group, a leftinvariant riemannian structure can be defined. Leftinvariant affine structures on reductive lie groups.

We study left invariant contact forms and left invariant symplectic forms on lie groups. Curvature of left invariant riemannian metrics on lie groups. More precisely, the contactization of a lie groups having a left invariant symplectic structure, yields a contact lie group with a center of dimension 1. Lie groups as riemannian homogeneous ksymmetric spaces. Similarly, a lie group is a group that is also a smooth manifold, such that the two. Left invariant contact structures on lie groups sciencedirect. Their study was inspired by the search for the geometric origins of a class of curves on stiefel manifolds, called quasigeodesics, that are important in interpolation theory. This talk will introduce a large class of leftinvariant subriemannian systems on lie groups that admit explicit solutions. These structures play an important role in the study of fundamental groups of a. Left invariant contact structures on lie groups core. Left invariant flat projective structures on lie groups and prehomogeneous vector spaces kato, hironao, hiroshima mathematical journal, 2012.