Conjugate points of left-invariant metrics on lie groups pdf

All the groups are unimodular, hence the hypoelliptic laplacian is the sum of squares. Invariant metrics with nonnegative curvature on compact. Geodesics of left invariant metrics on matrix lie groups. We obtain a partial result for n and the complete result for n with a onedimensional center. Finally, in the last section, we study the geodesic. We will rst begin with lie groups and some di erential geometry. International conference on mathematics and computer science, june 2628, 2014, bra. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract.

Also, you seem to be asking about the geodesic formula for more general lie groups and left invariant metrics. This permits us to extract the abstract structure from the case of the motion of the rigid body which we presented in section 1. Lie group that admits a biinvariant metric is a homogeneous riemannian. Riemannian manifold are homogeneous is a useful starting point in the classi. Curvatures of left invariant metrics on lie groups. In 3 the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate. For example, in 7 a classification of leftinvariant metrics on the heisenberg group is given. Next we will discuss some generalities about lie algebras.

Bibliography of pseudoriemannian nilpotent lie groups. There is a leftsymmetric algebraic approach with an explicit formula to the classification theorem given by milnor. But on a lie group itself there is an explicit description of jacobi fields available for right invariant metrics even on infinite dimensional lie groups in section 3 of. Curvatures of left invariant metrics 297 connected lie group admits such a biinvariant metric if and only if it is isomorphic to the cartesian product of a compact group and a commutative group. Left invariant vector fields of a lie group physics forums.

Globally symplectic manifolds may vary considerably, but. Harmonic tensors on threedimensional lie groups with left. Let g be a lie group which admits a flat left invariant metric. The rst optimal synthesis was obtained for the heisenberg group in 16, 17. V conjugate points of left invariant metrics on lie groups. Flow of a left invariant vector field on a lie group equipped with leftinvariant metric and the group s geodesics 12 uniqueness of biinvariant metrics on lie groups.

Conjugation, norm and division can be defined as natural. At every point p in our manifold we can define the riemannian exponential map expp. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. A remark on left invariant metrics on compact lie groups. In x5 we prove the existence of the levicivita connection. The beginning of this study goes back to auslander 3 and milnor 64. The curvature of a riemannian manifold at a point can be described most easily by the. Lie algebras for an arbitrary smooth manifold m, the space xm of smooth vector. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. Pdf left invariant geometry of lie groups researchgate.

Abstract 12 background leftinvariant riemannian metrics on lie group. While leftinvariant einstein metrics do exist on some semisimple and solvable lie groups, they cannot exist on nilpotent lie groups by a result of. The moduli space of leftinvariant metrics both riemannian and pseudoriemannian settings milnortype theorems one can examine all leftinvariant metrics this can be applied to the existence and nonexistence problem of distinguished e. Applications of jacobi fields and conjugate points cut locus and injectivity radius.

A leftinvariant riemannian metric on lie group is a special case of homogeneous riemannian manifold, and its differential geometry geodesics and curvature. In order to study a lie group with left invariant metric, it is best to choose an. Left invariant randers metrics on 3dimensional heisenberg. The simplest example is provided by the group e2 of rigid motions of the euclidean plane section 4. Conjugate and conformally conjugate parallelisms on finsler manifolds. Conjugate points in lie groups with leftinvariant metrics.

If g is a semigroup and p a metric on g, p will be called left invariant if pgx, gy px, y whenever g, x, y cg, right invariant if always pxg, yg px, y, and invariant if it is both right and left invariant. The most studied cases are those of left invariant subriemannian metrics see for instance 2 for a classi cation in the 3d case. Chapter 18 metrics, connections, and curvature on lie groups. Conjugate points on 2step nilpotent groups springerlink. Curvatures of left invariant metrics on lie groups core. Twostep nilpotent lie groups endowed with a left invariant riemannian metric cannot be also right invariant. 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. In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis. Classification of leftinvariant metrics on the heisenberg. Bibliography of pseudoriemannian nilpotent lie groups phillip e. On the set of homogeneous geodesics of a leftinvariant metric.

Section 4 geometry of lie groups with a left invariant metric. Let n be a simply connected 2step nilpotent lie group equipped with a leftinvariant metric. We study also the particular case of biinvariant riemannian metrics. Homogeneous geodesics of left invariant randers metrics on a threedimensional lie group dariush lati. Department of mathematics university of mohaghegh ardabili p. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Einstein metrics on lie groups 3 proof of theorem b. Intro case 1 case 2 case 3 summary leftinvariant metrics and submanifold geometry tamaru, hiroshi hiroshima university. For which lie groups that posses a biinvariant metric, this metric is unique up to scalar multiple. Rossiy abstract in this paper we study the carnotcaratheodory metrics on su2 s3, so3 and sl2 induced by their cartan decomposition and by the killing form. Specifically for solvable lie algebras of dimension up to and including six all algebras for which there is a compatible pseudoriemannian metric on the corresponding linear lie group are found. Lee is a group whose elements are organized continuously and smoothly, as opposed to discrete groups, where the elements are separatedthis makes lie groups differentiable manifolds. We study threedimensional lie groups with leftinvariant lorentz metric and almost harmonic with zero curl and divergence schoutenweyl tensor. Detecting orbits along subvarieties via the moment map.

Curvatures of left invariant metrics on lie groups john. In addition, we have already encountered the lie bracket, a bilinear map. Recall that a riemannian manifold m,g is called einstein if its ricci operator ric is a multiple of the identity map. In the third section, we study riemannian lie groups with.

Left invariant flat metrics on lie groups are revisited in terms of leftsymmetric algebras which correspond to affine structures. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2. When studying the optimality of extreme trajectories, the crucial role is played by symmetries of the exponen. Leftinvariant optimal control problems on lie groups are considered. We show that any nonflat left invariant metric on g has conjugate points and we describe how some of the conjugate points arise. Then complete optimal syntheses were obtained for the 3d simple lie groups. Thus there exist noncommutative lie groups with fiat left invariant metric, but they are all solvable of a very sharply restricted form. We apply this method to the problems described above, i.

Therefore, by bochners theorem, g is not locally isomorphic to a compact lie group. My question is whether this property characterizes biinvariant metrics among leftinvariant ones. We consider the characterizations of jacobi fields and conjugate points along geodesics emanating from the identity element in n. Curvature of left invariant riemannian metrics on lie. These structures play an important role in the study of fundamental groups of a. If gis a topological space and the maps are continuous, we get a topological group. On lifts of leftinvariant holomorphic vector fields in complex lie groups alexandru ionescu1 communicated to. Classi cation of complete manifolds of constant curvature. Pdf in this paper we study the geometry of lie groups with biinvariant finsler metrics. If gis a smooth manifold and the maps are smooth, we get a lie group. Homogeneous geodesics of left invariant randers metrics. Maximal isotropy groups of lie groups related to nilradicals of parabolic subalgebras bajo, ignacio, tohoku mathematical journal, 2000.

Barnet, on lie groups that admit leftinvariant lorentz metrics of constant sectional curvature, illinois j. Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the hamiltonian approach. Left invariant randers metrics on 3dimensional heisenberg group z. Biinvariant and noninvariant metrics on lie groups. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Define the riemannian distance dp, q for any pair of points p, q. In x1 we discuss the hessian and laplacian of a function. The rst and second variation of the energy and applications. A leftsymmetric algebraic approach to left invariant flat. How abundant are leftinvariant metrics on a compact lie group. On the moduli spaces of leftinvariant pseudoriemannian metrics on lie groups kubo, akira, onda, kensuke, taketomi, yuichiro, and tamaru, hiroshi, hiroshima mathematical journal, 2016. Pdf biinvariant finsler metrics on lie groups researchgate. On homogeneous geodesics and weakly symmetric spaces.

While there are few known obstruction for a closed manifold. I explain this in the following file, which i wrote in response to another persons questions about this. My question is more of a conceptual why do we do this rather than a technical how do we do this. Contracting the schoutenweyl tensor in an arbitrary direction, we introduce an antisymmetric 2tensor and study the structure of threedimensional lie groups and algebras with leftinvariant riemann metric in. Left invariant metrics on a lie group coming from lie algebras. In x4 we discuss lie groups and leftinvariant metrics. Lie groups are named after norwegian mathematician sophus lie, who laid the foundations of the theory of continuous. December 12, 2007 contents 1 examples of lie groups 2 2 carnotcarath eodory distance 4 3 l2spaces and weighted l2spaces 6 4 convolutions 10 5 the heat kernel 11 6. Invariant control systems on lie groups rory biggs claudiu c.