A remark on left invariant metrics on compact lie groups. Section 4 geometry of lie groups with a left invariant metric. Left invariant randers metrics on 3dimensional heisenberg. Lie algebras for an arbitrary smooth manifold m, the space xm of smooth vector.
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 flat metrics on lie groups are revisited in terms of leftsymmetric algebras which correspond to affine structures. Recall that a riemannian manifold m,g is called einstein if its ricci operator ric is a multiple of the identity map. The rst optimal synthesis was obtained for the heisenberg group in 16, 17. Leftinvariant metrics and submanifold geometry tamaru, hiroshi. Detecting orbits along subvarieties via the moment map. Chapter 18 metrics, connections, and curvature on lie groups.
Left invariant vector fields of a lie group physics forums. A leftinvariant riemannian metric on lie group is a special case of homogeneous riemannian manifold, and its differential geometry geodesics and curvature. Barnet, on lie groups that admit leftinvariant lorentz metrics of constant sectional curvature, illinois j. Also, you seem to be asking about the geodesic formula for more general lie groups and left invariant metrics. In addition, we have already encountered the lie bracket, a bilinear map. Left invariant randers metrics on 3dimensional heisenberg group z. 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. Pdf lie groups which admit flat left invariant metrics.
In 3 the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate. We obtain a partial result for n and the complete result for n with a onedimensional center. Let g be a lie group which admits a flat left invariant metric. V conjugate points of left invariant metrics on lie groups. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2. We consider the characterizations of jacobi fields and conjugate points along geodesics emanating from the identity element in n. 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 lorenz j. Twostep nilpotent lie groups endowed with a left invariant riemannian metric cannot be also right invariant.
The simplest example is provided by the group e2 of rigid motions of the euclidean plane section 4. Bibliography of pseudoriemannian nilpotent lie groups. While there are few known obstruction for a closed manifold. We will rst begin with lie groups and some di erential geometry. My question is whether this property characterizes biinvariant metrics among leftinvariant ones. In other words, in biinvariant metrics conjugate points always have even multiplicity in particular, due to the morse index theorem also the index of geodesics is always even. 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. My question is more of a conceptual why do we do this rather than a technical how do we do this. Bibliography of pseudoriemannian nilpotent lie groups phillip e. While leftinvariant einstein metrics do exist on some semisimple and solvable lie groups, they cannot exist on nilpotent lie groups by a result of. A leftsymmetric algebraic approach to left invariant flat.
Curvatures of left invariant metrics on lie groups core. Einstein metrics on lie groups 3 proof of theorem b. Abstract 12 background leftinvariant riemannian metrics on lie group. Conjugate and conformally conjugate parallelisms on finsler manifolds. 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. Riemannian manifold are homogeneous is a useful starting point in the classi. We study also the particular case of biinvariant riemannian metrics. Biinvariant and noninvariant metrics on lie groups. In order to study a lie group with left invariant metric, it is best to choose an. Pdf in this paper we study the geometry of lie groups with biinvariant finsler metrics. 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.
On the set of homogeneous geodesics of a leftinvariant metric. The curvature of a riemannian manifold at a point can be described most easily by the. Globally symplectic manifolds may vary considerably, but. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the hamiltonian approach. When studying the optimality of extreme trajectories, the crucial role is played by symmetries of the exponen. Curvatures of left invariant metrics on lie groups john. Invariant control systems on lie groups rory biggs claudiu c. Finally, in the last section, we study the geodesic. At every point p in our manifold we can define the riemannian exponential map expp.
Intro case 1 case 2 case 3 summary leftinvariant metrics and submanifold geometry tamaru, hiroshi hiroshima university. 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. Then complete optimal syntheses were obtained for the 3d simple lie groups. An elegant derivation of geodesic equations for left invariant metrics has been given by b. 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. Thus there exist noncommutative lie groups with fiat left invariant metric, but they are all solvable of a very sharply restricted form. Symmetries in leftinvariant optimal control problems. Homogeneous geodesics of left invariant randers metrics. We show that any nonflat left invariant metric on g has conjugate points and we describe how some of the conjugate points arise. Conjugate points on 2step nilpotent groups springerlink.
Classi cation of complete manifolds of constant curvature. Curvatures of left invariant metrics on lie groups. Geodesics of left invariant metrics on matrix lie groups. International conference on mathematics and computer science, june 2628, 2014, bra. On the moduli spaces of leftinvariant pseudoriemannian metrics on lie groups kubo, akira, onda, kensuke, taketomi, yuichiro, and tamaru, hiroshi, hiroshima mathematical journal, 2016. On lifts of leftinvariant holomorphic vector fields in complex lie groups alexandru ionescu1 communicated to. On the moduli space of leftinvariant metrics on a lie group. For example, in 7 a classification of leftinvariant metrics on the heisenberg group is given. In the third section, we study riemannian lie groups with. We apply this method to the problems described above, i. Pdf biinvariant finsler metrics on lie groups researchgate. 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. This permits us to extract the abstract structure from the case of the motion of the rigid body which we presented in section 1. Pdf left invariant geometry of lie groups researchgate.
The rst and second variation of the energy and applications. If gis a topological space and the maps are continuous, we get a topological group. 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. Lie group that admits a biinvariant metric is a homogeneous riemannian. If gis a smooth manifold and the maps are smooth, we get a lie group. Conjugation, norm and division can be defined as natural. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Conjugate points in lie groups with leftinvariant metrics.
We will discuss the classi cation of semisimple lie algebras, root systems, the weyl group, and dynkin diagrams. These structures play an important role in the study of fundamental groups of a. Define the riemannian distance dp, q for any pair of points p, q. Curvature of left invariant riemannian metrics on lie. Left invariant metrics on a lie group coming from lie algebras. On homogeneous geodesics and weakly symmetric spaces. I explain this in the following file, which i wrote in response to another persons questions about this. 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. In x5 we prove the existence of the levicivita connection. Invariant metrics with nonnegative curvature on compact.
In x1 we discuss the hessian and laplacian of a function. There is a leftsymmetric algebraic approach with an explicit formula to the classification theorem given by milnor. The beginning of this study goes back to auslander 3 and milnor 64. Leftinvariant optimal control problems on lie groups are considered. Harmonic tensors on threedimensional lie groups with left.
Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. Department of mathematics university of mohaghegh ardabili p. The most studied cases are those of left invariant subriemannian metrics see for instance 2 for a classi cation in the 3d case. Homogeneous geodesics of left invariant randers metrics on a threedimensional lie group dariush lati. How abundant are leftinvariant metrics on a compact lie group. For which lie groups that posses a biinvariant metric, this metric is unique up to scalar multiple.
Classification of leftinvariant metrics on the heisenberg. All the groups are unimodular, hence the hypoelliptic laplacian is the sum of squares. Let n be a simply connected 2step nilpotent lie group equipped with a leftinvariant metric. Applications of jacobi fields and conjugate points cut locus and injectivity radius. Next we will discuss some generalities about lie algebras. 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. Advances in mathematics 21,293329 1976 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. 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. Therefore, by bochners theorem, g is not locally isomorphic to a compact lie group. We study threedimensional lie groups with leftinvariant lorentz metric and almost harmonic with zero curl and divergence schoutenweyl tensor.