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. If the metric is bi invariant, then the geodesics are the left and right translates of 1parameter subgroups, but, in general, if all you have left invariance, this is far from the case. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. Start with any positive definite inner product on the lie algebra and ntranslate it to the rest of the group using left multiplication. Such a metric is called standard if the orthogonal complement of g. Let be a left invariant geodesic of the metric on the lie group and let be the curve in the lie algebra corresponding to it the velocity hodograph. 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. Theorem milnor if z belongs to the center of the lie algebra g, then for any left invariant metric the inequality kz. Lie groups which admit flat left invariant metrics 259 hence, for 1,2, the length of y.
Geodesics of left invariant metrics on matrix lie groups. Left invariant metrics on a lie group coming from lie. In the sequel, the identity element of the lie group, g, will be denoted by e or. Biinvariant and noninvariant metrics on lie groups. Department of mathematics university of mohaghegh ardabili p. Note also that riemannian metric is not the same thing as a distance function. This procedure is an analogue of the recent studies on left invariant riemannian metrics, and is based on the moduli space of left invariant pseudoriemannian metrics. Oct 10, 2007 a restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Pdf we give the explicit formulas of the flag curvatures of left invariant matsumoto and kropina metrics of berwald type. Compact simple lie groups admitting leftinvariant einstein. In this paper, for any left invariant riemannian metrics on any lie groups, we give a procedure to obtain an analogous of milnor frames, in the sense that the bracket relations.
Left invariant connections ron g are the same as bilinear. Homogeneous geodesics of left invariant randers metrics on a. Fourdimensional einstein lie groups department mathematics. When all the left translations lx are isometries, we call g a left invariant metric. Index formulas for the curvature tensors of an invariant metric on a lie group are.
Biinvariant means on lie groups with cartanschouten. Outline 1 introduction 2 equivalence of control systems 3 invariant optimal control 4 quadratic hamiltonpoisson systems 5 conclusion 6 references r. Combined with some known results in the literature, this gives a proof of the main theorem of this paper. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. Ricci curvatures of left invariant finsler metrics on lie groups. In this paper, we formulate a procedure to obtain a generalization of milnor frames for left invariant pseudoriemannian metrics on a given lie group. Metrics on solvable lie groups much is understood about left invariant riemannian einstein metrics with on solvable lie groups g. The moduli space of left invariant metrics both riemannian and pseudoriemannian settings milnortype theorems one can examine all left invariant metrics this can be applied to the existence and nonexistence problem of distinguished e. Every compact lie group admits one such metric see proposition 2.
This chapter deals with lie groups with special types of riemannian metrics. The geometry of any lie group g with left invariant riemannian metric re ects strongly the algebraic structure of the lie algebra g. Left invariant metrics and curvatures on simply connected three. In the third section, we study riemannian lie groups with. From now on elements of n are regarded as left invariant vector elds on n.
Homogeneous geodesics of left invariant randers metrics on a threedimensional lie group dariush lati. For all left invariant riemannian metrics on threedimensional unimodular lie groups, there exist particular left invariant orthonormal frames, socalled milnor frames. For lie groups, left or right invariant metric provide a nice setting as the lie group becomes a geodesically complete riemannian manifold, thus. Liang, leftinvariant pseudoeinstein metrics on lie groups. Chapter 18 metrics, connections, and curvature on lie groups. Curvatures of left invariant randers metrics on the ve. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Here we will examine various geometric quantities on a lie goup g with a leftinvariant or biinvariant metrics. Invariant metrics with nonnegative curvature on compact lie. Geodesics equation on lie groups with left invariant metrics. Let g be a lie group and g be the set of left invariant vector fields on g. In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis.
Geometrically a lie algebra g of a lie group g is the set of all left invariant vector. Our results improve a bit of milnors results of 7 in the three. Suppose, to begin with, that is a lie group acting on itself by left translations. A remark on left invariant metrics on compact 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.
In this section, we will show that the compact simple lie groups s u n for n. Left invariant randers metrics on 3dimensional heisenberg group. A remark on left invariant metrics on compact lie groups lorenz j. Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three. For example, in 9, ha and lee complete milnors classification of the available signatures of the ricci curvature of left invariant metrics on threedimensional lie groups, and in 12, kremlev. Left invariant metrics and curvatures on simply connected. Ricci curvatures of left invariant finsler metrics on lie. 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. Invariant metrics left invariant metrics these keywords were added by machine and not by the authors. Let h,i be a left invariant metric on g, and let x, y, z be left invariant vector. A riemannian metric on g is said to be biinvariant if it turns left and right translations into isometries. Note that any lie group admits a left or right invariant pseudometric.
Curvatures of left invariant metrics on lie groups core. An important role is played by the heisenberg mani. We have that every compact lie group admits a biinvariant. Notes on the riemannian geometry of lie groups rosehulman. Lie groups occupy a central position in modern differential geometry. On the moduli spaces of left invariant pseudoriemannian metrics on lie groups kubo, akira, onda, kensuke, taketomi, yuichiro, and tamaru, hiroshi, hiroshima mathematical journal, 2016 maximal isotropy groups of lie groups related to nilradicals of parabolic subalgebras bajo, ignacio, tohoku mathematical journal, 2000. In the first section, we consider general leftinvariant metrics on a lie group. The space of leftinvariant metrics on a generalization of. Computing biinvariant pseudometrics on lie groups for consistent. Curvature of left invariant riemannian metrics on lie groups. Well, then, it is a direct product of simple lie groups and of a torus afterwards, there can be a quotienting by a discrete subgroup.
A lie groupis a smooth manifold g with a group structure such that the map. G, where lx is the left translation satisfying lx y xy. A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Left invariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. When the manifold is a lie group and the metric is left invariant the curvature is also strongly related to the groups structure or equivalently to the lie algebras. 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. We study also the particular case of bi invariant riemannian metrics. On the moduli space of leftinvariant metrics on a lie group. Flow of a left invariant vector field on a lie group equipped with left invariant metric and the groups geodesics 12 uniqueness of bi invariant metrics on lie groups. In this paper, we prove several properties of the ricci curvatures of such spaces.
The space of leftinvariant metrics on a generalization. Curvatures of left invariant metrics on lie groups. This process is experimental and the keywords may be updated as the learning algorithm improves. A curvatures of left invariant metrics 297 connected lie group admits such a bi invariant metric if and only if it is isomorphic to the cartesian product of a compact group and a commutative group. 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. Milnor in the well known 2 gave several results concerning curvatures of left invariant riemannian metrics on lie groups. In the last post, geodesics of left invariant metrics on matrix lie groups part 1,we have derived arnolds equation that is a half of the problem of finding geodesics on a lie group endowed with leftinvariant metric. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular.
Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. For example, in 9, ha and lee complete milnors classification of the available signatures of the ricci curvature of leftinvariant metrics on threedimensional lie groups, and in 12, kremlev. While there are few known obstruction for a closed manifold. In particular, we give an example of a left invariant metric such that so3 is totally geodesic in gl3. Milnortype theorems have also been known for left invariant riemannian metrics on lie groups with dim. On lie groups with left invariant semiriemannian metric r. Remsing rhodes invariant control systems on lie groups icaamm 2015 58.