Left invariant vector fields on a lie 2 group 605 algebras. Lie groups are named after norwegian mathematician sophus lie, who. We give a new proof of a classical result of milnor on riemannian flat lie algebras. In the next paragraphs explicit involution formulas for. Einstein metrics on lie groups 3 proof of theorem b. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. We show that the double extension process can be used to construct all. Curvatures of left invariant metrics on lie groups john milnor. Birkhoff 22 and kakutani 5 proved that a hausdorff group admits a left invariant metric if and only if it satis.
Let g be a compact lie group, with biinvariant metric h0 and lie algebra g. Note also that riemannian metric is not the same thing as a distance function. 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 left invariant metric. Let g be a leftinvariant pseudoeinstein metric of a lie group g with the einstein constant then for any nonzero constant a, ag is also a leftinvariant pseudoeinstein metric with the constant. It would be related to the algebraic structure of g. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics.
A metric d on x is called intrinsic if any two points x and y in x can be joined by a curve with length arbitrarily close to dx, y. Set a lie group equipped with a left invariant metric n,h,i, endow its lie algebra n with the induced metric and denote by tnits tangent space, trivialized as tn. Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at. The complete classification of these metric lie groups is obtained in terms of the structure constants of corresponding lie algebras.
Let g be a compact connected lie group of dimension m. The reason for this is that the passage from the complex of all differential forms to the complex of left invariant differential forms uses an averaging process that only makes sense. However, depending on the choice of left invariant lorentz metric, the sign of the constant sectional curvature may be positive, negative, or zero. Apr 26, 2016 we consider left invariant degenerate metrics on the group equation. Curvatures of left invariant metrics on lie groups research. If h is a left invariant metric with nonnegative curvature on a compact lie group g, then the unique inverselinear path from any. An elegant derivation of geodesic equations for left invariant metrics has been given by b.
Lie groups with a cyclic leftinvariant metric are lie groups which are in some way far from being bi invariant lie groups, in a sense made explicit in terms of tricerri and vanheckes homogeneous. Left invariant metrics and curvatures on simply connected. Left invariant connections ron g are the same as bilinear. Left invariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. Razavi where z is a nonzero vector in the lie algebra g of g. Homogeneous geodesic in a lie group were studied by v.
Metrics, connections, and curvature on lie groups applying theorem 17. Therefore, by bochners theorem, g is not locally isomorphic to a compact lie group. At last, we study the left invariant pseudoriemannian metrics on compact lie groups and classify the pseudoeinstein metrics on the lowdimensional compact lie groups. Let g1 be a compact lie group with a leftinvariant einstein metric g1, the corresponding einstein constant. 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. I just read lie group methods from iserles, munthekaas, norsett and zanna and there they state. Once a biinvariant metric on g is fixed, leftinvariant metrics on g are in correspondence with m.
We prove that the isometry group of such a metric is equationitself unless the metric is. A riemannian metric that is both left and rightinvariant is called a biinvariant metric. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. Pdf diameter and laplace eigenvalue estimates for left. Let g1 be a compact lie group with a left invariant einstein metric g1, the corresponding einstein constant. Constructing a metric on a lie group mathematics stack. Further, every metric lie group is 1,cquasiisometric to a solvable lie group, and every simply connected metric lie group is 1,cquasiisometrically homeomorphic to a solvablebycompact metric lie group. 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 among them can be written with relatively smaller number of parameters. A metric on g that is both leftinvariant and rightinvariant is called biinvariant.
Curvatures of left invariant metrics on lie groups. Conjugate points in lie groups with leftinvariant metrics. 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. Finally it is proved that in low dimensions complete families of first integrals can be constructed with killing vector fields and symmetric killing 2tensor fields. The group g is a lie group with a left invariant metrics. In the sequel, the identity element of the lie group, g, will be denoted by e or. In the third section, we study riemannian lie groups with. Dec 29, 20 on the basis of a left invariant metric on g, left invariant vertical and horizontal distributions and a left invariant metric g on t 0 2 g are constructed. Leftinvariant einstein metrics on lie groups andrzej derdzinski august 28, 2012. Scalar curvatures of leftinvariant metrics on some. 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. As a general fact we show the following regularity result. Jul 02, 2014 lie groups with a cyclic left invariant metric are lie groups which are in some way far from being bi invariant lie groups, in a sense made explicit in terms of tricerri and vanheckes homogeneous. We estimate the diameter and the smallest positive eigenvalue of the laplacebeltrami operator associated to a leftinvariant metric on g in terms of the eigenvalues of the corresponding positive.
Let g be a real lie group of dimension n and g its lie algebra. We show that any nonflat left invariant metric on g has conjugate points and we describe how some of the conjugate points arise. In the presented paper left invariant pseudoriemannian metrics on fourdimensional lie groups with zero schoutenweyl tensor are investigated. Glg, we get our second criterion for the existence of a bi invariant metric on a lie group. We study also the particular case of bi invariant riemannian metrics. Furthermore, we show that the left invariant pseudoeinstein metric on sl2 is unique up to a constant. Also invariant first integrals are analyzed and new involution conditions are shown. Rightinvariant metrics on diffeomorphisms groups with. Here is an explicit although rather exceptional example of a left. I lagrangian mechanics is the geodesic ow on the group g. Leftinvariant and almost rightinvariant metric on a lie.
We reduce the study of lorentzian flat lie algebras to those with trivial center or those with degenerate center. The group g is a lie group with a leftinvariant metrics. Flow of a left invariant vector field on a lie group equipped with left invariant metric and the group s geodesics 12 uniqueness of bi invariant metrics on lie groups. Homogeneous geodesics of left invariant randers metrics on a. A compact lie group possesses a biinvariant metric. If t is a topological space and p a metric on t, we shall say that t admits p if the ptopology of t agrees with its original topology. Curvatures of left invariant metrics on lie groups core. Flow of a left invariant vector field on a lie group equipped with left invariant metric and the group s geodesics 2 proving smoothness of left invariant metric on a lie group. Symmetries in leftinvariant optimal control problems. Asymptotic directions on a surface in a 4dimensional metric lie group bayard, pierre and sanchezbringas, federico, 2018. Pdf on lie groups with left invariant semiriemannian metric. If h is a leftinvariant metric with nonnegative curvature on a compact lie group g, then the unique inverselinear path from any. The curvature on the heisenberg group which is computed via left invariant metric we call the lie group generalized heisenberg group which constitutes the matrices of following form. Invariant metrics with nonnegative curvature on compact.
Therefore, not every lie group is a metric space, where the distance is bi invariant. We call the lie algebra of a lie group with a left invariant pseudoriemannian flat metric pseudoriemannian flat lie algebra. Let k be a connected real compact lie group, and let lk denote the family of all leftinvariant riemannian metrics g on k. In this paper, we study some properties of the riemannian structures of a contact lie group and we give some conditions for a left invariant contact structure on a lie group to be the deformation of a foliation. Homogeneous geodesics of left invariant randers metrics on. In this paper, with the term metric lie group we mean a lie group equipped with a left invariant distance that induces the manifold topology. We have that every compact lie group admits a biinvariant. I have yet to read the paper itself, so thanks for the pointer. Leftinvariant lorentzian flat metrics on lie groups. Leftinvariant metrics on lie groups and submanifold geometry.
A riemannian metric on g is said to be biinvariant if it turns left and right translations into isometries. Left invariant degenerate metrics on lie groups springerlink. Index formulas for the curvature tensors of an invariant metric on a lie group are obtained. 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. Glg, we get our second criterion for the existence of a biinvariant metric on a lie group. The results are applied to the problem of characterizing invariant metrics of zero and nonzero constant curvature. Curvature of left invariant riemannian metrics on lie.
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. Lie groups are named after norwegian mathematician sophus lie, who laid the foundations of the theory of continuous transformation groups. A leftinvariant metric on a lie group or a differential manifold with transitive smooth group action can be constructed by. Every bi invariant metric is left invariant, and so can be constructed in a unique way from an inner product for t eg. We consider left invariant degenerate metrics on the group equation.
If i am not confused, milnors claim is that a lie group admits a bi invariant metric can be decomposed as a product group, not every connected lie group admits a bi invariant metric. Examine whether g admits a distinguished left invariant metric. Left invariant optimal control problems on lie groups are considered. On lie groups with left invariant semiriemannian metric r. Chapter 17 metrics, connections, and curvature on lie groups. Milnor in the well known 2 gave several results concerning curvatures of left invariant riemannian metrics on lie groups. Dynamics of geodesic ows with random forcing on lie groups.
Left invariant metrics on a lie group coming from lie. Suppose, to begin with, that is a lie group acting on itself by left translations. Left invariant metrics on lie groups and submanifold geometry hiroshi tamaru hiroshima university. A metric d on a group g written multiplicatively is said to be left invariant resp. 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. Given any lie group g, an inner product h,i on g induces a biinvariant metric on g i. For the case of biinvariant metrics, proposition 1 extends as follows. On lie groups with left invariant semiriemannian metric 11 and. We prove that the isometry group of such a metric is equationitself unless the metric is transversally riemannian in which. Given any lie group g, an inner product h,i on g induces a bi invariant metric on g i.
Computing biinvariant pseudometrics on lie groups for. Start with any positive definite inner product on the lie algebra and ntranslate it to the rest of the group using left multiplication. We show that there is a threedimensional unimodular lie group with a left invariant nonberwaldian randers metric which admits exactly one homogeneous geodesic through the identity element. Curvatures of left invariant metrics on lie groups john. Left invariant pseudoriemannian metrics on solvable lie. When studying the optimality of extreme trajectories, the crucial role is played by symmetries of the exponen. Geodesics of left invariant metrics on matrix lie groups. Kajzer in 3 where he proved that a lie group g with a left invariant metric has at least one homogeneous geodesic through the identity. Chapter 18 metrics, connections, and curvature on lie groups. For a given lie group g, examine whether g admit a distinguished left invariant metric.
Curvature of left invariant riemannian metrics on lie groups. Our procedure is based on the moduli space of left invariant riemannian metrics. First integrals on steptwo and stepthree nilpotent lie. Let g be a lie group which admits a flat left invariant metric.
Andrzej derdzinski left invariant einstein metrics. Considering a left invariant semiriemannian structure on g, let e 1. The geodesic motion on a lie group equipped with a left or right invariant riemannian metric is governed by the eulerarnold equation. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Classical mechanics for general group g of the symmetry of motion. Let g be a left invariant pseudoeinstein metric of a lie group g with the einstein constant then for any nonzero constant a, ag is also a left invariant pseudoeinstein metric with the constant. I other than the trivial group rn, we can consider much more general group g describing the symmetry of motion. While any contractible lie group may be made isometric to.
A more stringent requirement on a riemannian metric on a lie group is that it should be invariant under both left and right translations. Nomizu proved that every left invariant lorentz metric on such a lie group is also of constant sectional curvature. Left invariant metrics on a tensor bundle of type 2,0 over a lie group springerlink. A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. In other words, every left invariant metric on a unimodular lie group must possess some strictly positive sectional curvature unless it is completely flat as in 1. We obtain a partial description of the totally geodesic submanifolds of a 2step, simply connected nilpotent lie group with a left invariant metric. Thus there exist noncommutative lie groups with fiat left invariant. A left invariant riemannian metric on lie group is a special case of homogeneous riemannian manifold, and its differential geometry geodesics and curvature can be described in a quite compact form.
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