We can see these formulas are different from previous results given recently. Curvatures of left invariant metrics on lie groups core. Pdf left invariant contact structures on lie groups. Hence, denoting by rthe semiriemannian curvature tensor. 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. Curvatures of left invariant metrics 295 spanned by u and v. Andrzej derdzinski left invariant einstein metrics. Geometrically, k can be described as the gaussian curvature, at the point, of the surface swept out by all geodesics having a linear combination of u and v as tangent vector.
Leftinvariant metrics on lie groups and submanifold geometry. The most familiar nilpotent lie groups are matrix groups whose diagonal entries are. Ricci curvature of left invariant metrics on solvable. An abstract lie group g admits many left invariant metrics and it is well known that these metrics posess drastically different curvature properties. 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. Biinvariant and noninvariant metrics on lie groups. 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. Some of those results can be partial or totally generalized to indefinite metrics. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds.
We also study some lie groups whose spaces of left invariant metrics up to isometry and scaling are small. Left invariant metrics and curvatures on simply connected. Scalar curvatures of leftinvariant metrics on some. To begin with, we give some examples of pseudoeinstein metrics on lie groups. Computing biinvariant pseudometrics on lie groups for. A 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. Milnortype theorems for left invariant riemannian metrics on lie groups hashinaga, takahiro, tamaru, hiroshi, and terada, kazuhiro, journal of the mathematical society of japan, 2016. Let h,i be a left invariant metric on g, and let x, y, z be left invariant vector. 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. 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. Combined with some known results in the literature, this gives a proof of the main theorem of this paper. On lie groups with left invariant semiriemannian metric 11 and.
Lengyeln et oth1 1university of debrecen 2college of ny regyh aza symposium on finsler geometry, 20 sapporo. Our results improve a bit of milnors results of 7 in the three. In, milnor studied the curvatures of left invariant metrics on lie groups which outline what is the riemannian geometry of such a lie group. A leftsymmetric algebraic approach to left invariant flat.
Let g be a full connected semisimple isometry lie group of a connected riemannian symmetric space m gk with the stabilizer k. Alekseevskyconjecturedin1975that,whenever m gk is a simply connected. Department of mathematics university of mohaghegh ardabili p. 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. Our procedure is based on the moduli space of left invariant riemannian metrics.
Milnortype theorems for left invariant riemannian metrics on lie groups hashinaga, takahiro, tamaru, hiroshi, and terada, kazuhiro, journal of the mathematical society of japan, 2016 on the moduli spaces of left invariant pseudoriemannian metrics on lie groups kubo, akira, onda, kensuke, taketomi, yuichiro, and tamaru, hiroshi, hiroshima. A remark on left invariant metrics on compact lie groups lorenz j. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. 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. Left invariant pseudoriemannian metrics on solvable lie. The approach is to consider an orthonormal frame on the lie algebra, since all geometric information is gained considering an inner product on it vector space, once we have the correspondence between left invariant metrics and inner products on the lie algebra. Masumeh nejadahmad, hamid reza salimi moghaddam submitted on 2 mar 2019. Lorentzian left invariant metrics on three dimensional unimodular lie groups and their curvatures authors. Homogeneous geodesics of left invariant randers metrics on a threedimensional lie group dariush lati.
It thus defines a bilinear product b on g the lie algebra of g. G gk m the canonical projection which is a riemannian submersion for some gleft invariant and kright invariant riemannian metric on g, and d is a unique subriemannian metric on g defined by this metric and. 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. Mar 04, 2020 the most downloaded articles from advances in mathematics in the last 90 days. From this is easy to take information about levicivita connection, curvatures and etc. Lie group that admits a biinvariant metric is a homogeneous riemannian manifoldthere exists an isometry between that. This case is very special because the bi invariant metric has positive curvature, and therefore so does any. Curvatures of left invariant metrics on lie groups. An important role is played by the heisenberg mani. 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. In particular, as an extreme case, he gave a detailed structure theory on flat metrics, that is, the curvature is zero theorem 1.
There is a left symmetric algebraic approach with an explicit formula to the classification theorem given by milnor. Euler equations and totally geodesic subgroups 81 the paper is organised as follows. Lorentzian left invariant metrics on three dimensional unimodular lie groups and their curvatures preprint pdf available march 2019 with 195 reads how we measure reads. We study the spaces of left invariant riemannian metrics on a lie group up to isometry, and up to isometry and scaling. Using these formulas, we prove that at any point of an arbitrary connected noncommutative nilpotent lie group, the flag curvature of any left invariant matsumoto and kropina metrics of berwald type admits zero, positive and negative values, this is a generalization of wolfs theorem. In chapter 1 we introduce the necessary notions and state the basis results on the curvatures of lie groups. Ricci curvatures of left invariant finsler metrics on lie groups. For the convenience, we call such a lie group a lcs lie group. 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.
Geodesics of left invariant metrics on matrix lie groups. We establish the existence of solvable lie groups of dimension 4 and left invariant riemannian metrics with zero bach tensor which are neither conformally einstein nor half conformally flat. We begin by describing the general lie theoretic setup of arnold 1. Also we calculate the levicivita connection, and then ricci tensor associated with left invariant pseudoriemannian metrics on the unimodular lie groups of dimension three. In this article, we focus on left invariant pseudoeinstein metrics on lie groups. However, g admists a canonical metric if we view g as a flat and globalizable absolute parallelism w as in o1. In this paper, we formulate a procedure to obtain a generalization of milnor frames for left invariant pseudoriemannian metrics on a given lie group. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics.
Then there may be many g invariant riemannian metrics on m. A left invariant metric on a connected lie group is also right invariant if and only if adx is skewadjoint for every x g. 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. In this section, we will show that the compact simple lie groups s u n for n. While any contractible lie group may be made isometric to. In section 3, we classify the left invariant metrics on so3 with nonnegative curvature. Lie derivatives along antisymmetric tensors, and the m. In this paper, we explore the problem for smalldimensional lie groups. Section 2 begins with a brief presentation of geodesic ow on groups and the eulerarnold equation. Ricci curvature of left invariant metrics on solvable unimodular lie groups. Geodesics equation on lie groups with left invariant metrics.
More than two decades ago, it was formulated 10 as the gauging of a free differential algebra fda 4,5,10,19,22, an algebraic structure that extends the cartanmaurer equations of an ordinary lie algebra gby including pform potentials, besides the usual left invariant 1forms corresponding to the lie group. Left invariant randers metrics on 3dimensional heisenberg group. Left invariant randers metrics on 3dimensional heisenberg. Pdf the canonical geometry of a lie group semantic scholar. For each simply connected threedimensional lie group we determine the automorphism group, classify the left invariant riemannian metrics up to automorphism, and study the extent to which. Pdf on lie groups with left invariant semiriemannian metric. A remark on left invariant metrics on compact lie groups. Such map allows to parallel translate vectors along curves, con necting tangent spaces of m. Ricci curvatures of left invariant finsler metrics on lie. Index formulas for the curvature tensors of an invariant metric on a lie group are obtained. For a lie group, a natural choice is to take a leftinvariant metric.
In this paper, we prove several properties of the ricci curvatures of such spaces. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. Left invariant randers metrics on 3dimensional heisenberg group z. Let g be a left invariant metric on a connected lie group g. Leftinvariant einstein metrics on lie groups andrzej derdzinski august 28, 2012. Oct 10, 2007 a restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied.
From the above definition of homogeneous nilmanifolds, it is clear that any nilpotent lie group with left invariant metric is a homogeneous nilmanifold. Left invariant einstein metrics on lie groups andrzej derdzinski august 28, 2012 differential geometry seminar department of mathematics the ohio state university. Curvature of left invariant riemannian metrics on lie groups. Metric tensor on lie group for left invariant metric. Homogeneous geodesics of left invariant randers metrics on a. Curvatures of left invariant metrics on lie groups by j milnor.
Our results improve a bit of milnors results of 7 in the threedimensional case. Curvatures of left invariant randers metrics on the ve. On lie groups with left invariant semiriemannian metric. We study some surprising consequences of this shift in perspective. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. Lie groups which admit flat left invariant metrics 259 hence, for 1,2, the length of y.
Most downloaded advances in mathematics articles elsevier. In this paper, we see that such spaces can be identified with the orbit spaces of certain isometric actions on noncompact symmetric spaces. Centralizer of reeb vector field in contact lie groups hassanzadeh, babak, journal of geometry and symmetry in physics, 2018. 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. Curvatures of left invariant metrics on lie groups john.
While there are few known obstruction for a closed manifold. When all the left translations lx are isometries, we call g a left invariant metric. 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. 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. Milnor in the well known 2 gave several results concerning curvatures of left invariant riemannian metrics on lie groups. Jump to content jump to main navigation jump to main navigation.
We examine these three cases on a riemannian 3manifold, and prove the following. We study also the particular case of bi invariant riemannian metrics. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Mohamed boucetta, abdelmounaim chakkar submitted on 12 mar 2019. Pdf we find the riemann curvature tensors of all leftinvariant lorentzian metrics on 3dimensional lie groups. In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. In order to study a lie group with left invariant metric, it is best.
Curvature of left invariant riemannian metrics on lie. To dennis sullivan on the occasion of his 70th birthday. When the manifold is a lie group and the metric is left invariant the curvature is also strongly. 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. Left invariant flat metrics on lie groups are revisited in terms of left symmetric algebras which correspond to affine structures. Left invariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. The results are applied to the problem of characterizing invariant metrics of zero and nonzero constant curvature. Left invariant metrics on a lie group coming from lie.
Curvatures of left invariant metrics on lie groups john milnor. We investigate nontrivial mquasieinstein metrics on pseudoriemannian lcs lie group. 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 with nonnegative curvature on compact. On simple lie groups, we show that there is always an einstein biinvariant metric.
We give the explicit formulas of the flag curvatures of left invariant matsumoto and kropina metrics of berwald type. Leftinvariant lorentzian metrics on 3dimensional lie groups. Geodesics and curvatures of special subriemannian metrics. In the third section, we study riemannian lie groups with. When the manifold is a lie group and the metric is left invariant the curvature is also strongly related to the group s structure or equivalently to the lie algebra s structure. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Chapter 18 metrics, connections, and curvature on lie groups. We proved that although there exists only trivial ricci soliton on pseudoriemannian lcs lie group, any left invariant pseudoriemannian metric on lcs lie group is nontrivial mquasieinstein. Let m gk be an effective coset space of a connected lie group by a compact subgroup.
Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the threedimensional lie groups. Using these formulas, we prove that at any point of an arbitrary connected noncommutative nilpotent lie group, the flag curvature of any left invariant matsumoto and kropina metrics of. Bertrand mate of timelike biharmonic legendre curves in. The space of leftinvariant metrics on a lie group up to. For all left invariant riemannian metrics on threedimensional unimodular lie groups, there exist particular left invariant orthonormal frames, socalled milnor frames.
If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the biinvariant killing form. Sectional curvatures are therefore all nonnegative. As a corollary we show that all left invariant pseudoriemannian metrics of arbitrary signature on the lie groups of real hyperbolic spaces have constant sectional curvatures. The most familiar nilpotent lie groups are matrix groups whose diagonal entries are 1 and whose lower diagonal entries are all zeros. 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. On the left invariant randers and matsumoto metrics of. Geometrically a lie algebra g of a lie group g is the set of all left invariant vector. Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three.
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