Curvatures of left invariant metrics on lie groups john. In this article we study antikahler structures considering the distinguished cases where the complex structure j is abelian or bi invariant. We first show that every compact lie group admits a biinvariant finsler metric. We obtain a partial description of the totally geodesic submanifolds of a 2step, simply connected nilpotent lie group with a left invariant metric. On the riemannian geometry of tangent lie groups springerlink. I was reading the classical paper from milnor entitled curvature of left invariant metrics on lie groups. For instance, geodesics through the identity coincide with oneparameter groups, so the riemannian exponential map coincides with the lie group exponential. We investigate their geometry, especially holonomy groups and decomposability of these metrics. Classical yangbaxter equation and left invariant affine geometry on lie groups article pdf available in manuscripta mathematica 1144 april 2002 with 50 reads how we measure reads. Left invariant pseudoriemannian metrics on solvable lie groups.
Speci cally, many examples of left invariant hermitian fstructures on 6dimensional liform lie groups were presented 4. Pdf left invariant geometry of lie groups researchgate. The groups that are of greatest interest in geometry and topology typ ically consist. The second one is devoted to the preliminaries properties of left invariant riemannian metrics on lie groups. Finally, we dwell on the class of liform lie groups which also can be considered as a generalization in some sense of the classical 3dimensional heisenberg group. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. Perspectives 144 exercises and further results 147 notes 153 chapter iii structure of semisimple lie algebras 1. Left invariant einstein metrics on lie groups andrzej derdzinski august 28, 2012 differential geometry seminar department of mathematics the ohio state university. Invariant control systems on lie groups rory biggs claudiu c. The riemannian geometry of bi invariant metrics is very nice. Apart from the intrinsic interest, the theory of lie groups and their representations is used in various parts of mathematics.
Introduction to lie groups and lie algebras stony brook. Just as an abstract group is a coperent system of cyclic groups, a lie group is a very coherent system of oneparameter groups. Lie groups, lie algebra and left invariant vector fields. In the present article we consider a lie group g equipped with a left invariant riemannian metric g. Section 4 geometry of lie groups with a left invariant metric. The notes are selfcontained except for some details about topological groups for which we refer to chevalleys theory of lie. Then, we prove that every compact connected lie group is a symmetric finsler space with respect. 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. Lecture 3 lie groups and geometry july 29, 2009 1 integration of vector fields on lie groups let mbe a complete manifold, with a vector eld x.
On lie groups with left invariant semiriemannian metric cent. A metric on g that is both leftinvariant and rightinvariant is called biinvariant. Lorentz geometry of 4dimensional nilpotent lie groups. Differential geometry, lie groups, and symmetric spaces. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. A vector field x on a lie group g is left invariant if. Lie groups, physics, and geometry an introduction for physicists, engineers and chemists describing many of the most important aspects of lie group theory, this book presents the subject in a hands on way. 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. Anintroductiontoliegroups andsymplecticgeometry a series of nine lectures on lie groups and symplectic geometry delivered at the regional geometry institute in park city, utah, 24 june20 july 1991. For the case of biinvariant metrics, proposition 1 extends as follows. Section 1 basic properties and examples of symplectic. The geometry of left invariant structures on nilpotent lie groups vitaly v.
The purpose of the first two sections, therefore, is to provide. Each solution r of the classical yang baxter equation on g, corresponds to a connected lie subgroup h of tg such. Curvatures of left invariant metrics on lie groups john milnor. G its cotangent bundle considered as a lie group, where g acts on lieg via the coadjoint action. From now on until further notice gis a lie group of dimension n. Curvature of left invariant riemannian metrics on lie groups. The geometry of leftinvariant structures on nilpotent lie groups.
On the existence of biinvariant finsler metrics on lie groups. Notes on differential geometry and lie groups upenn cis. Proceedings of the 1st international meeting on geometry and topology braga, 1997, cent. This article is intended to be a presentation of a few results about the geometry of lie groups which admit biinvariant metrics, as well as a basic introduction the. We study also the particular case of bi invariant riemannian metrics. Section 8 other topics in the left invariant geometry of lie groups. Obviously gcan be made leftinvariant by placing a nondegenerate bilinear form gon gand then requiring g pv. Invariant control systems on lie groups rory biggs and claudiu c. In the complex geometry of lie groups, we have two distinguished classes of left invariant complex structures, namely, abelian and biinvariant. Contents section 1 basic properties and examples of symplectic structures 1. Citeseerx document details isaac councill, lee giles, pradeep teregowda. For a thorough treatment of the history of the subject, see. 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. The topology on homeom induced by any of these metrics is separable.
In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent lie groups equipped with left invariant riemannian metrics. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex lie groups admitting irreducible left. Invariant bilinear forms and semisimplicity of classical lie algebras. For all left invariant riemannian metrics on threedimensional unimodular lie groups, there exist particular left invariant orthonormal frames, socalled milnor frames. Notes on the riemannian geometry of lie groups semantic scholar. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex lie groups admitting irreducible left invariant flat complex projective structures. Speci cally, many examples of left invariant hermitian fstructures on. Bruhat, lectures on lie groups and representations of locally compact groups, notes by s. Obviously gcan be made left invariant by placing a nondegenerate bilinear form gon gand then requiring g pv. 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. Thanks for contributing an answer to mathematics stack. Lectures on lie groups and geometry imperial college london. We study also the particular case of biinvariant riemannian metrics.
Left invariant randers metrics of berwald type on tangent lie. It is clear that the lie bracket of left invariant vector fields is left invariant so we can use this as an alternative definition of the bracket on g, that is we make the proposition above a definition in the case of this action. Similarly, a lie group is a group that is also a smooth manifold, such that the two. The problem of finding which lie groups admit a left invariant contact structure contact lie groups, is then still open. Lie transformation groups and geometry 15 the set g of all left invariant vector. Rather than concentrating on theorems and proofs, the book shows the relation of lie groups with many branches of. 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. The geometry of leftinvariant structures on nilpotent lie groups vitaly v. There are some recent papers on invariant finsler metrics on homogeneous manifolds see e. But in general such structures are not invariant under left translations of the lie group. Pdf left invariant contact structures on lie groups. A compact lie group possesses a biinvariant metric.
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. Leftinvariant einstein metrics on lie groups andrzej derdzinski august 28, 2012 differential geometry seminar department of mathematics the ohio state university. Most lie groups do not have bi invariant metrics, although all compact lie groups do. Curvatures of left invariant metrics on lie groups. A riemannian metric on g is said to be biinvariant if it turns left and right translations into. In this paper, we study the geometry of lie groups with biinvariant finsler metrics. Curvatures of left invariant metrics on lie groups core. In this paper, for any leftinvariant riemannian metrics on any lie groups, we give a procedure to obtain an analogous of milnor frames, in the sense that the bracket relations. Balashchenko belarusian state university, minsk, belarus xix geometrical seminar zlatibor, serbia, august 28 september 4, 2016 main goals. Let g be a lie group of even dimension and let g, j be a left invariant antikahler structure on g. Helgasons books differential geometry, lie groups, and symmetric spaces and groups and geometric analysis, intermixed with new content created for the class. Geodesics equation on lie groups with left invariant metrics. The main result of this paper gives a characterization of leftinvariant almost.
Readings introduction to lie groups mathematics mit. The riemannian geometry of biinvariant metrics is very nice. Since it is invariant under left and right translations, it is called bi invariant. For all leftinvariant riemannian metrics on threedimensional unimodular lie groups, there exist particular leftinvariant orthonormal frames, socalled milnor frames. Lie transformation groups and geometry 15 the set g of all leftinvariant vector.
Theory, is based on using the methods of algebraic geometry see 17. A result from gromov ensures the existence of a contact structure on any connected noncompact odd dimensional lie group. Lie 18421899, who rst encountered lie groups and the associated lie algebras when studying symmetries of partial di erential equations. We show the correspondence between left invariant flat projective structures on lie groups and certain prehomogeneous vector spaces. The existence of a metric invariant under both left and right multiplication is a nontrivial question. The geometry of leftinvariant structures on nilpotent lie. The basic object mediating between lie groups and lie algebras is the oneparameter group. Leftinvariant metrics on lie groups and submanifold geometry. These lecture notes were created using material from prof. So we can identifyg with the set of left invariant vector fields. 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.
Geometry of 2 step nilpotent groups with a left invariant metric. Leftinvariant vector fields posted on october 1, 2010 by raghu mahajan a lie group is a group that is also a differentiable manifold such that the product and inverse maps of the group are continuous. Mis called the ow of xif, for any function fand any time. Sigurdur helgason, differential geometry, lie groups and symmetric spaces. Since it is invariant under left and right translations, it is called biinvariant. We find that if g admits a left invariant antikahler structure g, j where j is abelian then the lie algebra of g is unimodular and g, g is a flat. Precisely, we prove that there do not exist quasiisometric embeddings of such a nilpotent lie group into either a cat0 metric space or an alexandrov metric space with curvature bounded below. Pdf classical yangbaxter equation and left invariant.1520 892 623 1013 1420 988 886 1561 1202 546 1164 19 1445 533 506 455 278 1494 818 1156 50 1198 482 1248 407 896 422 620 1064 799 142 576