In this paper we show that every invariant finsler metric on a lie group g induces an invariant finsler metric on quotient group gh in the natural way, where h is a closed normal lie subgroup of g. Sep 01, 1976 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. Abstract riemannian submersions and lie groups william. Faculty of mathematics, kyushu university produced and listed by. Furthermore, we show that the left invariant pseudoeinstein metric on sl2 is unique up to a constant. For the case of bi invariant metrics, proposition 1 extends as follows.
For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. Considering a lie group g as a special homogeneous manifold, we have the following facts and results. Lie group that admits a bi invariant metric is a homogeneous riemannian manifoldthere exists an isometry between that. Every compact lie group admits one such metric see proposition 2. Ricci curvatures of left invariant finsler metrics on lie. Curvature of left invariant riemannian metrics on lie. Curvatures of left invariant metrics on lie groups, advances in mathematics, 21 1976, 293329. Invariant metrics with nonnegative curvature on compact lie.
I lagrangian mechanics is the geodesic ow on the group g. Classification of leftinvariant metrics on the heisenberg. In the third section, we study riemannian lie groups. In fact it is possible to show that g admit such a metric iff it is the. In the sequel, the identity element of the lie group, g, will be denoted by e or. The most familiar nilpotent lie groups are matrix groups whose diagonal entries are 1 and whose lower diagonal entries are all zeros.
This chapter deals with lie groups with special types of riemannian metrics. 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. Invariant metrics with nonnegative curvature on compact. In this paper, we prove several properties of the ricci curvatures of such spaces. However, groups with such metrics are always noncompact as a consequence of bochners vanishing theorem. The existencenonexistence problems of ice left invariant metrics are far from being wellunderstood. We demonstrate that there are no nontrivial isospectralities among the left invariant metrics on a compact lie group of rank one. A metric on g that is both left invariant and right invariant is called bi invariant. Constructing a metric on a lie group mathematics stack exchange. Proposition any compact or abelian lie group g admits a bi invariant metric. In 2, lauret showed that if a nonabelian lie group has a unique left invariant riemannian metric up to homothety, then it is isomorphic to either the hyperbolic space h n, or the direct. Ricci curvatures of left invariant finsler metrics on lie groups.
In fact it is possible to show that g admit such a metric iff it is the cartesian product of an abelian and a compact lie group, taking this idea forward one can show that. Any compact or abelian lie group g admits a bi invariant metric. Ricciflat left invariant lorentzian metrics on 2step nilpotent lie groups. John milnor, curvatures of left invariant metrics on lie groups, advances in mathematics, volume 21, issue 3, september 1976. If h is a left invariant metric with nonnegative curvature on a compact lie group g, then the unique inverselinear path from any. 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. Geodesics equation on lie groups with left invariant metrics.
For a given lie group g, examine whether g admits a distinguished left invariant metric. In the nonflat case one received by the editors november 27, 1962. I other than the trivial group rn, we can consider much more general group g describing the symmetry of motion. 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. Metrics of negative ricci curvature on project euclid. In chapter 1 we introduce the necessary notions and state the basis results on the curvatures of lie groups.
Semantic scholar extracted view of curvature of left invariant riemannian metrics on lie groups by m. Left invariant semi riemannian metrics on quadratic lie groups. Pdf curvature of left invariant riemannian metrics on lie groups. Let n,g be a connected nilpotent lie group with a left invariant riemannian metric and d be the induced distance function on n. Biinvariant means on lie groups with cartanschouten. Among all the riemannian metrics on a lie groups, those for which the left translations or the right translations are isometries are of particular interest because they take the group. Classification of leftinvariant metrics on the heisenberg group. For left invariant vector elds the rst three terms of the right hand side of 2. Chapter 17 metrics, connections, and curvature on lie groups. Right invariant metrics since a lie group, g,isasmoothmanifold,wecanendowg with a riemannian metric. Curvatures of left invariant randers metrics on the ve.
Pdf induced invariant finsler metrics on quotient groups. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. Itisfairlyeasytocheck that the above induces a left invariant metric on g. Advanced geometric methods in computer science, lecture notes 2018 pdf, web created on april 8, 2021 at 09. In this article we do not assume the quotient groups gh are reductive. We study also the particular case of bi invariant riemannian metrics. We show g has constant sectional curvature, and describe the irreducible unitary representations of g. Pdf lie groups as kaehler manifolds with killing norden. Mar 12, 2019 there are five unimodular simply connected three dimensional unimodular non abelian lie groups. Let m denote the set of left invariant metrics on g. Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three.
Sectional and ricci curvature for threedimensional lie groups. The main aim of this paper is the study of the flatness of left invariant semi riemannian metrics on quadratic lie groups i. We begin by describing the general lie theoretic setup of arnold 1. The thickness of leftinvariant metrics on compact connected lie. For any left invariant metric q on g, the tangent space tqm at q is left invariant symmetric, bilinear forms on g. An example of a kahler manifold with norden metric, illustrating the obtained results, is constructed on. Lie groups which admit flat left invariant metrics 259 hence, for 1,2, the length of y. Full text pdf 122k abstracts references1 full text pdf 122k. Dynamics of geodesic ows with random forcing on lie groups. Let n,g be a connected nilpotent lie group with a left invariant riemannian metric and d be the induced distance function on tv. Classical mechanics for general group g of the symmetry of motion. It is easy to see that every connected complex homogeneous manifold equipped with a free, transitive, holomorphic action by a real lie group is obtained this way. It thus defines a bilinear product b on g the lie algebra of g. Curvature of left invariant riemannian metrics on lie groups.
Pdf left invariant semi riemannian metrics on quadratic lie. In this case, i defines a left invariant complex structure on the corresponding lie group. Milnor in the well known 2 gave several results concerning curvatures of left invariant riemannian metrics. Curvatures of left invariant metrics on lie groups. G hwhere gand hare lie groups with left invariant metric. In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. Scalar curvatures of leftinvariant metrics on some. Leftinvariant metrics on lie groups and submanifold geometry. Curvatures of left invariant metrics on lie groups core. The group g is a lie group with a left invariant metrics.
Moreover, we give basic results on closed subgroups, bi invariant metrics. Geometry of 2step nilpotent groups with a left invariant metric. Chapter 17 discrete laplacians on polyhedral surfaces. Milnortype theorems for leftinvariant riemannian metrics. From now on elements of n are regarded as left invariant vector elds on n. In this paper we investigate the large scale geometry of connected nilpotent lie groups equipped with left invariant riemannian metrics by studying their quasiisometric embeddings into various metric spaces. Our results improve a bit of milnors results of 7 in the three. Basic results on bi invariant metrics and killing forms are then discussed, e. John milnor, curvatures of left invariant metrics on lie groups. Aug 01, 2015 the approach used in the present study is to classify all of the left invariant metrics on a particular lie group g, or on some class of lie groups.
Let h t be an inverselinear path of left invariant metrics on g beginning at a bi invariant metric h 0. All known examples of lie groups admitting left invariant riemannian einstein metrics with negative einstein constants lie group. Our results improve a bit of milnors results of 5 in the threedimensional case, and k owalski and nik. Ricciflat leftinvariant lorentzian metrics 4pt on 2step nilpotent lie. A riemannian metric that is both left and rightinvariant is called a bi invariant metric. To determine the lie groups that admit a flat eventually complete left invariant semiriemannian metric is an open and difficult problem. For every left invariant riemannian metrics on threedimensional unimodular lie groups, milnor 14 constructed certain orthonormal basis of the corresponding metric lie algebras. In the third section, we study riemannian lie groups with. Lie groups endowed with a bi invariant semi riemannian metric. Hence, in any lie group there exists a torsionfree connection, defined. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. In this paper we show that every invariant finsler metric on a lie group g induces an invariant finsler metric on quotient group gh in the natural way, where h is a closed normal. Constructing a metric on a lie group mathematics stack.
Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. To dennis sullivan on the occasion of his 70th birthday. Some properties of lie groups and lie algebras equipped with a bi invariant complex structure and a pair of killing norden metrics are studied. Although milnor used leftinvariant metrics we prefer to use rightinvariant metrics although. Such bases are nowadays called the milnor frames, and have played crucial roles in many branches of geometry. Pdf on lie groups with left invariant semiriemannian metric. If h,i be an inner product ong,andset hu,vi g hdl g 1 gu,dl g 1 gvi, for all u,v 2 t gg and all g 2 g. Let g be a left invariant metric on a connected lie group g. Leftinvariant pseudoriemannian metrics on some lie groups. We give a useful necessary and sufficient condition that.
Milnor, curvatures of left invariant metrics on lie groups, advances in math. Lie groups equipped with left invariant riemannian metrics by studying their quasiisometric embeddings into various metric spaces. Pdf on a lie group with constant negative sectional. For example, the heisenberg group is a 2step nilpotent lie. Milnortype theorems for leftinvariant riemannian metrics on. A compact lie group possesses a bi invariant metric. If the lie group is noncommutative and nilpotent, then there must be three directions whose ricci curvature is.
May 21, 2009 thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three. Curvatures of left invariant metrics on lie groups by j milnor. The value of h t at e is determined in terms of h 0 by some selfadjoint. Pdf left invariant metrics and curvatures on simply. On lie groups with left invariant semiriemannian metric r. Curvatures of left invariant metrics on lie groups john. Left invariant metrics and curvatures on simply connected. In particular, a theorem of wallace mentioned in milnors paper confirms your conjecture.
Such a manifold g, i is called a complex group manifold. Biinvariant means on lie groups with cartanschouten connections. Heber 9 and lauret 11 made substantial progress towards understanding the moduli and properties of such metrics on solvable lie groups. The construction method used in sc,pr exploits the fact that the lie groups in question have rank at least two. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. From this is easy to take information about levicivita connection, curvatures and etc. Then they studied bi invariant finsler metrics on lie groups and obtained a nec essary and sufficient condition for a lie group to have bi invariant finsler metrics. From the above definition of homogeneous nilmanifolds, it is clear that any nilpotent lie group with left invariant metric is a homogeneous nilmanifold. The purpose of this paper is to investigate ricciflatness of left invariant lorentzian metrics on 2step nilpotent lie groups.
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