The geometry of any lie group g with left invariant riemannian metric re ects strongly the algebraic structure of the lie algebra g. This procedure is an analogue of the recent studies on leftinvariant riemannian metrics, and is based on the moduli space of leftinvariant pseudoriemannian metrics. Homogeneous geodesics of left invariant randers metrics on a threedimensional lie group dariush lati. Left invariant randers metrics on 3dimensional heisenberg group. 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. Leftinvariant pseudoriemannian metrics on fourdimensional. Let gbe a lie group with connected subgroup h, and let gand h have lie algebras g and h, respectively. Computing biinvariant pseudometrics on lie groups for. The space of leftinvariant metrics on a generalization. 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. For this reason, lie groups form a class of manifolds suitable for testing general hypotheses and conjectures. Thanks for contributing an answer to mathematics stack exchange. The set of such frames becomes a homogeneous principal autfbundle by letting.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Department of mathematics university of mohaghegh ardabili p. Left invariant connection on lie groups mathoverflow.
Combined with some known results in the literature, this gives a proof of the main theorem of this paper. The analysis is carried out by classifying all left invariant lorentzian metrics on the connected, simplyconnected fivedimensional lie group having a lie algebra with basis vectors and and non. This notation is poor, though, since it doesnt indicate whether thats rightmultiplicationbyg or leftmultiplicationbyg. In the third section, we study riemannian lie groups with. G, where lx is the left translation satisfying lx y xy.
In this context, it is particularly interesting to investigate left invariant metrics on a compact connected lie group g with lie algebra g. Integrability of invariant metrics on the diffeomorphism. In all the following we will concentrate on the following left invariant, drift free control system on with 3 controls. Curvatures of left invariant metrics on lie groups core. We show that any nonflat left invariant metric on g has conjugate points and we describe how some of the conjugate points arise. 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. Index formulas for the curvature tensors of an invariant metric on a lie group are. 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. 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. Leftinvariant einstein metrics on lie groups andrzej derdzinski august 28, 2012.
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. Leftinvariant pseudoriemannian metrics on fourdimensional lie groups with nonzero schoutenweyl tensor p. Milnors paper gave many known results on the topic, proved and conjectured many new ones. Homogeneous geodesics of left invariant randers metrics. Invariant metrics with nonnegative curvature on compact. Invariant metrics with nonnegative curvature on compact lie. International conference on mathematics and computer science, june 2628, 2014, bra. There is a onetoone correspondence between ginvariant lorentz metrics on gh, and inner products on gh that are adhinvariant i. Let h,i be a left invariant metric on g, and let x, y, z be left invariant vector.
Curvatures of left invariant metrics on lie groups. 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. In this paper, we formulate a procedure to obtain a generalization of milnor frames for leftinvariant pseudoriemannian metrics on a given lie group. Here we will examine various geometric quantities on a lie goup g with a leftinvariant or biinvariant metrics. A remark on left invariant metrics on compact lie groups. Left invariant metrics on a lie group coming from lie algebras. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. Then, using left translations defines a left invariant. Note that any lie group admits a left or right invariant pseudometric. Integrability of invariant metrics on the diffeomorphism group of the circle adrian constantin and boris kolev abstract. Dynamics of geodesic flows with random forcing on lie.
Since the span of the set of lie brackets generated by coincides with, the proposition is a consequence of chows theorem. Left invariant randers metrics on 3dimensional heisenberg. If the metric is biinvariant, then the geodesics are the left and right translates of 1parameter subgroups, but, in general, if all you have leftinvariance, this is far from the case. For all left invariant riemannian metrics on threedimensional unimodular lie groups, there exist particular left invariant orthonormal frames, socalled milnor frames. Let g be a lie group which admits a flat left invariant metric. Combined with some known results in the literature, this. 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 this is easy to take information about levicivita connection, curvatures and etc. Left invariant metrics on a lie group coming from lie. Biinvariant and noninvariant metrics on lie groups. Leftinvariant pseudoeinstein metrics on lie groups. 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 control systems on lie groups rory biggs and claudiu c. Jul 19, 2017 leftinvariant pseudoriemannian metrics on fourdimensional lie groups with nonzero schoutenweyl tensor p.
A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. To determine the lie groups that admit a flat eventually complete left invariant semiriemannian metric is an open and difficult. Milnortype theorems for leftinvariant riemannian metrics on lie groups hashinaga, takahiro, tamaru, hiroshi, and terada, kazuhiro, journal of the mathematical society of japan, 2016. The analysis is carried out by classifying all left invariant lorentzian metrics on the connected, simplyconnected fivedimensional lie group having a. We find the riemann curvature tensors of all left invariant lorentzian metrics on 3dimensional lie groups.
Conjugate points in lie groups with leftinvariant metrics. 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. We study also the particular case of bi invariant riemannian metrics. In this section, we will show that the compact simple lie groups s u n for n. Biinvariant means on lie groups with cartanschouten.
Klepikov 1 russian mathematics volume 61, pages 81 85 2017 cite this article. These group geodesics correspond to the ones of a leftinvariant metric for. Lie groups which admit left inv ariant riemannian metrics of constant neg ative curv ature, admit left inv ariant lorentzian metrics of constan t positive curvature. 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. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Left invariant vector fields of a lie group physics forums. The moduli space of leftinvariant metrics both riemannian and pseudoriemannian settings milnortype theorems one can examine all leftinvariant metrics this can be applied to the existence and nonexistence problem of distinguished e.
We consider stochastic perturbations of geodesic flow for leftinvariant metrics on finitedimensional lie groups and study the h\ormander condition and some properties of the solutions of the corresponding fokkerplanck equations. The same remarks apply to homogeneous spaces, which are certain quotients of lie groups. Theorem milnor if z belongs to the center of the lie algebra g, then for any left invariant metric the inequality kz. Left invariant semi riemannian metrics on quadratic 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. Presumably, tex\frac\partial g\partial xitex is supposed to be the partial derivative in the ith direction of the multiplicationbyg automorphism of g. We consider stochastic perturbations of geodesic flow for left invariant metrics on finitedimensional lie groups and study the h\ormander condition and some properties of the solutions of the corresponding fokkerplanck equations. Whatevers being done, its just working through the definition of leftinvariant.
We do not require the inner products to be positive. Homogeneous geodesics of left invariant randers metrics on a. For left invariant vector elds the rst three terms of the right hand side of 2. Geodesics of left invariant metrics on matrix lie groups. M, with velocity t is a finslerian geodesic if d t t ft 0, with reference vector t. On lie groups with left invariant semiriemannian metric r. Milnor in the well known 2 gave several results concerning curvatures of left invariant riemannian metrics on lie groups. 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.
Intro preliminaries case 1 case 2 summary abstract 12 background leftinvariant riemannian metrics on lie group. A lie groupis a smooth manifold g with a group structure such that the map. 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. On the moduli space of leftinvariant metrics on a lie group. Invariant metrics left invariant metrics these keywords were added by machine and not by the authors. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2. 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. The space of leftinvariant metrics on a generalization of. Geodesics equation on lie groups with left invariant metrics. In particular, we give an example of a left invariant metric such that so3 is totally geodesic in gl3. Invariant control systems on lie groups rory biggs claudiu c. In this paper, we formulate a procedure to obtain a generalization of milnor frames for left invariant pseudoriemannian metrics on a given lie group. Left invariant connections ron g are the same as bilinear. 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.
Ricci curvature of left invariant metrics on solvable. Left invariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. A driftfree left invariant control system on the lie group. Left invariant metrics and curvatures on simply connected three. More precisely, decomposing endg into the direct sum of the subspaces consisting of all endomorphisms of g which are selfadjoint or, respec. From now on elements of n are regarded as left invariant vector elds on n. Lie groups endowed with left invariant riemannian metrics. For all leftinvariant riemannian metrics on threedimensional unimodular lie groups, there exist particular leftinvariant orthonormal frames, socalled milnor frames. Pdf leftinvariant lorentzian metrics on 3dimensional lie.
Curvature of left invariant riemannian metrics on lie. Leftinvariant lorentzian metrics on 3dimensional lie groups. On lifts of leftinvariant holomorphic vector fields in complex lie groups alexandru ionescu1 communicated to. Curvature of left invariant riemannian metrics on lie groups. Ricci curvature of left invariant metrics on solvable unimodular lie groups. Metric tensor on lie group for left invariant metric. Conjugate points in lie groups with left invariant metrics 14 construction of the lie functor. Compact simple lie groups admitting leftinvariant einstein.
We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. 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. Curvatures of left invariant randers metrics on the ve. In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. Browse other questions tagged metricspaces liegroups liealgebras or ask your own question.
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