Curvature of left invariant metrics on lie groups pdf

Milnor in the well known 2 gave several results concerning curvatures of left invariant riemannian. Lie algebra g of a lie group g is the set of all left invariant vector. Glg, we get our second criterion for the existence of a biinvariant metric on a lie group. In this article, we focus on leftinvariant pseudoeinstein metrics on lie groups. Metrics, connections, and curvature on lie groups applying theorem 18. Let h,i be a left invariant metric on g, and let x, y, z be left invariant vector. We classify determine the moduli space of leftinvariant pseudoriemannian metrics on some particular lie groups. Geodesics equation on lie groups with left invariant metrics. If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the biinvariant killing form. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Invariants constructed using covariant derivatives up to order n are called nth order differential invariants the riemann tensor is a multilinear operator of. Metrics, connections, and curvature on lie groups applying theorem 17. For example, in 9, ha and lee complete milnors classification of the available signatures of the ricci curvature of left invariant metrics on threedimensional lie groups, and in 12, kremlev.

Curvature spectra of simple lie groups andrzej derdzinski swiatoslaw r. Curvature of left invariant riemannian metrics on lie groups. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. Einstein metrics on lie groups 3 proof of theorem b. The connection between the lie group structure of the manifold and the leftinvariant riemannian structure is most easily seen by using a leftinvariant frame. A riemannian metric that is both left and rightinvariant is called a biinvariant metric. Ricci curvatures of left invariant finsler metrics on lie. Invariant metrics with nonnegative curvature on compact. Left invariant metrics and curvatures on simply connected. Every compact lie group admits a biinvariant metric, which has nonnegative sectional curvature. This problem has been studied mainly in the low dimensions.

Lie algebra g of a lie group g is the set of all left invariant vector fields on the lie. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Glg, we get our second criterion for the existence of a bi invariant metric on a lie group. 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. Here we will examine various geometric quantities on a lie goup g with a leftinvariant or biinvariant metrics. Milnortype theorems for leftinvariant riemannian metrics on lie groups hashinaga, takahiro, tamaru, hiroshi, and terada, kazuhiro, journal of the mathematical society of japan, 2016. 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. Pdf we give the explicit formulas of the flag curvatures of left invariant matsumoto and kropina metrics of berwald type. 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. 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. These are polynomials constructed from contractions such as traces. Pdf on lie groups with left invariant semiriemannian metric. For the convenience, we call such a lie group a lcs lie group.

In the following we will use these two views interchangably. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. Biinvariant finsler metrics on lie groups a finsler metric on a. Invariant metrics left invariant metrics these keywords were added by machine and not by the authors.

Invariant metrics with nonnegative curvature on compact lie groups. For a lie group, a natural choice is to take a leftinvariant metric. This process is experimental and the keywords may be updated as the learning algorithm improves. In this paper, we prove several properties of the ricci curvatures of such spaces. Metrics, connections, and curvature on lie groups it will be convenient to say that an inner product on g is biinvariant i. Also we calculate the levicivita connection, and then ricci tensor associated with leftinvariant pseudoriemannian metrics on the unimodular lie groups of dimension three. When the manifold is a lie group and the metric is left invariant the curvature. Curvatures of left invariant metrics on lie groups. This paper will consider lie groups in dimension two and three and will focus on the solutions of killings equations. Most lie groups do not have biinvariant metrics, although all compact lie groups do. Pdf leftinvariant lorentzian metrics on 3dimensional. The conjecture is known to be true when g is one of so3 or u2 this follows from the complete classi. Milnors paper gave many known results on the topic.

Metric tensor on lie group for left invariant metric. To begin with, we give some examples of pseudoeinstein metrics on lie groups. This article is the first in a series that will investigate symmetry and curvature properties of a rightinvariant metric on a lie group. Chapter 17 metrics, connections, and curvature on lie groups. In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. 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. 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. While there are few known obstruction for a closed manifold. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.

Second degree examples are called quadratic invariants, and so forth. 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. Therefore, by bochners theorem, g is not locally isomorphic to a compact lie group. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. For a connected lie group g, determine all the signatures of the ricci operators for all leftinvariant riemannian metrics on g. Then, using left translations defines a left invariant. We investigate nontrivial mquasieinstein metrics on pseudoriemannian lcs lie group. In and, problem 1 has been solved, respectively, in the case of 3dimensional lie groups and 4dimensional lie groups. Buttsworth extended this result in but19 to all signatures of t and all unimodular lie groups of dimension 3. Left invariant lorentzian metrics on 3dimensional lie groups rend. In this case, the equation may fail to have a solution.

Geodesics of left invariant metrics on matrix lie groups. We find the riemann curvature tensors of all left invariant lorentzian metrics on 3dimensional lie groups. Pdf biinvariant and noninvariant metrics on lie groups. Chapter 18 metrics, connections, and curvature on lie groups. Curvatures of left invariant metrics on lie groups core. For all leftinvariant riemannian metrics on threedimensional unimodular lie groups, there exist particular leftinvariant orthonormal frames, socalled milnor frames. For a lie group, a natural choice is to take a left invariant metric.

On the existence of biinvariant finsler metrics on lie. 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. Index formulas for the curvature tensors of an invariant metric on a lie group are. On the moduli spaces of leftinvariant pseudoriemannian metrics on lie groups kubo, akira, onda, kensuke, taketomi, yuichiro, and tamaru, hiroshi, hiroshima mathematical journal, 2016. Given any lie group g, an inner product h,i on g induces a biinvariant metric on g i. Leftinvariant metrics on lie groups and submanifold geometry.

In this paper, we consider a special class of solvable lie groups such that for any x, y in its lie algebra, x, y is a linear combination of x and y. In order to generalize this fundamental starting point, we address the question. Curvature of left invariant riemannian metrics on lie. The invariants most often considered are polynomial invariants. I was reading the classical paper from milnor entitled curvature of left invariant metrics on lie groups. The following page shows two surfaces comprised of all the equatorial geodesics emanating.

We find the riemann curvature tensors of all leftinvariant lorentzian metrics on 3dimensional lie groups. The curvature is zero if and only if the lie algebra g of g is nilpotent of order two. Furthermore, we show that the leftinvariant pseudoeinstein metric on sl2 is unique up to a constant. Given any lie group g, an inner product h,i on g induces a bi invariant metric on g i. Metrics on solvable lie groups much is understood about leftinvariant riemannian einstein metrics with metrics, connections, and curvature on lie groups applying theorem 18. A striking result is that several of the threedimensional lie groups turn out to be spaces of constant curvature. Pdf the i signature of the ricci curvature of left. In the sequel, the identity element of the lie group, g, will be denoted by e or. Scalar curvatures of leftinvariant metrics on some. 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. Constant mean curvature surfaces in metric lie groups.

A remark on left invariant metrics on compact lie groups. Killings equations for invariant metrics on lie groups. Ricci curvature of left invariant metrics on solvable. Finally, we show that if g is a lie group endowed with a biinvariant finsler metric, then there exists a biinvariant riemanninan metric on g such that its levicivita connection coincides the connection of f. Curvatures of left invariant metrics on lie groups sciencedirect. If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the bi invariant killing form. Let w denote the multiple of its curvature operator, acting on symmetric 2tensors, with the factor chosen. In the nonflat case one received by the editors november 27, 1962. Conjugate points in lie groups with leftinvariant metrics. A remark on left invariant metrics on compact lie groups lorenz j.

Leftinvariant pseudoriemannian metrics on some lie groups. The signature of the ricci curvature of leftinvariant. Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at. Left invariant pseudoriemannian metrics on solvable lie.