For what we have seen above, with the left invariant metric induced by this scalar product, g has constant sectional curvature k. Curvatures of left invariant metrics on lie groups core. G is a compact subgroup, admits aginvariant riemannian metric of positive ricci curvature if and only if. Research article sectional and ricci curvature for three. Negatively curved leftinvariant metrics on lie groups. Metrics, connections, and curvature on lie groups the center zgofaliealgebrag is the set of all elements u 2 g such that u,v0forallv 2 g,or equivalently, such that adu0. Ricci curvatures of left invariant finsler metrics on lie. An elementary introduction to lie group theory with an emphasis on concrete.
Hilbertschmidt groups as lie groups and their riemannian. Only certain signatures of t are allowed in three dimensions. For unit tangent vector x, rx rx, x is called the ricci curvature in the direction x. Negative ricci curvature on lie groups with a compact levi factor. Curvatures of left invariant metrics on lie groups john milnor.
Negative ricci curvature and isometry group xianzhe daiy zhongmin shenz guofang wei x abstract we show that for ndimensional manifolds with ricci curvature bounded between two negative constants the order of their isometry groups is uniformly bounded by the ricci curvature bounds, the volume, and the injectivity radius. Until a couple of years ago, the only known examples of lie groups admitting leftinvariant metrics with negative ricci curvature were either solvable or semisimple. Chapter 18 metrics, connections, and curvature on lie groups. Lie brackets and lie derivatives, the frobenius theorem, tensors, di. After some preliminaries section 2, we prove in section 3 the following theorem, which gives some necessary conditions and some su. A homogeneous space with negative ricci curvature is noncompact. Given nand dthere are positive constants 0 and csuch that the following holds. In differential geometry, the ricci curvature tensor, named after gregorio riccicurbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved riemannian manifold deviates from that of the standard ball in euclidean space. Curvature of left invariant riemannian metrics on lie.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Although a great progress has been made lately, specially in the solvable case, the general case seems to be far from being completely understood. Ricci curvature is a linear operator on tangent space at a point, usually denoted by ric. If g is a lie group equipped with a leftinvariant metric, then it is possible to express the levicivita connection and the sectional curvature in terms of quantities. Let ei, 62, 63 be the pseudoorthonormal basis used in j2. In chapter 1 we introduce the necessary notions and state the basis results on the curvatures of lie groups. On the other hand, i do not assume any previous acquaintance with riemannian metrics, or even with the classical theory of curves and surfaces in r3. Formulas for the riemann and ricci curvature tensors of an invariant metric on a lie group are determined. Arroyo, artem pulemotov, and wolfgang ziller dedicated to the memory of yuri berezansky abstract. Ricci curvature plays an important role in general relativity, where it is the key term in the einstein field equations. Moreover, there is no direct relation between compactness and scalar curvature. Lie groups admit a leftinvariant metric with negative ricci scalar. Lectures on di erential geometry math 240bc john douglas moore department of mathematics university of california santa barbara, ca, usa 93106 email.
Apr 02, 2001 we prove that on any connected unimodular lie group g, the space l p. Finally it would be interesting to know which groups admit left invariant metrics with ricci curvature identically zero,x or with constant or nonconstant strictly negative ricci curvature. Curvature in mathematics and physics septdec 2019, western university instructor. The signature of the ricci curvature of leftinvariant. Scalar curvature of lie groups henglung lai and hueishyong lue1 abstract. It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or, and is a one. We refer to the nice survey lott and the references therein for a discussion of the geometric interest of lower bounds on ricci curvature and the need for a generalized notion of positive ricci curvature for metric measure spaces. Chapter 17 metrics, connections, and curvature on lie groups. Jan 11, 2015 we study solvable lie groups which admit a leftinvariant metric of strictly negative ricci curvature. It was proved in 6 that to every left invariant metric on a lie group with lie. One motivation comes from the connection between representation theory and homotopy theory. In particular, the ricci curvature, which is understood as the limit of the ricci curvature of. The isometry group of a compact riemannian manifold with negative ricci curvature is discrete.
Dec 24, 20 pdf we consider the question of whether a given solvable lie group admits a leftinvariant metric of strictly negative ricci curvature. Curvature of left invariant riemannian metrics on lie groups. The prescribed ricci curvature problem on threedimensional unimodular lie groups timothy buttsworth the university of queensland august 4, 2016. We obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or standard filiform and discuss some open questions. Notions of differential geometry are introduced for these groups. Curvatures of left invariant metrics on lie groups john. We obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or. In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. Nonsolvable lie groups with negative ricci curvature.
So far the only known obstructions to have positive ricci curvature come from obstructions to have positive scalar curvature, see li and rs, and from the classical bonnetmyers theorem, which implies that a closed manifold with positive ricci curvature must have nite fundamental group. Research article sectional and ricci curvature for threedimensional lie groups gerardthompson 1 andgirirajbhattarai 2 department of mathematics,e university of toledo, toledo, oh, usa. Ricci curvature of markov chains on polish spaces revisited gong, fuzhou, liu, yuan, and wen, zhiying, osaka journal of mathematics, 20. The idea of curvature plays an important role in many areas of mathematics and physics. Anyunimodularlie group which admits a leftinvariant metric with negative ricci curvature is noncompactsemisimple dlm. On solvable lie groups of negative ricci curvature springerlink. So when we want to perform calculations in g we can use the elements of teg, or the other way around. In this paper, we prove several properties of the ricci curvatures of such spaces. The necessary and sufficient condition is that g g, g rx for some x such that all eigenvalues of adx l g, g have positive real part. On lie groups with left invariant semiriemannian metric r. Lie triple systems 224 exercises and further results 226 notes 227 chapter v decomposition of symmetric spaces 1. We describe the exponential map from an infinitedimensional lie algebra to an infinitedimensional group of operators on a hilbert space. If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the biinvariant killing form.
Those lie groups which admit left invariant metrics of strictly negative sectional curvature have been classified by heintze. Curvature of riemannian manifolds wikipedia, the free encyclopedia 33110 1. We use a general construction from a previous article of the second named author to produce a great amount of examples with compact levy factor. In each case the metric is reduced by using the automorphism group of the associated lie algebra.
Negative ricci curvature on lie groups with a compact levi. We study the problem of prescribing the ricci curvature in the class of naturally reductive metrics on a compact lie group. Differential geometry, lie groups, and symmetric spaces. Prescribing leftinvariant ricci curvature let g be a lie group. Invariant nonholonomic riemannian structures on threedimensional lie groups, j. Ricci curvature also appears in the ricci flow equation, where a timedependent riemannian metric is deformed in the. Solvable extensions of negative ricci curvature of filiform lie groups nikolayevsky, y. All known examples of homogeneous einstein manifolds of negative ricci curvature are riemannian solvmanifolds of iwasawa type.
The atiyahsegal completion theorem relates representations of a compact lie group gto gequivariant ktheory, and one hopes that representations of the loop group lgis related to gequivariant elliptic cohomology. In particular, the ricci curvature, which is understood as the limit of the ricci curvature of finitedimensional. Project muse sobolev algebras on lie groups and riemannian. Resultspreliminaries u2 negative ricci curvature on lie groups with a compact levi factor cynthia will, famaf and ciem, c ordoba, argentina vi workshop on di erential geometry. In particular, the maximum and minimum values of the sectional.
The ricci curvature is a tensor that contracts the curvature tensor and is computable when one has the contravariant riemannian curvature tensor. Lie groups with biinvariant metric in this section we describe indecomposable lie groups with biinvariant inde. Curvatures of left invariant metrics on lie groups. As its title suggests, this paper deals with the curvature properties of the large class of so called riemannian lie groups, i. So in the 3dimensional case the ricci curvature gives us as much information about the manifold as the sectional cruvature. The results are applied to a systematic study of the curvature properties of invariant metrics on threedimensional lie groups. Given a leftinvariant candidate t, can we nd a leftinvariant g such that ricg t on g. Ricci signature underestimates in nilpotent riemannian lie groups and a proof of theorems 1 and 3. Some recent progress and open questions je cheeger courant institute september 9, 2016. On topology of some riemannian manifolds of negative curvature with a compact lie group of isometries mirzaie, r. Full text of algebraic ricci solitons of threedimensional. We also introduce an open and convex cone cn of derivations attached to each nilpotent lie algebra n, which is defined as the image of certain moment map and parametrizes a set of solvable lie algebras with. Solvable extensions of negative ricci curvature of.
The lie algebra g of a lie group g is isomorphic to the tangent space teg at the identity element of g, thus g is. Let us consider the sign of the ricci curvature scalar of g. Full text of algebraic ricci solitons of threedimensional lorentzian lie groups see other formats algebraic ricci solitons of threedimensional lorentzian lie groups w. Necessary conditions on the lie algebra of a nilpotent lie group for the existence of an einstein metric of nonzero scalar curvature are given in. The structure and meaning of ricci curvature aaron naber structure of limit spaces, lower ricci curvature.
In fact, the calculations we have done already allow us to compute one further object of interest for a riemannian or pseudoriemannian manifold. Negative ricci curvature on some nonsolvable lie groups. Since a lie group g is a smooth manifold as well as a group, it is customary to use riemannian metrics that link the geometry of g with the group structure. May be it will be a basis of further discussion on this. Onda april 3, 2012 abstract we classify algebraic ricci solitons of threedimensional lorentzian lie groups. Let d,dbe a lie group with biinvariant semiriemannian metric dof signature p,q,letd be the lie algebra of dand b d its killing form. We give an overview of what is known on lie groups admitting a leftinvariant metric of negative ricci curvature, including many natural questions and conjectures in the solvable case.
Future directions in ricci curvature will involve more than a regularity theory. We prove that a homogeneous effective spacemgh, whereg is a connected lie group andh. We study solvable lie groups which admit a leftinvariant metric of strictly negative ricci curvature. Sectional and ricci curvature for threedimensional lie groups. Many ways to interpret the meaning of ricci curvature bounds. The prescribed ricci curvature problem on threedimensional. This generalizes results due to strichartz in the euclidean case, to bohnke in the case of stratified groups, and others. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. Let w denote the multiple of its curvature operator, acting on symmetric 2tensors, with the factor chosen.
Solvable extensions of negative ricci curvature of filiform. We prove that no curvature dimension bound cdk,n holds in any heisenberg group hn. Nikonorov, the signature of the ricci curvature of leftinvariant riemannian metrics on fourdimensional lie groups, the nonunimodular case, siberian adv. In this section, we will show that lemma 1 turn out to be very useful in the case of nilpotent riemannian lie groups that permits us to prove theorem 1, theorem 3. For a lie group, a natural choice is to take a leftinvariant metric. Looking at bases, nilpotency of g implies that g is the group of all ordinary translations of m.
Abstract the killing form b of a real or complex semisimple lie group g is a leftinvariant pseudoriemannian or, respectively, holomorphic einstein metric. In this talk we will be interested in lie groups admitting a metric with negative ricci curvature. Readings introduction to lie groups mathematics mit. For a lie group with a biinvariant riemannian metric the riemannchristoffel connection is half the lie algebra, i. Although individually, the weyl tensor and ricci tensor do not in general determine the full curvature tensor, the riemann curvature tensor can be decomposed into a weyl part and a ricci part. We give necessary and sufficient conditions of the existence of a left. Find materials for this course in the pages linked along the left. Compact lie group actions on closed manifolds of non. Invariant nonholonomic riemannian structures on three. On solvable lie groups of negative ricci curvature. Ricci curvature, a logarithmic sobolev inequality holds.
In this course we shall look at some of its definitions and manifestations in geometry, topology, and mathematical physics. If g is a connected lie group, then g admits left invariant metric of positive scalar curvature if and only if the universal covering space g of g is not homeomorphic to the euclidean space. Then by the result of connerraymond, the compact lie groupg must be a. Curvature of riemannian manifolds uc davis mathematics. Negatively curved leftinvariant metrics on lie groups arxiv. In this work we study the geometry of lie groups, with a left invariant metric, analyzing the sectional curvature, ricci curvature and scalar curvature.