This paper has been motivated by various problems and results in differential geometry. Ricci curvatures of left invariant finsler metrics on lie groups. Hence, in order to complete the classi cation it is necessary to classify leftinvariant einstein metrics on s3. Let qn be a lie group, and let g be the lie algebra of all left invariant vector fields on q. The curvature properties of such metrics on various kinds of lie groups are mainly investigated in classical works of milnor see 7. The signature of the ricci curvature of leftinvariant. The most familiar nilpotent lie groups are matrix groups whose diagonal entries are.
The set e of levicivita connections of left invariant pseudoriemannian einstein metrics on a given semisimple lie group always includes d, the levicivita connection of the killing form. Curvature of left invariant riemannian metrics on lie groups. 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 among them can be written with relatively smaller number of parameters. We show that any nonflat left invariant metric on g has conjugate points and we describe how some of the conjugate points arise. On the existence of biinvariant finsler metrics on lie. Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three. The large scale geometry of nilpotent lie groups scott d. Combined with some known results in the literature, this gives a proof of the main theorem of this paper. Here and in the following we will consider s3 as the group of unit quaternions.
While there are few known obstruction for a closed manifold. Invariant metrics with nonnegative curvature on compact lie. When all the left translations lx are isometries, we call g a left invariant metric. 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. In riemannian geometry and pseudoriemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. 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. Research into left invariant riemannian metrics on lie groups is an active subject of research and this topic is mentioned among many authors works so far. Using these formulas, we prove that at any point of an arbitrary connected noncommutative nilpotent lie group, the flag curvature of any left invariant matsumoto and kropina metrics of. Research into leftinvariant riemannian metrics on lie groups is an active subject of research and this topic is mentioned among many authors works so far. Let kbe a connected real compact lie group, and let lk denote the family of all left invariant riemannian metrics gon k.
Index formulas for the curvature tensors of an invariant metric on a lie group are obtained. Department of mathematics university of mohaghegh ardabili p. Using this fact, we show that its lie algebra is obtained by the double extension process from a flat lorentzian unimodular lie algebra. 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. We classify determine the moduli space of leftinvariant pseudoriemannian metrics on some particular lie groups. Curvatures of left invariant metrics on lie groups. Metrics on solvable lie groups much is understood about left invariant riemannian einstein metrics with on solvable lie groups g. Then there may be many g invariant riemannian metrics on m.
Milnor in the well known 2 gave several results concerning curvatures of left invariant riemannian. 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. 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. B left invariant metrics and curvatures on simply connected three dimensional lie groups. A general program is set out to describe the geodesic in terms of the. Geometrically a lie algebra g of a lie group g is the set of all left invariant vector.
In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. Research into leftinvariant riemannian metrics on lie groups. A left invariant metric on a connected lie group is also right invariant if and only if adx is skewadjoint for every x g. Lengyeln et oth1 1university of debrecen 2college of ny regyh aza symposium on finsler geometry, 20 sapporo. We are led to a different proof of thurstons 16 and part of walls 17 classification of 3 and 4 geometries. The signature of the ricci curvature of leftinvariant riemannian metrics on nilpotent lie groups. Scalar curvatures of leftinvariant metrics on some. Precisely, we prove that there do not exist quasiisometric embeddings of such a. Curvature of left invariant riemannian metrics on lie. A curvatures of left invariant metrics 297 connected lie group admits such a biinvariant metric if and only if it is isomorphic to the cartesian product of a compact group and a commutative group. In this paper, we prove several properties of the ricci curvatures of such spaces. For a lie group, a natural choice is to take a leftinvariant metric.
On the moduli spaces of leftinvariant geometric structures. In chapter 1 we introduce the necessary notions and state the basis results on the curvatures of lie groups. Using these formulas, we prove that at any point of an arbitrary connected noncommutative nilpotent lie group, the flag curvature of any left invariant matsumoto and kropina metrics of berwald type admits zero, positive and negative values, this is a generalization of wolfs theorem. Metric tensor on lie group for left invariant metric. Kodi archive and support file community software vintage software apk msdos cdrom software cdrom software library console living room software sites tucows software library shareware cdroms software capsules compilation cdrom images zx spectrum doom level cd. 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 left invariant metric. Let g be a lie group which admits a flat left invariant metric. When the manifold is a lie group and the metric is left invariant the curvature is also strongly related to the group s structure or equivalently to the lie algebra s structure. On the left invariant randers and matsumoto metrics of. We classify determine the moduli space of left invariant pseudoriemannian metrics on some particular lie groups.
We study metric contraction properties for metric spaces associated with left invariant subriemannian metrics on carnot groups. The signature of the ricci curvature of left invariant riemannian. Centralizer of reeb vector field in contact lie groups hassanzadeh, babak, journal of geometry and symmetry in physics, 2018. Curvatures of left invariant metrics on lie groups john. Lie groups are named after norwegian mathematician sophus lie, who laid the foundations of the theory of continuous transformation groups. Ricci curvature of left invariant metrics on solvable unimodular lie groups. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics. Homogeneous geodesics of left invariant randers metrics on a. Left invariant randers metrics on 3dimensional heisenberg group z. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j.
Note that, for a lower curvature bound, these distinctions are not necessary. Our procedure is based on the moduli space of left invariant riemannian metrics. Flow of a left invariant vector field on a lie group equipped with leftinvariant metric and the group s geodesics 12 uniqueness of biinvariant metrics on lie groups. As a corollary we show that all left invariant pseudoriemannian metrics of arbitrary signature on the lie groups of real hyperbolic spaces have constant sectional curvatures. More precisely, decomposing endg into the direct sum of the subspaces consisting of all endomorphisms of g which are selfadjoint or, respec. Left invariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. For left invariant or right invariant metrics, this paper of arnold gives a formula for the sectional and riemannian curvatures, in terms of the adjoint of the lie bracket operation in the metric. Leftinvariant connections ron g are the same as bilinear mappings g g. Mathematical sciences on the existence of bi invariant finsler metrics on lie groups dariush latifi 0 megerdich toomanian 1 0 department of mathematics, university of mohaghegh ardabili, ardabil, 56199167, iran 1 department of mathematics, islamic azad university, karaj branch, karaj, 3148635731, iran in this paper, we study the geometry of lie groups with bi invariant finsler metrics. 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. 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.
Geodesics equation on lie groups with left invariant metrics. Curvatures of left invariant metrics on lie groups core. Pdf we find the riemann curvature tensors of all leftinvariant lorentzian metrics on 3dimensional lie groups. 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. 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 among them can be. We give the explicit formulas of the flag curvatures of left invariant matsumoto and kropina metrics of berwald type. Left invariant metrics on a lie group coming from lie algebras. Here we will examine various geometric quantities on a lie goup g with a leftinvariant or biinvariant metrics. Curvatures of left invariant metrics on lie groups john milnor. For each simply connected threedimensional lie group we determine the automorphism group, classify the left invariant riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras.
A classification of riemannian 3manifolds with constant. We can see these formulas are different from previous results given recently. Our results improve a bit of milnors results of 7 in the three. A remark on left invariant metrics on compact lie groups lorenz j. For some distinguish geometric structures, left invariant ones on lie groups provide several nice examples. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. Invariant metrics with nonnegative curvature on compact. Biinvariant and noninvariant metrics on lie groups. Flow of a left invariant vector field on a lie group equipped with left invariant metric and the groups geodesics. Leftinvariant pseudoriemannian metrics on some lie groups. From this is easy to take information about levicivita connection, curvatures and etc. An important role is played by the heisenberg mani.
In this section, we will show that the compact simple lie groups s u n for n. Ams proceedings of the american mathematical society. Index formulas for the curvature tensors of an invariant metric on a lie group are. Curvatures of left invariant randers metrics on the ve. We show that ideal subriemannian structures on carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas. Then we can write g gh with h e and the action is the left translation of g. Left invariant metrics on a lie group coming from lie. Pdf on lie groups with left invariant semiriemannian metric. 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. Suppose, to begin with, that is a lie group acting on itself by left translations. On simple lie groups, we show that there is always an einstein biinvariant metric. This manuscript covers some of the material given in three lectures by the.
Mohamed boucetta, abdelmounaim chakkar submitted on 12 mar 2019. In chapter i we consider equations of geodesic motion on a lie group. Ricci curvature of left invariant metrics on solvable. The main motivation is the study of curvature homogeneous riemannian spaces initiated in 1960 by i. Let be a left invariant geodesic of the metric on the lie group and let be the curve in the lie algebra corresponding to it the velocity hodograph. Compact simple lie groups admitting leftinvariant einstein. Ode and pde, but the emphasis is on pde, and in such cases the lie groups involved are in. The results are applied to the problem of characterizing invariant metrics of zero and nonzero constant curvature. Sectional curvatures are therefore all nonnegative.
Oct 10, 2007 a restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Lee is a group whose elements are organized continuously and smoothly, as opposed to discrete groups, where the elements are separatedthis makes lie groups differentiable manifolds. Lorentzian left invariant metrics on three dimensional unimodular lie groups and their curvatures authors. If the connected lie group is unimodular, then we show that if admits a flat left invariant pseudoriemmanian metric of signature such that is degenerate, then for any, where is the levicivita connection of. Geometry of left invariant randers metric on the heisenberg group. 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.
It is important to examine whether given lie groups admit some distinguished left invariant geometric structures or not. Milnortype theorems for left invariant riemannian metrics on lie groups hashinaga, takahiro, tamaru, hiroshi, and terada, kazuhiro, journal of the mathematical society of japan, 2016 on the moduli spaces of left invariant pseudoriemannian metrics on lie groups kubo, akira, onda, kensuke, taketomi, yuichiro, and tamaru, hiroshi, hiroshima. B left invariant metrics and curvatures on simply connected threedimensional lie groups. Left invariant metrics and curvatures on simply connected. Let m gk be an effective coset space of a connected lie group by a compact subgroup. The signature of the ricci curvature of left invariant riemannian metrics on 4dimensional lie groups. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. We study also the particular case of bi invariant riemannian metrics.
Lie groups which admit flat left invariant metrics 259 hence, for 1,2, the length of y. Left invariant randers metrics on 3dimensional heisenberg. The large scale geometry of nilpotent lie groups 953 metric spaces which are locally cat k or cbb k. Let h,i be a left invariant metric on g, and let x, y, z be left invariant vector. From the above definition of homogeneous nilmanifolds, it is clear that any nilpotent lie group with left invariant metric is a homogeneous nilmanifold. Geodesics of left invariant metrics on matrix lie groups. Leftinvariant metrics on lie groups and submanifold geometry. Homogeneous geodesics of left invariant randers metrics on a threedimensional lie group dariush lati. Pdf leftinvariant lorentzian metrics on 3dimensional lie. Left invariant randers metrics on 3dimensional heisenberg group. In this paper, we formulate a procedure to obtain a generalization of milnor frames for left invariant pseudoriemannian metrics on a given lie group. In this section, we consider invariant finsler metrics on lie groups.
If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the biinvariant killing form. This is the simplest case of a natural problem we now describe. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Left invariant degenerate metrics on lie groups springerlink. In the third section, we study riemannian lie groups with. Pauls 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. Ricci curvatures of left invariant finsler metrics on lie. The signature of the ricci curvature of leftinvariant riemannian. Curvatures of left invariant metrics 297 connected lie group admits such a biinvariant metric if and only if it is isomorphic to the cartesian product of a compact group and a commutative group. These tensors are usually the riemann tensor, the weyl tensor, the ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations. We shall denote by 0x the orthocomplement of g with respect to the killing form. A remark on left invariant metrics on compact lie groups. 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. Such a metric is called standard if the orthogonal complement of g.
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