He computed the connections of these metrics and obtained the formula for geodesics and curvatures. 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. All left invariant nonberwaldian randers metrics of. In this paper, we prove several properties of the ricci curvatures of such spaces. For left invariant vector elds the rst three terms of the right hand side of 2. Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in riemannian geometry. Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at.
Namely, we give all the metrics with constant curvature, semisymmetric. Milnortype theorems for leftinvariant riemannian metrics on lie groups hashinaga, takahiro, tamaru, hiroshi, and terada, kazuhiro, journal of the mathematical society of japan, 2016. An elegant derivation of geodesic equations for left invariant metrics has been given by b. 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. In the third section, we study riemannian lie groups with. Static skt metrics on lie groups static skt metrics on lie groups enrietti, nicola 20120616 00. Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the threedimensional lie groups. Left invariant degenerate metrics on lie groups springerlink. We study also the particular case of biinvariant riemannian metrics. 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. Moreover, for any lorentzian left invariant metric in this list we give its ricci curvature, scalar curvature, the signature of the ricci curvature and we exhibit some special features of these curvatures.
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. In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis. Which lie groups have lie algebras admitting an ad. While there are few known obstruction for a closed manifold. From this is easy to take information about levicivita connection, curvatures and etc. Let g be a full connected semisimple isometry lie group of a connected. Alekseevskyconjecturedin1975that,whenever m gk is a simply connected. 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. Nomizus research on the properties of invariant riemannian metrics on gh obtained many interesting and significant results. Mathematical sciences on the existence of biinvariant 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 biinvariant finsler metrics. Static skt metrics on lie groups, manuscripta mathematica.
Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. Full text pdf 122k abstracts references1 full text pdf 122k. Ricci curvature of left invariant metrics on solvable. Curvature of left invariant riemannian metrics on lie. This means that all lie groups that have the same lie algebra as some compact group also satisfy the condition. Flow of a left invariant vector field on a lie group equipped with leftinvariant metric and the group s geodesics 2 proving smoothness of leftinvariant metric on a lie group. 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. Ricci curvature of left invariant metrics on solvable unimodular lie groups. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. Invariant finsler metrics on homogeneous manifolds. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Curvatures of left invariant randers metrics on the ve.
Milnor j curvatures of leftinvariant metrics on lie groups. Left invariant metrics and curvatures on simply connected. Index formulas for the curvature tensors of an invariant metric on a lie group are obtained. Index formulas for the curvature tensors of an invariant metric on a lie group are. We study metric contraction properties for metric spaces associated with leftinvariant subriemannian metrics on carnot groups. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2.
Compact lie group biinvariant metric stack exchange. Geodesics and curvatures of special subriemannian metrics on. For a lie group, a natural choice is to take a leftinvariant metric. Curvatures of left invariant metrics on lie groups sciencedirect. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Milnor conjectured that these are the only possible ways to obtain positive scalar curvature on a lie group with left invariant. 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. On the moduli spaces of leftinvariant pseudoriemannian metrics on lie groups kubo, akira, onda, kensuke, taketomi, yuichiro, and tamaru, hiroshi, hiroshima mathematical journal, 2016. On the existence of biinvariant finsler metrics on lie. A remark on left invariant metrics on compact lie groups lorenz j. Our results improve a bit of milnors results of 7 in the three. But, i want to understand given a specific vector field what should i do to test if it is left. 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. Faculty of mathematics, kyushu university produced and listed by.
Homogeneous geodesics of left invariant randers metrics on. A remark on left invariant metrics on compact lie groups. Scalar curvatures of leftinvariant metrics on some. Lorentzian left invariant metrics on three dimensional unimodular lie groups and their curvatures.
M, with velocity t is a finslerian geodesic if d t t ft 0, with reference vector t. First of all, all compact lie groups satisfy this condition because one can average an arbitrary inner product on the algebra over the group. This article outlines what is known to the author about the riemannian geometry of a lie group which has been. In chapter 2 and 3 we calculate the sectional and ricci curvatures of the 3 and 4dimensional lie groups with standard metrics. From now on elements of n are regarded as left invariant vector elds on n. The signature of the ricci curvature of leftinvariant riemannian metrics on nilpotent lie groups. Invariant metrics with nonnegative curvature on compact. The results are applied to the problem of characterizing invariant metrics of zero and nonzero constant curvature.
Hence, denoting by rthe semiriemannian curvature tensor. Curvatures of left invariant metrics on lie groups. Curvature of left invariant riemannian metrics on lie groups. Curvatures of left invariant metrics on lie groups john milnor. The signature of the ricci curvature of leftinvariant riemannian. The curvature is zero if and only if the lie algebra g of g is nilpotent of order two. Pdf on lie groups with left invariant semiriemannian metric. The theory of lie groups involves many areas of mathematics.
Geodesics equation on lie groups with left invariant metrics. Ricci curvatures of left invariant finsler metrics on lie. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. We study also the particular case of bi invariant riemannian metrics.
Geodesics of left invariant metrics on matrix lie groups. Pdf biinvariant and noninvariant metrics on lie groups. Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three. But after reading it, students will be wellprepared to continue with more advanced, graduatelevel topics in differential geometry or the theory of lie groups. Curvatures of left invariant metrics on lie groups john. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. On the moduli space of leftinvariant metrics on a lie group.
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. Curvatures of left invariant metrics on lie groups core. Biinvariant and noninvariant metrics on lie groups. Metrics on solvable lie groups much is understood about leftinvariant riemannian einstein metrics with 0 for. If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the biinvariant killing form.