Curvatures of left invariant metrics on lie groups pdf merge

Curvatures of left invariant metrics on lie groups. Remark sectional curvatures associated with a biinvariant metric can be computed by the explicit formula ku. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. When all the left translations lx are isometries, we call g a left invariant metric. Let h,i be a left invariant metric on g, and let x, y, z be left invariant vector. A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. 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. 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. Index formulas for the curvature tensors of an invariant metric on a lie group are obtained. For a lie group, a natural choice is to take a leftinvariant metric. Browse other questions tagged metricspaces liegroups liealgebras or ask your own question. Curvatures of left invariant metrics on lie groups john milnor. They give new insights into the behaviour of metric spaces. A lie groupis a smooth manifold g with a group structure such that the map. For this reason, lie groups form a class of manifolds suitable for testing general hypotheses and conjectures. Invariant metrics left invariant metrics these keywords were added by machine and not by the authors. A leftsymmetric algebraic approach to left invariant flat metrics on lie groups. A remark on left invariant metrics on compact lie groups.

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. 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. We show that any nonflat left invariant metric on g has conjugate points and we describe how some of the conjugate points arise. Therefore b possesses a biinvariant metric, that is, one satisfying the conditions of 7.

Thanks for contributing an answer to mathematics stack exchange. The same remarks apply to homogeneous spaces, which are certain quotients of lie groups. Our results improve a bit of milnors results of 7 in the three. The invariants most often considered are polynomial invariants. Left invariant randers metrics on 3dimensional heisenberg. We can see these formulas are different from previous results given recently. Curvatures of left invariant randers metrics on the ve. The subject is in close link with subriemannian lie groups because curvatures could be classi. Invariant metrics with nonnegative curvature on compact. For all leftinvariant riemannian metrics on threedimensional unimodular lie groups, there exist particular leftinvariant orthonormal frames, socalled milnor frames. 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 other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis.

We find the riemann curvature tensors of all leftinvariant lorentzian metrics on 3dimensional lie groups. Geodesics equation on lie groups with left invariant metrics. 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. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the biinvariant killing form. 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.

On lifts of leftinvariant holomorphic vector fields in complex lie groups alexandru ionescu1 communicated to. Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at. Second degree examples are called quadratic invariants, and so forth. Metrics on solvable lie groups much is understood about leftinvariant riemannian einstein metrics with solution of a problem of banach v. If g is a semigroup and p a metric on g, p will be called left invariant if pgx, gy px, y whenever g, x, y cg, right invariant if always pxg, yg px, y, and invariant if it is both right and left invariant. For left invariant vector elds the rst three terms of the right hand side of 2. Let g be a lie group which admits a flat left invariant metric. Homogeneous geodesics of left invariant randers metrics on. In this talk, we introduce our framework, and mention some. Invariants constructed using covariant derivatives up to order n are called nth order differential invariants the riemann tensor is a multilinear operator of. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. Curvatures of left invariant metrics on lie groups john.

Left invariant metrics on a lie group coming from lie algebras. Let be a leftinvariant geodesic of the metric on the lie group and let be the curve in the lie algebra corresponding to it the velocity hodograph. From now on elements of n are regarded as left invariant vector elds on n. International conference on mathematics and computer science, june 2628, 2014, bra. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Here we will examine various geometric quantities on a lie goup g with a leftinvariant or biinvariant metrics.

We expect that nice leftinvariant metrics such as einstein or ricci soliton are corresponding to nice submanifolds. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. Intro preliminaries case 1 case 2 summary abstract 12 background leftinvariant riemannian metrics on lie group. These are polynomials constructed from contractions such as traces. As we have indicated in the introduction, the study of left invariant flat metrics on lie groups can be reduced to the study of real leftsymmetric algebras with positive definite symmetric left invariant bilinear forms. On the moduli space of leftinvariant metrics on a lie group. More precisely, decomposing endg into the direct sum of the subspaces consisting of all endomorphisms of g which are selfadjoint or, respec. Geodesics of left invariant metrics on matrix lie groups. In this context, it is particularly interesting to investigate left invariant metrics on a. For a leftinvariant metric on a given lie group, we can construct a submanifold, where the ambient space is the space of all leftinvariant metrics on that lie group. This process is experimental and the keywords may be updated as the learning algorithm improves. 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.

If z belongs to the center of the lie algebra g, then for any left invariant metric the inequality kz. Left invariant metrics on a lie group coming from lie. Since on is the lie algebra of a compact group on, it possesses a curvatures of left invariant metrics 327 biinvariant metric. The results are applied to the problem of characterizing invariant metrics of zero and nonzero constant curvature. From this is easy to take information about levicivita connection, curvatures and etc. M, with velocity t is a finslerian geodesic if d t t ft 0, with reference vector t. 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. A leftsymmetric algebraic approach to left invariant flat.

In this paper, we formulate a procedure to obtain a generalization of milnor frames for leftinvariant pseudoriemannian metrics on a given lie group. 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 study also the particular case of biinvariant riemannian metrics. Ricci curvature of left invariant metrics on solvable.

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. This procedure is an analogue of the recent studies on leftinvariant riemannian metrics, and is based on the moduli space of leftinvariant pseudoriemannian metrics. G, where lx is the left translation satisfying lx y xy. 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.

Then, using left translations defines a left invariant. We can compute the left invariant vector elds on h. Leftinvariant lorentzian metrics on 3dimensional lie groups. Metric tensor on lie group for left invariant metric. Ricci curvatures of left invariant finsler metrics on lie. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics. Curvature of left invariant riemannian metrics on lie. In this paper, we prove several properties of the ricci curvatures of such spaces. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. Geometrically a lie algebra g of a lie group g is the set of all left invariant vector. 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 give the explicit formulas of the flag curvatures of left invariant matsumoto and kropina metrics of berwald type. 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. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. 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 metrics and curvatures on simply connected. Scalar curvatures of leftinvariant metrics on some.