The invariants most often considered are polynomial invariants. 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. 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. Scalar curvatures of leftinvariant metrics on some. When all the left translations lx are isometries, we call g a left invariant metric. The subject is in close link with subriemannian lie groups because curvatures could be classi. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. Intro case 1 case 2 case 3 summary abstract 12 background leftinvariant riemannian metrics on lie group.
Invariants constructed using covariant derivatives up to order n are called nth order differential invariants the riemann tensor is a multilinear operator of. The results are applied to the problem of characterizing invariant metrics of zero and nonzero constant curvature. If z belongs to the center of the lie algebra g, then for any left invariant metric the inequality kz. Curvatures of left invariant metrics on lie groups john milnor. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j. Homogeneous geodesics of left invariant randers metrics on. Metric tensor on lie group for left invariant metric. G, where lx is the left translation satisfying lx y xy.
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. International conference on mathematics and computer science, june 2628, 2014, bra. We expect that nice leftinvariant metrics such as einstein or ricci soliton are corresponding to nice submanifolds. Left invariant randers metrics on 3dimensional heisenberg. A leftsymmetric algebraic approach to left invariant flat metrics on lie groups. Geodesics of left invariant metrics on matrix lie groups. Browse other questions tagged metricspaces liegroups liealgebras or ask your own question. We study also the particular case of biinvariant riemannian metrics. Index formulas for the curvature tensors of an invariant metric on a lie group are obtained. Thereby we obtain the principal ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three. In this paper, we prove several properties of the ricci curvatures of such spaces. We can see these formulas are different from previous results given recently.
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. Ricci curvature of left invariant metrics on solvable. Then, using left translations defines a left invariant. Invariant metrics with nonnegative curvature on compact. Curvatures of left invariant metrics on lie groups john. We give the explicit formulas of the flag curvatures of left invariant matsumoto and kropina metrics of berwald type. Therefore b possesses a biinvariant metric, that is, one satisfying the conditions of 7. Since on is the lie algebra of a compact group on, it possesses a curvatures of left invariant metrics 327 biinvariant metric. They give new insights into the behaviour of metric spaces.
Geometrically a lie algebra g of a lie group g is the set of all left invariant vector. These are polynomials constructed from contractions such as traces. A leftsymmetric algebraic approach to left invariant flat. 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. 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. 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. For all leftinvariant riemannian metrics on threedimensional unimodular lie groups, there exist particular leftinvariant orthonormal frames, socalled milnor frames. Leftinvariant lorentzian metrics on 3dimensional lie groups. A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Curvatures of left invariant metrics on lie groups. Invariant metrics left invariant metrics these keywords were added by machine and not by the authors.
This procedure is an analogue of the recent studies on leftinvariant riemannian metrics, and is based on the moduli space of leftinvariant pseudoriemannian metrics. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. On lifts of leftinvariant holomorphic vector fields in complex lie groups alexandru ionescu1 communicated to. In this talk, we introduce our framework, and mention some. Left invariant metrics and curvatures on simply connected. 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. 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. Suppose, to begin with, that is a lie group acting on itself by left translations. Remark sectional curvatures associated with a biinvariant metric can be computed by the explicit formula ku. 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. If you are interested in the curvature of pseudoriemannian metrics, then in the semisimple case you can also consider the biinvariant killing form. 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.
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. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. For this reason, lie groups form a class of manifolds suitable for testing general hypotheses and conjectures. 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. We can compute the left invariant vector elds on h. Curvatures of left invariant randers metrics on the ve. 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. From now on elements of n are regarded as left invariant vector elds on n.
Left invariant metrics on a lie group coming from lie. Metrics on solvable lie groups much is understood about leftinvariant riemannian einstein metrics with solution of a problem of banach v. Biinvariant and noninvariant metrics on lie groups. In this context, it is particularly interesting to investigate left invariant metrics on a. On the moduli space of leftinvariant metrics on a lie group. Let g be a lie group which admits a flat left invariant metric. In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis. Intro preliminaries case 1 case 2 summary abstract 12 background leftinvariant riemannian metrics on lie group. For left invariant vector elds the rst three terms of the right hand side of 2. Left invariant metrics on a lie group coming from lie algebras. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied.
Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. For a lie group, a natural choice is to take a leftinvariant metric. We find the riemann curvature tensors of all leftinvariant lorentzian metrics on 3dimensional lie groups. Curvature of left invariant riemannian metrics on lie. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. A lie groupis a smooth manifold g with a group structure such that the map. This process is experimental and the keywords may be updated as the learning algorithm improves. Second degree examples are called quadratic invariants, and so forth.
An elegant derivation of geodesic equations for left invariant metrics has been given by b. 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. More precisely, decomposing endg into the direct sum of the subspaces consisting of all endomorphisms of g which are selfadjoint or, respec. Ricci curvatures of left invariant finsler metrics on lie. Here we will examine various geometric quantities on a lie goup g with a leftinvariant or biinvariant metrics. Thanks for contributing an answer to mathematics stack exchange. Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at.
Geodesics equation on lie groups with left invariant metrics. We show that any nonflat left invariant metric on g has conjugate points and we describe how some of the conjugate points arise. Let h,i be a left invariant metric on g, and let x, y, z be left invariant vector. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2. A remark on left invariant metrics on compact lie groups. In this paper, we formulate a procedure to obtain a generalization of milnor frames for leftinvariant pseudoriemannian metrics on a given lie group. Our results improve a bit of milnors results of 7 in the three. 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. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics. From this is easy to take information about levicivita connection, curvatures and etc.
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