Geodesics of left invariant metrics on matrix lie groups. Curvature of left invariant riemannian metrics on lie groups. I am reading these lines from a text which shows why the bracket of two leftinvariant vector fields is also a leftinvariant vector field. Ricci curvature of left invariant metrics on solvable. A remark on left invariant metrics on compact lie groups. Left invariant semi riemannian metrics on quadratic lie groups.
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. This process is experimental and the keywords may be updated as the learning algorithm improves. Metrics, connections, and curvature on lie groups applying theorem 17. 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 g is a semi group 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. Invariant metrics with nonnegative curvature on compact lie groups. The case of quadratic 2 step nilpotent lie groups is also addressed. Here we will derive these equations using simple tools of matrix algebra and differential geometry, so that at.
On lie groups with left invariant semiriemannian metric 11 and. Thanks for contributing an answer to mathematics stack exchange. Invariant lorentzian orders on simply connected lie groups. More precisely, decomposing endg into the direct sum of the subspaces consisting of all endomorphisms of g which are selfadjoint or, respec. Curvatures of left invariant metrics on lie groups. Index formulas for the curvature tensors of an invariant metric on a lie group are. In this paper, we formulate a procedure to obtain a generalization of milnor frames for leftinvariant pseudoriemannian metrics on a given lie group. A left invariant metric on a connected lie group is also right invariant if and only if adx is skewadjoint for every x g. The geometry of any lie group g with left invariant riemannian metric re ects strongly the algebraic structure of the lie algebra g.
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 section, we will show that the compact simple lie groups s u n for n. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. A riemannian metric that is both left and rightinvariant is called a biinvariant metric. 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. Rightinvariant sobolev metrics of fractional order on the di eomorphism group of the circle. Curvatures of left invariant metrics on lie groups core. Left invariant randers metrics on 3dimensional heisenberg. Invariant metrics left invariant metrics these keywords were added by machine and not by the authors. If the metric is biinvariant, then the geodesics are the left and right translates of 1parameter subgroups, but, in general, if all you have leftinvariance, this is far from the case. Geodesics equation on lie groups with left invariant metrics. Let g be a lie group and n be a closed normal subgroup of g. Note also that riemannian metric is not the same thing as a distance function. Curvature of left invariant riemannian metrics on lie.
Invariant lorentzian orders on simply connected lie groups 111 we will now use this lemma to show that under certain circumstances it suffices to establish the existence of subsemigroups on a quotient group of g. In the third section, we study riemannian lie groups with. 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. Invariant metrics with nonnegative curvature on compact. Biinvariant and noninvariant metrics on lie groups. This procedure is an analogue of the recent studies on leftinvariant riemannian metrics, and is based on the moduli space of leftinvariant pseudoriemannian metrics. Left invariant metrics and curvatures on simply connected. Curvatures of left invariant randers metrics on the ve. Rightinvariant sobolev metrics of fractional order on the. While there are few known obstruction for a closed manifold.
A curvatures of left invariant metrics 297 connected lie group admits such a bi invariant 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. Rightinvariant sobolev metrics of fractional order on the di eomorphism group of the circle joachim escher, boris kolev to cite this version. Two definitions of leftinvariant vector fields of a lie. Theorem milnor if z belongs to the center of the lie algebra g, then for any left invariant metric the inequality kz. For example, if all the ricci curvatures are nonnegative, then the underlying lie group must be unimodular. Lie algebra g of a lie group g is the set of all left invariant vector fields on the lie. For left invariant vector elds the rst three terms of the right hand side of 2. Combined with some known results in the literature, this gives a proof of the main theorem of this paper. Left invariant metrics on a lie group coming from lie.
We study also the particular case of bi invariant riemannian metrics. Start with any positive definite inner product on the lie algebra and ntranslate it to the rest of the group using left multiplication. For example, the area of a triangle is an invariant with. A metric induces a topology on a set, but not all topologies can be generated by a metric. M, with velocity t is a finslerian geodesic if d t t ft 0, with reference vector t. Metrics on solvable lie groups much is understood about leftinvariant riemannian einstein metrics with solution of a problem of banach v. Ricci curvatures of left invariant finsler metrics on lie. Left invariant finsler metrics on lie groups provide an important class of finsler manifolds. For all leftinvariant riemannian metrics on threedimensional unimodular lie groups, there exist particular leftinvariant orthonormal frames, socalled milnor frames. A remark on left invariant metrics on compact lie groups lorenz j. Chapter 17 metrics, connections, and curvature on lie groups. Invariantmetrics with nonnegativecurvatureon compact lie groups 27 proof let gl be a leftinvariant metric on so3.
Invariant control systems on lie groups rory biggs claudiu c. 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. We study also the particular case of biinvariant riemannian metrics. A topological space whose topology can be described by a metric is called metrizable an important source of metrics in differential. In order to study a lie group with left invariant metric, it is best to choose. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Given any lie group g, an inner product h,i on g induces a bi invariant metric on g i. The same remarks apply to homogeneous spaces, which are certain quotients of lie groups. Leftinvariant lorentz metrics on lie groups katsumi nomizu received october 7, 1977 with j.
Pdf on lie groups with left invariant semiriemannian metric. We expect that nice leftinvariant metrics such as einstein or ricci soliton are corresponding to nice submanifolds. Homogeneous geodesics of left invariant randers metrics on. We classify the leftinvariant metrics with nonnegative sectional curvature on. Leftinvariant and biinvariant metrics since a lie group g is a. In mathematics, an invariant is a property of a mathematical object or a class of mathematical objects which remains unchanged, after operations or transformations of a certain type are applied to the objects. In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. 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. Left invariant connections ron g are the same as bilinear. In this paper, we prove several properties of the ricci curvatures of such spaces. From now on elements of n are regarded as left invariant vector elds on n.
Suppose, to begin with, that is a lie group acting on itself by left translations. Left invariant metrics on a lie group coming from lie algebras. Then, using left translations defines a left invariant. The three other classes can be characterized as follows. A riemannpoisson lie group is a lie group endowed with a left invariant riemannian metric and a left invariant poisson tensor which are compatible in the sense introduced in c. When the manifold is a lie group and the metric is left invariant the curvature. Leftinvariant pseudoriemannian metrics on fourdimensional lie groups with nonzero schoutenweyl tensor p. 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. An elegant derivation of geodesic equations for left invariant metrics has been given by b. A riemannian metric on a lie group g is called leftinvariant if. 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. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras.
Browse other questions tagged metricspaces liegroups liealgebras or ask your own question. The space of leftinvariant metrics on a generalization. For this reason, lie groups form a class of manifolds suitable for testing general hypotheses and conjectures. In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis. In this talk, we introduce our framework, and mention some. A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Glg, we get our second criterion for the existence of a bi invariant metric on a lie group. 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.
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