Invariant metrics with nonnegative curvature on compact. Lie algebra and the inner product g is biinvariant. 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. Finally, we show that if g is a lie group endowed with a biinvariant. Curvatures of left invariant metrics on lie groups john milnor. The existence of a biinvariant riemannian metric is stronger than that of a pseudoriemannian metric, and implies that the lie algebra is the lie algebra of a compact lie group. The basic facts about such biinvariant metrics can be summarized as follows. Below i outline a more elementary proof in the case you care about. Dec 16, 2017 bi invariant metrics of linear growth on the free group 16 december, 2017 in math. The cone of bi invariant metrics is made up of positive linear combinations of the bi invariant metrics of the factors in this direct product. In particular, we prove that the hofer metric on hamm.
For example, z is an index two unbounded subgroup in the in. Although in symplectic topology a famous biinvariant metric, the hofer metric, has been studied since more than 20 years ago, it is only. 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. Pdf on the existence of biinvariant finsler metrics on lie. Isometric actions of simple lie groups on pseudoriemannian manifolds. Jun 30, 2011 this is a sequel to the previous post on the exponential maps of lie groups. Biinvariant metrics on the group of symplectomorphisms zhigang han abstract. 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.
Some lie groups may possess a metric which is invariant not only under eft translation but also under right translation. This is a special case of lemma 3 in oneill 44 chapter 11. On the existence of biinvariant finsler metrics on lie groups. Unfortunately, biinvariant riemannian metrics do not exist on most non compact and noncommutative lie groups. Lie algebras with biinvariant pseudometric were known to exist since the. Sectional curvatures associated with a biinvariant metric can be. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2. This is done concisely due to the use of riemannian geometry techniques.
We first show that every compact lie group admits a bi invariant finsler metric. Biinvariant and noninvariant metrics on lie groups article pdf available in journal of mathematical physics 4810 october 2007 with 3 reads how we measure reads. Biinvariant metrics of linear growth on the free group. In this note we record some results which are derived using kens theorems. Thus, for example, the connected, nonabelian lie group of dimension 2 does not have such a metric or even a bi invariant volume form, for that matter. The cone of biinvariant metrics is made up of positive linear combinations of the biinvariant metrics of the factors in this direct product. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. They are intimately related to symplectic manifolds, i. We first show that every compact lie group admits a biinvariant finsler metric. A left invariant metric on a connected lie group is also. Proposition 1 if is a lie group with a biinvariant metric, let be its levicivita connection, then for any leftinvariant vector field.
In general, lie groups do not have a biinvariant metric, though all connected semisimple or reductive lie groups do. In the sequel, the identity element of the lie group, g, will be denoted by e or. Contact manifolds are odddimensional smooth manifolds endowed with a maximally nonintegrable field of hyperplanes. Pdf biinvariant and noninvariant metrics on lie groups. Not all lie groups carry bi invariant metrics of any signature. In particular we impose a topological restriction on lie groups with bi invariant metric. Biinvariant metrics and quasimorphisms on groups of. In section 3, we present mostly well known tools used to prove unboundedness of. Biinvariant and noninvariant metrics on lie groups.
Computing biinvariant pseudometrics on lie groups for. As a special case, we study biinvariant finsler metrics on lie groups and obtain a necessary and sufficient condition for a lie group to have biinvariant finsler metrics. This chapter deals with lie groups with special types of riemannian metrics. Metrics, connections, and curvature on lie groups it will be convenient to say that an inner product on g is biinvariant i. Euler equations and totally geodesic subgroups 81 the paper is organised as follows. Left invariant metrics induced by the pnorms of the trace in the matrix algebra are studied on the general lineal group. Biinvariant metrics on the contactomorphism groups. In this paper, we study the geometry of lie groups with biinvariant finsler metrics.
The autonomous metric is a particular case of such metrics. In section 3, we present mostly well known tools used to prove unboundedness of biinvariant word metrics. This paper studies the extension of the hofer metric and general finsler metrics on the hamiltonian symplectomorphism group hamm. Finally, we show that if g is a lie group endowed with a bi invariant. Every compact lie group admits one such metric see proposition 2. Moreover, we give basic results on closed subgroups, biinvariant metrics, killing forms and splitting in simple ideals. Chapter 17 metrics, connections, and curvature on lie groups. The proof uses classification of compact lie groups, and some other lie theory. An elegant derivation of geodesic equations for left invariant metrics has been given by b. Lie group that admits a biinvariant metric is a homogeneous riemannian manifoldthere exists an isometry between that takes any point to any other point. Now suppose we have a compact, connected lie group g with a biinvariant metric and an involution. Biinvariant metrics of linear growth on the free group, ii. Left invariant randers metrics on 3dimensional heisenberg group. Curvature of left invariant riemannian metrics on lie groups.
Chapter 18 metrics, connections, and curvature on lie groups. In this paper, we study the geometry of lie groups with bi invariant finsler metrics. How abundant are left invariant metrics on a compact lie group. Section 2 begins with a brief presentation of geodesic ow on groups and the eulerarnold equation. The left invariant metric in the general linear group. Leftinvariant einstein metrics on lie groups andrzej derdzinski august 28, 2012.
This work was supported by the sfb 676 of the deutsche forschungsgemeinschaft. The space of bi invariant affine connections is determined for arbitrary compact lie groups. In other cases, such as di erential operators on sobolev spaces, one has to deal with convergence on a casebycase basis. Then, we prove that every compact connected lie group is a symmetric finsler space with respect to the biinvariant absolute homogeneous finsler metric. Well, then, it is a direct product of simple lie groups and of a torus afterwards, there can be a quotienting by a discrete subgroup.
Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Curvatures of left invariant metrics on lie groups. Invariant metrics with nonnegative curvature on compact lie. 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. Isometric actions of simple lie groups on pseudoriemannian. A riemannian metric that is both left and rightinvariant is called a biinvariant metric. Special hermitian metrics and lie groups sciencedirect. On the existence of biinvariant finsler metrics on lie. The killing form let t g 2 be the space of realvalued symmetric bilinear. Curvatures of left invariant metrics on lie groups john. 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. Then, we prove that every compact connected lie group is a symmetric finsler space with respect to the bi invariant absolute homogeneous finsler metric.
In particular, such metrics do not exist in any dimension for rigidbody transformations, which form the most simple lie. In particular we impose a topological restriction on lie groups with biinvariant metric. On the existence of biinvariant finsler metrics on lie groups article pdf available in mathematical sciences 71 july 20 with 111 reads how we measure reads. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Compact lie group biinvariant metric stack exchange. Left invariant randers metrics on 3dimensional heisenberg. Introduction invariant connections on homogeneous spaces have been well understood. The basic facts about such bi invariant metrics can be summarized as follows. In particular, there is a surprising new family of such connections on sun.