Then, we prove that every compact connected lie group is a symmetric finsler space with respect to the biinvariant absolute homogeneous finsler metric. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. Computing bi invariant pseudo metrics on lie groups for consistent statistics. Theclaimaboutunbounded adg orbitsnowfollowsfromlemmaa. In particular, such metrics do not exist in any dimension for rigidbody transformations, which form the most simple lie group involved in biomedical image registration. For instance, geodesics through the identity coincide with oneparameter groups, so the riemannian exponential map coincides with the lie group exponential. In this paper we study the geometry of lie groups with biinvariant finsler metrics.
Every compact lie group admits one such metric see proposition 2. On lie groups with left invariant semiriemannian metric r. When studying relationships between curvature of a complete. In the sequel, the identity element of the lie group, g, will be denoted by e or. Also, homogeneous spaces are occasionally tackled in more advanced textbooks of differential geometry.
The basic facts about such bi invariant metrics can be summarized as follows. A riemannian metric that is both left and rightinvariant is called a biinvariant metric. On biinvariant word metrics journal of topology and analysis. Biinvariant and noninvariant metrics on lie groups. In both cases, a parametrisation of the set of such lie groups is known. Curvatures of left invariant metrics on lie groups core. Biinvariant metrics on the group of symplectomorphisms zhigang han abstract. Pdf biinvariant finsler metrics on lie groups researchgate. Let g be a real lie group of dimension n and g its lie algebra. Leftinvariant and biinvariant metrics since a lie group g is a smooth manifold as well as a group, it is customary to use riemannian metrics that link the geometry of g with the group structure. However, it is known that lie groups which are not direct product of compact and abelian groups have no bi invariant metric. The riemannian geometry of bi invariant metrics is very nice. Inria computing biinvariant pseudometrics on lie groups. Homogeneous geodesic introduction lie groups are the most beautiful and most important manifolds.
We consider the effects of vacuum polarization and particle creation of a scalar field on lie groups with a nonstationary biinvariant metric of the robertsonwalker type. In this paper, we study the geometry of lie groups with bi invariant finsler metrics. Exponential barycenters of the canonical cartan connection. Metrics, connections, and curvature on lie groups it will be convenient to say that an inner product on g is biinvariant i.
We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. Chapter 17 metrics, connections, and curvature on lie groups. However, since so3 is a compact lie group, it has a unique biinvariant integration measure which can be used to average metrics that are not invariant, to produce ones that are invariant. Spectral isolation of biinvariant metrics on compact lie groups. Aug 11, 2018 in this short note, we prove that a biinvariant riemannian metric on spn is uniquely determined by the spectrum of its laplacebeltrami operator within the class of leftinvariant metrics on spn. Mathematical sciences on the existence of bi invariant 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 bi invariant finsler metrics. We prove that lie groups with biinvariant absolutely. 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. Lie groups that have a locally unique biin variant mean do not possess a biinvariant pseudometric. Pdf computing biinvariant pseudometrics on lie groups for.
To overcome the lack of existence of bi invariant riemannian. Einstein metrics on lie groups 3 proof of theorem b. Einstein and conformally einstein biinvariant semi. We first show that every compact lie group admits a biinvariant finsler metric. It is known that for some nonuniform lattices in semisimple lie groups e. Vacuum quantum effects on lie groups with biinvariant metrics. Splitting lie groups with biinvariant metrics 40 ii. Invariant metrics with nonnegative curvature on compact lie. This paper studies the extension of the hofer metric and general finsler metrics on the hamiltonian symplectomorphism group hamm.
W e conclude that, for the purpose of statistics on general real lie groups and, more. In this paper, we study the geometry of lie groups with biinvariant finsler metrics. Lie groups and geometric aspects of isometric and hamiltonian. Oct 10, 2007 a restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. We study also the particular case of bi invariant riemannian metrics. In general, the problem of understanding the biinvariant geometry of lattices in higher rank semisimple lie groups is widely open.
Then, we prove that every compact connected lie group is a symmetric finsler space with respect to the bi invariant absolute homogeneous finsler 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. I own do carmos book as well, and the proposition in the hint is in fact proved on page 41 chapter 1, section 2, in example 2. This is a special case of lemma 3 in oneill 44 chapter 11.
In particular, we prove that the hofer metric on hamm. For \straight paths of left invariant metrics starting at bi invariant metrics and ending at nonnegatively curved. Spectral isolation of biinvariant metrics on compact lie groups by carolyn s. 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. We show that a biinvariant metric on a compact connected lie group g is spec. Biinvariant word metrics are at present not well understood. Pdf computing biinvariant pseudometrics on lie groups. In order to generalize this fundamental starting point, we address the question. A left invariant metric on a connected lie group is also. Computing biinvariant pseudo metrics on lie groups for consistent statistics. Computing biinvariant pseudometrics on lie groups for.
Left invariant metrics on compact lie groups jack huizenga abstract. Curvature of left invariant riemannian metrics on lie groups. Some lie groups may possess a metric which is invariant not only under eft translation but also under right translation. This notion coincides with the usual euclidean mean associated with the novel vector space structure. In other words, on any of these compact simple lie groups, every leftinvariant metric which is not rightinvariant cannot be isospectral to a biinvariant metric. 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. Finally, we show that if g is a lie group endowed with a biinvariant finsler metric, then, there exists a biinvariant riemanninan metric on g such that its levicivita connection coincides the connection of f. The present book is designed to provide an introduction to sev eral aspects of the geometry of lie groups and homogeneous spaces. On the existence of biinvariant finsler metrics on lie. Lie group that admits a biinvariant metric is a homogeneous riemannian manifoldthere exists an isometry between that takes any point to any other point.
On bi invariant word metrics journal of topology and analysis. A condition for biinvariance of a metric can be expressed entirely in terms of the lie algebra, as follows. We prove that bi invariant word metrics are bounded on certain chevalley groups. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Since it is invariant under left and right translations, it is called bi invariant. 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. As a second step, we prove that compact lie groups with biinvariant metrics does not admit riemannian foliations of totally geodesic 7spheres. Isometric and adjoint actions and some generalizations 43 chapter 3. On the existence of biinvariant finsler metrics on lie groups.
The thesis then considers bi invariant metrics on lie groups when the lie algebra is created through a double extension procedure, as described initially by medina 25. Conjugate points in lie groups with leftinvariant metrics. Pdf in this paper we study the geometry of lie groups with biinvariant finsler metrics. Curvatures of left invariant metrics on lie groups. 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. We prove that lie groups with biinvariant absolutely homogeneous finsler metrics are symmetric finsler space. As an application we provide restrictions on hamiltonian actions of such groups. We show two examples of bi invariant metrics on nonreductive lie groups that are bachat but not conformally einstein, however, we show that all lorentzian bi invariant metrics.
Finally, we show that if g is a lie group endowed with a bi invariant. Existence of cocompact lattices in lie groups with a bi. Left invariant einstein metrics on lie groups andrzej derdzinski august 28, 2012 differential geometry seminar department of mathematics the ohio state university. However, it is known that lie groups, which are not a direct product of compact and abelian groups, have no bi invariant metric. This chapter deals with lie groups with special types of riemannian metrics. We provide techniques for studying the nonnegatively curved left invariant metrics on a compact lie group. In the third section, we study riemannian lie groups with. Free products of groups with biinvariant metrics sciencedirect. Splitting lie groups with bi invariant metrics 40 ii. Therefore, by bochners theorem, g is not locally isomorphic to a compact lie group. Curvatures of left invariant metrics on lie groups john milnor.
The lie algebras are taken from tables compiled originally by mubarakzyanov izv. Spectral isolation of biinvariant metrics on compact lie. Unfortunately, bi invariant riemannian metrics do not exist for most non compact and noncommutative lie groups. Lie algebras with biinvariant pseudometric were known to exist since the.
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