In general, lie groups do not have a biinvariant metric, though all connected semisimple or reductive lie groups do. Lie groups with biinvariant metrics request pdf researchgate. Every compact lie group admits one such metric see proposition 2. Unfortunately, biinvariant riemannian metrics do not exist on most non compact and noncommutative lie groups. Oct 10, 2007 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. This allows us to obtain geometric conditions which are necessary and sufficient for the submanifold m to be curvature adapted to g. Invariant metrics with nonnegative curvature on compact lie groups nathan brown, rachel finck, matthew spencer, kristopher tapp and zhongtao wu abstract. Most lie groups do not have biinvariant metrics, although all compact lie groups do. Spectral isolation of biinvariant metrics on compact lie groups by carolyn s. The existence of a bi invariant 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. Given two groups g 1, g 2 with biinvariant metrics, there are many open questions concerning the norm topology on g 1. Curvatures of left invariant metrics on lie groups. Then it is one of the lie groups, or a commutative group, and the following statements hold see 6, 12.
Start with any positive definite inner product on the lie algebra and ntranslate it to the rest of the group using left multiplication. Conversely, let v, be the lie algebra of a pseudoriemannian lie group of dimension n with biinvariant metric h. In this instance, one considers a simple lie group which is the reali cation of a complex simple lie group. Curvature of left invariant riemannian metrics on lie groups. G bounded with respect to the biinvariant word metric. We define the lorentz force of a magnetic field in a lie group g, and then, we give the lorentz force equation for the associated magnetic trajectories that are curves in g. This is because bi invariant metrics on a compact lie group have nonnegative sectional curvature. If g is a connected lie group, it admits a bi invariant nondegenerate symmetric bilinear form if and only if its lie algebra admits a nondegenerate symmetric bilinear inner product, also called a bi invariant pseudo metric. The metric is induced by the imaginary part of the killing form from the complex lie group. If one takes the metric on gto be bi invariant, the quotient in any case will be seen to admit a metric of nonnegative sectional curvature. If a lie group g has a bi invariant metric then each adjoint matrix in the adjoint r epr esentation has even rank. Partial biinvariance of se3 metrics rpk laboratory. If one is lucky, this quotient metric has positive sectional curvature. Let g be a real lie group of dimension n and g its lie algebra.
We first show that every compact lie group admits a bi invariant finsler metric. A restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied. Lie algebras with biinvariant pseudometric were known to exist since the 1910s with the classification of simple lie. Thus, ifagroup admits ahomogeneousquasimorphism that is bounded on a conjugation invariant generating set then the group is automatically unbounded with respect to the biinvariant word metric associated with this set. However, it is known that lie groups, which are not a direct product of compact and abelian groups, have no bi invariant metric. 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. Pdf in this paper we study the geometry of lie groups with bi invariant randers metric. Curvature of left invariant riemannian metrics on lie. Lie group the isometrygroup of a metric space resp. Free products of groups with biinvariant metrics sciencedirect. This paper studies the extension of the hofer metric and general finsler metrics on the hamiltonian symplectomorphism group hamm.
In this paper, we study the geometry of lie groups with bi invariant finsler metrics. The first question concerns the fundamental group of g. In this section, we will analyze fermiwalker derivative along the curve s which is a parametrized curve on g. Riemannian metric, like compact lie groups such as the group of rotations. This is an example of a biinvariant metric on a simple lie group that is not einstein. Conversely, let v, be the lie algebra of a pseudoriemannian lie group of dimension n with bi invariant metric h.
In this paper we give explicit calculations of the laplace operators spectrum for smooth real or complex functions on all connected compact simple lie groups of rank 3 with biinvariant riemannian metric and establish a connection of these formulas with the number theory and ternary and binary quadratic forms. Flow of a left invariant vector field on a lie group equipped with leftinvariant metric and the groups geodesics 2 proving smoothness of leftinvariant metric on a lie group. However, it is known that lie groups which are not direct product of compact and abelian groups have no biinvariant metric. 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. 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. Existence of cocompact lattices in lie groups with a bi. Finally, we prove that if the lie algebra of a compact lie group g is simple, then the bi invariant metric on g is unique up to rescaling proposition 2. Invariant metrics with nonnegative curvature on compact. Computing biinvariant pseudometrics on lie groups for.
The main tool for proving unboundedness of biinvariant word metrics are homogeneousquasimorphisms. Lecture 2 lie groups, lie algebras, and geometry january 14, 20. The value of ht at e is determined in terms of h0 by some selfadjoint t. 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. In this paper we give explicit calculations of the laplace operators spectrum for smooth real or complex functions on all connected. The bi invariant metric restricts to an ad invariant scalar product on the lie algebra.
Pdf curvature adapted submanifolds of biinvariant lie. 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 value of h t at e is determined in terms of h 0 by some selfadjoint. Geometric aspects of a compact lie group here we will examine various geometric quantities on a lie goup g with a left invariant or bi invariant metrics.
Schurs lemma says that, in an irreducible representation, there is only one invariant symmetric bilinear form, up to a scalar multiple, this is just one version. The 0connection is levicivita with the associated metric the bi invariant metric. Every compact lie group admits a biinvariant metric, which has nonnegative sec tional curvature. We classify the left invariant metrics with nonnegative sectional curvature on so3 and u2. Einstein and conformally einstein biinvariant semi. Metrics, connections, and curvature on lie groups applying theorem 18. We classify the leftinvariant metrics with nonnegative sectional curvature on so3 and u2. Curvatures of left invariant metrics on lie groups core. Let be a 3dimensional lie group with a biinvariant metric. In the in nite dimensional case of di eomorphism groups, we are lead to biinvariant forms.
Given any lie group g, an inner product h,i on g induces a bi invariant metric on g i. In particular, we prove that the hofer metric on hamm. Statistics on riemannian manifolds have been well studied, but to use the statistical riemannian framework on lie groups, one needs to define a riemannian metric compatible with the group structure. Hence, within the class of leftinvariant metrics on a compact lie group g, any metric g 6 g0 that is isospectral to a biinvariant metric g0 must be su. In physics, the leftinvariance and rightinvariance correspond to the independence of the choices of the inertial frame and the body. On the existence of biinvariant finsler metrics on lie. Then, we prove that every compact connected lie group is a symmetric finsler space with respect to the bi invariant absolute homogeneous finsler metric. Glg, we get our second criterion for the existence of a biinvariant metric on a lie group. Let g0 be a biinvariant metric on a compact lie group g. Chapter 17 metrics, connections, and curvature on lie groups. Metrics, connections, and curvature on lie groups applying theorem 17. Invariant metrics with nonnegative curvature on compact lie. A riemannian metric that is both left and rightinvariant is called a biinvariant metric.
Any commutative group certainly admits a biinvariant metric, and any compact group can be given a biinvariant metric by starting with an arbitrary metric on the lie algebra and then averaging adgz as g varies over g. This allows us to obtain geometric conditions which are necessary and sufficient for the submanifold m to be curvature. We also prove that a simplyconnected lie group admits a bi invariant metric if and only if it is a product of a compact lie group with a vector space theorem 2. But riemannian biinvariant metrics do not always exist. Rigid shape registration based on extended hamiltonian. However, there is no bi invariant metric on sen 36. Then, we prove that every compact connected lie group is a symmetric finsler space with respect to the biinvariant absolute homogeneous finsler metric. The main tool for proving unboundedness of bi invariant word metrics are homogeneousquasimorphisms. On lie groups with left invariant semiriemannian metric r.
For exact divergence free and hamiltonian vector elds respectively, a biinvariant nondegenerate bilinear symmetric form has been given by smolentsev 16, 17, 18. When the manifold is a lie group g equipped with bi. Invariant metrics with nonnegative curvature on compact lie groups. Pdf biinvariant and noninvariant metrics on lie groups. Their argument realizes such a metric via its eigenspace. However, it is known that lie groups, which are not a direct product of compact and abelian groups, have no biinvariant metric. The biinvariant metric restricts to an adinvariant scalar product on the lie algebra. We first show that every compact lie group admits a biinvariant finsler metric.
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. A lie group g is semisimple if its lie algebra gis semisimple, i. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. If one takes the metric on gto be biinvariant, the quotient in any case will be seen to admit a metric of nonnegative sectional curvature. Biinvariant means on lie groups with cartanschouten. Let h t be an inverselinear path of leftinvariant metrics on g beginning at a biinvariant metric h 0. Because any other biinvariant metric gives rise to an associative bilinear form on sl 2, and.
Biinvariant finsler metrics on lie groups a finsler metric on a manifold m is a continuous function, f. Conversely, let v, be the lie algebra of a pseudoriemannian lie group of dimension n. Such a metric is characterized by being both leftinvariant, and having leftinvariant elds as killing elds. Geometric aspects of a compact lie group here we will examine various geometric quantities on a lie goup g with a leftinvariant or biinvariant metrics. Spectral isolation of biinvariant metrics 3 neighborhood u of g0 in m leftg such that no g. Every biinvariant metric is leftinvariant, and so can be constructed in a unique way from an inner product for t eg. Abstract riemannian submersions and lie groups william. On any compact simple lie group, is a biinvariant riemannian einstein metric with 0. However, it is known that lie groups which are not direct product of compact and abelian groups have no bi invariant metric. In general, lie groups do not have a bi invariant metric, though all connected semisimple or reductive lie groups do.
Suppose that g is semisimple real lie groups of higher rank and with. Given any lie group g, an inner product h,i on g induces a biinvariant metric on g i. Because any other bi invariant metric gives rise to an associative bilinear form on sl 2, and because any two associative forms are equal up to constant multiplication which must be real if it comes from a metric, it follows that sl2. Most lie groups do not have bi invariant metrics, although all compact lie groups do. When studying relationships between curvature of a complete. In the third section, we study riemannian lie groups with. Note also that riemannian metric is not the same thing as a distance function. In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. G is abelian, then the normal jacobi operator of m equals the square of its invariant shape operator. 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. Walker derivative in lie groups with leftinvariant metric. Finally, we show that if g is a lie group endowed with a bi invariant.
Inria computing biinvariant pseudometrics on lie groups. In particular, we show that, if the normal bundle of m. This is because biinvariant metrics on a compact lie group have nonnegative sectional curvature. Biinvariant and noninvariant metrics on lie groups. Datri and ziller proved in 1979 that every compact simple lie group other than su2 and so3 admits a leftinvariant riemannian einstein metric necessarily, with 0 which is not a multiple of. This observation is the starting point of almost all known con.
We study submanifolds of arbitrary codimension in a lie group g equipped with a biinvariant metric. 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. If g and h are compact lie groups and g h is a group homomorphism then. Thus, ifagroup admits ahomogeneousquasimorphism that is bounded on a conjugation invariant generating set then the group is automatically unbounded with respect to the bi invariant word metric associated with this set. Lie algebras with bi invariant pseudo metric were known to exist since the 1910s with the classification of simple lie. In this paper, we study the geometry of lie groups with biinvariant finsler metrics. We study magnetic trajectories in lie groups equipped with bi. This chapter deals with lie groups with special types of riemannian metrics. Schurs lemma says that, in an irreducible representation, there is only one invariant symmetric bilinear form, up to a scalar multiple, this is just one version of schurs lemma, which has many other uses. The other two connections arent levicivita due to the presence of torsion. Curvatures of left invariant metrics on lie groups john. We first give a necessary and sufficient condition for a left.
Let ht be an inverselinear path of leftinvariant metrics on g beginning at a biinvariant metric h0. Lie group that admits a biinvariant metric is a homogeneous riemannian manifoldthere exists an isometry between that takes any point to any other point. In the sequel, the identity element of the lie group, g, will be denoted by e. Analogously, a metric lie algebra is called indecomposable if it is not the direct sum of nontrivial metric lie algebras. In the case of a connected group g, a left invariant metric is actually bi invariant if and only if the linear transformation adx is skewadjoint for every x in the lie algebra gof g. Curvatures of left invariant metrics on lie groups john milnor. Notice that certain lattices in groups of rank 1 admit nontrivial homogeneous quasihomomorphisms which implies that their biinvariant. Biinvariant finsler metrics on lie groups article pdf available in australian journal of basic and applied sciences 512. Flow of a left invariant vector field on a lie group equipped with left invariant metric and the group s geodesics 2 proving smoothness of left invariant metric on a lie group. If g is a connected lie group, it admits a biinvariant nondegenerate symmetric bilinear form if and only if its lie algebra admits a nondegenerate symmetric bilinear inner product, also called a biinvariant pseudometric. We study also the particular case of bi invariant riemannian metrics.
The lie algebras are taken from tables compiled originally by mubarakzyanov izv. If a lie group g with the lie algebra g admits a leftinvariantriemannianmetric. A bi invariant metric means it is both left invariant and right invariant. Glg, we get our second criterion for the existence of a bi invariant metric on a lie group. Finally, we prove that if the lie algebra of a compact lie group g is simple, then the biinvariant metric on g is unique up to rescaling proposition 2. Spectral isolation of biinvariant metrics on compact lie. We also prove that a simplyconnected lie group admits a biinvariant metric if and only if it is a product of a compact lie group with a vector space theorem 2. Chapter 18 metrics, connections, and curvature on lie groups.
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