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