Riemannian Lie Groups Research Articles

Existence de l'application exponentielle riemannienne d'un groupe de difféomorphismes muni d'une métrique de Sobolev

We prove the existence of geodesics of the weak Riemannian Lie group ( Diff H ∞ ( R n ) , g H k ) = ( Diff ( R n ) ∩ ( Id + ⋂ k ∈ N H k ( R n ; R n ) ) , g H k ) , where g H k is the weak Sobolev metric of order k. Next, we study the Riemannian exponential mapping induced by this metric. For k = 1 , the result immediately gives the local existence of solution in C ∞ ( ; H ∞ ( R n ; R n ) ) of the n-dimensional analog of the Camassa–Holm's equation on the Euclidean space R n .

Chern-Simons Forms Associated to Homogeneous Pseudo-Riemannian Structures

Forms of Chern-Simons type associated to homogeneous pseudo-Riemannian structures are considered. The corresponding secondary classes are a measure of the lack of a homogeneous pseudo-Riemannian space to be locally symmetric. Explicit computations are done for some pseudo- Riemannian Lie groups and their compact quotients. © 2005 Rocky Mountain Mathematics Consortium.

On isometries of some Riemannian Lie groups

We study isometry groups of Lie groups endowed with left-invariant Riemannian metrics. We mainly consider triangular Lie groups. By the familiar Gordon-Wilson theorem, calculating the isometry groups of left-invariant metrics on such groups is reduced to calculating the automorphism groups of the corresponding Lie algebras and to distinguishing compact subgroups of these groups. We consider nilpotent Lie groups in more detail, with special attention to filiform Lie groups and their relatives (prefiliform, quasifiliform). As a rule, we state the main results in terms of the automorphism groups of Lie algebras and then give their geometric interpretation. Special attention is paid to finding the group of connected components of the isometry group (in particular, it is calculated for all filiform Lie groups) and to conditions guaranteeing that the group of rotations (that is, isometries preserving a given point) is finite for certain classes of Riemannian Lie groups.

Maximal isotropy groups of Lie groups related to nilradicals of parabolic subalgebras

We consider Lie groups whose Lie algebra is the nilradical of a parabolic subalgebra of a complex simple Lie algebra, endowed with left-invariant Hermitian metrics. For such Riemannian Lie groups, we describe the Lie algebras of their maximal isotropy groups.

Geometrical Aspects in Hydrodynamics and Integrable Systems

The motion of fluid particles of an inviscid incompressible fluid on a bounded domain is formulated from a Lagrangian point of view. This is accomplished by observing that Euler's equation of motion is a geodesic equation on a group of volume-preserving diffeomorphisms with the metric defined by the kinetic energy. This formulation is based on Riemannian geometry and Lie group theory, first developed by Arnold (1966). Behaviors of the geodesics are characterized by Riemannian (sectional) curvatures, which are shown to be mostly negative (with some exceptions). This property is related to the mixing and ergodicity of the fluid motions. Free rotation of a rigid body fixed at a point gives a simplest example of the dynamical systems which are integrable and represented with such formulation. The same method is applied to the other integrable systems such as the vortex-filament equation or the KdV equation. In contrast to the hydrodynamic system, sectional curvatures are found to be mostly positive (with exceptions). Thus it is found that integrable systems are more stable in the behavior of geodesics than the hydrodynamic system governed by the Euler's equation of motion.

Isotropy of non-nilpotent Riemannian solvable Lie groups

We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra.