Isometries of nilpotent metric groups

@article{Kivioja2016IsometriesON,
  title={Isometries of nilpotent metric groups},
  author={Ville Kivioja and Enrico Le Donne},
  journal={arXiv: Metric Geometry},
  year={2016}
}
We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equipped with Carnot-Carath\'eodory distances. We study the regularity of isometries, i.e., distance-preserving homeomorphisms. Our first result is the analyticity of such maps… Expand
From homogeneous metric spaces to Lie groups
We study connected, locally compact metric spaces with transitive isometry groups. For all ε ∈ R, each such space is (1, ε)-quasi-isometric to a Lie group equipped with a left-invariant metric.Expand
Topics in the geometry of non-Riemannian lie groups
This dissertation consists of an introduction and four papers. The papers deal with several problems of non-Riemannian metric spaces, such as sub-Riemannian Carnot groups and homogeneous metricExpand
Isometries of almost-Riemannian structures on Lie groups
A simple Almost-Riemannian Structure on a Lie group G is defined by a linear vector field (that is an infinitesimal automorphism) and dim(G) − 1 left-invariant ones. It is first proven that twoExpand
On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups
This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results fromExpand
A primer on Carnot groups: homogenous groups, CC spaces, and regularity of their isometries
Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphismsExpand
Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups
Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine theExpand
Isometric embeddings into Heisenberg groups
We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitraryExpand
Metric Lie groups admitting dilations
We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$,Expand
Control systems on nilpotent Lie groups of dimension ≤4: Equivalence and classification☆
Abstract Left-invariant control affine systems on simply connected nilpotent Lie groups of dimension ≤4 are considered. First, we classify these control systems under two natural equivalenceExpand
Infinite Geodesics and Isometric Embeddings in Carnot Groups of Step 2
  • Eero Hakavuori
  • Mathematics, Computer Science
  • SIAM J. Control. Optim.
  • 2020
TLDR
It is proved that all infinite geodesics are lines exactly when the induced norm on the horizontal space is strictly convex, and it is shown that all isometric embeddings between such homogeneous groups are affine. Expand
...
1
2
...

References

SHOWING 1-10 OF 36 REFERENCES
From homogeneous metric spaces to Lie groups
We study connected, locally compact metric spaces with transitive isometry groups. For all ε ∈ R, each such space is (1, ε)-quasi-isometric to a Lie group equipped with a left-invariant metric.Expand
Strongly solvable spaces
This work builds on the foundation laid by Gordon and Wilson in the study of isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a transitive solvable group of isometries. WeExpand
The Group of Isometries of a Riemannian Manifold
In the classical theory of groups of motions (isometries) of a Riemannian space,1 neither "whole" groups nor "whole" spaces were ever considered, but only "group germs" and spaces in the neighborhoodExpand
Regularity properties of spheres in homogeneous groups
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group withExpand
Isometry groups of Riemannian solvmanifolds
A simply connected solvable Lie group R together with a leftinvariant Riemannian metric g is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds (R,g) and (R',g') may beExpand
THE COHOMOLOGY OF QUOTIENTS OF CLASSICAL GROUPS
THE HOMOLOGY and cohomology rings of the classical compact Lie groups so(n), SU(n), Sp(n) are well known (for example see Bore1 [3]). Most of these groups have non-trivial centers, and in [4], Bore1Expand
Isometries of Carnot Groups and Sub-Finsler Homogeneous Manifolds
We show that isometries between open sets of Carnot groups are affine. This result generalizes a result of Hamenstädt. Our proof does not rely on her proof. We show that each isometry of aExpand
Besicovitch Covering Property on graded groups and applications to measure differentiation
We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admitsExpand
Homogeneous Ricci solitons
In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to aExpand
Smoothness of subRiemannian isometries
We show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof isExpand
...
1
2
3
4
...