346 research outputs found
A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries
No full textCarnot 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 automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks.We consider them as special cases of graded groups and as homogeneous metric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups
Properties of isometrically homogeneous curves
No full textThis paper is devoted to the study of isometrically homogeneous spaces from the view point of metric geometry. Mainly we focus on those spaces that are homeomorphic to lines. One can reduce the study to those distances on that are translation invariant. We study possible values of various metric dimensions of such spaces. One of the main results is the equivalence of two properties: the first one is linear connectedness and the second one is one-dimensionality, with respect to Nagata dimension. Several concrete pathological examples are provided. © 2012 The Author(s) 2012
Metric spaces with unique tangents
No full textWe are interested in studying doubling metric spaces with the property that at some of the points the metric tangent is unique. In such a setting, Finsler-Carnot-Carathéodory geometries and Carnot groups appear as models for the tangents. The results are based on an analogue for metric spaces of Preiss's phenomenon: tangents of tangents are tangents. In fact, we show that, if X is a general metric space supporting a doubling measure μ, then, for μ-almost every x ∈ X, whenever a pointed metric space (Y; y) appears as a Gromov-Hausdorff tangent of X at x, then, for any y' ∈ Y , also the space (Y; y') appears as a Gromov-Hausdorff tangent of X at the same point x. As a consequence, uniqueness of tangents implies their homogeneity. The deep work of Gleason-Montgomery-Zippin and Berestovskiĭ leads to a Lie group homogeneous structure on these tangents and a characterization of their distances
Geodesic manifolds with a transitive subset of smooth biLipschitz maps
No full textThis paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let X = G/H be a homogeneous manifold of a Lie group G and let d be a geodesic distance on X inducing the same topology. Suppose that there exists a subgroup GS of G that acts transitively on X such that each element g ∈ GS induces a locally biLipschitz homeomorphism of the metric space (X, d). Then the metric is locally biLipschitz equivalent to a sub-Riemannian metric. Any such metric is defined by a bracket generating G S-invariant sub-bundle of the tangent bundle. The result is a consequence of a more general fact that requires a transitive family of uniformly biLipschitz diffeomorphisms with a control on their differentials. It will be relevant that the group acting transitively on the space is a Lie group and so it is locally compact, since, in general, the whole group of biLipschitz maps, unlikely the isometry group, is not locally compact. Our method also gives an elementary proof of the following fact. Given a Lipschitz subbundle of the tangent bundle of a Finsler manifold, both the class of piecewise differentiable curves tangent to the sub-bundle and the class of Lipschitz curves almost everywhere tangent to the sub-bundle give rise to the same Finsler-Carnot- Carathéodory metric, under the condition that the topologies induced by these distances coincide with the manifold topology. © European Mathematical Society
Doubling property for bilipschitz homogeneous geodesic surfaces
No full textIn this paper we discuss general properties of geodesic surfaces that are locally biLipschitz homogeneous. In particular, we prove that they are locally doubling and that there exists a special doubling measure analogous to the Haar measure for locally compact groups. © 2010 Mathematica Josephina, Inc
Isometries of nilpotent metric groups
No full textWe consider Lie groups equipped with arbitrary distances. We only assume that
the distances are left-invariant and induce the manifold topology. For brevity, we call such
objects metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are subRiemannian
Lie groups, homogeneous groups, and, in particular, Carnot groups equipped with
Carnot–Carathéodory distances. We study the regularity of isometries, i.e., distance-preserving
homeomorphisms. Our first result is the analyticity of such maps between metric Lie groups. The
second result is that if two metric Lie groups are connected and nilpotent then every isometry
between the groups is the composition of a left translation and an isomorphism. There are
counterexamples if one does not assume the groups to be either connected or nilpotent. The
first result is based on a solution of the Hilbert’s fifth problem by Montgomery and Zippin.
The second result is proved, via the first result, reducing the problem to the Riemannian case,
which was essentially solved by Wolf.Nous considérons des groupes de Lie
munis de distances arbitraires. Nous supposons seulement que ces distances sont invariantes
à gauche et induisent la topologie de la variété sous-jacente. Nous appelons groupes de Lie
métriques de tel objets. Mis à part les groupes de Lie riemanniens, des exemples remarquables
sont donnés par les groupes de Lie sous-riemanniens, les groupes homogènes et, en particulier, les
groupes de Carnot munis de distances de Carnot–Carathéodory. Nous montrons la régularité des
isométries, c’est-à-dire des homéomorphismes qui préservent la distance. Notre premier résultat
est l’analyticité de telles applications entre des groupes de Lie métriques. Le second résultat est
que, si deux groupes de Lie métriques sont connexes et nilpotents, alors toute isométrie entre
ces groupes est la composition d’une translation à gauche et d’un isomorphisme. Il y a des
contre-exemples si on ne suppose pas que les groupes sont connexes ou nilpotents. Le premier
résultat repose sur la solution du cinquième problème de Hilbert par Montgomery et Zippin. Le
second résultat est démontré à l’aide du premier, en réduisant le problème au cas riemannien,
cas qui a été essentiellement résolu par Wolf.peerReviewe
A metric characterization of Carnot groups
No full textWe give a short axiomatic introduction to Carnot groups and their subRiemannian and subFinsler geometry. We explain how such spaces can be metrically described as exactly those proper geodesic spaces that admit dilations and are isometrically homogeneous
Measure of submanifolds in the Engel group
No full textWe find all the intrinsic measures for smooth submanifolds in the Engel group whose tangent space has Lipschitz regularity, showing that they are equivalent to the corresponding d-dimensional spherical Hausdorff measure restricted to the submanifold. The integer d is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique
Lipschitz and path isometric embeddings of metric spaces
No full textWe prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C 1 Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map. © 2012 Springer Science+Business Media Dordrecht
Lusin approximation for horizontal curves in step 2 Carnot groups
No full textA Carnot group G admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve γ in G and ε> 0 , there is a C1horizontal curve Γ such that Γ = γ and Γ′= γ′outside a set of measure at most ε. We verify this property for free Carnot groups of step 2 and show that it is preserved by images of Lie group homomorphisms preserving the horizontal layer. Consequently, all step 2 Carnot groups admit Lusin approximation for horizontal curves
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