Search results for "Lien ryhmät"
showing 5 items of 5 documents
Affine decomposition of isometries in nilpotent Lie groups
2015
Tässä työssä esitetään uusi tulos koskien isometrioiden säännöllisyyttä nilpotenttien yhtenäisten metristen Lien ryhmien välillä. Termillä metrinen Lien ryhmä tarkoitamme Lien ryhmää, joka on varustettu etäisyysfunktiolla siten, että ryhmän (vasen) siirtokuvaus on isometria, ja etäisyysfunktio indusoi topologian, joka Lien ryhmällä on monistona alun perin olemassa. Todistamme, että isometriat tässä tilanteessa ovat välttämättä affiinikuvauksia: jokainen isometria voidaan esittää yhdistettynä kuvauksena siirrosta ja isomorfismista. Tämän seurauksena kaksi isometrista ryhmää ovat välttämättä isomorfiset. Klassisesti isometrioiden lineaariaffiinisuus on…
A Cornucopia of Carnot groups in Low Dimensions
2022
Abstract Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invaria…
Lipschitz Functions on Submanifolds of Heisenberg Groups
2022
Abstract We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation of Lipschitz functions on ${\mathbb {H}}$-rectifiable sets and a coarea formula on ${\mathbb {H}}$-rectifiable sets that completes the program started in [18].
Intrinsic rectifiability via flat cones in the Heisenberg group
2022
We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones. peerReviewed
Nilpotent Groups and Bi-Lipschitz Embeddings Into L1
2022
We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an L1 space, then it is abelian. We reach this conclusion by proving that every Carnot group that bi-Lipschitz embeds into L1 is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into L1 and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on “generic” tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipsch…