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 metric spaces. The research has been carried out between the University of Trento (Italy) and the University of Jyväskylä (Finland) under the supervision of prof. F. Serra Cassano and E. Le Donne, respectively. In the following we present the abstracts of the four papers. 1) REGULARITY PROPERTIES OF SPHERES IN HOMOGENEOUS GROUPS E. Le Donne AND S. Nicolussi Golo 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 with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of every homogeneous distance on the Heisenberg group. In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres present cusps. 2) ASYMPTOTIC BEHAVIOR OF THE RIEMANNIAN HEISENBERG GROUP AND ITS HOROBOUNDARY E. Le Donne, S. Nicolussi Golo, AND A. Sambusetti The paper is devoted to the large scale geometry of the Heisenberg group H equipped with left-invariant Riemannian metrics. We prove that two such metrics have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian metric d on H there is a unique sub-Riemanniann metric d' for which d − d' goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group. 3) FROM HOMOGENEOUS METRIC SPACES TO LIE GROUPS M. G. Cowling, V. Kivioja, E. Le Donne, S. Nicolussi Golo, AND A. Ottazzi We study connected, locally compact metric spaces with transitive isometry groups. For all $\epsilon\in R_+$, each such space is $(1,\epsilon)$- quasi-isometric to a Lie group equipped with a left-invariant metric. Further, every metric Lie group is $(1,C)$-quasi-isometric to a solvable Lie group, and every simply connected metric Lie group is $(1,C)$-quasi-isometrically homeomorphic to a solvable-by-compact metric Lie group. While any contractible Lie group may be made isometric to a solvable group, only those that are solvable and of type (R) may be made isometric to a nilpotent Lie group, in which case the nilpotent group is the nilshadow of the group. Finally, we give a complete metric characterisation of metric Lie groups for which there exists an automorphic dilation. These coincide with the metric spaces that are locally compact, connected, homogeneous, and admit a metric dilation. 4) SOME REMARKS ON CONTACT VARIATIONS IN THE FIRST HEISENBERG GROUP S. Nicolussi Golo We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if $f : R^2 \rightarrow R^2$ is a $C^1$-intrinsic function, and $\nabla^f\nabla^ff = 0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.