2017/08/29 by Csaba Vincze
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Affine space #Combinatorics #Euclidean distance matrix #Euclidean geometry #Euclidean group #Euclidean space #Geometric Analysis and Curvature Flows #Geometry #Group (periodic table) #Mathematics #Minkowski space #Orbit (dynamics) #Point processes and geometric inequalities #Pure mathematics #Rank (graph theory) #Transitive relation #Unit (ring theory) #Unit sphere #math.DG #math.MG #msc:51B20 #msc:52A40
paper · pdf · doi:10.1016/j.geomphys.2017.10.018
arxiv created 2017/08/29 · openalex publication_date 2017/11/08 · arxiv updated 2018/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Non-transitive subgroups of the orthogonal group play an important role in the non-Euclidean geometry. If G is a closed subgroup in the orthogonal group such that the orbit of a single Euclidean unit vector does not cover the (Euclidean) unit sphere centered at the origin then there always exists a non-Euclidean Minkowski functional such that the elements of G preserve the Minkowskian length of vectors. In other words the Minkowski geometry is an alternative of the Euclidean geometry for the subgroup G. It is rich of isometries if G is "close enough" to the orthogonal group or at least to one of its transitive subgroups. The measure of non-transitivity is related to the Hausdorff distances of the orbits under the elements of G to the Euclidean sphere. Its maximum/minimum belongs to the so-called lazy/busy orbits, i.e. they are the solutions of an optimization problem on the Euclidean sphere. The extremal distances allow us to characterize the reducible/irreducible subgroups. We also formulate an upper and a lower bound for the ratio of the extremal distances. As another application of the analytic tools we introduce the rank of a closed non-transitive group G. We shall see that if G is of maximal rank then it is finite or reducible. Since the reducible and the finite subgroups form two natural prototypes of non-transitive subgroups, the rank seems to be a fundamental notion in their characterization. Closed, non-transitive groups of rank n-1 will be also characterized. Using the general results we classify all their possible types in lower dimensional cases n=2, 3 and 4. Finally we present some applications of the results to the holonomy group of a metric linear connection on a connected Riemannian manifold.