Lazy orbits: An optimization problem on the sphere

摘要

AbstractNon-transitive subgroups of the orthogonal group play an important role in the non-Euclidean geometry. IfGis 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 ofGpreserve the Minkowskian length of vectors. In other words the Minkowski geometry is an alternative of the Euclidean geometry for the subgroupG. It is rich of isometries ifGis “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 ofGto 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 groupG. We shall see that ifGis 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 rankn?1will be also characterized. Using the general results we classify all their possible types in lower dimensional casesn=2,3and4.Finally we present some applications of the results to the holonomy group of a metric linear connection on a connected Riemannian manifold.]]>