Lazy orbits: an optimization problem on the sphere

TL;DR

This paper explores the geometric and algebraic properties of non-transitive subgroups of the orthogonal group, introducing optimization problems on the sphere to characterize subgroup types and their applications to Riemannian geometry.

Contribution

It introduces the concept of lazy orbits and formulates an optimization problem on the sphere to classify non-transitive subgroups, including bounds and the notion of rank.

Findings

Characterization of lazy and busy orbits via extremal distances

Classification of non-transitive groups of maximal and rank n-1

Applications to holonomy groups in Riemannian geometry

Abstract

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.