Homogeneous Metrics on Spheres
Martin Kerin, David J. Wraith · Irish Mathematical Society Bulletin · 2003
This article is a summary of the work carried out by the first author towards a Master's thesis under the direction of the second author.Our aim is to investigate certain aspects of the geometry of spheres, especially the spheres S 3 and S 5 .Usually, when one imagines a sphere, one imagines a round sphere, and of course the geometry of such objects is well understood.However, there are many different geometries with which a sphere can be equipped.A sphere is first and foremost a smooth manifold.It only becomes a geometric object -in other words assumes some kind of shape (such as being round) -when we equip it with a Riemannian metric.Recall that a Riemannian metric is a smoothly varying choice of inner product on each tangent space.Of course there are uncountably many Riemannian metrics we can equip any sphere with, and each will provide the sphere with a certain geometry.However, we will focus on certain very special kinds of Riemannian metrics, namely those which are homogeneous.A round sphere is clearly highly symmetric.More than just having lots of symmetry, it in fact looks the same at every point!Spaces with this property are known as homogeneous spaces.More formally, if M is a smooth manifold, we say that M is a homogeneous space if there is a (Lie) group G of self-diffeomorphisms of M which acts transitively.In other words, given any two points x and y in M , there is an element g ∈ G such that gx = y.Any homogeneous space M can be regarded as a space of cosets in the following way.Choose a point x ∈ M .Let H be the subgroup of G which fixes the point x (that is, hx = x for all h ∈ H).H is called the isotropy subgroup of G at the point x.(The isotropy at any other point in M is conjugate in G to H.) It is not difficult