Laplacian eigenmodes for the three-sphere

Abstract

The vector space of the eigenfunctions of the Laplacian on the three-sphere S3, corresponding to the same eigenvalue ?k = ?k(k + 2), has dimension (k + 1)2. After recalling the standard bases for , we introduce a new basis B3, constructed from the reductions to S3 of a peculiar homogeneous harmonic polynomial involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of S3 and SO(4), we are able to write explicitly the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility of selecting those functions of which remain invariant under a chosen rotation of SO(4). When the rotation is a holonomy transformation of a spherical space S3/?, this gives a method for calculating the eigenmodes of S3/?, which remains an open problem in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a companion paper, we present the derivation for dodecahedral space.