Laplacian eigenmodes for the three-sphere

The vector space V k of the eigenfunctions of the Laplacian on the three-sphere S 3 , corresponding to the same eigenvalue λk =− k( k +2 ), has dimension (k +1 ) 2 . After recalling the standard bases for V k , we introduce a new basis B3, constructed from the reductions to S 3 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 S 3 andSO(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 V k which remain invariant under a chosen rotation of SO(4). When the rotation is a holonomy transformation of a spherical space S 3 /� , this gives a method for calculating the eigenmodes of S 3 /� , 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.