Laplacian eigenmodes for the three-Sphere

Abstract

The vector space $V^k$ of the eigenfunctions of the Laplacian on the three sphere $S^3$, corresponding to the same eigenvalue $lambda_k = -k (k +2)$, 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 peculiar homogeneous harmonic polynomia involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of $S^3$ and SO(4), we are able to write explicitely the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility to select those functions of $ V^k$ which remain invariant under a chosen rotation of SO(4). When the rotation is an holonomy transfor- mation of a spherical space $S^3/Gamma$, this gives a method to calculates the eigenmodes of $S^3/Gamma$, which remains an open problem in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a forthcoming paper, we derive the eigenmodes of dodecahedral pace.