Laplacian eigenmodes for the three-Sphere

Original · EN

The vector space Vᵏ of the eigenfunctions of the Laplacian on the three sphere S³, corresponding to the same eigenvalue lambdaₖ = -k (k +2), has dimension (k + 1)². After recalling the standard bases for Vᵏ, we introduce a new basis B3, constructed from the reductions to S³ 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 S³ 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 Vᵏ which remain invariant under a chosen rotation of SO(4). When the rotation is an holonomy transfor- mation of a spherical space S³/Gamma, this gives a method to calculates the eigenmodes of S³/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.