2008/10/31 by Peter Krämer, Peter Kramer
Engineering · Mathematics · Physics and Astronomy · #Advanced Materials and Mechanics #Geometric and Algebraic Topology #Geometry #Harmonic #Mathematics #Mathematics and Applications #Physics #Polyhedron #Quantum mechanics #Theoretical physics #gr-qc #math-ph #math.DG #math.MP
paper · pdf · doi:10.1088/0031-8949/79/04/045008
published as Phys.Scripta 79:045008,2009 · 28 pages, 2 figures, minor corrections
arxiv created 2009/01/02 · openalex publication_date 2009/03/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of reduction of group representations. A basis for the harmonic analysis on the (n-1)-sphere is given by the spherical harmonics which transform according to irreducible representations of the orthogonal group. The deck transformations form a subgroup, and so the representations of the orthogonal group can be reduced to those of this subgroup. Upon reducing to the identity representation of the subgroup, the reduced subset of spherical harmonics becomes periodic on the tiling and tunes the harmonic analysis on the (n-1)-sphere to the manifold. A particular class of spherical 3-manifolds arises from the Platonic polyhedra. The harmonic analysis on the Poincare dodecahedral 3-manifold was analyzed along these lines. For comparison we construct here the harmonic analysis on simplicial spherical manifolds of dimension n=1,2,3. Harmonic analysis applied to the cosmic microwave background by selection rules can provide evidence for multiply connected cosmic topologies.