• Fourier transform and Plancherel Theorem

When we try to transfer these three concepts to a “curved manifold” M , the situation becomes instantly very complicated. Let us look for an appropriate class of M ’s for which we can expect satisfactory answers. We need the requirement to measure distances, so M should be Riemannian. In order to have a reasonable theory of selfadjoint operators on M we also need completeness. But still the class of complete Riemannian manifolds is far too big for our purpose. Let us assume that M “looks everywhere the same”; i.e., M allows a transitive action by a Lie group G of isometries. Hence