Spectral Analysis and Zeta Determinant on the Deformed Spheres

We consider a class of singular Riemannian manifolds, the deformed spheres , defined as the classical spheres with a one parameter family gk] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian , we study the associated zeta functions . We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in . An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular and . We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2,3, thus generalizing a result of Dowker 25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k. Partially supported by FAPESP: 2005/04363-4