Spectral zeta function of a sub-Laplacian on product sub-Riemannian manifolds and zeta-regularized determinant

We analyze the spectral zeta function for sub-Laplace operators on product manifolds M×N. Starting from suitable conditions on the zeta functions on each factor, the existence of a meromorphic extension to the complex plane and real analyticity in a zero neighbourhood is proved. In the special case of N=S1 and using the Poisson summation formula, we obtain expressions for the zeta-regularized determinant. Moreover, we can calculate limit cases of such determinants by inserting a parameter into our formulas. This is a generalization of results in Furutani and de Gosson (2003) [1] and in particular it applies to an intrinsic sub-Laplacian on U(2)≅S3×S1U(2)≅S3×S1 induced by a sum of squares of canonical vector fields on S3S3; cf. Bauer and Furutani (2008) [2]. Finally, the spectral zeta function of a sub-Laplace operator on Heisenberg manifolds is calculated by using an explicit expression of the heat kernel for the corresponding sub-Laplace operator on the Heisenberg group; cf. Beals et al. (2000) [18] and Hulanicki (1976) [19].