The Polynomial Associated with the BFK-Gluing Formula of the Zeta-Determinant on a Compact Warped Product Manifold

摘要

In the proof of the BFK-gluing formula of the zeta-determinant of a Laplacian there appears a polynomial of degree less than half of the dimension of an underlying manifold. This polynomial is determined completely by some data on a collar neighborhood of a cutting compact hypersurface. In this paper we compute the polynomial in terms of a warping function when a collar neighborhood of a cutting hypersurface is a warped product manifold. We also use a similar method to compute the values of a relative zeta function and a zeta function associated to the Dirichlet-to-Neumann operator at zero on a warped product manifold.