The polynomial associated with the BFK-gluing formula of the zeta-determinant on a compact warped product manifold
The gluing formula for the zeta-determinant of a Laplacian on a compact-oriented Riemannian manifold was proven by \textit{D. Burghelea} et al. in [J. Funct. Anal. 107, No. 1, 34--65 (1992; Zbl 0759.58043)]. In the proof of this gluing formula, there appears a real polynomial whose degree is less than half of the dimension of an underlying manifold. This polynomial is determined by some data on an arbitrarily small collar neighborhood of a cutting compact hypersurface. The constant term of this polynomial appears in the BFK-gluing formula as one ingredient. In [J. Math. Phys. 56, No. 12, 123501, 19 p. (2015; Zbl 1334.58022)], the authors computed the constant term of this polynomial in terms of a warping function when a collar neighborhood is a warped product manifold using methods in [\textit{P. B. Gilkey}, Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Wilmington, Delaware: Publish or Perish, Inc (1984; Zbl 0565.58035)]. In this interesting paper, authors compute this polynomial itself, and 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 in terms of a warping function.
scientific article; zbMATH DE number 7002691
scientific article