The curvature tensors associated with the gluing formula of the zeta-determinants for the Robin boundary condition

Abstract

The gluing formula for the zeta-determinants of Laplacians with respect to the Robin boundary condition was proved in [15]. This formula contains a constant which is expressed by some curvature tensors on the cutting hypersurface including the scalar and principal curvatures. In this paper we compute this constant explicitly when the cutting hypersurface is a 2-dimensional closed submanifold in a closed Riemannian manifold, and discuss some related topics. We next use the conformal rescaling of the Riemannian metric to compute the value of the zeta function at zero associated to the generalized Dirichlet-to-Neumann operator defined by the Robin boundary condition on this cutting hypersurface.