  We use the BFK-gluing formula for zeta-determinants to compute the
zeta-determinant and analytic torsion of a metric mapping torus induced from an
isometry. As applications, we compute the zeta-determinants of the Laplacians
defined on a Klein bottle ${\mathbb K}$ and some compact co-K\"ahler manifold
${\mathbb T}_{\varphi}$. We also show that a metric mapping torus and a
Riemannian product manifold with a round circle have the same heat trace
asymptotic expansions. We finally compute the analytic torsion of a metric
mapping torus for the Witten deformed Laplacian and recover the result of J.
Marcsik in \cite{Ma}.
