The zeta-determinants and anlaytic torsion of a metric mapping torus

TL;DR

This paper computes zeta-determinants and analytic torsion of metric mapping tori using the BFK-gluing formula, with applications to Klein bottles, co-Kähler manifolds, and Witten deformed Laplacians.

Contribution

It introduces a method to compute zeta-determinants and analytic torsion of metric mapping tori, extending previous results and applying to specific geometric structures.

Findings

Computed zeta-determinants for Klein bottle and co-Kähler manifolds.

Established equivalence of heat trace asymptotics for certain manifolds.

Determined analytic torsion for Witten deformed Laplacian on mapping tori.

Abstract

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}.