Gluing formula of real analytic torsion forms and adiabatic limit (Q502982)

The author considers the gluing problem of Bismut-Lott torsion form. Suppose that \(X\) is a compact hypersurface in \(M\) such that \(M=M_{1}\cup_{X}M_{2}\) and \(M_{1},M_{2}\) are manifolds with the common boundary \(X\). Let \(T^{H}M\) be a horizontal tangent bundle of \(M\). The main theorem (Theorem 0.2) gives a gluing formula -- an equation relating the Bismut-Lott torsion forms of \(T^{H}M\), \(T^{H}M_{1}\), and \(T^{H}M_{2}\). The proofs use the adiabatic limit method, which is a ``limiting process that one stretches the original manifold along the normal direction of certain hypersurface into two manifolds with cylinder ends of infinite length''. The main motivation for this work is that such a gluing formula may lead to a relation between the Bismut-Lott torsion and the Igusa-Klein torsion.