Relative Zeta Functions, Determinants, Torsion, Index Theorems and Invariants for Open Manifolds

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\text{gencomp}(E)$ as the set of Clifford bundles $E^{\mathrm{\prime }}$ which have finite distance to $E$. If $D$, $D^{\mathrm{\prime }}$ are the associated generalized Dirac operators, we prove for the pair $(D,D^{\mathrm{\prime }})$ relative index theorems, define relative $\zeta$– and $\eta$–functions, relative determinants and in the case of $D=\mathrm{\Delta }$ relative analytic torsion. To define relative $\zeta$– and $\eta$–functions, we assume additionally that the essential spectrum of $D^2$ has a gap above zero.