Sub-signature operators, \(\eta\)-invariants and a Riemann-Roch theorem for flat vector bundles (Q1429994)

Let \(M\) be an odd dimensional closed oriented Riemannian manifold. Let \(E\) be an even dimensional oriented subbundle of the tangent bundle and let \(F\) be a real vector bundle over \(M\) equipped with a flat connection. The author uses these data to construct a twisted signature operator and discusses the eta invariant of this setting. He also discusses the eta invariant of a smooth fibration \(Z\rightarrow M\rightarrow B\) and uses methods of Bismut-Cheeger to study the adiabatic limit. Analogues are also discussed when \(TM/E\) is spin.