Existence of a positive scalar curvature (PSC) metric has been one of the central topics in differential topology of higher dimensional manifolds. Index theory provides a topological invariant whose vanishing is an effective necessary condition of the existence of a PSC metric. Recently, Chang-Weinberger-Yu introduce a new index theoretic invariant called the relative higher index detecting the non-existence of a PSC metric on spin manifolds with boundary. The aim of this paper is to understand this invariant from Riemannian gemetric point of view. Firstly, we introduce a new definition of the relative higher index and prove that it is equivalent to the existing definitions. Secondly, by using our new definition, we relate the relative higher index with a Riemannian geometry of vector bundles; the index pairing with almost flat vector bundles. This is a relative analogue of the theory of Gromov-Lawson for closed spin manifolds.

Yosuke Kubota
"The relative Mishchenko--Fomenko higher index and almost flat bundles I: The relative Mishchenko--Fomenko index"
arXiv: 1807.03181