We introduce partial secondary invariants associated to complete Riemannian
metrics which have uniformly positive scalar curvature outside a prescribed
subset on a spin manifold. These can be used to distinguish such Riemannian
metrics up to concordance relative to the prescribed subset. We exhibit a
general external product formula for partial secondary invariants, from which
we deduce product formulas for the higher rho-invariant of a metric with
uniformly positive scalar curvature as well as for the higher relative index of
two metrics with uniformly positive scalar curvature. Our methods yield a new
conceptual proof of the secondary partitioned manifold index theorem and a
refined version of the delocalized APS-index theorem of Piazza-Schick for the
spinor Dirac operator in all dimensions. We establish a partitioned manifold
index theorem for the higher relative index. We also show that secondary
invariants are stable with respect to direct products with aspherical manifolds
that have fundamental groups of finite asymptotic dimension. Moreover, we
construct examples of complete metrics with uniformly positive scalar curvature
on non-compact spin manifolds which can be distinguished up to concordance
relative to subsets which are coarsely negligible in a certain sense. A
technical novelty in this paper is that we use Yu's localization algebras in
combination with the description of K-theory for graded C*-algebras due to
Trout. This formalism allows direct definitions of all the invariants we
consider in terms of the functional calculus of the Dirac operator and enables
us to give concise proofs of the product formulas.Comment: 37 pages; v2: added new results and many minor revisions; v3: Minor
revision following the referees' suggestions. To appear in Journal of
Topolog
