How do metrics behave under joining along a manifold embedded in the boundary?
This is, more-or-less, part of Problem 4.66 in Kirby's List:
Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, constant curvature) behave under standard topological constructions such as connected sum, plumbing, handle addition? Same question for $\eta$-invariants, moduli spaces, etc.
So, in theory it is an open problem. However, Kirby states of the problem:
Much has probably been done on this open ended problem, and the editor has not attempted to update it.
So, it would seem that this question may indeed have an answer in the literature. My question is: What is that answer and where does one find it?