Decompositions of the space of Riemannian metrics on a compact manifold with boundary

In this paper, for a compact manifold M with non-empty boundary partial derivative M, we give a Koisotype decomposition theorem, as well as an Ebin-type slice theorem, for the space of all Riemannian metrics on M endowed with a fixed conformal class on partial derivative M. As a corollary, we give a characterization of relative Einstein metrics.