  In this article, we consider an orthogonal decomposition of the traceless
part of the Ricci tensor of a compact Riemannian manifold and study its
application to the geometry of compact almost Ricci solitons. In addition, we
consider an orthogonal expansion of the traceless part of the second
fundamental form of a compact spacelike hypersurface in a Lorentzian manifold
and study its application to the problem of constructing solutions of general
relativistic constraint equations in vacuum. In these two cases, we use the
well-known Ahlfors Laplacian.
