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In present article, we consider a $L^2$-orthogonal decomposition of the second fundamental form of a closed spacelike hypersurface in a Lorentzian spacetime and its applications to the study of some algebraic-differential properties of the general relativistic vacuum constraint equations. For the study we will use the well-known Ahlfors Laplacian., Comment: arXiv admin note: text overlap with arXiv:2302.02799

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.

In the present paper we prove Liouville-type theorems: non-existence theorems for conformal mappings of complete Riemannian manifolds. In addition, we give an application of these results to the theory of conharmonic transformations. A part of these results was announced in our reports on the conferences "Differential Geometry and its Applications" (July 11-15, 2016, Brno, Czech Republic) and "International Conference on Algebra, Analysis and Geometry" (June 26-July 2, 2016, Kazan, Russia).

In current paper we refer to the geometrical classification of the Einstein equations which has been developed by one of the authors of this paper. This classification was based on the classical theory for decomposition of the tensor product of representations into irreducible components, which is studied in the elementary representation theory for orthogonal groups. We return to this result for more detailed investigation of classes of the Einstein equations., Comment: 9pages