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In this work, two particular orthogonal and conformal decompositions of the3+1 dimensional Einstein equation and Arnowitt-Deser-Misner (ADM) formalism forgeneral relativity are obtained. In order to do these, the 3+1 foliation of thefour-dimensional spacetime, the fundamental conformal transformations and theHamiltonian form of general relativity that leads to the ADM formalism, definedfor the conserved quantities of the hypersurfaces of the globally-hyperbolicasymptotically flat spacetimes, are reconstructed. All the calculations up tochapter 7 are just a review.  We propose a method in chapter 7 which gives an interesting relation betweenthe Cotton (Conformal) soliton and the static vacuum solutions. The formulationthat we introduce can be extended to find the gradient Cotton soliton and thesolutions of Topologically Massive Gravity (TMG) as well as the gradient Riccisoliton.

3+1 Orthogonal and Conformal Decomposition of the Einstein Equation and the ADM Formalism for General Relativity