Goedesics Completeness and Cauchy Hypersurfaces of Ricci Solitons on Pseudo-Riemannian Hypersurfaces at the Fictitious Singularity: Schwarzschild-Soliton Geometries and Generalized-Schwarzschild-Soliton Ones

The methodology is developed here to write Ricci solitons on the newly found structure of the pseudo-spherical cylinder. The methodology is specified for Schwarzschild solitons and for Generalized-Schwarzschild solitons. Accordingly, a new classification is written for the Schwarzschild solitons and for the Generalized-Schwarzschild solitons. The rotational field is spelled out. The potential for a tangent vector field is used. The conditions are recalled to discriminate which submanifold of a Ricci manifold is a soliton or is an almost-Ricci soliton. It is my aim to prove that a concurrent vector field is uniquely determined after the 4-velocity vector of a Schwarzschild soliton. As a result, the analytically specified manifold, which is a spacelike submanifold of the Schwarzschild spacetime that admits Ricci solitons. The rotational killing fields are tangent to the event horizon. The conditions that are needed to match the new aspects are spelled out analytically. As a result, the two manifolds described in the work of Bardeen et al. about the requested mass of a stationary, axisymmetric solution of the Einstein Field Equations of the spacetime, which contains a blackhole surrounded with matter from the new results obtained after correcting the work of Hawking 1972 about would-be point ’beyond the conjugate point’ on the analytic continuation of the would-be geodesics: they are proven here to become the tangent manifold (which is expressed from the tangent bundle in General-Relativistic notation). The prescription here is based on one of the books of Landau et al., that the matter is not put into the metric tensor, not even in the ultra-Relativistic limit. This way, the pseudo-spherical cylinder is one implemented from the Minkowskian description and whose asymptotical limit is proven. The new methodology allows one to describe the outer region of the blackhole as one according to which the (union of the trapped) regions is one with null support. For the purpose of the present investigation, the definition of concurrent vector fields in General-Relativity is newly developed. As a further new result, the paradigm is implemented for the shrinking case, which admits as subcase the Schwarzschild manifolds and the Generalized-Schwarzschild manifolds. The Penrose 1965 Theorem is discussed for the framework outlined here; in particular, the presence of trapped hypersurfaces is discarded. The no-hair theorem can now be discussed.