Schwarzschild solitons and Generalized Schwarzschild solitons have recently been studied on the new metrized smooth metric space of convex functions.

The completeness of geodesics and the Cauchy hypersurfaces are established for Schwarzschild spacetimes after correction [Hawking, S.W.: Commun. Math. Phys. 25, 152–166 (1972).].

The solitons are newly posed on the pseudo-spherical General Relativistic cylinder, with new equations set and uniquely solved.

The uniqueness of Schwarzschild solitons is newly investigated.

The concurrent vector field in General Relativity is newly defined, and the 4-position vector is newly indicated as a concurrent vector field in General Relativity; the Killing vector field is newly determined accordingly to the 4-velocity vector when the configuration of the observer frame is chosen as one solidal with the photon.

New isopoertimetric inequalities and new curvature estimates are provided to hold. New thoerems are proven about these geometries.

The Penrose tipping lightcones are newly written for these spacetimes; the cross-sections are determined accordingly. Accordingly, the Yamabe flow on the S2 sphere is newly proven to converge for submersion from a Schwarzschild spacetime: geodesics spheres are taken into account.

Submersions and immersions are newly discussed when these spacetimes are considered according to the pullback from the metric tensor.

The elements are also gathered for discussion of the Penrose 1965 Theorem.