The new Generalized Schwarzschild-spacetimes trivial Ricci solitons and the new smooth metric space

The Ricci flow of the Generalized-Schwarzschild spacetimes is newly studied. The soliton configurations are newly stated as trivial Ricci soliton of (Generalized)-Schwarzschild spacetimes. The new smooth metric space is written; the majorization theorem for the distance is given. The application of harmonic maps is presented. The definition of topological soliton as a Schwarzschild soliton of complete Riemannian manifold is newly provided with. New theorems about Generalized-Schwarzschild solitons which are extended from those about the Kaehler solitons are proven; the new theorems are given, which allow one to establish the differences with respect to Kaehler solitons. The new properties of the Generalized Schwarzschild metric are studied. As results, smooth metric spaces are newly exposed as ones endowed with bounded Bakry-Emery curvature; the initial conditions are newly studied: the weight function is majorized as consisting of a polynomial function of the distance(s) (from the initial condition) at most.The Generalized-Schwarzschild metric is now newly proven to be descending from a smooth function. The initial conditions are newly studied to depend only on the spherical neighborhood of the point. The trivial expanding Ricci Kaehler soliton is newly proven to be a Generalized-Schwarzschild soliton; accordingly, this soliton is newly proven to have only one end.