  In this pedagogical note, the differences between the Schwarzschild and the
Hilbert-Droste solutions of Einstein equation are scrutinized through a
rigorous mathematical approach, based on the idea of warped product of
manifolds. It will be shown that those solutions are indeed different because
the topologies of the manifolds corresponding to them are different. After
establishing this fact beyond any doubt, the maximal extension of the
Hilbert-Droste solution (the Kruskal-Szekeres spacetime) is derived with
details and its topology compared with the ones of the Schwazschild and the
Hilbert-Droste solution. We also study the problem of the imbedding of the
Hilbert-Droste solution in a vector manifold, hopefully clarifying the work of
Kasner and Fronsdal on the subject. In an Appendix, we present a rigorous
discussion of the Einstein-Rosen Bridge. A comprehensive bibliography of the
historical papers involved in our work is given at the end.
