Exact spherically symmetric and hyperbolically symmetric static solutions and Kantowski-Sachs universes for conformally coupled scalar fields

We study static spherically and hyperbolically symmetric solutions of the Einstein equations in the presence of a conformally coupled scalar field and compare them with those in the space filled with a minimally coupled scalar field. We also study the Kantowski-Sachs cosmological solutions, which are connected with the static solutions by some duality relations. A brief discussion of questions such as the relation between the Jordan and the Einstein frames and the description of the singularity crossing is also presented.