Rotational Surfaces With Constant Gauss Curvature In Product Spaces

In this paper the author generalizes Liebmann theorem and Hilbert theorem in the product spaces2 H ?M and2 S ?M, where M is defined later in the text. More precisely, this paper mainly focuses on some properties of complete constant Gauss curvature rotational surfaces.First, in the given space2 H ?M, the author proves that the Gauss curvature K(I) of any rotational surface with constant Gauss curvature must satisfy7()9K I ? ? and this surface can be given by parameters. Moreover, for any positive constant0 c, the author shows that there exists a complete rotational surface with Gauss curvature0K(I) ?c, which is unique up to isometrics in2 H ?M.Secondly, in the space2 S ?M, the author proves that the Gauss curvature K(I) of any rotational surfaces must satisfy7()9K I ?, and this surface can also be given by parameters. For any constant0 c,satisfying079c ?, the author is able to show that there exists a unique complete rotational surface with Gauss curvature0K(I) ?c, up to isometrics in2 S ?M.Finally, the author discusses in the last chapter the relationship between the Codazzi pairs and theire induced metrics on surfaces with constant Gauss curvature. As the result, the author obtains the generalized Hilbert type theorem: there exists no complete immersion of constant Gauss curvature97IK)( ?? in2 H ?M and2 S ?M.