  In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere,
with constant mean curvature in the Euclidean space are the round (geometrical)
spheres. These results were generalized by S. S. Chern, and then by Eschenburg
and Tribuzy, for surfaces, homeomorphic to the sphere, in Riemannian manifolds
with constant sectional curvature whose mean curvature function satisfies some
bound on its differential. In this paper, using techniques partial differential
equations in the complex plane which generalizes the notion of holomorphy, we
extend these results for surfaces in a wide class of warped product manifolds,
which includes, besides the classical space forms of constant sectional
curvature, the de Sitter-Schwarzschild manifolds and the Reissner-Nordstrom
manifolds, which are time slices of solutions of the Einstein field equations
of the general relativity.
