Some characterizations of totally umbilical surfaces in three-dimensional warped product spaces

Surfaces with positive definite second fundamental form in a Riemannian, three-dimensional warped product space are considered. A formula expressing the Gaussian curvature with respect to this new metric on the surface in terms of the Gaussian and mean curvature of the first fundamental form is presented. This formula is then used to give some characterizations of compact, totally umbilical surfaces.