Spacelike surfaces with positive definite second fundamental form in 3D spacetimes

For a spacelike surface with positive definite second fundamental form in any 3-dimensional Lorentzian manifold, a new formula relating its mean and Gauss curvature with the Gauss curvature of the second fundamental form is obtained. As an application, necessary and sufficient conditions are established in order to prove that such a compact spacelike surface is totally umbilical.