  We study the effects of the null energy condition on totally umbilic
hypersurfaces in a class of static spacetimes, both in the spacelike and the
timelike case, respectively. In the spacelike case, we study totally umbilic
warped product graphs and give a full characterization of embedded surfaces
with constant spacetime mean curvature using an Alexandrov Theorem by Brendle
and Borghini--Fogagnolo--Pinamonti. In the timelike case, we achieve a
characterization of photon surfaces with constant umbilicity factor similar to
a result by Cederbaum--Galloway.
