I recently attended a talk where the person stated a uniqueness result for static vacuum spacetimes whereby he came to a conclusion about a type of spacetime (a 4-manifold) by studying 3-manifolds which are embedded as hypersurfaces in the 4-manifold (similar to the analysis by Schoen and Yau for the positive mass theorem).
However, he made the assumption that the 3-manifold in the spacetime always has a vanishing second fundamental form (similar to Part I of the Schoen-Yau proof). I believe in the literature that this a special case known as the time-symmetric case, but when I asked if his argument could then be generalized to the case where this is not assumed (perhaps using a PDE), he stated that it could not as the spacetime being static implies that it contains a 3-manifold with zero second fundamental form.
I would like to confirm if that is true. Surely a manifold can be 'time-symmetric' in some sense without being 'static'. Time-symmetric is just talking about symmetry under reversal of time, whereas static means it does not change at all: it cannot even rotate as with stationary state metrics like the Kerr metric.