Rigidity of time-flat surfaces in the Minkowski spacetime

A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to the constant torsion condition for curves in a three-dimensional space and has been studied in [2, 5, 6, 13, 14]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In this paper, we address the question in the title and prove several local and global rigidity theorems for such surfaces in the Minkowski and Schwarzschild spacetimes. Higher-dimensional generalizations are also considered.