Yesterday, in Brazilian School on Differential Geometry, a friend asked me the question:
Given an (non-trivial) initial data set $(M,g,k)$ for the Cauchy problem in General Relativity. Is there some instant of time in your development, namely $t \neq 0$, such that the slice $(M_t,g_t, k_t)$ is isomorphic to $(M,g,k)$, in the sense of that are isometric Riemannian manifolds and has the same second fundamental form?
Unfortunately, I do not know any answer for this question. I will be grateful with any answer or reference that can clarify this question.