Let $(M,g)$ be spacetime. A spacelike hypersurface $\Sigma\subset M$ is a submanifold of dimension $3$ with a timelike normal vector field. They have many uses: to define conserved quantities by integrating over them, to define an inner product on the solutions to the Klein-Gordon equation, and so forth.
I have the impression though that those submanifolds are a generalization of the idea of "space at a fixed time" and hence of simultaneous events. I just don't know how to justify that and that's the main point of the question.
The situation seems simpler when we have a family of hypersurfaces obtained by a timelike future-directed vector field $\xi$ satisfying the integrability condition, i.e., letting $\omega$ the physically equivalent one-form $\omega_\mu = g_{\mu\nu}\xi^ \nu$, the condition $\omega\wedge d\omega = 0$.
In that case each integral line of $\xi$ is an observer and, if I'm not wrong, each hypersurface would consist of 'space with fixed time' for the reference frame of those observers.
But this is a family of hypersurfaces, not one alone.
What is the physical meaning of a spacelike hypersurface in General Relativity? Is it really a generalization of the idea of "space with time fixed"? If so, how can we justify this way to understand this mathematical object?