This is not implied by a Riemannian ( i.e. curved ) time-space continuum. It can be thought of as "all that there is" just as well as a traditional "flat" time-space continuum.
Conversely, a flat 3-space can be thought of as a slice of a 4-space, and in fact this was a very popular notion in Victorian times as a physical justification of the "spirit world", which shared a universal time measure with familiar 3-space.
I’ll try to work on that as soon as I’m home. Thanks for the explanation!