The theory that the universe would be curled up into itself isn't new, and
is actually one of the many theories proposed for a unified field theory that would merge quantum mechanics and general relativity.
The model of a curved universe, as you describe it, is called
Brane Theory, and it is a variant of String Theory. Brane Theory proposes that the universe would indeed be slightly curved (beyond our current ability to measure any significant curvature), and that it must therefore be curved either in a concave way (i.e. a saddle-shaped brane) or a convex way (i.e. a spherical brane). It then also by definition follows that our universe would itself be a four-dimensional hologram as a delimiting membrane around a presumably five-dimensional hyperspace, which could itself then be only a membrane around a six-dimensional hyperspace, and so on.
A very interesting thing to note is that the observable universe ─ currently calculated to be approximately 76 billion light-years across, and delineated by the visually impenetrable barrier of cosmic background radiation from the time of the Big Bang ─ is not the complete universe. The universe is still expanding, and it is actually doing so at an accelerating rate. This means that the edges of the observable universe are by now moving away from us at velocities greater than the speed of light ─ note: the speed of light is only the highest speed anything can attain
inside of spacetime, but it does not pose any limitations of the speed by which spacetime itself can expand.
Now, if the outer edges of the expanding universe are accelerating away from us at speeds greater than the speed of light, then it follows that any light/radiation coming our way from those regions of spacetime will never reach us. And in addition to that, the accelerated expansion of the universe also holds that there will eventually be a time when even the nearby galaxies are starting to move away from us at ever greater speeds, and eventually at speeds greater than the speed of light, so that when we look up at the night sky ─ which we would of course be doing from another planet than Earth, because Mercury, Venus, Earth and possibly Mars will be destroyed by the expansion of the sun in about a billion years from now ─ we won't be seeing any stars anymore. It'll be pitch black, and we would truly be feeling alone in the universe, because everything is so far away from us and moving away so fast that its light will never reach us. Any burgeoning society that would come into existence in the galaxy at that point in time would genuinely come to believe that the galaxy is all there is, and that there isn't anything else anymore beyond that. The galaxy would itself become the observable universe, surrounded by nothing but blackness.
What's also interesting to note here is that this is actually the complete opposite of what happens beyond the event horizon of a black hole. Beyond the event horizon, the gravitational pull of the black hole is so strong that the escape velocity of the singularity at the core of the black hole becomes greater than the speed of light, which means that no light can ever be emanated from inside the event horizon of a black hole. Which is why they are termed black, of course, even though there definitely
is light there, but that light is trapped by the infinite distortion of spacetime. (
Note: The proposed Hawking radiation ─ which has so far never been observed, mind you ─ does not come from the inside of the black hole, but from virtual particle pairs created by the vacuum of space, whereby one half of the normally short-lived virtual particle pair falls inside the event horizon while the other one is able to escape and survive, whereas it would normally have been annihilated again by its anti-particle sibling within nanoseconds.)
Either way, the interesting thing here is that the expansion of the universe as we perceive it is actually the exact opposite of what is going on inside a black hole, but with a similar result, namely that there is light ─ or more generically, microwave radiation ─ being emanated in our direction, but that we'll never see it. Light from beyond the event horizon of a black hole is simply following the trajectory of the infinitely curved-in spacetime, and light from beyond the cosmic background radiation horizon is too slow to reach us due to the faster-than-light acceleration of the expansion of the universe in those regions. So in a way, you could say that the universe is actually an inside-out black hole, or that a black hole is an inside-out universe.
Also, with regard to time, there is still some contention within the scientific world on whether time existed when the Big Bang took place, or whether time is an emergent phenomenon of the Big Bang. Nevertheless, time is very real, and we also know from special relativity that the future is indeed ─ and I'm sorry for those who wish to believe that we have free will ─ effectively "written in stone". The future has already happened, even though we haven't reached it yet. And that is what brings us to the definition of time, i.e. time is a dimension along which everything is moving from event to event. And inside a black hole, spacetime breaks down. As such, once you pass the event horizon of a black hole, and as a precursor to the total breakdown of all dimensions, time becomes a
spatial dimension. At that point, it is still a dimension that you cannot travel along in reverse ─ unlike with the normal spatial dimensions we experience outside of a black hole ─ because the escape velocity is greater than the speed of light, but it'll be a spatial dimension, rather than a temporal one.
Given the similarity between the phenomenon of the spacetime distortion inside a black hole and the accelerated expansion of the universe at speeds greater than the speed of light, some theorists have therefore proposed that if you were to cross the boundary of the observable universe, time would similarly become a spatial dimension again. But that's just a theory, and for that matter, one of many about which there is no consensus yet. Anything of which we are certain regarding what lies beyond the observable segment of the universe is by definition determined
mathematically only, and this in and of itself leaves the door open for multiple theories ─ even conflicting ones ─ so long as the math is sound and the model works.
After all, that is why general relativity works at the macroscopic scale and why quantum mechanics works at the microscopic scale, while both theories are in contradiction to one another. Or to put it differently, they both work, but their contradiction means that neither is a complete model of the universe. Einstein was aware of this, and he continued working on a unified field theory until his death, but he never found the answer. And even today, almost sixty years later, we, too, still don't have that answer.