[VernonNemitz (OP)]

This notion concerns a way by which the Universe can have lots more mass than we observe.

First, the very concept of "curved space" means that we have to talk about geometry in four dimensions.  Cosmologists say that if the Universe has enough mass it will be "closed", and its geometry will be equivalent to that of a sphere (except it will be a 4D sphere and not a 3D sphere).

I'm thinking of an analogy between the normal 3D sphere and a 4D sphere (hypersphere): where is the horizon?  If we were FlatLanders at some location on the surface of a 3D sphere, things that are some distance away could be distorted, due to the curvature. Suppose the Universe does indeed have the shape of a hypersphere, and we are in its 3D "surface" --and we can only see part-way around it, because red-shift and other effects creates at least an imitation of that purely geometric horizon.  Then all the "missing mass", needed to ensure that the Universe is both "closed" and hyperspherical, would simply be out of sight beyond the horizon, filling the rest of its "surface" in a manner very much as we see in our Observable section of the Universe.

[LeeE]

From our point of view, we're only directly aware of four dimensions, and one of those, being time, is difficult enough to get our heads around as it is.  A complete dimensional model really needs to be applicable to n-dimensional environments though, so you shouldn't get too tied up with just four dimensional solutions, or indeed any specific number of dimensions.

Another key factor is change.  If you try to think in terms of purely spatial dimensions there's no scope for change, resulting in a totally static universe.

If our universe is closed, then a sphere, or even better still a torus, is an applicable model.  However, our universe does not seem to be closed, so a better model would have been that of a four-dimensional elliptic paraboloid, which may be 'open' in both the spatial and temporal dimensions; that the expansion of the universe does not seem to be slowing down at all, but is actually accelerating, indicates that it is open spatially, and although we believe there to have been a start, or origin, of time, but no obvious end, suggests that there's no end to the temporal axis of the elliptic defining curve.  In fact, if the acceleration of expansion turns out to be true, then we're not in an elliptic paraboloid shaped universe but a Hyperboloid shaped one, the essential difference being that we're on the opposite surface to the one we thought we were on i.e. outside rather than inside, and instead of the gradient of the curve approaching zero in one dimension it's increasingly deviating from it and actually approaching zero in another dimension, at right-angles to the one we were thinking of.

All fun stuff, but in terms of seeing the curvature of space, you really need go no further than gravitational lensing - have a look at some of the excellent photos that show it.  This shows, in a very real way, the curvature of space; the light that we see lensed has followed a straight path through space.  Gravitational lensing seems to be a paraboloid type function though []

Regarding mass, well logically, if you imagine looking at the 3D spatial matter in our four-dimensional universe from a five-dimensional point of view you'd be seeing a hypermass, which would be the product of the 3D mass multiplied by (our) time.  As to whether the hypermass <-> energy equivalence would simply be e=mc^3, or whether the 'c' factor is finite at all, due to time appearing to be unbounded and so possibly even infinite (or infinitely small), is an interesting question.