No.

Space-time is most certainly curved.

That is what the general theory of relativity is all about. It describes the curvature tensor that applies to the space-time manifold very precisely as being proportinional to the the stress-energy tensor, which is the source of Wheeler's famous quote that "Matter tells space how to curve and space tells matter how to move".

The curvature of space-time is quite real and it is not merely an analogy. It is in fact a central facet of modern physics.

I will have to disagree with you here, DrRocket, although not completely.

If you remember the principle of equivalence, then being in a gravitational field is no different from acceleration. By a simple thought experiment, it is easy to show that an observer in acceleration will observe a different value for pi. In order to make myself clearer, I will describe the exact nature of the thought experiment:

Imagine a circular chamber, in which tow observers are enclosed. The chamber begins to accelerate. Armed with rulers, the observers set out to measure the circumference and diameter respectively. The first observer measures the circumference; because his ruler will contract in the direction of motion, he will measure a larger circumference. The second observer's ruler, however, is perpendicular to the direction of motion, and hence will not contract; obviously, the observer will measure the same diameter as he would at rest.

The chamber stops. The two observers meet, and with their measurements, begin to calculate the value of pi, which will turn out different, because of an observed change in the circumference without a corresponding change in the diameter. However, pi can only change when measured on a curved surface, and from this it is possible to deduce the notion of a curved space-time in a gravitational field.

My bone of contention, if you will, is based on this. Obviously, the principle of equivalence demands that pi be observed to change in a gravitational field too. Yet if a curved space-time operates as a real, physical entity in a gravitational field, surely it should do the same in accelerated motion. However, we do not observe any physical 'curvature' of space or time in accelerated motion; while time dilation and Fitzgerald contraction do occur, I doubt you can label that as the curvature of space and time. You do not measure time to suddenly curve; nor do you observe space do the same.

In the thought experiment I described above, the reason pi came out different was because of a difference in the actual observation of the circumference when it was at rest and at motion. And the principle of equivalence allows us to substitute the notion of a curved space-time with instead a circular chamber, undergoing acceleration. It is obvious from this that we cannot conclude that space-time is an actual physical thing, just as acceleration is not an actual physical thing: both are effects that can be observed and experienced yet cannot be touched and hence have no physical status. If it was, we could not replace this space-time with an accelerating circular chamber as easily as we can.

Since space-time cannot be said to be an actual physical thing, or something we can touch, but only something we can experience, it follows that the notion of its curvature is also only something that can be experienced; in no other way is it physical. While we can experience its effects, this does not automatically mean that it is a physical, tangible thing.

That is why I said that the curvature of space-time is only an analogy, because space-time does not curve in a physical sense. We can experience the effects of such a curvature, but in no way can we conclude that it is actually because such a curvature actually exists, in a physical sense, because we can just as easily replace the notion with an accelerating circular chamber. Mathematically, yes, space-time does curve, and is the reason why general relativity works so well; physically, space-time has no business being curved, or at least we can't see, or feel, or in any way sense them curve, because they are not actual physical things.

Most people, however, conclude that space-time actually curve in a physical sense, instead of mathematically, and this can be the basis of questions like kojax's.

I hope you understand, DrRocket, why I stated that space-time curvature is only a useful analogy to explain general relativity and not actually a physical event. You are not wrong in saying that it is curved, but this is only inb a mathematical sense; physically, we have no way of determining if they are actually curving.

Oh, and kojax: since space-time does not actually curve in a physical sense, it follows that it does not need to expand, again in a physical sense. Mathematically, however, is another question.