>I'd need an update here.
>
>Some 25 years ago, when I was at college, it was stated that the geometry of the universe is hyperbolic, i.e. has a negative curvature. IOW, it's a Lobachevsky space - which Lobachevsky himself proved in mid-nineteenth century by measuring a defect in a triangle (total of its angles being less than 180 degrees) between some stars when seen in the summer and in the winter.

In an astronomy textbook from some 30 years ago, it was still not clear whether the curvature of space - on a large scale - was positive, negative, or zero (on the average).

I don't know whether the large-scale structure of the Universe is confirmed yet; from my recent readings, I have the impression that it is not.

On a large scale, however, it seems that the curvature of space is many, many times less than it was thought a few decades ago. This may also indicate that the Universe is much, much bigger than was previously thought.

This leads to the fascinating inflationary theory; you may want to do some research on this. It seems that during a fraction of a second after the Big Bang, the Universe went through an incredible exponential expansion, until it eventually sort of stabilized.