When I was reading about de Sitter it said that the we looked in 5-dimensional Minkowski metric and that we had immersed in it a hyperboloid, and with some coordinate transformation we get some metric that we see resembles a known metric.
But how to visualize this? And why did we took the hyperboloid in the first place? None of the books I read on the subject really gave some simple explanation to that :\
I'm kinda puzzled about this :\ How do I 'see' things with given metric?
For instance, I have, for de Sitter:
$$ds^2~=~-dt^2+l^2 \cosh^2(t/l) d\Omega_3^2$$
And this describes a space tri-sphere that shrinks to a minimum at t=0, and then expands.
Now, this means that the $d\Omega_3^2$ is a three-sphere, contracts because of the hyperbolic cosine, and time is just normal like in Minkowski. Am I right in interpreting this?