I'm about to be taking an undergrad course in GR. I'm trying to get ahead of the game, and I've hit a problem.
The metric for flat Minkowski space (using the $-+++$ signature and $c=1$) is:
$$ds^2 = -dt^2+dx^2 +dy^2 +dz^2$$
which in polar coordinates is: $$ds^2 = -dt^2+dr^2 +r^2(d\theta^2 +d\phi^2\sin^2\theta)$$
Firstly, the $r^2(d\theta^2 +d\phi^2\sin^2\theta)$ part of the metric is identical to the metric for a 2-sphere, which to me suggests that the Minkowski metric above should signify curved spacetime - why is this part of the metric there if the space the metric describes is flat? In my mind, the metric containing "something curved" should mean that the space it describes is curved.
Also, why does simply doing this:
$$ds^2 = -dt^2+dr^2 +(r^2+b^2)(d\theta^2 +d\phi^2\sin^2\theta)$$
to the metric produce a wormhole geometry ($b$ a constant - the size of the wormhole's throat)?
I suppose my questions are summarised by: what does a 2-sphere have to do with flat Minkowski spacetime or the geometry of a wormhole?
This is my first question on stackexchange - sorry if I've not been clear enough!