Intuitive interpretation of these differential forms Let $\pi: S^2-\{N\}\to \mathbb R^2$ be the stereographic projection map.
Let $\sigma:\mathbb R^2\to S^2-\{N\}$ be its inverse.
Let $p\in S^2-\{N\}$ and $x_1,x_2\in$ the tangent space of $S^2$
Would someone be nice enough to explain to me then what the following mean intuitively? And hopefully also a way to visualize them/gain some sort of physical intuition on them?
1) $(d\sigma)_{\pi(p)}$
2) $d\pi_p$
3) Why $x_1\cdot x_2=(d\pi_p(x_1),d\pi_p(x_2))_{\pi(p)}$
4) $d\pi_p\circ (d\sigma)_{\pi(p)}=$ identity
I have read the differential forms article on Wikipedia in hope to learn more, but I still don't quite get the idea. I know for example that (1) is the differential of $\sigma$ at the point $\pi(p)$ but I don't understand what that means. I hope that someone could give me a geometric picture of some kind. And if there should be such a saint out there, I would like to thank you very much (in advance).
 A: I will explain answer only geometrically as required:
What does $\pi $ does:  $\pi $ takes (great circle- N)  to  line passes through origin. $\sigma $ take line passes through origin to (great circle -N) in a smooth manner.
Now for $p\in (S^2- N)$, and $v\in T_p (S^2-N)$,  there is a great circle $\gamma$ which passes through $p$, $\gamma(0)= p$ and $\gamma'(0)= v$.  Via map $\pi$, $\gamma$ will be map to some line $l$ passes through origin.  $d\pi_p$ maps vector $v$ to speed of $l$ at $\pi(p)$.
And conversely $d\sigma_{\pi(p)}$ maps some speed vector of particular line passing through origin to the speed of the corresponding great circle at $p$.
Now in your 3rd question you are defining metric on $S^2$, This matrix is induced metric...  Actually this expression just says that map $\pi$ is conformal map.   Angle between two line passes through origin is same as angle between their corresponding image that is great circles.
As $\pi$ and $\sigma$ are inverse of each other hence we have $\pi \circ\sigma = Id$. Now take the derivative and use composition rule of rule of the derivative you will get your 4th one... Geometrically it says that Tangent space at $p$ is isomorphic to $\mathbb R^2$
