I am trying to understand contact structures. To this end, as an exercise, I intend to define a contact form on $S^3$. Here is what I have so far:
Since $S^3$ is in $\mathbb R^4$ one can specify a point on $S^3$ by a $4$-tuple $(x,y,z,w)$. If one endows $S^3$ with stereographic coordinate charts one obtain coordinates as follows: $({2X\over X^2 + Y^2 + Z^2 + 1}, {2Y\over X^2 + Y^2 + Z^2 + 1}, {2Z\over X^2 + Y^2 + Z^2 + 1}, {X^2 + Y^2 + Z^2 - 1\over X^2 + Y^2 + Z^2 + 1})$ where $(X,Y,Z)$ is a point in $\mathbb R^3$.
The standard contact structure on $S^3$ is given by $xdz - z dx + y dw - w dy$.
My question is:
Does $xdz - z dx + y dw - w dy$ mean the contact structure is ${2X\over X^2 + Y^2 + Z^2 + 1}d{2Z\over X^2 + Y^2 + Z^2 + 1} - \dots$ and so on?
Yes, if you use the stereographic projection then it is the form you indicated.
To be precise, $\alpha = x dz - zdx + y dw - wdy$ is a one form on $\mathbb R^4$. So the contact structure on $\mathbb S^3$ is really $\iota^* \alpha$, where $\iota : \mathbb S^3 \to \mathbb R^4$ is the inclusion.