Charts, spheres and determinants Here are some things I don't understand, I would be very grateful for any help!


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*I am trying to find a chart on the unit sphere that preserves area. The most natural map that springs to my mind is taking the inverse of $\phi(x,y)=(\sin x\cos y, \sin x\sin y, \cos x)$. But the first fundamental form of this map is $EG-F^2=\sin ^2 x$, so presumably the inverse map has $E'G'-F'^2={1\over \sin^2 x}$. So it doesn't preserve area? What is a chart that does?

*Since the unit sphere cannot be fully represented by one chart, I will need another one, which I believe we can just take the same map but change the domain so that we exclude a disjoint  half great circle? I want the transition functions relating the charts to have derivative s with determinant 1 but I don't understand what that means. Presumably it implies area- and orientation- preservation.
I am very sorry about the original version of my question.
 A: (1) To find a chart on the unit sphere that preserves area, you might try checking out Wikipedia and going from there.  I believe cartographers have been worrying about this for centuries!
To produce one yourself, write down the volume form on $\mathbb{S}^2$ and make sure that it is preserved under your chart $\phi$.
(2) You are exactly correct; a matrix with determinant $1$ preserves both area and orientation.  Saying that the change-of-coordinate map "has determinant $1$" is exactly saying that it preserves both infinitesimal area and orientation. (More precisely, it preserves the volume form on each tangent space.)
To produce such a chart, since I believe you are operating on the round sphere (i.e., the metric induced from $\mathbb{S}^2\hookrightarrow\mathbb{R}^3$, you could simply precompose your chart with an appropriate rotation of the sphere.  For instance, if your chart $\phi_n$ is well-defined everywhere but the south pole, define a chart about the south pole by first rotating $180$ degrees about an axis through the equator, then using $\phi_n$.  
More generally, if $\phi$ is an area-preserving chart of a Riemannian manifold $M$ and $f$ is an isometry of $M$, $\phi\circ f$ is also an area-preserving chart.  To see that the change-of-coordinate map has determinant $1$, just note that $\det d(\phi f\phi^{-1}) = \det d\phi \det df (\det d\phi)^{-1} = 1$ by the chain rule and properties of determinants, and because $f$ is an isometry (hence is, in particular, volume-preserving).
A: Introduce polar coordinates $(\theta,\phi)$ around the north pole $(\theta=0)$ and polar coordinates $(\rho,\phi)$ in the $(\xi,\eta)$-plane. You want a map $$(\theta,\phi)\mapsto\bigl(\rho(\theta),\phi\bigr) \qquad(\theta\geq0)$$ that preserves areas. Looking at an infinitesimal annulus of width $d\theta$ at geographic latitude $\theta$ and its image in the $(\xi,\eta)$-plane we should have
$$2\pi \sin\theta\ d\theta = 2\pi \rho(\theta)d\rho\ .$$
This is a differential equation for the unknown function $\theta\mapsto\rho(\theta)$, and as $\rho(0)=0$ we obtain
$$\rho(\theta)=2\sin{\theta\over2}\ .$$
This is the basic formula that guarantees preserving of area. Now you have to convert this somehow to the cartesian coordinates $(\xi,\eta)$ in the parameter plane and to the  coordinates $(x,y,z)$ describing points of $S^2$.
