I am studying differential geometry on my own by reading Do Carmo's book. I have trouble understanding how atlases are constructed. For the sphere I understand why we have to use 6 parametrizations with cartesian coordinates, but for the torus I don't understand why we need 3 with the following parametrization:
\begin{equation}
x(u,v) = ((r\cos{u}+a)\cos{v},(r\cos{u}+a)\sin{v},r\sin{u})
\end{equation}
for $0 < u < 2\pi$ and $0 < v < 2\pi$.
I have already showed that this is indeed a parametrization and how it is constructed geometrically. Now, I understand that with this parametrization, the circle of radius $a+r$ and the circle of radius $a$ are not covered, so the question is if the atlas for the torus consists of this parametrization plus the parametrization of these two circles or is there any other detail I am missing?
Best Answer
You cannot cover a circle with one single parametrization, but you are right, you have to cover the circles which are omitted by your parametrization. By using the same parametrization as before on $c< u < 2\pi+c$ and $ c < v < 2\pi + c$ and on $d< u < 2\pi+d$ and $ d < v < 2\pi + d$ for some small $c \neq d$ you will acchieve this. The first one, with $c>0$ will cover the circles you omitted up to now with the exception of two points, and the last one will also catch those two points. Of course, even smaller patches would do.
(This does, admittedly, not prove that you cannot do better)