Exercise 2.03

Show that the two-dimensional torus $T^2$ is a manifold, by explicitly constructing an appropriate atlas. (Not a maximal one, obviously.)

We will work with the following torus as an example. 

As we have seen in exercise 2.01, the cylinder can be covered with one chart. We will map the torus onto the cylinder or radius 1 for our first chart $\phi_{1}$ and onto the cylinder of radius 3 for our second chart $\phi_{2}$, then apply the map described in exercise 2.01 from the cylinders to the 2D plane. 

Here is an illustration of the entire procedure. $$R\equiv\sqrt{ (x^{1})^{2}+(x^{2})^{2} }$$

First chart

$$ \begin{align} \begin{pmatrix} x^{1} \\ x^{2} \\ x^{3} \end{pmatrix} &\to \begin{pmatrix} x^{1}/R \\ x^{2}/R \\ 2x^{3}/\big(3-R\big) \end{pmatrix} \\& \swarrow \\ \begin{pmatrix} x^{1}\exp\Big[2x^{3}/\big(3-R\big)\Big]/R \\ x^{2}\exp\Big[2x^{3}/\big(3-R\big)\Big]/R \\ 2x^{3}/(3-R) \end{pmatrix} &\to \begin{pmatrix} x^{1}\exp\Big[2x^{3}/\big(3-R\big)\Big]/R \\ x^{2}\exp\Big[2x^{3}/\big(3-R\big)\Big]/R \\ 0 \end{pmatrix} \end{align} $$ $$ \boxed{ \phi_{1}(x^{1},x^{2},x^{3})\equiv \begin{pmatrix} x^{1}\exp\Big[2x^{3}/(3-R)\Big]/R \\ x^{2}\exp\Big[2x^{3}/(3-R)\Big]/R \end{pmatrix} } $$

Second chart

$$ \begin{align} \begin{pmatrix} x^{1} \\ x^{2} \\ x^{3} \end{pmatrix} &\to \begin{pmatrix} 3x^{1}/R \\ 3x^{2}/R \\ 2x^{3}/\big(R-1\big) \end{pmatrix} \\& \swarrow \\ \begin{pmatrix} 3x^{1}\exp\Big[2x^{3}/\big(R-1\big)\Big]/R \\ 3x^{2}\exp\Big[2x^{3}/\big(R-1\big)\Big]/R \\ 2x^{3}/(3-R) \end{pmatrix} &\to \begin{pmatrix} 3x^{1}\exp\Big[2x^{3}/\big(R-1\big)\Big]/R \\ 3x^{2}\exp\Big[2x^{3}/\big(R-1\big)\Big]/R \\ 0 \end{pmatrix} \end{align} $$ $$ \boxed{ \phi_{2}(x^{1},x^{2},x^{3})\equiv \begin{pmatrix} 3x^{1}\exp\Big[2x^{3}/\big(R-1\big)\Big]/R \\ 3x^{2}\exp\Big[2x^{3}/\big(R-1\big)\Big]/R \end{pmatrix} } $$