I am currently undergoing a first course on Differentiable Manifolds. An an example I was trying to construct a differential structure for a 2D unit sphere $\mathbb{S}^2=\{(x,y,z):x^2+y^2+z^2=1\}$. By definition, the coordinate charts are as follows: \begin{array}{ c c} U_1=\{(x,y,z):z>0\} & \phi_1(x,y,\sqrt{1-x^2-y^2})=(x,y) \\ U_2=\{(x,y,z):z<0\} & \phi_2(x,y,\sqrt{1-x^2-y^2})=(x,y) \\ U_3=\{(x,y,z):x>0\} & \phi_3(x,y,\sqrt{1-x^2-y^2})=(y,\sqrt{1-x^2-y^2}) \\ U_4=\{(x,y,z):x<0\} & \phi_4(x,y,\sqrt{1-x^2-y^2})=(y,\sqrt{1-x^2-y^2}) \\ U_5=\{(0,1,0)\} & \phi_5(0,1,0)=(0,1) \\ U_6=\{(0,-1,0)\} & \phi_6(0,-1,0)=(0,-1) \end{array} Have I done this correctly ? Now I want to find a differential structure $\mathscr{F}$ on $\mathbb{S}^2$. For this I need to show that each composition map $\phi_j\circ\phi_i^{-1},\: i\neq j \in\{1,2,3,4,5,6\}$ is $C^\infty$, $X=\bigcup_\alpha U_\alpha$ and $\mathscr{F}$ is maximal. I am very confused about how to proceed ? Any help will be appreciated. Thanks.
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