Lie Groups

Sphere as a 4-Manifold

In the same way that we found two coordinate patches that completely covered the circle, we need to find coordinate patches that cover the sphere. Parametrization gives us a function from 2-space to the sphere (S2). The variables in $$\mathbb{R}^2$$ are $$\theta$$ and $$\phi$$:

\begin{eqnarray} \gamma:\mathbb{R}^2\rightarrow\mathbb{S}^2 \\ {\theta \choose \phi} \mapsto \left( \begin{matrix} \sin\theta\cos\theta \\ \sin\theta\sin\phi \\ \cos\theta \end{matrix}\right) \end{eqnarray}

Now the challenge is to determine how to restrict the domain in $$\mathbb{R}^2$$ to ensure that the function is injective, while keeping the domain open and covering the whole sphere. The two obvious patches are:

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\begin{align} \gamma_1:\left(0,\pi\right)\times\left(-\pi,\pi\right)\rightarrow\mathbb{S}^2 \\ {\theta_1 \choose \phi_1} \mapsto \left(\begin{matrix} \sin\theta_1\cos\theta_1 \\ \sin\theta_1\sin\phi_1 \\ \cos\theta_1 \end{matrix}\right) \\ \gamma_2:\left(0,\pi\right)\times\left(0,2\pi\right)\rightarrow\mathbb{S}^2 \\ {\theta_2 \choose \phi_2} \mapsto \left(\begin{matrix} \sin\theta_2\cos\theta_2 \\ \sin\theta_2\sin\phi_2 \\ \cos\theta_2 \end{matrix}\right) \end{align}

As with the circle, the domains of the coordinate patches have to be open sets in order to be differentiable This creates the problem of "incomplete coverage" of the manifold. The first coordinate system does not address the arc defined where $$\phi = \pi$$ and the second does not address the arc $$\phi = 0$$. Both patches neglect to include the north and south poles.

The problem of covering the north and south poles still remains, but is easily solved. All that is needed is a "cap" for each hemisphere. If we define D as the open unit disc in the xy-plane, then the points on the two hemispheres can be defined based on D. If

D := \{ {x \choose y} \in \mathbb{R}^2 : x^2 + y^2 = 1 \}

then only two more coordinate patches are needed:

\begin{align} \gamma_3 : D\rightarrow\mathbb{S}^2 \\ {x_1 \choose y_1} \mapsto \left(\begin{matrix} x_1 \\ y_1 \\ \sqrt{1-x_1^2 - y_1^2} \end{matrix}\right) \\ \gamma_4 : D\rightarrow\mathbb{S}^2 \\ {x_2 \choose y_2} \mapsto \left(\begin{matrix} x_2 \\ y_2 \\ -\sqrt{1-x_2^2 - y_2^2} \end{matrix}\right) \end{align}

Conditions one and two are satisfied by these four patches. It is up to you to prove that the sphere is a smooth manifold by showing that the transition functions between the patches are infinitely differentiable.