how to prove a torus in $\mathbb{R}^3$ is a smooth manifold

Question

I want to prove that $T^2=\{(x,y,z)\in\mathbb{R}^3:(\sqrt{x^2+y^2}-R)^2+z^2=r^2\}\ (r<R)$ is a smooth manifold. Since I am a student in physics and haven't learned much math, so I want to prove it by constructing an atlas with smooth transition functions.

I tried taking something like $U_1=T^2\backslash\{(R\cos\theta,R\sin\theta,r):\theta\in[0,2\pi)\}\cup\{(R+r\cos\theta,0,R+r\sin\theta):\theta\in[0,2\pi)\}$ (namely a torus deleting the circle on the top and the circle in half plane $y=0,x>0$) and $\phi_1:U_1\to(0,2\pi)\times(0,2\pi):(R\cos\theta+r\cos\theta\sin\phi,R\sin\theta+r\sin\theta\sin\phi,r\cos\phi)\to(\theta,\phi)$. Then I found that it seems I need to construct 2 more similar charts to get an atlas. Am I correct? Can we construct it more elegantly?

Answer 1

A more elegant approach is to use the preimage theorem. Consider the smooth function $$ f: (x,y,z)\to \left(R-\sqrt{x^2+y^2}\right)^2+z^2 $$ from $\mathbb R^3$ to $\mathbb R$ and show that $r^2\in \mathbb R$ is a regular value.

Answer 2

If you are comfortable with $S^1$ being a manifold ( a submanifold of $\mathbb{R}^2$ also), use the fact that $T^2$ is the image of the map you wrote down:

$$(\theta,\phi) \mapsto (R\cos\theta+r\cos\theta\sin\phi,R\sin\theta+r\sin\theta\sin\phi,r\cos\phi)$$

which is an injective immersion from a compact manifold $S^1\times S^1$ , so in fact an embedding.

Now if you want an atlas, you can get that from an atlas of $S^1$, by taking products.

For an atlas of $S^1$: let $a\in \mathbb{R}$. Consider the chart who is the inverse of the bijection

$$\theta \mapsto (\cos \theta, \sin \theta)$$

for $\theta \in (a-\pi, a+ \pi)$. This is a chart around the point $(\cos \theta, \sin theta)$.

Note that there are countably many charts like this centered around a point on $S^1$, since the argument function is multivalued.

Answer 3

Let \begin{align*} A_1 &= \{ 0 < x^2 + y^2 < 4\pi^2 \} \subset \mathbb{R}^2\\ A_2 &= \{ \pi^2 < x^2 + y^2 < 9\pi^2 \} \subset \mathbb{R}^2. \end{align*} Define two coordinate maps $\phi_1: A_1 \rightarrow T$ and $\phi_2: A_2 \rightarrow T$, where \begin{align*} \phi_1(x,y) &= \left(\frac{x}{\sqrt{x^2+y^2}}(R+r\sin \sqrt{x^2+y^2}),\right.\\ &\quad\left.\frac{y}{\sqrt{x^2+y^2}} (R+r\sin \sqrt{x^2+y^2}),r\cos \sqrt{x^2+y^2}\right)\\ \phi_2(x,y) &= \left(\frac{x}{\sqrt{x^2+y^2}}(R+r\sin (\sqrt{x^2+y^2}),\right.\\ &\quad \left.\frac{y}{\sqrt{x^2+y^2}} (R+r\sin (\sqrt{x^2+y^2})),r\cos (\sqrt{x^2+y^2})\right) \end{align*}