n-Dimensional Toruses as Manifolds

# n-Dimensional Toruses as Manifolds

Recall from the n-Dimensional Spheres as Manifolds page that the $n$-dimensional sphere in $\mathbb{R}^{n+1}$ is the following set of points:

(1)
\begin{align} \quad S^n = \{ (x_0, x_1, ..., x_n) \in \mathbb{R}^{n+1} : x_0^2 + x_1^2 + ... + x_n^2 = 1 \} \end{align}

We will now look at another famous class of examples of $n$-dimensional manifolds called the $n$-dimensional toruses.

 Definition: The $n$-Dimensional Torus in $\mathbb{R}^{n+1}$ is $\displaystyle{T^n = \prod_{i=1}^{n} S_i = S_1 \times S_2 \times ... \times S_n}$.

The $1$-dimensional torus is the same space as $S^1$. However, the $2$-dimensional torus is much more interesting and has the following graph:

There are a few other ways to represent the $2$-dimensional torus. One way is with the following formula where $r, R \in \mathbb{R}$ and $0 < r < R$:

(2)
\begin{align} \quad (\sqrt{x^2 + y^2} - R^2)^2 + z^2 = r^2 \end{align}

The value of $R$ represents the inner radius of the circular hole in the torus, while $r$ represents the radius of the circular cross sections of the torus as illustrated below:

Alternatively, we can represent the $2$-dimensional torus topological by associating the red edges and blue edges as depicted below with the corresponding orientations indicated by the arrows:

This certainly fits with our intuition of the $2$-dimensional torus locally looking like $\mathbb{R}^2$ (a plane)!