^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.1 - Definition of a Manifold|2.1 Definition of a Manifold]] | [[Schutz - 2.3 - Other Examples of Manifolds|2.3 Other Examples of Manifolds]]>> # The Sphere as a Manifold > **The two sphere $S^2$** > The set of points in $\pmb{x} \in \mathbb{R}^3$ such that $\{\pmb{x} \in \mathbb{R}^3 \,:\, |x| = r\}$ where $r \in \mathbb{R}$ is a constant known as the radius of the sphere. It is said to be *embedded* in $\mathbb{R}^3$. An example of a map for the sphere is the so-called stereographic projection, demonstrated in the figure below. Note that this map only fails at a single point $N$, thus, we need at least two charts to cover the whole sphere. ![[Pasted image 20210123183150.png]]