The \(n\)-sphere \(\mathbb{S}^n_R \subset \mathbb{R}^{n+1}\) with radius \(R>0\) is

\begin{equation*} \mathbb{S}^{n}_R:= \{x\in \mathbb{R}^{n+1}\mid \lVert x\rVert_2=R\}. \end{equation*}

If \(R=1\), then \(\mathbb{S}^n_1=:\mathbb{S}^n\), and \(\mathbb{S}^n\) is called the \(n\)-unit-sphere. In particular, \(\mathbb{S}^1\) is the unit circle .

See also Link to heading

• topological properties
• sphere as a manifold
• covering multiplicity of \(\mathbb{S}^n\)
• UCP on a sphere