Tangent Space of the Sphere

Since the sphere \(\mathbb{S}^n\) can be considered as a regular level set of \(\mathbb{R}^{n+1}\) the tangent space can be computed using the defining function \(f(x)=\lvert x\rvert^2\).

Then for \(p\in \mathbb{S}^n\) we have

\begin{equation*} T_p\mathbb{S}^n = \{v\in \mathbb{R}^{n+1}\mid 2 \sum_{i=1}^{n+1} p^iv^i = 0 \}=\{v\in \mathbb{R}^{n+1}\mid v\perp p\}, \end{equation*}

where \(p\) must be seen in euclidean coordinates. That means tangent space at each point contains all vectors which are orthogonal to it.