# Statement(s) > [!NOTE] Statement 1 (....) > $\text{Isom}(S^n,d_{S^n})\cong O(n+1)$ # Proof(s) ###### Proof of statement 1 (MA243): To see that $O(n+1) \subset \text{Isom}(S^n,d_{S^n})$ we need that show that if $T$ is a linear isometry of $\mathbb{R}^{n+1}$, then the restriction of $T|_{S_{n}}$ to $S_{n}$ is a spherical isometry. Since $T$ preserves Euclidean distance and fixes $\mathbf{0}$, it must map $S_{n}$ to $S_{n}$ since it is the set of points distance $1$ from $\mathbf{0}$. Since $T$ has an inverse which also maps $S_{n}$ to $S_{n}$ and so $T|_{S^n}$ is a bijection. It follows from [[Isometry group of real n-space with standard metric]] that $T$ preserves inner product, and thus preserves spherical distance $\cos^{-1}(\langle x,y\rangle)$ on $S^n$. Hence $T|_{S_{n}}$ is a spherical isometry. Lemma 14: Define $\begin{align} O(n+1) &\to \text{Isom}(S^n, d_{S^n}) \\ A &\mapsto T_{A}:S^n \to S^n \\ & \quad\quad\quad\quad \underline{v} \mapsto A\underline{v} \end{align}$check that it's well defined; bijection and $T_{A}$ is distance preserving. $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography