If there is a smooth isometric embedding $f: (S^2, g)\rightarrow \mathbb{R}^n$, where $(S^2, g)$ is a sphere with Riemannian metric such that the corresponding sectional curvature is equal to $1$, and $n\geq 4$. Is $f(S^2)$ always a round $2$-dim unit sphere in $\mathbb{R}^3\subseteq \mathbb{R}^n$?

If there is a smooth isometric embedding $f: (S^2, g)\rightarrow \mathbb{R}^n$, where $(S^2, g)$ is a sphere with Riemannian metric such that the corresponding sectional curvature is equal to $1$, and $n\geq 4$. Is Is $f(S^2)$ always $\mathbb{S}^2\subseteq \mathbb{R}^n$ modulo an isometry of $\mathbb{R}^n$?