I would like to prove that the sphere of radius $R>0$, $S^2(R)\subset \mathbb{R}^3$, with the induced metric is isometric to the sphere with radius $1$, $S^2\subset \mathbb{R}^3$, furnished with the metric $g(X,Y):=R^2 \cdot \langle X,Y\rangle$. Here the scalar product $\langle \cdot, \cdot \rangle$ denotes the standard metric induced from the euclidean space. Is the following proof correct?
Claim: $$\phi:S^2 \ni p\mapsto R\cdot p\in S^2(R)$$ is an isometry.
The map is a diffeomorphism. I feel comfortable with this part of the proof, therefore I will omit the details.
Since $\langle d\phi X,d\phi Y \rangle=\langle R\cdot X, R\cdot Y \rangle=R^2\langle X,Y \rangle=g(X,Y),$ the map is a Riemannian isometry.
Is the argument correct?:
Best wishes
Yes, that looks right. ${}{}{}{}$