The original problem is：Prove that circles on $\mathbb{R}^2$ with different radii are not isometric.
But someone told me that one could generalize the exercise problem into the following proposition:
Suppose we have two smooth spheres：$(\mathbb{S}^n,g_1)$ and $(\mathbb{S}^n,g_2)$, with different metrics $g_1,g_2$.
Could we prove that these two manifolds are not isometric?
That is to prove: There doesn't exists a diffeomorphism $F:(\mathbb{S}^n,g_1) \rightarrow (\mathbb{S}^n,g_2)$ such that $F$ pulls metric $g_2$ back to $g_1$.
Even though the metrics are different, it is quite possible that these two manifolds are isometric.
For example, take $g_1$ to be the standard metric on $\mathbb{S}^n$.
Now, choose any diffeomorphism $F : \mathbb{S}^n \to \mathbb{S}^n$ which is not an isometry from $(\mathbb{S}^n,g_1)$ to $(\mathbb{S}^n,g_1)$.
Next, define a new metric $g_2$ on $\mathbb{S}^n$ to be the pushforward of $g_1$ using the map $F$.
Since $F$ is not an isometry from $(\mathbb{S}^n,g_1)$ to $(\mathbb{S}^n,g_1)$, it follows that $g_1 \ne g_2$. But by definition of $g_2$, it follows that $F$ is an isometry from $(\mathbb{S}^n,g_1)$ to $(\mathbb{S}^n,g_2)$.