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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 $g_1$.

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    So what does "different" mean? – 2017-01-29
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    For example, if one of the metrics is $g$ , then the other one is $\lambda g$, where $\lambda \neq 1$is a positive real number. – 2017-01-29
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    But there are many, many more metrics than just those obtained by scaling, and infinitely many of them will be isometric to any given one, as Lee's answer describes. If you want to insist they're truly "different" in the sense of having different geometries, different means non-isometric and your conclusion is a tautology. – 2017-01-29

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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 $(\mathbb{S}^n,g_2)$.

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    No quite sure about this. This problem comes from an exercise asking me to prove that circles on $\mathbb{R}^2$ with different radii are not isometric. The teacher told me that I could generalize the exercise problem into this one. – 2017-01-29
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    Your generalization does not match the circle problem in the exercise. On the other hand, your generalization did make for a good question. – 2017-01-29