Let $M$ and $N$ be two spheres (of different radius) in $\mathbb{R}^n$ of dimension $n-1$. Suppose there is a Riemannian isometry between them (so a diffeomorphism and isometry). Then distances must be preserved in some way... but does this not constrain the radiuses of the spheres?
Consider the two spheres as being hypersurfaces in $\mathbb{R}^{n+1}$.
For example, consider two intervals. Then I don't think there can be a isometry between them unless the intervals are of the same size. Am I right?
Does requiring a diffeomorphism to be an isometry restrict the range of manifolds we work with massively?
An isometry must preserve intrinsic quantities like scalar and sectional curvature, which are proportional to some negative power of the radius, so spheres of the same dimension which are isometric, even locally, must have the same radius. An exception is 1-spheres (circles); there are no local invariants, so they are all locally isometric, but not globally, because the total length (circumference), which is intrinsic, is proportional to the radius.