I am interested in smooth Riemannian metrics on $n$-sphere, $n\geq 3$, which can be imbedded isometrically both to $n+1$-dimensional Euclidean space and $n+1$-dimensional standard sphere of radius $r$.
How many there are such metrics? Are there non-trivial examples? (A trivial example is the standard metric on the sphere of certain radius.)
ADD: I prefer that under both imbedding one gets convex hypersurfaces.
