# Non-trivial isometric embedding of the standard sphere into $\mathbb{R}^n$? [closed]

Let $S$ be the unit sphere in $\mathbb{R}^3$. Is it possible to embed $S$ isometrically into some $\mathbb{R}^n$, $n>3$, such that the image does not lie on any 3-dimensional affine subspace?

More generally, if $S$ is a compact surface with positive Gaussian curvature, we know it can be isometrically embedded into $\mathbb{R}^3$ uniquely, can there be a non-trivial isometric embedding into higher dimensional Euclidean space? Thanks!

After seeing the comments I realized the question is trivial. While it looks like a HW question, I'm actually thinking about a research problem and thought that a unique isometric embedding may help, so I set up this as a toy problem (without realising that $\mathbb R^3$ has a simple but non-trivial embedding into $\mathbb R^4$). Anyway, although not related to my problem in mind, I am curious to know if the same is true for $\mathbb {H}^n$ or $\mathbb S^n$? I don't see how e.g. $\mathbb S^n$ can be non-trivially into $\mathbb S^{n+1}$. It seems that for such an embedding to exist, all except one of the principal curvatures vanish. I don't know how to construct such a submanifold globally.
• I agree with Robert. Hint: $\mathbb{R}^3$ has a non-trivial isometric imbedding into $\mathbb{R}^4$. – Willie Wong May 5 '14 at 10:44
• Any embedding $\mathbb R\to\mathbb R^2$ is isometric, so by taking products $\mathbb R^n$ isometrically embeds into $\mathbb R^{n+1}$. – Igor Belegradek May 5 '14 at 12:15
• Your modified question is not clear. Are you asking about 'nontrivial' isometric embeddings of the $2$-sphere into $\mathbb{H}^n$ or $\mathbb{S}^n$, or are you asking about 'nontrivial' isometric embeddings of $\mathbb{H}^n$ or $\mathbb{S}^n$ into higher dimensional hyperbolic or elliptic spaces? The answer to the former question is 'yes, they always exist', the answer to the latter question depends on the codimension of the embedding and the relative curvatures, etc., as shown by Élie Cartan in his famous 1919 and 1920 papers on isometric embeddings of space forms into space forms. – Robert Bryant May 9 '14 at 11:04