Let $(S^n, g)$ be an $n$-dimensional positively curved sphere. Assume the smoothness of the metric, does it admits an isometric embedding into $\mathbb R^{n+1}$?


for $n=2$ it is proved by A.D Alexandrov, also by H. Weyl



Does positively curved sphere admit an isometric embedding as hypersurface in Euclidean space?
When $n\geq 3$, the condition for a metric to be even locally isometrically embeddable in $R^{n+1}$ is nontrivial, so you may wish to add such an additional condition. Then this paper ["A Priori Bounds for Co-dimension One Isometric Embeddings" by Yanyan Li and Gilbert Weinstein](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.65.7559&rep=rep1&type=pdf) states as a conjecture:



> 
> **Conjecture.** Let $g$ be a smooth metric of non-negative sectional curvature and positive scalar curvature on $S^n$ which is locally isometrically embeddable in
>  $R^{n+1}$. Then $(S^n,g)$ admits a smooth global isometric embedding $X:(S^n,g)\rightarrow R^{n+1}$.
> 
> 
> 


Section 5 of the paper discusses the positive sectional curvature case, and in particular Theorem 7 gives a result in the case $n=3$ for strictly positive curvature.


