I just read Azov's article in the considered two classes of Riemannian metrics, \begin{align*} ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\ ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2 ,&g>0\end{align*} and solved the problem of their immersibility into $\mathbb R^n$ and $\mathbb S^n$, $n>l$. He proves the following theorems

• The metrics admit isometric immersions into the Euclidean space $\mathbb R^{4l-3}$.
• The metrics admit isometric immersions into the Euclidean space $\mathbb S^{4l-3}$.

But in Surfaces of Negative Curvature of Rozendorn, he mentions the following I can't understand how Rozendorn comes to that conclusion, can someone help me? On the other hand, is there already an improvement for that lower bound for isometric immersions?

I just read Azov's article in the considered two classes of Riemannian metrics, \begin{align*} ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\ ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2 ,&g>0\end{align*} and solved the problem of their immersibility into $\mathbb R^n$ and $\mathbb S^n$, $n>l$. He proves the following theorems

• The metrics admit isometric immersions into the Euclidean space $\mathbb R^{4l-3}$.
• The metrics admit isometric immersions into the Euclidean space $\mathbb S^{4l-3}$.

But in Surfaces of Negative Curvature of Rozendorn, he mentions the following I can't understand how Rozendorn comes to that conclusion, can someone help me? On the other hand, is there already an improvement for that lower bound for isometric immersions?

This question is crossed with: https://math.stackexchange.com/q/4277758/691503