Thank you very much. Very nice example! Does anyone know if the lower bound on nir can be obtained by assuming that the area of $N$ is bounded by $K$?

Thank you for your answer. Maybe the lower bound can be obtained by introducing some other dependency, such as the area of $M$?

Yes, I require that. My goal is to have a suitable generalization of space forms.

I think that once we have that $\cos\theta+\frac{|v|^2}2\ge 1$, then can conclude easily by using that $$|v|^2 \leq |x|^2 + (1-\sqrt{1-|x|^2})^2,$$ for $|x|\leq1$, which follows by writing the touching spheres at $O$. Indeed we find that $\cos\theta \geq \sqrt{1-|x|^2}$ and we conclude. Thank you again for your answer!