2 problems

Let $\Sigma^{n}$ be a smoothly embedded $n$-torus in $\mathbf{S}^{2n-1}$, and let $\mathds{T}_{Cl}^{n}$ denote the Clifford torus. The focal radius of $\Sigma$ is the largest $r>0$…

Let $(M^n,g)$ be a $n$-sphere with $\operatorname{sec}(g)\geq1$, where $n=p+q+1$ for $p,q\in\mathbf N_+$. Suppose that $i:\Sigma\approx\mathbf S^p\times\mathbf S^q\to M$ is an imme…