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A major open problem in submanifold geometry is to determine whether there exists a nontotally geodesic isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$. Every isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^{n+p}$ is totally geodesic if $p<n$, and nontotally geodesic examples are known for $p>n$ and also for $p=n=2$. Case $p=n\geq3$ remains open. It can be shown that the existence of a nontotally geodesic isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^m$ is equivalent to the existence of a nonconstant solution $Q:\mathbb{R}^{n+1}\setminus\{0\}\to O\left(n+1,m+1\right)$ to the linear first-order PDE system $$\frac{\partial Q}{\partial x_i}x=0,\ 1\leq i\leq n+1,$$ where $O\left(n+1,m+1\right)=\left\{Q\in\mathbb{R}^{\left(m+1\right)\times\left(n+1\right)}:Q^TQ=I\right\}$. What parametric representation for $O\left(n+1,m+1\right)$ might be useful to addressing this problem?

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