Let $\Sigma$ be an $n-1$ dimensional ellipsoid in $\textbf{R}^{n}$ and $S$ the unit sphere.

I would like to understand the connected components $C$ of the intersection of $\Sigma$ and $S$.

In my application, I know that $C \cap H$ is non-empty for all hyperplanes $H$ and moreover that $C$ is not contained in any halfspace. I would like to conclude that $C$ is the entire sphere, ie the ellipsoid $\Sigma$ was the unit sphere and their intersection was trivial.