Let $S$ be the unitary sphere of $\mathbb{R}^3$ and for $0<a<b<c$ and $A>0$ let $G(x,y,z)=ax^2+by^2+cz^2 $ and the ellipsoid $E_A$ : $G^{-1}(A)$.
How many connected components does $\Gamma _A := S \cap E_A$ have ?
I showed that $\Gamma _A $ is non empty if and only if $A \in [a,c]$ and, in this case I want to compute the number of connected components for $A \in ]a,b[ \cap ]b,c[$ of $\Gamma _A$
First of all I showed that for those $A$, $\Gamma _A$ is a one dimensional compact manifold.
It remain to me impossible to count the connected components. Of course a drawing is not an appropriate answer for me...
Another question : I know that the connected components of $\Gamma _A$ are periodical orbits for the EDO : $X'=X \wedge G(X)$ (or $X'=X \times G(X)$, US notation) with $X=(x,y,z)$.
How can I prove that the period of the solution does not depend of the connected component where X(0) is. (But depend of $A$ of course)
If you have any idea feel free to comment or answer.
Thanks.