Let $F:\mathbb{R}^{n+1} \to \mathbb{R}$, $n\geq 4$, be given by $$F(x_1, \dots , x_{n+1})=\left( \sum_{i=1}^{n+1} x_i ^2 -5 \right)^2 - 16\left(1-\sum_{i=3} ^{n+1} x_i ^2 \right)$$ and let $M=F^{-1}(0)$. Consider the height function $h$ with respect of $e_1 = (1,0, \dots, 0)$. That is $h:M \to \mathbb{R}$ given by $h(p)=\langle e_1 , p \rangle$, where $\langle,\rangle$ denotes the usual inner product on $\mathbb{R}^n$.
Show that $h$ is a Morse function (this means that all of its critical points are non-degenerate) with exactly four critical points. Compute the index of $h$ at the critical points and use it to compute the homology of $M$.
Show that $M$ is the image of the map $g: \mathbb{R} \times \mathbb{S}^1 \times \mathbb{S} ^{n-1} \to \mathbb{R}^{n+1}$ such that $$g(t,u,v)=((\sin t +2)u, \cos t).$$ Conclude that $M$ is a tube of radius one around the circle $(2u,0)$.
I'm having a lot of trouble with this problem. To find the critical points of $h$ I should solve for all $i\in \{1, \dots, n\}$ $0=\dfrac{\partial h}{\partial x_i}(p)= \dfrac{\partial h\circ \varphi}{\partial x_i}(x)=\dfrac{\partial \varphi_1}{\partial x_i}(x) $, where $\varphi$ is a local parametrization of $M$ around $p$, but I don't really know how to calculate the critical points with this abstract parametrization. If you could help me with this or any part of this problem I would be so thankful. Thanks for your help!