How to show that $(x_1,\dots,x_{n+1})\in\mathbf S^n\mapsto x_{n+1}$ is a Morse function?
According to my course definition, I have to demonstrate two things:
- $f$ is $\mathcal C^2$,
- all critical points are non-degenerate.
The first part is trivial so I start by looking for the critical points. For all $x,h\in\mathbf S^n$, $$f(x+h)=x_{n+1}+h_{n+1}$$ hence $df_x\colon h\mapsto h_{n+1}$. I think that the critical points are the $x\in\mathbf S^n$ such that $df_x=0$. So it seems like there are no critical points.
Is this correct? Then I would like to know if these points are degenerate.
You are looking for the critical points of the restriction of $f$ to the sphere. Those are the points $x\in \mathbb{S}^n$ where the differential $df_x$ annihilates every vector $v$ tangent to $\mathbb{S}^n$ at $x$.
One can prove that $x$ is a critical point of $f$ (restricted to the sphere) if and only if $df_x = \lambda dg$ for some $\lambda\in \mathbb{R}$, where $g(x_1,\dots, x_{n+1}) = x_1^2 + \dots +x^2_{n+1} - 1$ is the function defining the sphere. (The number $\lambda$ is then called a Lagrange multiplier)
In this case you are looking for points where $dx_{n+1} = 2\lambda x_1dx_1 + \dots + 2\lambda x_{n+1}dx_{n+1}$ for some $\lambda$, and it's obvious that this can happen only when $x_1= x_2 = \dots = x_n = 0$ (because otherwise the differentials are linearly independent) which for a point $x$ on the sphere implies that $x_{n+1} = \pm 1$ (and then $\lambda = \pm 1/2$). The critical points of $f$ are the north and south poles (this is obvious considering that the function $x_{n+1}$ takes extreme values there).
Now, to check non-degeneracy, I think you had to compute the hessian of $f - \lambda g$ at the poles, and show that its restriction to vectors tangent to the sphere is non-singular (I'm guessing it will be positive for the south pole and negative for north pole, because those are a minimum and a maximum)