On a similar note to my previous question, I am still thinking about functions on the sphere.

Assume I have a $C^2$ function on the sphere $ f : S^{d-1} \mapsto \mathbb{R} $ which is Morse (i.e. all critical points are non-degenerate and thus all critical points are isolated) and assume that this function has at least one saddle point.

Now, if we say $ d = 3 $, if I am not mistaken, then we will actually have at least two saddles, since by the Poincare-Hopf theorem, the sum of the indices of any vector field must sum up to its Euler characteristic, which in this case is 2.

Now, my question is, are there any restrictions regarding where these critical points are exactly on the sphere (when $ d = 3 $, but also in general) ? Can they be arbitrarily close to each other? If there are some restrictions, what exactly is causing them?

Again, apologies if my question is really basic, as I am absolutely unknowledgeable about differential topology. Any references or comments are appreciated!


1 Answer 1


Theorem: For any $v, f \in\{1, 2,\cdots \}$, $e\in \mathbb \{0, 1, \cdots\}$ such that $v-e + f = 2$, and for any distinct $p_1, \cdots, p_v , q_1, \cdots, q_e, r_1, \cdots ,r_f \in \mathbb S^2$, there is a Morse function $F$ on $\mathbb S^2$ so that $$\text{Crit}(F) = \{p_1, \cdots, p_v , q_1, \cdots, q_e, r_1, \cdots r_f\}$$ and $$\text{Ind}(p_i) = 0, \ \text{Ind}(q_i) = 1, \ \text{Ind}(r_i) = 2.$$

Thus, there is absolutely no constraint on the location of the critical points, and no constraint on the number of saddle points (besides the one coming from the Euler formula)

Proof of Theorem: Given any $k\ge 1$ let $g_k$ be a function on $[-1,1]$ with the following properties:

  • $g_k$ is continuous and smooth and positive in $(-1, 1)$,
  • $g_k(t) = \sqrt{1-t^2}$ away from $[-1/2, 1/2]$
  • $g_k$ has exactly $k$ local maximum and $k-1$ local minimum (excluding endpoints $\pm 1$), and $g''_k\neq 0$ at each critical points.

Now for any $v, f\in \{1, 2, \cdots\}$ let \begin{align} r(t)=r_{v,f}(t) &= \frac{g_f(t) + g_v(t)}{2},\\ c(t)=c_{v,f}(t) &= g_{f}(t)- r_{v,f}(t) \\ &= -g_v(t)+ r_{v,f}(t). \end{align} Note that both $r_{v,f}$, $c_{v,f}$ are smooth functions on $[-1,1]$, $r_{v,f}(t) = \sqrt{1-t^2}$ and $c_{v,f} = 0$ away from $[-1/2,1/2]$.

Now define the function $F = F_{v.f}$ on $\mathbb S^2$ as follows: for each $(t, y, z)\in \mathbb S^2$, write $$(t, y, z) = (t , \sqrt{1-t^2 }\cos\theta, \sqrt{1-t^2}\sin\theta)$$ for some $\theta\in \mathbb S^1$. Then define $$ \tilde F(t, y, z) = c(t) + r(t)\sin\theta. $$

The proof of the following is direct and skipped.

Claim 1: $\tilde F$ is a morse function. All critical points occurs at $\sin \theta = \pm 1$. There are $v$ local minimum, $f$ local maximum, and $e:= v+f-2$ saddle points.

Thus we have found, for every $v, f\ge 1$ and $e\ge 0$ such that $v-e+f=2$, a morse function $\tilde F$ so that $$\text{Crit}(\tilde F) = \{\tilde p_1, \cdots, \tilde p_v , \tilde q_1, \cdots, \tilde q_e, \tilde r_1, \cdots \tilde r_f\}$$ and $$\text{Ind}( \tilde p_i) = 0, \ \text{Ind}( \tilde q_i) = 1, \ \text{Ind}(\tilde r_i) = 2$$ for some distinct $\tilde p_i, \tilde q_i, \tilde r_i \in \mathbb S^2$.

The proof of the theorem is finished using the following claim:

Claim 2: Let $M$ be a connected smooth manifold of dimension $n\ge 2$, and let $$S = \{s_1, \cdots, s_K\}, \ \ \ \tilde S = \{\tilde s_1, \cdots, \tilde s_K\}$$ be two finite subsets of $M$. Then there is a diffeomorphism $\phi : M\to M$ so that $\phi(s_i) = \tilde s_i$ for $i=1, \cdots, K$.

Proof of Claim 2 (Sketch): Since $M$ is connected, there is a smooth regular curve $\gamma$ in $M$ joining $\gamma(0)=s_1$ to $\gamma(1)=\tilde s_1$, and there is an open neighborhood $U$ of the image of the curve that does not pass through $s_2, \cdots, s_K, \tilde s_2, \cdots, \tilde s_K$ (this we need $n\ge 2$). Now let $X$ be a smooth vector field compactly supported in $U$ so that $X(\gamma(\tau ))= \gamma'(\tau )$ for all $\tau \in [0,1]$. Let $\phi^t_X:M\to M$ be the one parameter group of diffeomorphism generated by $X$. Then $\phi_1 :=\phi^1_X$ satisfies $$\phi_1(s_1) =\tilde s_1$$ and $\phi_1$ fixes every other elements in $S\cup \tilde S\setminus\{s_1, \tilde s_1\}$ since the support of $X$ does not intersect $S\cup \tilde S\setminus\{s_1, \tilde s_1\}$. Then one can similarly construct $\phi_2, \cdots, \phi_K$ so that $\phi_i (s_i) = \tilde s_i$ and $\phi_i$ fixes elements in $S\cup \tilde S\setminus\{s_i, \tilde s_i\}$. Then $$\phi := \phi_K\circ \phi_{K-1}\circ\cdots \circ \phi_1$$ is a diffeomorphism which satisfies the conclusion of claim 2.

Lastly, to finish the proof of the theorem: let $\phi$ be a diffeomorphism on $\mathbb S^2$ so that $$ \phi (p_i) = \tilde p_i, \ \ \phi (q_j) = \tilde q_j, \ \ \phi (r_k) = \tilde r_k$$ for all $i=1, \cdots, v$, $j=1, \cdots, e$, $k=1, \cdots, f$. Then $F:= \tilde F\circ \phi$ is a Morse function which satisfies the conclusion of the theorem.


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