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A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function can be modified to produce a self-indexing Morse function (c.f Milnor's "Notes on the h-cobordism theorem").

Let $x_1:\Bbb R^3 \to \Bbb R$ be the projection onto the first coordinate. Let us call $f:\Bbb T^2 \to \Bbb R$ a height function, if there exists an embedding $i:\Bbb T^2 \to \Bbb R^3$ with $x_1\circ i=f$. In my experience the first examples of Morse (and Morse-Smale) functions one sees are usually height functions on the torus.

Does there exist a self-indexing Morse function on $\Bbb T^2$ which is also a height function? It seems geometrically obvious that there exists no function on the torus which satisfies both of these conditions (if one allows the height function to come from an embedding into $\Bbb R^4$ it is easy to construct such an $f$), but I have never run into a proof of such a result.

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    $\begingroup$ Just rotate like in this picture until the two heights of the index 1 critical points coincide. $\endgroup$
    – user98602
    Jan 27, 2016 at 17:25

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Start with the standard Morse embedding of the torus into $\Bbb R^3$ (that is, a donut standing up on its side). Rotate this in the plane "perpendicular to the torus" until you're laying flat on the table. An interim stage looks like this picture, and for every time $t \in [0,1)$, the height function is Morse. One of the critical points (when we started, the bottom index 1 critical points) will never have its height change, and the index 0 critical point of height zero also never has its height change. Consider $h_2(t)$, the height of the second index 1 critical point, as a function of time. For $t=1-\varepsilon$, $h_2(t)$ is just above zero, and for $t=0$, $h_2(t)=2$. So there is some $t$ such that $h_2(t)=1$.

This is almost the desired height function; the only problem being that I don't know precisely what the height of the index 2 critical point is. But whatever, just pick a diffeomorphism $\Bbb R \to \Bbb R$ that fixes 0, 1, and sends whatever the height of the index 2 critical point is to 2. Composing the embedding with this diffeomorphism in the $z$-direction gets you the desired self-indexing embedding.

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