# Existence of a proper Morse function

Given a manifold $$M$$, I know there exists a proper function $$f: M \to \mathbb R$$ (using the usual partition of unity argument) and a Morse function $$g: M \to \mathbb R$$ (genericity of Morse functions). However, I am not sure how to prove the existence of a proper Morse function.

My initial idea is that, if $$f$$ has isolated critical points, then we can perturb $$f$$ locally whenever the critical points are degenerate so that the resulting $$f$$ is Morse. Moreover local perturbations by some bounded quantity preserves properness. I haven't been able to prove that proper functions have isolated critical points though, so I was wondering if I was on the right track.

I will write a very fancy and unnecessarily complicated proof. However, it is my expectation that this proof will be illuminating. Similar ideas are used in other contexts, such as Morse theory on infinite-dimensional manifolds. Instead of your finite-dimensional space of functions coming from the dot product, I will construct an infinite-dimensional space of functions which covers every way you might think of perturbing $$f$$.

For every point $$x \in M$$, pick a chart $$(U_x, \varphi_x)$$ with $$\varphi_x(x) = 0 \in \Bbb R^n$$ so that $$f(U_x) \subset (f(x) - \delta, f(x) + \delta)$$ for a predetermined uniform constant $$\delta$$, and pick a bump function $$\rho$$ supported in the unit ball of $$\Bbb R^n$$ and the identity near zero. For every $$v \in T_0 \Bbb R^n$$, define $$g_{x,v}(p) = \rho\left(\varphi_x(p)\right) \left(\varphi_x(p) \cdot v\right),$$ where the dot means the dot product. This function has $$(dg_{x,v})_x(w) = d\varphi_x(w) \cdot v$$. In particular, as $$v$$ varies, we see that the $$(dg_{x,v})_x$$ runs through the space of all functionals on $$T_x M$$.

Pick a countable set $$(x_i, v_i)$$ which is dense in $$TM$$. Let $$C_n = \sum \|g_{x_i, v_i}\|_{C^n} + 2^n.$$ Write $$\mathcal P = \ell^1(C_n)$$ for the Banach space whose elements are sequences $$(a_1, \cdots)$$ such that $$\sum C_n |a_i| < \infty$$. To each element $$\pi = (a_1, \cdots)$$ of $$\mathcal P$$ is associated a function $$g_\pi: M \to \Bbb R$$, given as $$g_\pi(x) = \sum a_i g_{x_i,v_i}(x).$$

The bounds on the $$C^n$$ norms imply that $$g_\pi$$ is smooth (in fact, the map $$g: \mathcal P \times M \to \Bbb R$$ is smooth), and further each $$g_\pi$$ is bounded (this comes from $$C_n \geq 2^n$$). So any $$f + g_\pi$$ is a proper smooth function.

Now consider the map $$F: \mathcal P \times M \to TM$$ given by $$(\pi, x) \mapsto \nabla(f + g_\pi)$$. You can quickly check that this map is transverse to the zero section (essentially because $$\mathcal P$$ is so big that it makes up all of those directions).

In particular, the "parameterized critical set" $$\mathcal C \subset \mathcal P \times M$$, given as $$F^{-1}(0)$$, is a smooth manifold; furthermore, $$\mathcal C \cap \{\pi\} \times M$$ is the critical set of the function $$f + g_\pi$$; this critical set is cut out transversely (that is,

Now apply the Sard-Smale theorem to the projection $$p: \mathcal C \to \mathcal P$$ to find a regular value $$\pi$$, and hence a smooth proper Morse function $$f + g_\pi$$.

Initially I was looking for something that used a little less machinery, but the problem is pretty easy once we assume Whitney embedding.

Let $$f: M \to \mathbb R^m$$ be an embedding, which we can find for $$m$$ sufficiently large, so now we just think of $$M \subseteq \mathbb R^m$$. We can translate $$M$$ so that it does not hit the origin, in which case the norm map $$g: x \mapsto |x|$$ is smooth on $$M$$ and proper. By theorem in Guillemin and Pollack, the map $$\begin{equation*} g_a = g + a \cdot x \end{equation*}$$ is Morse for almost ever $$a \in \mathbb R^m$$. Choose some $$a$$ with sufficiently small norm, say $$|a|<1/2$$, such that $$g_a$$ is Morse, then $$|g_a (x)| \geq |x|/2$$, so $$g_a$$ is also proper.

I guess the question still stands, is there a way of doing this without Whitney embedding?