Transversality and Morse function

Question

Let $f:\mathbb{R}^{k} \to \mathbb{R}^{k}$, and, for each $a \in \mathbb{R}^{k}$, define $$f_a(x)=f(x) + a_1x_1+...+a_kx_k$$ Prove that for almost all $a\in \Re^k$, $f_a$ is a Morse function.

Answer 1

First off, you have a typo. A Morse function is a smooth function $f : X \to \Bbb R$, your codomain should have been $\Bbb R$, not $\Bbb R^k$.

In any case, understand the following: if $f : \Bbb R^n \to \Bbb R$ is a Morse function, then $0$ is a regular value of the gradient $\nabla f : \Bbb R^n \to \Bbb R^n$. This is because critical points of $f$ correspond to the zeroes of $\nabla f$, and nondegeneracy of those critical points imply $Hf = D\nabla f \neq 0$ around the critical points.

That being said, if $f : \Bbb R^n \to \Bbb R$ is some arbitrary function, then the function $f_{\mathbf{a}}(\mathbf{x}) = f(\mathbf{x}) + \mathbf{a} \cdot \mathbf{x}$ has $\nabla f_{\mathbf{a}} = \nabla f + \mathbf{a}$. The critical points of $f_\mathbf{a}$ are points where $\nabla f = - \mathbf{a}$. By Sard's theorem, pick $\mathbf{a}$ such that $-\mathbf{a}$ is a regular value of $\nabla f$. Since the Hessian matrix of $f$ is the same as that of $f_\mathbf{a}$, we can conclude $0$ is a regular value of $\nabla f_\mathbf{a}$, that is, $f_\mathbf{a}$ is Morse.

We invoked Sard's theorem to choose $\mathbf{a}$ in the proof above; so it's clear from the statement of Sard's theorem that every $\mathbf{a} \in \Bbb R^n$ outside some measure $0$ set works.

Footnote: An interesting way to parse this fact is that the space of Morse functions $\Bbb R^n\to \Bbb R$ is dense in $C^\infty(\Bbb R^n, \Bbb R)$ (in some appropriate topology on the function space); any smooth function on $\Bbb R^n$ can be approximated by Morse functions arbitrarily close to it, and even by a linear homotopy. With some more work this is true for space of functions on any manifold whatsoever.