Question

The following question is probably classic in Morse theory, so a reference to an existing result should be sufficient. I don't know much about Morse theory and I am dealing with the following situation. I have a compact manifold $X$ and I have a Morse function $f$ on it with a saddle point at $x_0$. I don't like the properties of $f$(for some "mysterious" reason) so I take a parametrization around $x_0$, taking $x_0$ to the origin and containing no additional critical points, and add to $f$ a function of the type $\epsilon \rho Q$, where $Q$ is a quadratic polynomial, $\epsilon$ is a small number and $\rho$ is a bump function living in the coordinate patch that is identically equal to 1 in a neighborhood of the origin.

Clearly this new function still has a critical point at $x_0$. My question is: for small enough $\epsilon$ is this new function still Morse, with the same critical points as the original function $f$?

Answer 1

Sure. For any sequence $\epsilon_n\to 0$ a sequence of critical points of $f_n:=f+\epsilon_n \rho Q$ would have a subsequence which converges to a critical point of $f$, so in particular would eventually lie in a neighborhood on which the function equals either the original $f$ (if the limit isn't $x_0$) or $f+\epsilon_n Q$ (if the limit is $x_0$). But in such neighborhoods the only critical points are the critical points of the original function $f$, at least if $\epsilon_n$ is small enough.

If the answer to your question was "no" then you could find $\epsilon_n\to 0$ contradicting the above.

Answer 2

Lemma B on page 12 of Milnor's Lectures on h-cobordism theorem. I am sure it is also in his book on Morse functions, I just do not have it with me now.

Answer 3

There is classic result (I think going back to H. Whitney) that states that if $f\to \mathbb{R}$ is a Morse funtion on a compact smooth manifold $M$, thent there exist a neighborhood $\mathscr{N}$ of $f$ in $C^\infty(M)$ sucht that if $g\in M$, then $g$ is equivalent to $f$, i.e., one can obtain $g$ from $f$ by conjugating $f$ with a diffeomorphism $\Phi$ of $M$ and a diffeomorphim $\varphi$ of $\mathbb{R}$. More precisely, this means that

$$ g = \varphi \circ f \circ \Phi. $$

For a proof see the book of Golubitski and Guillemin Stability of smooth mappings. Loosely speaking this result says that if $g$ is not too far from $f$, then $g$ is obtained from $f$ via a change of "coordinates" on $M$ and a change in the target space $\mathbb{R}$.

Answer 4

Are you sure you cannot achieve what you want by simply changing coordinates? Answers are about adding general small function but you want to add a quadratic polynomial only on a neighborhood of $x_0$.