Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function. My goal is to better understand gradient-like vector fields for $f$.
Question: Do any two gradient-like vector fields necessarily coincide on a sufficiently small neighborhood of any critical point of $f$ ?
This seems to be a natural question to ask.
So I'm quite surprised that it's not addressed in any book about Morse theory that I've seen so far.
To eliminate any misunderstanding I will recall the definition of gradient-like vector fields:
Definition: A vector field $X\in$Vect$(M)$ is called gradient-like for $f$ if
1. $\forall q\in M\setminus Crit(f):\; df(q)X(q)<0$.
2. For each critical point $p\in Crit(f)$ of $f$, $\exists$ Morse coordinate chart $\phi:U\to\mathbb{R^n}$ (i.e. $\phi(p)=0$ and $f\circ\phi^{-1}(x)=f(p)-\sum_{i=1}^k x_i^2+\sum_{i=k+1}^n x_i^2$ for $x\in\phi(U)$, $k=index(p)$) such that $$\phi_*X(x)=d\phi(\phi^{-1}(x))X(\phi^{-1}(x))=(2x_1,\ldots,2x_k,-2x_{k+1},\ldots,-2x_n)^T$$ for $x\in\phi(U)$ (i.e. in this Morse-chart $X$ coincides with the negative gradient of $f$ w.r.t. the euclidean metric on $\mathbb{R^n}$).
Some motivation: I'm interested whether the space of all gradient-like vector fields for $f$ is contractible. If the answer to my question is affirmative then clearly any convex combination of a gradient-like vector field is still gradient-like and hence the space of gradient-like vector fields is contractible.
Any relevant references are also much appreciated.
Thanks for any contribution.
Edit: I've posted a related question on MathOverflow.
Best Answer
No. Since your question is about a neighborhood of a critical point, we can work over $\mathbb{R}^n$ instead of the compact manifold $M$.
Consider $\mathbb{R}^2$ with the following two coordinate charts in a neighborhood of 0. First we have the standard $x,y$ coordinates. Next we have the coordinates
$$ z = x \cos r^2 + y \sin r^2 \qquad w = y \cos r^2 - x \sin r^2 $$
where $r^2 = x^2 + y^2$. We easily verify that $z^2 + w^2 = x^2 + y^2 = r^2$. So that both $(x,y)$ and $(z,w)$ are Morse charts for $f = r^2$.
Let the vector field $X$ be $- x\partial_x - y\partial_y$ in the $(x,y)$ coordinates, and $X'$ be $- z\partial_z - w\partial_w$ in the $(z,w)$ coordinates. You can compute the change of variables explicitly and see that $X \neq X'$ except at the origin.
(It may be easier to see in standard polar coordinates, where $X = r\partial_r$ and $X' = r\partial_r + 2r^2\partial_\theta$. With this you also see that by adding a cut-off at finite $r$ for the perturbation, we can also directly extend this example to any two dimensional manifold. Higher dimensional analogues are also immediate.)