Let $M$ be a compact manifold (without boundary) and let $f:M\to \mathbb{R}$ be a fixed Morse-function.

Question:Is the space $GVect(M,f)$ of all gradient-like vector-fields for $f$ contractible or even convex?

To eliminate any misunderstanding, this is the definition of gradient-like vector fields that I'm using:

Definition:A vector field $X\in Vect(M)$ is calledgradient-likefor $f$ 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 $\mathbb{R^n}$).

Thank you in advance for any insights.