Let $M$ be a compact smooth $3$-manifold, and $h: M\to \mathbb{R}$ a function such that $\{0\}$ is a regular value of $h$, and define $\Sigma = h^{-1}(0).$ Moreover, we will denote $\mathfrak{X}^r(M)$ as the space of the smooth vector-fields over $M$ provided with the Whitney $\mathcal{C}^r$ topology (once $M$ is compact there exists a norm compatible with the $\mathcal{C}^r$ topology that makes $\mathfrak{X}^r(M)$ a banach space).
Notation Let $X \in \mathfrak{X}^r(M)$, then $Xh(p) := \nabla h(p)\cdot X(p)$
I trying to prove that the function
\begin{align*} \mathcal{F}: \mathfrak{X}^r(M)\times \mathfrak{X}^r(M) &\to \mathfrak{X}^r(\Sigma)\\ (X,Y) &\mapsto Yh(\cdot) X(\cdot) - Xh(\cdot) Y(\cdot), \end{align*} is an open mapping. Does anyone know if it is true and how to prove it?
It is clear that $\mathcal{F}$ is well defined because $\forall p \in \Sigma$, $\nabla h(p) \cdot \mathcal{F}(X,Y)(p) =0,$ then $\mathcal{F}(X,Y)(p) \in T_p\Sigma$