Suppose $(M,g)$ is a three dimensional smooth compact simply connected Riemannian manifold with boundary and suppose that $\Sigma$ is a smooth simply connected hypersurface in $M$ with a smooth boundary $\partial \Sigma \subset \partial M$. Let $D_X Y$ denote the Levi-Civita connection on $(M,g)$. Let $Z$ be a smooth vector field on $\Sigma$ and let $f:\Sigma \to \mathbb R$ be a smooth function on $\Sigma$. Does there exist a smooth function $\phi$ in $M$ such that $\Sigma=\{\phi=0\}$ and such that $$ (D_{\nabla \phi} \nabla \phi)\big|_{\Sigma} = f\, Z.$$ Note that the left hand side is just the restriction of $D_{\nabla \phi}\nabla \phi$ to $\Sigma$ and that $\nabla \phi$ is the gradient of $\phi$ with respect to $g$.

If the answer is no, would it make a difference if the latter condition is replaced with $$(D_{\nabla \phi} \nabla \phi)\big|_{\Sigma} - f\, Z \in \textrm{Span}\{(\nabla \phi)|_{\Sigma}\}$$

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