Hessian matrix of level set function

Question

For a given level set function $\phi : \mathbb{R}^3 \rightarrow \mathbb{R}$ whose zero level set presents a closed and smooth surface, what is a geometrical meaning of the following function? \begin{align*} f(\mathbf{x}) = \mathbf{m}(\mathbf{x})^{\text{T}} \frac{Hess(\phi(\mathbf{x}))}{|\nabla \phi(\mathbf{x})|} \frac{\nabla \phi(\mathbf{x})}{|\nabla \phi(\mathbf{x})|}, \end{align*} where $\mathbf{m}(\mathbf{x}) \perp \nabla \phi(\mathbf{x})$ for all $\mathbf{x} \in \mathbb{R}^3$ and $Hess(\phi(\mathbf{x}))$ is a Hessian matrix of $\phi$ at $\mathbf{x}$. How should it be interpreted on zero level set in a differential geometry?