I am reading a book that defines a curvature tensor as $\boldsymbol{K} = -\nabla_\pi \boldsymbol{\hat{n}}$ where $\boldsymbol{\hat{n}}$ is the unit normal vector of a surface and
\begin{align*} \nabla(\cdot) = \nabla_\pi(\cdot) + \boldsymbol{\hat{n}}\dfrac{\partial(\cdot)}{\partial n} \end{align*}
What are the components of the tensor $\boldsymbol{K}$ and can they be written in terms of curvature $\boldsymbol{\kappa} = \dfrac{\partial \boldsymbol{\hat{t}}}{ds}$ and torsion $\boldsymbol{\tau} = -\dfrac{\partial \boldsymbol{\hat{b}}}{ds}$?