Vector that is tangent to the surface but normal to the boundary

Question

In a paper I read about a conormal $\mu$ which is a vector normal to the surface boundary $\partial\Sigma$ but tangent to the surface $\Sigma$. How is it geometrically possible for a vector to be normal to the boundary but tangent to the surface?

Answer 1

Take $\mathcal H^2=\{(x,y,z)\in \mathbb R^3:z=0\wedge y\ge 0\}.$ Then, for any point $p$ lying on the $x$- axis, $v=-\left(\frac{\partial}{\partial y} \right)_p\in T_p\mathcal H^2$ and is normal to the surface:

Answer 2

Just to visualize it, assume $\Sigma\subseteq \Bbb R^3$. Then the conormal $\mu = N_{\partial \Sigma}$ can be given, for example, by $$N_{\partial \Sigma}(p) = T_{\partial\Sigma}(p)\times N_\Sigma(p),\qquad p\in\partial\Sigma.$$Here $\times$ is the usual cross product, $T_{\partial\Sigma}$ is a unit tangent field to $\partial \Sigma$ and $N_\Sigma$ is a unit normal field to $\Sigma$. There is a difference between a tangent vector field to $\partial \Sigma$ ($\Gamma(T(\partial\Sigma))$) and a tangent vector field to $\Sigma$ along $\partial\Sigma$ ($\Gamma(T\Sigma|_{\partial\Sigma})$).