Appendix A: Results from differential geometry

Abstract

We assume $\Gamma \subset \mathbb{R}^d$ to be an oriented$C^2$ -hypersurface, i.e., for $x^* \epsilon \Gamma$ there exists an open set $U_{x*} \subset \mathbb{R}^d$ with $x^* \epsilon U_{x*}$ and a scalar function$\psi \epsilon C^2(U_{x*})$ such that

$$U_{x^*} \cap \Gamma = {x \epsilon U_{x^*} : \psi (x) = 0 }, and \nabla \psi (x) \neq 0 for all x \epsilon U_{x^*} \cap \Gamma.$$

(1)