Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ be antipodal points that are regular values for $\gamma$. Since $n$ is even, there exists a vector field $u$ on the sphere that has exactly $a$ and $b$ as singularities, each one with index 1. Now define a vector field $v$ on $M$ by $v(p) = u(\gamma(p))$. This makes sense if we identify $T_p M$ with $T_{\gamma(p)} S^n$. Then the singularities of $v$ are the points in $\gamma^{-1}(a) \cup \gamma^{-1}(b) = \{a_1, \dots, a_r \} \cup \{ b_1, \dots, b_s \}$.
The fact that $a$ and $b$ are regular values imply that $\gamma$ is a diffeomorphism on a neighbourhood of these points. Why does that imply that they are simple singularities of $v$?
Thank you very much!