Let $M^n \subset \mathbb{R}^{n+1}$ be a compact hypersurface, oriented with the smooth normal vector field $N(X) \perp T_xM$. Let $G: M^n \to S^n$ be the corresponding Gauss map. Does it follow that $\deg(G) = \chi(M)$, the Euler number of $M$?
Thoughts so far. A generic point $a \in S^n$ induces a vector field $v_a$ on $M$, by $v_a(x) =$ tangential component of $a$ at $x \in M$. But I am not sure what do from here on out. Could anybody help?