# Question about the proof that the Gauss map on a compact surface is surjective

I'm reading through Kristopher Tapp's Differential Geometry of Curves and Surfaces, and it has a proof (first suggested in one of the exercises, then later written out in the proof of one of the propositions) that the Gauss map $$N$$ on a compact surface is surjective. The proof given involves choosing an arbitrary unit vector $$v$$, taking a plane orthogonal to that vector far away from the surface, then moving it toward the surface until it just touches the surface at a point $$p$$ (i.e. taking the maximum of $$H_v(p) = \langle p, v\rangle$$). We will then have $$N(p) = v$$. The same proof appears in several of the questions here on StackExchange, but neither the book nor those questions fully address the possibility that $$N(p) = -v$$ (except, in some, by way of mentioning the Jordan-Brouwer theorem, which the book does not cover). One questioner did ask whether the minimum would produce the necessary point if $$N(p) = -v$$, but never had their concern addressed. Thus, how do you guarantee you get $$N(p) = v$$ and not just $$N(p) = \pm v$$?

Given a unit vector $$v$$ and the boundedness of the surface $$S$$, you can pick two hyperplanes perpendicular to $$v$$, $$\langle v,x\rangle=a$$ and $$\langle v,x\rangle=b$$ with $$a far away from the surface in such a way that your surface is in between, $$a<\langle v,x\rangle for all $$x\in S$$. They have normal vectors $$v$$ and $$-v$$ pointing towards the outside (not towards the region where $$S$$ is located. Then you start moving the one with normal vector $$v$$. Eventually you touch the surface at a point with normal vector $$v$$ (not $$-v$$).
• Without the Jordan-Brouwer theorem, there's no guarantee you can talk about "outside" for a general compact surface, so my question is how you justify "eventually you touch the surface at a point with normal vector $v$ (not $-v$)". Commented Mar 21, 2022 at 6:57
• There is no need for Jordan-Brouwer theorem here: the existence of $a$ and $b$ is given by compactness of $S$ (hence the linear form $x \mapsto \langle v,x\rangle$ is bounded on $S$). You now have to consider the first contact point on the right / on the left and show that $v$ is normal to your surface at these points. Commented Mar 21, 2022 at 8:24
• @Didier The existence of $a$ and $b$ isn't the problem; guaranteeing that the two first contact points give you both $v, -v$ is. Without being able to talk about an outside, the only immediate (i.e. without some further argument) conclusion about $N$ at these contact points is that it's $\pm v$, and it might be $-v$ at both. Commented Mar 21, 2022 at 20:36