If $M$ is a submanifold of $\mathbb R^3$ and the normal space $N$ on $M$ at $p$ is one-dimensional, can we choose an unique "outer" normal from $N$?

Question

Let $M$ be a two-dimensional submanifold of $\mathbb R^3$. Then, the normal space $N_p(M)$ on $M$ at $p\in M$ is one-dimensional. So, there are only two unit normal vectors $n_1$ and $n_2$ on $M$ at $p$.

Can we somehow choose an "outer" one $\nu(p)\in\left\{n_1,n_2\right\}$?

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I only know the definition of the outer normal field $\nu:\partial K\to\mathbb R^n$ for a compact space $K\subseteq\mathbb R^n$ with a smooth boundary $\partial K$. In this case, there is an open environment $U$ of $p\in\partial K$ and some $\psi\in C^1(U)$ such that

• $K\cap U=\left\{\psi\le 0\right\}$
• $\psi'\ne 0$

and it's easy to see, that $$\nu(p):=\frac{\nabla\psi(p)}{\left\|\nabla\psi(p)\right\|}\tag 1$$ is the unique $n\in N_p(\partial K)$ such that $\left\|n\right\|=1$ and $$p+tn\not\in K\;\;\;\text{for all }t\in (0,\varepsilon)$$ for some $\varepsilon>0$. Moreover, it's easy to see from $(1)$ that $\nu:\partial K\to\mathbb R^n$ is continuous.

Can we find a similiar mapping $\nu:M\to\mathbb R^3$ which is at least Borel-measurable?

Answer 1

What you are seeking for is called the Gauss map. Although the wikipedia article is pretty informative, it fails to mention that it is the outer normal which the Gauss map produces, so it does answer your question.

Answer 2

You said in a comment (which has been deleted) that you're most interested in the case of flat polygons in space. Any flat polygon is a piece of a plane, and all planes in space have an equation of the form $Ax + By + Cz = D$, where $A$, $B$, $C$, and $D$ are constants. Moreover, if you know this equation, it is super easy to cook up a (not necessarily unit) normal vector, namely the vector $\langle A, B, C\rangle$.

Or, if you don't have the equation for the plane, but instead have three non-colinear points $P$, $Q$, and $R$ on the plane, you can determine a (not necessarily unit) normal vector to the plane with a cross product: $\vec{PQ}\times \vec{PR}$