Let $M\subseteq\mathbb R^3$. I've got a very basic question, but since I'm not very familiar with differential geometry, I'm really lost by all the concepts for the moment.
I'd like to know which regularity assumptions we need to impose in order for $M$ being a $2$-dimensional manifold satisfying:
- There is a well-defined surface measure on the Borel $\sigma$-algebra on $M$;
- There is a unique outer normal field on $M$.
Maybe 2. needs to be phrased differently. For example, $M$ being a nondegenerate triangle should be allowed, but I guess then there is a problem with the normal field at the border. On the other hand, if we take only the interior of the triangle, this issue should disappear, shouldn't it?
I'd really appreciate any advice. I don't want to dive deep into this topic. 1. and 2. are just high-level assumptions I need to impose and I'm curious what exactly we would need to assume in order to guarantee them.
It is sufficient for $M$ to be an oriented embedded $C^1$-submanifold of $\mathbb{R}^3$ (possibly with boundary or even corners). Being $C^1$ gives a well-defined tangent plane at each point, and then with an orientation you get a well-defined outer unit normal vector to the tangent plane at each point. To get a surface measure, you just integrate the surface area $2$-form, which is just the $2$-form given by taking determinants with the normal vector. Note that there isn't really any issue with the boundary here--the tangent plane is still perfectly well-defined at the boundary, even if half of the tangent plane is pointing "outside" the boundary.