📂Geometry

Differentiable Surfaces and Boundaries of Regions in Differential Geometry

Region¹

Consider a subset $\mathscr{R}$ of a surface $M$. If $\mathscr{R}$ is an open set, and for any two points in $\mathscr{R}$ there exists a curve on $\mathscr{R}$ containing both, then $\mathscr{R}$ is called a region of $M$.

Boundary

For a region $\mathscr{R}$ of a surface $M$, the following set $\partial \mathscr{R}$ is called the boundary of $\mathscr{R}$.

$$\partial \mathscr{R} = \left\{ p \notin \mathscr{R} : \exists \left\{ p_{j} \right\} \subset \mathscr{R} \text{ such that } \lim\limits_{j \to \infty} p_{j} = p \right\}$$

Curve Enclosing a Region

If the image of curve $\boldsymbol{\gamma}$ is the boundary of region $\mathscr{R}$, and the intrinsic normal $\mathbf{S}$ of $\boldsymbol{\gamma}$ points inside while $-\mathbf{S}$ points outside $\mathscr{R}$, then it is said that curve $\boldsymbol{\gamma}$ bounds a region $\mathscr{R}$.

Explanation

The intrinsic normal $\mathbf{S}$ pointing inside means that the curve must rotate counterclockwise. For example, for a surface $M = \mathbb{R}^{2}$, $\mathscr{R} = \left\{ (x,y) \in \mathbb{R}^{2} : x^{2} + y^{2} \lt 1 \right\}$ is a region of $M$. Also, curve $\boldsymbol{\alpha}(\theta) = (\cos \theta, \sin \theta)$ is the boundary of $\mathscr{R}$.

On the other hand, let’s say the surface $M$ is a torus $T^{2}$ as in the right picture below. If all points except the image of $\boldsymbol{\gamma}$ are considered the region $\mathscr{R}$, then the boundary of $\mathscr{R}$ becomes the image of $\boldsymbol{\gamma}$. However, since $-\mathbf{S}$ of curve $\boldsymbol{\gamma}$ does not point outside but inside $\mathscr{R}$, it is said that curve $\boldsymbol{\gamma}$ does not bound $\mathscr{R}$.