1D manifolds

The set ${\mathbb{R}}$ of reals is the most obvious example of a 1D manifold because ${\mathbb{R}}$ certainly looks like (via homeomorphism) ${\mathbb{R}}$ in the vicinity of every point. The range can be restricted to the unit interval to yield the manifold $(0,1)$ because they are homeomorphic (recall Example 4.5).

Another 1D manifold, which is not homeomorphic to $(0,1)$, is a circle, ${\mathbb{S}}^1$. In this case ${\mathbb{R}}^m = {\mathbb{R}}^2$, and let

$${\mathbb{S}}^1 = \{ (x,y) \in {\mathbb{R}}^2 \;\vert\; x^2 + y^2 = 1\} .$$ (4.5)

If you are thinking like a topologist, it should appear that this particular circle is not important because there are numerous ways to define manifolds that are homeomorphic to ${\mathbb{S}}^1$. For any manifold that is homeomorphic to ${\mathbb{S}}^1$, we will sometimes say that the manifold is ${\mathbb{S}}^1$, just represented in a different way. Also, ${\mathbb{S}}^1$ will be called a circle, but this is meant only in the topological sense; it only needs to be homeomorphic to the circle that we learned about in high school geometry. Also, when referring to ${\mathbb{R}}$, we might instead substitute $(0,1)$ without any trouble. The alternative representations of a manifold can be considered as changing parameterizations, which are formally introduced in Section 8.3.2.