Manifold definition

A topological space $M \subseteq {\mathbb{R}}^m$ is a manifold^4.4 if for every $x \in M$, an open set $O \subset M$ exists such that: 1) $x \in O$, 2) $O$ is homeomorphic to ${\mathbb{R}}^n$, and 3) $n$ is fixed for all $x \in M$. The fixed $n$ is referred to as the dimension of the manifold, $M$. The second condition is the most important. It states that in the vicinity of any point, $x \in M$, the space behaves just like it would in the vicinity of any point $y \in {\mathbb{R}}^n$; intuitively, the set of directions that one can move appears the same in either case. Several simple examples that may or may not be manifolds are shown in Figure 4.4.

One natural consequence of the definitions is that $m \geq n$. According to Whitney's embedding theorem [449], $m \geq 2n+1$. In other words, ${\mathbb{R}}^{2n+1}$ is ``big enough'' to hold any $n$-dimensional manifold.^4.5Technically, it is said that the $n$-dimensional manifold $M$ is embedded in ${\mathbb{R}}^m$, which means that an injective mapping exists from $M$ to ${\mathbb{R}}^m$ (if it is not injective, then the topology of $M$ could change).

As it stands, it is impossible for a manifold to include its boundary points because they are not contained in open sets. A manifold with boundary can be defined requiring that the neighborhood of each boundary point of $M$ is homeomorphic to a half-space of dimension $n$ (which was defined for $n=2$ and $n=3$ in Section 3.1) and that the interior points must be homeomorphic to ${\mathbb{R}}^n$.

Figure 4.4: Some subsets of ${\mathbb{R}}^2$ that may or may not be manifolds. For the three that are not, the point that prevents them from being manifolds is indicated.

The presentation now turns to ways of constructing some manifolds that frequently appear in motion planning. It is important to keep in mind that two manifolds will be considered equivalent if they are homeomorphic (recall the donut and coffee cup).