8.3.2 Smooth Manifolds

The manifold definition given in Section 4.1.2 is often called a topological manifold. A manifold defined in this way does not necessarily have enough axioms to ensure that calculus operations, such as differentiation and integration, can be performed. We would like to talk about velocities on the configuration space ${\cal C}$ or in general for a continuous state space $X$. As seen in Chapter 4, the configuration space could be a manifold such as ${\mathbb{RP}}^3$. Therefore, we need to define some more qualities that a manifold should possess to enable calculus. This leads to the notion of a smooth manifold.

Figure 8.8: Intuitively, the tangent space is a linear approximation to the manifold in a neighborhood around $p$.

Assume that $M$ is a topological manifold, as defined in Section 4.1.2. For example, $M$ could represent $SO(3)$, the set of all rotation matrices for ${\mathbb{R}}^3$. A simpler example that will be helpful to keep in mind is $M = {\mathbb{S}}^2$, which is a sphere in ${\mathbb{R}}^3$. We want to extend the concepts of Section 8.3.1 from ${\mathbb{R}}^n$ to manifolds. One of the first definitions will be the tangent space ${\mathbb{T}}_p(M)$ at a point $p \in M$. As you might imagine intuitively, the tangent vectors are tangent to a surface, as shown in Figure 8.8. These will indicate possible velocities with which we can move along the manifold from $p$. This is more difficult to define for a manifold than for ${\mathbb{R}}^n$ because it is easy to express any point in ${\mathbb{R}}^n$ using $n$ coordinates, and all local coordinate frames for the tangent spaces at every $p \in {\mathbb{R}}^n$ are perfectly aligned with each other. For a manifold such as ${\mathbb{S}}^2$, we must define tangent spaces in a way that is not sensitive to coordinates and handles the fact that the tangent plane rotates as we move around on ${\mathbb{S}}^2$.

First think carefully about what it means to assign coordinates to a manifold. Suppose $M$ has dimension $n$ and is embedded in ${\mathbb{R}}^m$. For $M = SO(3)$, $n=3$ and $m=9$. For $M = {\mathbb{S}}^2$, $n=2$ and $m = 3$. The number of coordinates should be $n$, the dimension of $M$; however, manifolds embedded in ${\mathbb{R}}^m$ are often expressed as a subset of ${\mathbb{R}}^m$ for which some equality constraints must be obeyed. We would like to express some part of $M$ in terms of coordinates in ${\mathbb{R}}^n$.