Manifolds

A manifold, $M$, is a collection of points which have a differential structure. In plain terms, this means that one is able to take derivatives of some reasonable class of functions, the $C^\infty$ functions, defined on $M$. What this class of differentiable functions is can be somewhat arbitrary, though some technical consistency conditions must be satisfied. We will not discuss those conditions in this section.

We consider the manifold $\mbox{Stief}(n,k)$ (for purposes of clearer explanation, we restrict our discussion to the real version of the Stiefel manifold) of points which are written as $n \times k$ matrices satisfying the constraint $Y^*Y=I$. We will select for our set of $C^\infty$ functions those real-valued functions which are restrictions to $\mbox{Stief}(n,k)$ of functions of $nk$ variables which are $C^\infty({\cal R}^{n \times k})$ about $\mbox{Stief}(n,k)$ in the ${\cal R}^{n \times k}$ sense. It should not be difficult to convince oneself that the set of such functions must satisfy any technical consistency condition one could hope for.

Additionally, $M = \mbox{Stief}(n,k)$ is a smooth manifold. This means that about every point of $\mbox{Stief}(n,k)$ we can construct a local coordinate system which is $C^\infty$. For example, consider the point

$$Y = \left[ \begin{array}{c} I \cr 0 \end{array} \right]$$

and a point in the neighborhood of $Y$,

$$\tilde{Y} = \left[ \begin{array}{c} A \cr B \end{array} \right].$$

For small $B$, we can solve for $A$ in terms of the components of $B$ and the components of an arbitrary small antisymmetric matrix $\Delta$ by solving $S^*S = I - B^*B$ for symmetric $S$ and letting $A = e^{\Delta} S$. This can always be done smoothly and in a locally 1-to-1 fashion for small enough $B$ and $\Delta$. Since the components of $B$ are all smooth functions (being restrictions of the coordinate functions of the ambient ${\cal R}^{n \times k}$ space), and since the solution for $A$ is $C^\infty({\cal R}^{(n-k) \times k} \oplus {\cal R}^{k \times k})$ for small enough $B$ and $\Delta$, we have shown that any point of $\mbox{Stief}(n,k)$ can be expressed smoothly as a function of $(n-k)k+k(k-1)/2 = nk - k(k+1)/2$ variables. The only difference between $[{I \atop 0}]$ is a Euclidean rigid motion; therefore, this statement holds for all points in $\mbox{Stief}(n,k)$.

Summarizing, a manifold is a set of points with a set of differentiable functions defined on it as well as a sense of local coordinatization in which the dependent coordinates can be represented as smooth functions of the independent coordinates. Specifically, we see that there are always $nk - k(k+1)/2$ independent coordinates required to coordinatize neighborhoods of points of $\mbox{Stief}(n,k)$. This number is called the dimension of the manifold, and it should be the same for every point of a manifold (if not, then the manifold is either disconnected or not actually smooth).