** Next:** The Difference Between a
** Up:** Geometric Technicalities
** Previous:** Geometric Technicalities
** Contents**
** Index**

###

Manifolds

A manifold, , 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
functions, defined on . 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
(for purposes of clearer explanation, we restrict our discussion to
the real version of the Stiefel manifold) of points which are
written as matrices
satisfying the constraint . We will select
for our set of functions those real-valued functions which
are restrictions to
of functions of variables which
are
about
in
the
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,
is a smooth manifold. This means that about
every point of
we can construct a local coordinate system which
is . For example, consider the point

and a point in the neighborhood of ,

For small , we can solve for
in terms of the components of and the components of an
arbitrary small antisymmetric matrix by solving
for symmetric and letting
. This can
always be done smoothly and in a locally 1-to-1 fashion for small
enough and . Since the components of are all smooth
functions (being restrictions of the coordinate functions of the
ambient
space), and since the solution for is
for small enough
and , we have shown that any point of
can be expressed smoothly as a function of
variables. The only difference between
is a Euclidean rigid motion; therefore, this statement holds for all
points in
.
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 independent coordinates required to
coordinatize neighborhoods of points of
.
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).

** Next:** The Difference Between a
** Up:** Geometric Technicalities
** Previous:** Geometric Technicalities
** Contents**
** Index**
Susan Blackford
2000-11-20