7.1.1 Differentiable manifolds

We say that
is a
-dimensional differentiable manifold embedded in
, or simply a *manifold*, if it is given
by

where , are functions from to such that the gradient vectors , are linearly independent for each . This is actually not a new object for us: it is just a -dimensional surface in already considered in Section 1.2.2 and later in the formulation of the Basic Variable-Endpoint Control Problem in Chapter 4. The linear independence assumption imposed on the gradients means, in the terminology of Section 1.2.2, that all points of are regular.

Intuitively speaking, a
-dimensional
manifold
is a subset of
that locally ``looks like"
. This idea can be made precise by equipping
with *local coordinates*, as follows. Fix an arbitrary
. Since by assumption the (nonsquare) Jacobian matrix

is of rank , it has linearly independent columns; shuffling the variables if necessary, we can assume that these are the last columns. Using the Implicit Function Theorem, we can then solve the equations in (7.1) for the variables in terms of in some neighborhood of . Consequently, the components can be used to describe points in near , i.e., they provide a local coordinate chart for . A simple example of a 1-dimensional manifold in is the unit circle, for which we invite the reader to work out the above construction.

The standard definition of a manifold is actually stated in terms of the existence of local coordinate charts satisfying suitable compatibility conditions, without any explicit reference to the ambient space . However, it is known that every manifold defined in this way can be embedded in for some large enough. The above more concrete notion of an embedded manifold is sufficient for our purposes, but the concept of local coordinates will be useful for us as well. Basically, we have the choice of representing points in as -vectors or locally as -vectors (cf. the discussion of holonomic constraints in Section 2.5.2).

Let
be a
-dimensional manifold. We know that at each point
we can define the
*tangent space*
(see page ).
This is a
-dimensional linear vector space, spanned by the tangent vectors
associated with all possible curves in
passing through
. Recall that for a curve
lying in
, with
, the corresponding tangent vector is
. In local coordinates, if the curve has components
then the components of the tangent vector
are
.
It is also useful to consider the union

which is called the

Given a point
on a manifold
,
the linear vector space of linear
-valued functions on
is called
the *cotangent space* to
at
, and is denoted by
. This is the *dual space*^{7.1} to
; its elements are called *cotangent vectors*, or simply *covectors*.
The simplest example of a covector is the differential of a
function. Let
be a
function. For each
, the linear function
is defined as follows. Given a tangent vector
and an arbitrary
curve
in
with
having
as its tangent vector (so that
), let

In local coordinates this is given by , which clearly does not depend on the choice of a specific curve . The map assigns to each tangent vector the derivative of in the direction of . If are local coordinates on , then the corresponding differentials form a basis for and hence yield local coordinates on the cotangent space. For a tangent vector , the numbers give the components of . One also defines the

which is itself a manifold, of dimension . Combining local coordinates on with local coordinates on relative to the basis given by , one obtains local coordinates on the cotangent bundle known as