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7.1.1 Differentiable manifolds

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

 (7.1)

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 tangent bundle of . This is a manifold of dimension . We note that in general, is not globally diffeomorphic to the direct product (well-known counterexamples are obtained by taking to be the two-dimensional sphere or the Mobius band). It is convenient to think of vector fields on as sections of the tangent bundle, i.e., functions such that for all .

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 space7.1 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 cotangent bundle

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 canonical coordinates.

Next: 7.1.2 Re-interpreting the maximum Up: 7.1 Maximum principle on Previous: 7.1 Maximum principle on   Contents   Index
Daniel 2010-12-20