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Overview
Let be a differentiable manifold. Analogously to the construction of the tangent bundle, we can make the set of covectors on a given manifold into a vector bundle over , denoted and called the cotangent bundle of .
Rigorous Definition
To make this definition precise it is convenient to use the classical definition of a manifold. Let be an -dimensional differentiable manifold, let
(each is an open subset of
) be an atlas of with transition functions
.
As an atlas for , we may take
. We may construct transition functions
as follows:
For these to be valid transition functions, they must satisfy the three criteria. For a verification that these criteria are satisfied, please see the attachment.
Bundle Structure
The cotangent bundle is a vector bundle over the manifold . To substantiate this claim, we must specify a projection map onto the manifold and local trivializations and transition functions and verify that they satisfies the defining properties of a bundle. In terms of the local coordinates used above, it is easy to describe the projection map :
The local trivializations are also somewhat trivial:
Finally, the transition functions are given as follows:
For a verification that
satisfies the three criteria for a bundle, please see the attachment.
Properties
The cotangent bundle is the vector bundle dual to the tangent bundle . On any differentiable manifold,
(for example, by the existence of a Riemannian metric), but this identification is by no means canonical, and thus it is useful to distinguish between these two objects.
The cotangent bundle to any manifold has a natural symplectic structure given in terms of the Poincaré 1-form, which is in some sense unique. This is not true of the tangent bundle. The existence of a symplectic structure implies that the cotangent bundle is always orientable, even if the original manifold is not.
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