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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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