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=                          Cotangent bundle                          =
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                             Introduction
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In mathematics, especially differential geometry, the cotangent bundle
of a smooth manifold is the vector bundle of all the cotangent spaces
at every point in the manifold. It may be described also as the dual
bundle to the tangent bundle. This may be generalized to categories
with more structure than smooth manifolds, such as complex manifolds,
or (in the form of cotangent sheaf) algebraic varieties or schemes. In
the smooth case, any Riemannian metric or symplectic form gives an
isomorphism between the cotangent bundle and the tangent bundle, but
they are not in general isomorphic in other categories.


                         The cotangent sheaf
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Smooth sections of the cotangent bundle are differential one-forms.


 Definition of the cotangent sheaf
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