Editing Cotangent bundle (section)

{{short description|Vector bundle of cotangent spaces at every point in a manifold}}
In [[mathematics]], especially [[differential geometry]], the '''cotangent bundle''' of a [[smooth manifold]] is the [[vector bundle]] of all the [[cotangent space]]s at every point in the manifold. It may be described also as the [[dual bundle]] to the [[tangent bundle]]. This may be generalized to [[Category (mathematics)|categories]] with more structure than smooth manifolds, such as [[complex manifold]]s, or (in the form of cotangent sheaf) [[Algebraic variety|algebraic varieties]] or [[Scheme (mathematics)|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.