Cotangent bundle

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.
Formal Definition.
Let "M" be a smooth manifold and let "M"×"M" be the Cartesian product of "M" with itself. The diagonal mapping Δ sends a point "p" in "M" to the point ("p","p") of "M"×"M". The image of Δ is called the diagonal. Let formula_1 be the sheaf of germs of smooth functions on "M"×"M" which vanish on the diagonal. Then the quotient sheaf formula_2 consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is defined as the pullback of this sheaf to "M":
By Taylor's theorem, this is a locally free sheaf of modules with respect to the sheaf of germs of smooth functions of "M". Thus it defines a vector bundle on "M": the cotangent bundle.
Smooth sections of the cotangent bundle are called (differential) one-forms.
Contravariance Properties.
A smooth morphism formula_4 of manifolds, induces a pullback sheaf formula_5 on "M". There is an induced map of vector bundles formula_6.
Examples.
The tangent bundle of the vector space formula_7 is formula_8, and the cotangent bundle is formula_9, where formula_10 denotes the dual space of covectors, linear functions formula_11. 
Given a smooth manifold formula_12 embedded as a hypersurface represented by the vanishing locus of a function formula_13 with the condition that formula_14 the tangent bundle is
where formula_16 is the directional derivative formula_17. By definition, the cotangent bundle in this case is
where formula_19 Since every covector formula_20 corresponds to a unique vector formula_21 for which formula_22 for an arbitrary formula_23
The cotangent bundle as phase space.
Since the cotangent bundle "X" = "T"*"M" is a vector bundle, it can be regarded as a manifold in its own right. Because at each point the tangent directions of "M" can be paired with their dual covectors in the fiber, "X" possesses a canonical one-form θ called the tautological one-form, discussed below. The exterior derivative of θ is a symplectic 2-form, out of which a non-degenerate volume form can be built for "X". For example, as a result "X" is always an orientable manifold (the tangent bundle "TX" is an orientable vector bundle). A special set of coordinates can be defined on the cotangent bundle; these are called the canonical coordinates. Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out.
The tautological one-form.
The cotangent bundle carries a canonical one-form θ also known as the symplectic potential, "Poincaré" "1"-form, or "Liouville" "1"-form. This means that if we regard "T"*"M" as a manifold in its own right, there is a canonical section of the vector bundle "T"*("T"*"M") over "T"*"M".
This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose that "x""i" are local coordinates on the base manifold "M". In terms of these base coordinates, there are fibre coordinates "p""i": a one-form at a particular point of "T"*"M" has the form "p""i" "dx""i" (Einstein summation convention implied). So the manifold "T"*"M" itself carries local coordinates ("x""i", "p""i") where the "x"'s are coordinates on the base and the "p's" are coordinates in the fibre. The canonical one-form is given in these coordinates by