# Cotangent bundle

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Vector bundle of cotangent spaces at every point in a manifold

In [mathematics](/source/Mathematics), especially [differential geometry](/source/Differential_geometry), the **cotangent bundle** of a [smooth manifold](/source/Smooth_manifold) is the [vector bundle](/source/Vector_bundle) of all the [cotangent spaces](/source/Cotangent_space) at every point in the manifold. It may be described also as the [dual bundle](/source/Dual_bundle) to the [tangent bundle](/source/Tangent_bundle). This may be generalized to [categories](/source/Category_(mathematics)) with more structure than smooth manifolds, such as [complex manifolds](/source/Complex_manifold), or (in the form of cotangent sheaf) [algebraic varieties](/source/Algebraic_variety) or [schemes](/source/Scheme_(mathematics)). 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 via [diagonal morphism](/source/Diagonal_morphism)

There are several equivalent ways to define the cotangent bundle. [One way](/source/Cotangent_sheaf#Construction_through_a_diagonal_morphism) is through a [diagonal mapping](/source/Diagonal_mapping) Δ {\displaystyle \Delta } and [germs](/source/Germ_(mathematics)).

Let M {\displaystyle M} be a [smooth manifold](/source/Differentiable_manifold) and let M × M {\displaystyle M\times M} be the [Cartesian product](/source/Cartesian_product) of M {\displaystyle M} with itself. The [diagonal mapping](/source/Diagonal_mapping) Δ {\displaystyle \Delta } sends a point p {\displaystyle p} in M {\displaystyle M} to the point ( p , p ) {\displaystyle (p,p)} of M × M {\displaystyle M\times M} . The image of Δ {\displaystyle \Delta } is called the diagonal. Let I {\displaystyle {\mathcal {I}}} be the [sheaf](/source/Sheaf_(mathematics)) of [germs](/source/Germ_(mathematics)) of smooth functions on M × M {\displaystyle M\times M} which vanish on the diagonal. Then the [quotient sheaf](/source/Sheaf_(mathematics)#Operations) I / I 2 {\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}} consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The [cotangent sheaf](/source/Cotangent_sheaf) is defined as the [pullback](/source/Inverse_image_functor) of this sheaf to M {\displaystyle M} :

Γ T ∗ M = Δ ∗ ( I / I 2 ) . {\displaystyle \Gamma T^{*}M=\Delta ^{*}\left({\mathcal {I}}/{\mathcal {I}}^{2}\right).}

By [Taylor's theorem](/source/Taylor's_theorem), this is a [locally free sheaf](/source/Locally_free_sheaf) of modules with respect to the sheaf of germs of smooth functions of M {\displaystyle M} . Thus it defines a [vector bundle](/source/Vector_bundle) on M {\displaystyle M} : the **cotangent bundle**.

[Smooth](/source/Smooth_function) [sections](/source/Section_(fiber_bundle)) of the cotangent bundle are called (differential) [one-forms](/source/One-form).

## Contravariance properties

A smooth morphism ϕ : M → N {\displaystyle \phi \colon M\to N} of manifolds induces a [pullback sheaf](/source/Pullback_(differential_geometry)) ϕ ∗ T ∗ N {\displaystyle \phi ^{*}T^{*}N} on *M*. There is an [induced map](/source/Pullback_(differential_geometry)#Pullback_of_cotangent_vectors_and_1-forms) of vector bundles ϕ ∗ ( T ∗ N ) → T ∗ M {\displaystyle \phi ^{*}(T^{*}N)\to T^{*}M} .

## Examples

The tangent bundle of the vector space R n {\displaystyle \mathbb {R} ^{n}} is T R n = R n × R n {\displaystyle T\,\mathbb {R} ^{n}=\mathbb {R} ^{n}\times \mathbb {R} ^{n}} , and the cotangent bundle is T ∗ R n = R n × ( R n ) ∗ {\displaystyle T^{*}\mathbb {R} ^{n}=\mathbb {R} ^{n}\times (\mathbb {R} ^{n})^{*}} , where ( R n ) ∗ {\displaystyle (\mathbb {R} ^{n})^{*}} denotes the [dual space](/source/Dual_space) of covectors, linear functions v ∗ : R n → R {\displaystyle v^{*}:\mathbb {R} ^{n}\to \mathbb {R} } .

Given a smooth manifold M ⊂ R n {\displaystyle M\subset \mathbb {R} ^{n}} embedded as a [hypersurface](/source/Hypersurface) represented by the vanishing locus of a function f ∈ C ∞ ( R n ) , {\displaystyle f\in C^{\infty }(\mathbb {R} ^{n}),} with the condition that ∇ f ≠ 0 , {\displaystyle \nabla f\neq 0,} the tangent bundle is

- T M = { ( x , v ) ∈ T R n : f ( x ) = 0 , d f x ( v ) = 0 } , {\displaystyle TM=\{(x,v)\in T\,\mathbb {R} ^{n}\ :\ f(x)=0,\ \,df_{x}(v)=0\},}

where d f x ∈ T x ∗ M {\displaystyle df_{x}\in T_{x}^{*}M} is the [directional derivative](/source/Directional_derivative) d f x ( v ) = ∇ f ( x ) ⋅ v {\displaystyle df_{x}(v)=\nabla \!f(x)\cdot v} . By definition, the cotangent bundle in this case is

- T ∗ M = { ( x , v ∗ ) ∈ T ∗ R n : f ( x ) = 0 , v ∗ ∈ T x ∗ M } , {\displaystyle T^{*}M={\bigl \{}(x,v^{*})\in T^{*}\mathbb {R} ^{n}\ :\ f(x)=0,\ v^{*}\in T_{x}^{*}M{\bigr \}},}

where T x ∗ M = { v ∈ T x R n : d f x ( v ) = 0 } ∗ . {\displaystyle T_{x}^{*}M=\{v\in T_{x}\mathbb {R} ^{n}\ :\ df_{x}(v)=0\}^{*}.} Since every covector v ∗ ∈ T x ∗ M {\displaystyle v^{*}\in T_{x}^{*}M} corresponds to a unique vector v ∈ T x M {\displaystyle v\in T_{x}M} for which v ∗ ( u ) = v ⋅ u , {\displaystyle v^{*}(u)=v\cdot u,} for an arbitrary u ∈ T x M , {\displaystyle u\in T_{x}M,}

- T ∗ M = { ( x , v ∗ ) ∈ T ∗ R n : f ( x ) = 0 , v ∈ T x R n , d f x ( v ) = 0 } . {\displaystyle T^{*}M={\bigl \{}(x,v^{*})\in T^{*}\mathbb {R} ^{n}\ :\ f(x)=0,\ v\in T_{x}\mathbb {R} ^{n},\ df_{x}(v)=0{\bigr \}}.}

## The cotangent bundle as phase space

Since the cotangent bundle X = T ∗ M {\displaystyle X=T^{*}M} is a [vector bundle](/source/Vector_bundle), it can be regarded as a manifold in its own right. Because at each point the tangent directions of M {\displaystyle M} can be paired with their dual covectors in the fiber, X {\displaystyle X} possesses a canonical one-form θ {\displaystyle \theta } called the [tautological one-form](/source/Tautological_one-form), discussed below. The [exterior derivative](/source/Exterior_derivative) of θ {\displaystyle \theta } is a [symplectic 2-form](/source/Symplectic_manifold), out of which a non-degenerate [volume form](/source/Volume_form) can be built for X {\displaystyle X} . For example, as a result X {\displaystyle X} is always an [orientable](/source/Orientable) manifold (the tangent bundle T X {\displaystyle TX} is an orientable vector bundle). A special set of [coordinates](/source/Coordinates) can be defined on the cotangent bundle; these are called the [canonical coordinates](/source/Canonical_coordinates). Because cotangent bundles can be thought of as [symplectic manifolds](/source/Symplectic_manifold), any real function on the cotangent bundle can be interpreted to be a [Hamiltonian](/source/Symplectic_vector_space); thus the cotangent bundle can be understood to be a [phase space](/source/Phase_space) on which [Hamiltonian mechanics](/source/Hamiltonian_mechanics) plays out.

### The tautological one-form

Main article: [Tautological one-form](/source/Tautological_one-form)

The cotangent bundle carries a canonical one-form θ {\displaystyle \theta } also known as the [symplectic potential](/source/Symplectic_potential), *Poincaré* *1*-form, or *Liouville* *1*-form. This means that if we regard T ∗ M {\displaystyle T^{*}M} as a manifold in its own right, there is a canonical [section](/source/Section_(fiber_bundle)) of the vector bundle T ∗ ( T ∗ M ) {\displaystyle T^{*}(T^{*}M)} over T ∗ M {\displaystyle T^{*}M} .

This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose that x i {\displaystyle x^{i}} are local coordinates on the base manifold M {\displaystyle M} . In terms of these base coordinates, there are fibre coordinates p i {\displaystyle p_{i}} : a one-form at a particular point of T ∗ M {\displaystyle T^{*}M} has the form p i d x i {\displaystyle p_{i}{\text{d}}x^{i}} ([Einstein summation convention](/source/Einstein_summation_convention) implied). So the manifold T ∗ M {\displaystyle T^{*}M} itself carries local coordinates ( x i , p i ) {\displaystyle (x^{i},p_{i})} where the x {\displaystyle x} 's are coordinates on the base and the p {\displaystyle p} 's are coordinates in the fibre. The canonical one-form is given in these coordinates by θ ( x , p ) = ∑ i = 1 n p i d x i . {\displaystyle \theta _{(x,p)}=\sum _{i=1}^{n}p_{i}\,dx^{i}.}

Intrinsically, the value of the canonical one-form in each fixed point of T ∗ M {\displaystyle T^{*}M} is given as a [pullback](/source/Pullback_(differential_geometry)). Specifically, suppose that π : T ∗ M → M {\displaystyle \pi :T^{*}M\rightarrow M} is the [projection](/source/Projection_(mathematics)) of the bundle. Taking a point in T x ∗ M {\displaystyle T_{x}^{*}M} is the same as choosing of a point x {\displaystyle x} in M {\displaystyle M} and a one-form ω {\displaystyle \omega } at x {\displaystyle x} , and the tautological one-form θ {\displaystyle \theta } assigns to the point ( x , ω ) {\displaystyle (x,\omega )} the value θ ( x , ω ) = π ∗ ω . {\displaystyle \theta _{(x,\omega )}=\pi ^{*}\omega .}

That is, for a vector v {\displaystyle v} in the tangent bundle of the cotangent bundle, the application of the tautological one-form θ {\displaystyle \theta } to v {\displaystyle v} at ( x , ω ) {\displaystyle (x,\omega )} is computed by projecting v {\displaystyle v} into the tangent bundle at x {\displaystyle x} using d π : T ( T ∗ M ) → T M {\displaystyle {\text{d}}\pi :T(T^{*}M)\rightarrow TM} and applying ω {\displaystyle \omega } to this projection. Note that the tautological one-form is not a pullback of a one-form on the base M {\displaystyle M} .

### Symplectic form

The cotangent bundle has a canonical [symplectic 2-form](/source/Symplectic_manifold) on it, as an [exterior derivative](/source/Exterior_derivative) of the [tautological one-form](/source/Tautological_one-form), the [symplectic potential](/source/Symplectic_potential). Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on R n × R n {\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}} . But there the one form defined is the sum of y i d x i {\displaystyle y_{i}\,dx_{i}} , and the differential is the canonical symplectic form, the sum of d y i ∧ d x i {\displaystyle dy_{i}\land dx_{i}} .

### Phase space

If the manifold M {\displaystyle M} represents the set of possible positions in a [dynamical system](/source/Dynamical_system), then the cotangent bundle T ∗ M {\displaystyle T^{*}\!M} can be thought of as the set of possible *positions* and *momenta*. For example, this is a way to describe the [phase space](/source/Phase_space) of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is constant). The entire state space looks like a cylinder, which is the cotangent bundle of the circle. The above symplectic construction, along with an appropriate [energy](/source/Energy) function, gives a complete determination of the physics of system. See [Hamiltonian mechanics](/source/Hamiltonian_mechanics) and the article on [geodesic flow](/source/Geodesic_flow) for an explicit construction of the Hamiltonian equations of motion.

## See also

- [Legendre transformation](/source/Legendre_transformation)

## References

- [Abraham, Ralph](/source/Ralph_Abraham_(mathematician)); [Marsden, Jerrold E.](/source/Jerrold_E._Marsden) (1978). *Foundations of Mechanics*. London: Benjamin-Cummings. [ISBN](/source/ISBN_(identifier)) [0-8053-0102-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8053-0102-X).

- [Jost, Jürgen](/source/J%C3%BCrgen_Jost) (2002). *Riemannian Geometry and Geometric Analysis*. Berlin: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [3-540-63654-4](https://en.wikipedia.org/wiki/Special:BookSources/3-540-63654-4).

- Singer, Stephanie Frank (2001). [*Symmetry in Mechanics: A Gentle Modern Introduction*](/source/Symmetry_in_Mechanics). Boston: Birkhäuser.

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Adapted from the Wikipedia article [Cotangent bundle](https://en.wikipedia.org/wiki/Cotangent_bundle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cotangent_bundle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.