cotangent bundle in nLab

Contents

Contents

Idea

Given a differentiable manifold XX, the cotangent bundle T *(X)T^*(X) of XX is the dual vector bundle over XX dual to the tangent bundle TxT x of XX.

A cotangent vector or covector on XX is an element of T *(X)T^*(X). The cotangent space of XX at a point aa is the fiber T a *(X)T^*_a(X) of T *(X)T^*(X) over aa; it is a vector space. A covector field on XX is a section of T *(X)T^*(X). (More generally, a differential form on XX is a section of the exterior algebra of T *(X)T^*(X); a covector field is a differential 1-form.)

Given a covector ω\omega at aa and a tangent vector vv at aa, the pairing ⟨ω,v⟩\langle{\omega,v}\rangle is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for T *(X)T^*(X) to be the dual vector bundle to T *(X)T_*(X). More globally, given a covector field ω\omega and a tangent vector field vv, the paring ⟨ω,v⟩\langle{\omega,v}\rangle is a scalar function on XX.

Given a point aa in XX and a differentiable (real-valued) partial function ff defined near aa, the differential d af\mathrm{d}_a f of ff at aa is a covector on XX at aa; given a tangent vector vv at aa, the pairing is given by

⟨d af,v⟩=v[f], \langle{\mathrm{d}_a f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions defined near aa. (It is really the germ at aa of ff that matters here.) More globally, given a differentiable function ff, the de Rham differential df\mathrm{d}f of ff is a covector field on XX; given a vector field vv, the pairing is given by

⟨df,v⟩=v[f], \langle{\mathrm{d}f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions.

One can also define covectors at aa to be germs of differentiable functions at aa, modulo the equivalence relation that d af=d ag\mathrm{d}_a f = \mathrm{d}_a g if f−gf - g is constant on some neighbourhood of aa. In general, a covector field won't be of the form df\mathrm{d}f, but it will be a sum of terms of the form hdfh \mathrm{d}f. More specifically, a covector field ω\omega on a coordinate patch can be written

ω=∑ iω idx i \omega = \sum_i \omega_i\, \mathrm{d}x^i

in local coordinates (x 1,…,x n)(x^1,\ldots,x^n). This fact can also be used as the basis of a definition of the cotangent bundle.

Properties

Symplectic structure

Every cotangent bundle T *XT^\ast X carries itself a canonical differential 1-form

θ∈Ω 1(T *X) \theta \in \Omega^1(T^* X)

with the property that under the isomorphism

j:Γ(T *X)→≃Ω 1(X) j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)

between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section σ∈Γ(T *X)\sigma \in \Gamma(T^* X) the identification

σ *θ=j(σ) \sigma^* \theta = j(\sigma)

between the pullback of θ\theta along σ\sigma and the 1-form corresponding to σ\sigma under jj.

This unique differential 1-form θ∈Ω 1(T *X)\theta \in \Omega^1(T^* X) is called the Liouville-Poincaré 1-form or canonical form or tautological form on the cotangent bundle.

The de Rham differential ω≔dθ\omega \coloneqq d \theta is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.

On a coordinate chart ℝ n\mathbb{R}^n of XX with canonical coordinate functions denoted (x i)(x^i), the cotangent bundle over the chart is T *ℝ n≃ℝ n×ℝ nT^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n with canonical coordinates ((x i),(p j))((x^i), (p_j)). In these coordinates the canonical 1-form is (using Einstein summation convention)

θ=p idx i \theta = p_i d x^i

and hence the symplectic form is

ω=dp i∧dq i. \omega = d p_i \wedge d q^i \,.

• tangent bundle

• differential form

• exterior bundle

• Liouville-Poincaré 1-form

• principal symbol, bicharacteristic flow

• wavefront set, microlocal analysis

• microsupport?, microlocal sheaf theory