nLab cotangent bundle

Idea

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

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

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

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

$$\langle{\mathrm{d}_a f, v}\rangle = v[f] ,$$

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

$$\langle{\mathrm{d}f, v}\rangle = v[f] ,$$

thinking of $v$ as a derivation on differentiable functions.

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

$$\omega = \sum_i \omega_i\, \mathrm{d}x^i$$

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

Properties

Symplectic structure

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

$$\theta \in \Omega^1(T^* X)$$

with the property that under the isomorphism

$$j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)$$

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

$$\sigma^* \theta = j(\sigma)$$

between the pullback of $\theta$ along $\sigma$ and the 1-form corresponding to $\sigma$ under $j$.

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

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

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

$$\theta = p_i d x^i$$

and hence the symplectic form is

$$\omega = d p_i \wedge d q^i \,.$$

Related concepts

• tangent bundle

• differential form

• exterior bundle

• Liouville-Poincaré 1-form

• principal symbol, bicharacteristic flow

• wavefront set, microlocal analysis

• microsupport?, microlocal sheaf theory