nLab
cotangent bundle

Idea

Given a manifold (or generalized smooth space) $X$, the cotangent bundle $T^{*}(X)$ of $X$ is the 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 $⟨\omega ,v⟩$ 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 dual to $T_{*}(X)$. More globally, given a covector field $\omega$ and a tangent vector field $v$, the paring $⟨\omega ,v⟩$ 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

$$⟨{\mathrm{d}}_{a}f,v⟩=v[f],$$\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 differential $\mathrm{d}f$ of $f$ is a covector field on $X$; given a vector field $v$, the pairing is given by

$$⟨\mathrm{d}f,v⟩=v[f],$$\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

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

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

Related concepts

• tangent bundle

• differential form

• exterior bundle

• Liouville-Poincaré 1-form