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\begin{document}

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\section*{cotangent bundle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SymplecticStructure}{Symplectic structure}\dotfill \pageref*{SymplecticStructure} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

A \textbf{cotangent vector} or \textbf{covector} on $X$ is an element of $T^*(X)$. The \textbf{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 \textbf{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 \textbf{[[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 \textbf{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

\begin{displaymath}
\langle{\mathrm{d}_a f, v}\rangle = v[f] ,
\end{displaymath}
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 \textbf{[[de Rham differential]]} $\mathrm{d}f$ of $f$ is a covector field on $X$; given a vector field $v$, the pairing is given by

\begin{displaymath}
\langle{\mathrm{d}f, v}\rangle = v[f] ,
\end{displaymath}
thinking of $v$ as a derivation on differentiable functions.

One can also \emph{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

\begin{displaymath}
\omega = \sum_i \omega_i\, \mathrm{d}x^i
\end{displaymath}
in local coordinates $(x^1,\ldots,x^n)$. This fact can also be used as the basis of a definition of the cotangent bundle.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SymplecticStructure}{}\subsubsection*{{Symplectic structure}}\label{SymplecticStructure}

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

\begin{displaymath}
\theta
   \in
  \Omega^1(T^* X)
\end{displaymath}
with the property that under the [[isomorphism]]

\begin{displaymath}
j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)
\end{displaymath}
between [[differential 1-forms]] and smooth [[sections]] of the [[cotangent bundle]] we have for every smooth section $\sigma \in \Gamma(T^* X)$ the identification

\begin{displaymath}
\sigma^* \theta = j(\sigma)
\end{displaymath}
between the [[pullback of a differential form|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 \textbf{[[Liouville-Poincaré 1-form]]} or \textbf{canonical form} or \textbf{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]])

\begin{displaymath}
\theta = p_i d x^i
\end{displaymath}
and hence the [[symplectic form]] is

\begin{displaymath}
\omega = d p_i \wedge d q^i
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[tangent bundle]]


\item [[differential form]]


\item [[exterior bundle]]


\item [[Liouville-Poincaré 1-form]]


\item [[principal symbol]], [[bicharacteristic flow]]


\item [[wavefront set]], [[microlocal analysis]]


\item [[microsupport]], [[microlocal sheaf theory]]



\end{itemize}
[[!redirects cotangent vector]] [[!redirects cotangent vectors]] [[!redirects covector]] [[!redirects covectors]]

[[!redirects cotangent space]] [[!redirects cotangent spaces]] [[!redirects cotangent vector space]] [[!redirects cotangent vector spaces]]

[[!redirects covector field]] [[!redirects covector fields]] [[!redirects cotangent vector field]] [[!redirects cotangent vector fields]] [[!redirects differential 1-form]] [[!redirects differential 1-forms]] [[!redirects smooth differential 1-form]] [[!redirects smooth differential 1-forms]] [[!redirects 1-form]] [[!redirects 1-forms]]

[[!redirects cotangent bundle]] [[!redirects cotangent bundles]] [[!redirects Cotangent bundle]] [[!redirects Cotangent bundles]]



\end{document}