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\begin{document}

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\section*{contact manifold}

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

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic 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{darboux_theorem}{Darboux theorem}\dotfill \pageref*{darboux_theorem} \linebreak
\noindent\hyperlink{relation_to_principal_connections}{Relation to $U(1)$-principal connections}\dotfill \pageref*{relation_to_principal_connections} \linebreak
\noindent\hyperlink{History}{History}\dotfill \pageref*{History} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{characterization}{Characterization}\dotfill \pageref*{characterization} \linebreak
\noindent\hyperlink{further}{Further}\dotfill \pageref*{further} \linebreak


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

A \emph{contact manifold} is a [[smooth manifold]] $P$ of odd [[dimension]] $2n+1$ which is equipped with a [[differential form|differential 1-form]] $A$ that is non-degenerate in the sense that the [[wedge product]] $A \wedge (d_{dR} A)^{\wedge^n}$ does not vanish.

The special case of [[closed manifold|closed]] \emph{regular} contact manifolds $(P,A)$ are essentially equivalent to the total spaces of [[circle bundles]] $P \to X$ over an $2n$-dimensional manifold equipped with a [[connection on a bundle|connection]] such that $A$ is the corresponding [[Ehresmann connection]] 1-form on the total space (\hyperlink{BoothbyWang}{BoothbyWang (1958)}).

If in this case the [[curvature]] 2-form $\omega$ on $X$ makes the base space $X$ into a [[symplectic manifold]], then $(P,A)$ is a corresponding [[prequantum circle bundle]] that provides a [[geometric prequantization]] of $(X,\omega)$.

A [[diffeomorphism]] $f : P \to P$ of a contact manifold $(P,A)$ is called a \emph{contactomorphism} (in analogy with \emph{[[symplectomorphism]]}) if it preserves the 1-form $A$ up to multiplication by a function. If $(P,A)$ is regular and hence a [[prequantum line bundle]] a contactomorphism that strictly preserves the connection 1-form is called a \emph{[[quantomorphism]]}. The [[Lie algebra]] of the [[quantomorphism group]] is that of the [[Poisson algebra]] of the base symplectic manifold $(X,\omega)$.

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

\hypertarget{darboux_theorem}{}\subsubsection*{{Darboux theorem}}\label{darboux_theorem}

There is a [[Darboux theorem]] for contact structures, stating how they are locally equivalent to a standard contact structure (e.g. \hyperlink{Arnold78}{Arnold 78, page 362})

\hypertarget{relation_to_principal_connections}{}\subsubsection*{{Relation to $U(1)$-principal connections}}\label{relation_to_principal_connections}

\begin{theorem}
\label{}\hypertarget{}{}
If $X$ is a [[closed manifold|closed]] [[smooth manifold]], $P \to X$ a smooth [[circle bundle]] ($U(1)$-[[principal bundle]]) and $\omega \in \Omega^2(X)$ a [[differential 2-form]] representing its [[Chern class]] in [[de Rham cohomology]], then there is a corresponding [[Ehresmann connection]] 1-form $A \in \Omega^1(P)$ with [[curvature]] $\omega$ and such that

\begin{enumerate}%
\item $A$ is a regular contact form on $P$;


\item the [[Reeb vector field]] of $A$ generates the given $U(1)$-[[action]] on $P$.



\end{enumerate}
Moreover, every regular contact form $A$ on a closed manifold $P$ arises this way, up to rescaling by a constant.

\end{theorem}
This is due to (\hyperlink{BoothbyWang}{Boothby-Wang 58}). The proof is recalled (and completed) in (\hyperlink{Geiges}{Geiges 08, theorem 7.2.4, 7.2.5}).

\hypertarget{History}{}\subsection*{{History}}\label{History}

\begin{quote}%
The following is taken from (\hyperlink{Lin}{Lin}).


\end{quote}
Originating in [[Hamiltonian mechanics]] and [[geometric optics]], contact geometry caught geometers' attention much earlier. In 1953, [[Shiing-shen Chern]] showed that the structure group of a contact manifold $M^{2n+1}$ can be reduced to the [[unitary group]] $U(n)$ and therefore all of its odd [[characteristic classes]] vanish. Since all the characteristic classes of a closed, orientable 3-manifold vanish, Chern in 1966 posed the questions of whether such a manifold always admits a contact structure and whether there are non-isomorphic contact structures on one manifold.

One of the milestones in the study of contact geometry is Bennequin's proof of the existence of exotic contact structures (i. e., contact structures not isomorphic to the standard one) on $\mathbb{R}^3$. Bennequin recognized that the induced singular [[foliation]] on a [[surface]] [[embedding|embedded]] in a contact 3-manifold plays a crucial role for the classification of contact structures. This role was further explored in the work of Eliashberg, who distinguished two classes of contact structures in dimension 3, overtwisted and tight, and gave a homotopy classification for overtwisted contact structures on 3-manifolds. Eliashberg also proved that on $\mathbb{R}^3$ and $S^3$, the standard contact structure is the only tight contact structure.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Legendrean submanifold]]


\item [[symplectic manifold]]


\item [[symplectic field theory]]


\item [[generalized contact geometry]]


\item [[Sasakian manifold]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Monographs and introductions include

\begin{itemize}%
\item [[Hansjörg Geiges]], \emph{Contact geometry}, in F.J.E. Dillen and L.C.A. Verstraelen, (eds.), \emph{Handbook of Differential Geometry vol. 2 North-Holland, Amsterdam (2006), pp. 315-382 (\href{http://arxiv.org/abs/math/0307242}{arXiv:math/0307242})}


\item [[Hansjörg Geiges]], \emph{An Introduction to Contact Topology}, Cambridge Studies in Advanced Mathematics 109, Cambridge University Press, Cambridge, (2008) (\href{http://www.cmi.ac.in/~aneesh/textbooks/Geiges.pdf}{pdf})


\item Xiao-Song Lin, \emph{An introduction to 3-dimensional contact geometry} (\href{http://math.ucr.edu/~xl/contact.pdf}{pdf})


\item John Etnyre, \emph{Introductory lectures on contact geometry} Proc. Sympos. Pure Math. 71 (2003), 81-107. (\href{http://people.math.gatech.edu/~etnyre/preprints/papers/contlect.pdf}{pdf})



\end{itemize}
A [[higher differential geometry]]-generalization of contact geometry in line with [[multisymplectic geometry]]/[[n-plectic geometry]] is discussed in

\begin{itemize}%
\item Luca Vitagliano, \emph{L-infinity Algebras From Multicontact Geometry} (\href{http://arxiv.org/abs/1311.2751}{arXiv.1311.2751})


\item [[Vladimir Arnol'd]], \emph{[[Mathematical methods of classical mechanics]]}, Graduate texts in Mathematics 60 (1978)



\end{itemize}
\hypertarget{characterization}{}\subsubsection*{{Characterization}}\label{characterization}

The observation that regular contact manifolds are prequantum circle bundles is due to

\begin{itemize}%
\item W. M. Boothby and H. C. Wang, \emph{On contact manifolds}, Ann. of Math. (2) 68 (1958) 721--734. (\href{http://www.jstor.org/stable/10.2307/1970165}{JSTOR})

\end{itemize}
A modern review of this is in (\hyperlink{Geiges}{Geiges, section 7.2}).

An analogous result for a weaker form of regularity is discussed in

\begin{itemize}%
\item C. Thomas, \emph{Almost regular contact manifolds}, J. Diff. Geom. 11 (1976) (\href{http://projecteuclid.org/euclid.jdg/1214433722}{Euclid})

\end{itemize}
A characterization of $S^1$-invariant contact structures on [[circle bundles]] is in

\begin{itemize}%
\item Fan Ding, [[Hansjörg Geiges]], \emph{Contact structures on principal circle bundle}, Bull. London Math. Soc, (\href{http://arxiv.org/abs/1107.4948}{arXiv:1107.4948})

\end{itemize}
For the special case of 2-dimensional base manifolds ($n = 1$) this was obtained before in

\begin{itemize}%
\item R. Lutz, \emph{Structures de contact sur les fibr\'e{}s principaux en cercles de dimension trois}, Ann. Inst. Fourier (Grenoble) 27 (1977) no. 3, 1--15.

\end{itemize}
See also

\begin{itemize}%
\item John Bland, Tom Duchamp, \emph{The Group of Contact Diffeomorphisms for Compact Contact Manifolds} (\href{http://arxiv.org/abs/1007.2036}{arXiv:1007.2036})

\end{itemize}
\hypertarget{further}{}\subsubsection*{{Further}}\label{further}

See also

\begin{itemize}%
\item [[Hansjörg Geiges]], \emph{Contact structures on 1-connected 5-manifolds}, Mathematika 38 (1991), 303-311;


\item [[Hansjörg Geiges]], \emph{Contact structures on $(n-1)$-connected $(2n+1)$-manifolds}, Pacific J. Math. 161 (1993), 129-137;


\item [[Hansjörg Geiges]], \emph{Constructions of contact manifolds}, Math. Proc. Cambridge Philos. Soc. 121 (1997), 455-464;


\item [[Hansjörg Geiges]], \emph{Normal contact structures on 3-manifolds}, T\^o{}hoku Math. J. 49 (1997), 415-422.


\item [[Hansjörg Geiges]], J. Gonzalo, \emph{Moduli of contact circles}, J. Reine Angew. Math. 551 (2002), 41-85; \emph{Contact geometry and complex surfaces}, Invent. Math. 121 (1995), 147-209.



\end{itemize}
[[!redirects contact manifolds]]

[[!redirects contactomorphism]] [[!redirects contactomorphisms]]

[[!redirects contact structure]] [[!redirects contact structures]]

[[!redirects regular contact manifold]] [[!redirects regular contact manifolds]]

[[!redirects contact geometry]] [[!redirects contact geometries]]

[[!redirects contactomorphism]] [[!redirects contactomorphisms]]

[[!redirects contact form]] [[!redirects contact forms]]



\end{document}