An infinitesimal structure of order one on a smooth manifold of odd dimension that is determined by defining on a -form for which . The form is then called a contact form on . A contact structure exists only on an orientable and defines a unique vector field on for which and for any vector field ; the field is called the dynamical system on corresponding to the contact form . Contact structures find applications in analytic mechanics due to the fact that on any level submanifold of the Hamiltonian, defined in phase space, there arises a natural contact structure.
|||C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)|
More precisely, the notion defined above is a strict contact structure or exact contact structure, [a1], [a2]. Let be a Pfaffian equation on , i.e. a one-dimensional subbundle of the cotangent bundle . Let be a -form (i.e. a Pfaffian form) in a neighbourhood of that defines over , i.e. is a section of over that is everywhere non-zero on . Then there is an integer such that and . This does not depend on the choice of . The odd integer is called the class of the Pfaffian equation at . A contact structure on is now given by a Pfaffian equation on which is everywhere of class . The pair is called a contact manifold. If there exists a Pfaffian form on which defines the contact structure everywhere, i.e. if there exists a global everywhere non-zero section of (so that is a trivial bundle, or, as is also said, transversally orientable), then defines a strict contact structure and is a strict contact manifold with contact form . In that case is a volume form on making orientable. The unique vector field satisfying the contraction conditions (i.e. ) and (i.e. for all vector fields ) also satisfies (and this is equivalent). It is sometimes called the Reeb vector field of . By Darboux's theorem (cf. Pfaffian equation) a contact form can be written locally in the form
where are local coordinates on . The Reeb vector field in these coordinates is then given by .
For more details on the above and the role of contact structures in mechanics, cf. [a2], Chapt. V. Contact structures on circle bundles over a symplectic manifold play an important role in the quantization theory of B. Kostant and J.-M. Souriau, cf. [a3]–[a5].
|[a1]||R. Abraham, "Foundations of mechanics" , Benjamin (1967)|
|[a2]||P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French)|
|[a3]||N.E. Hurt, "Geometric quantization in action" , Reidel (1983)|
|[a4]||B. Kostant, "Quantization and representation theory. Part 1: prequantization" , Lect. in Modern Anal. and Applications , 3 , Springer (1970)|
|[a5]||J.-M. Souriau, "Structures des systèmes dynamiques" , Dunod (1969)|
Contact structure. Ãœ. Lumiste (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Contact_structure&oldid=19137