Contact structure

An infinitesimal structure of order one on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254701.png" /> of odd dimension that is determined by defining on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254702.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254703.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254704.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254705.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254706.png" /> is then called a contact form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254707.png" />. A contact structure exists only on an orientable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254708.png" /> and defines a unique vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c0254709.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547010.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547012.png" /> for any vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547013.png" />; the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547014.png" /> is called the dynamical system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547015.png" /> corresponding to the contact form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547016.png" />. 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.

References

[1]	C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)

Comments

More precisely, the notion defined above is a strict contact structure or exact contact structure, [a1], [a2]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547017.png" /> be a Pfaffian equation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547018.png" />, i.e. a one-dimensional subbundle of the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547019.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547020.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547021.png" />-form (i.e. a Pfaffian form) in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547023.png" /> that defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547024.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547025.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547026.png" /> is a section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547027.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547028.png" /> that is everywhere non-zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547029.png" />. Then there is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547032.png" />. This does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547033.png" />. The odd integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547034.png" /> is called the class of the Pfaffian equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547035.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547036.png" />. A contact structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547037.png" /> is now given by a Pfaffian equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547039.png" /> which is everywhere of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547040.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547041.png" /> is called a contact manifold. If there exists a Pfaffian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547043.png" /> which defines the contact structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547044.png" /> everywhere, i.e. if there exists a global everywhere non-zero section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547046.png" /> (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547047.png" /> is a trivial bundle, or, as is also said, transversally orientable), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547048.png" /> defines a strict contact structure and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547049.png" /> is a strict contact manifold with contact form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547050.png" />. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547051.png" /> is a volume form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547052.png" /> making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547053.png" /> orientable. The unique vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547054.png" /> satisfying the contraction conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547055.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547056.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547057.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547058.png" /> for all vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547059.png" />) also satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547060.png" /> (and this is equivalent). It is sometimes called the Reeb vector field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547061.png" />. By Darboux's theorem (cf. Pfaffian equation) a contact form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547062.png" /> can be written locally in the form

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547063.png" />

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547064.png" /> are local coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547065.png" />. The Reeb vector field in these coordinates is then given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025470/c02547066.png" />.

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].

References

[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)