Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-11T01:01:17.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Contact Schwarzian Derivatives

Published online by Cambridge University Press:  11 January 2016

Daniel J. F. Fox
Daniel J. F. Fox*
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry St. Atlanta, GA 30332-0160, U.S.A.fox@math.gatech.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

H. Sato introduced a Schwarzian derivative of a contactomorphism of ℝ3 and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a contact projective structure is defined to be a cocycle of the contactomorphism group taking values in the space of sections of a certain vector bundle associated to the contact structure, and measuring the extent to which a contactomorphism fails to be an automorphism of the contact projective structure. For the flat model contact projective structure, this gives a contact Schwarzian derivative associating to a contactomorphism of ℝ2n−1 a tensor which vanishes if and only if the given contactomorphism is an element of the linear symplectic group acting by linear fractional transformation.

Keywords

53B1553D10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 179 , 2005 , pp. 163 - 187
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[1] Baston, R. J. and Eastwood, M. G., Invariant operators, Twistors in mathematics and physics, London Math. Soc. Lecture Note Ser., vol. 156, Cambridge Univ. Press, Cambridge (1990), pp. 129–163.Google Scholar
[2] Bouarroudj, S. and Ovsienko, V., Schwarzian derivative related to modules of differential operators on a locally projective manifold, Poisson geometry (Warsaw, 1998), Banach Center Publ., vol. 51, Polish Acad. Sci., Warsaw (2000), pp. 15–23.Google Scholar
[3] Calderbank, D. M. J. and Diemer, T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math., 537 (2001), 67–103.Google Scholar
[4] Čap, A., Automorphism groups of parabolic geometries, ESI preprint 1478, available at www.esi.ac.at/preprints/ESI-Preprints.html (2004), 1–7.Google Scholar
[5] Chern, S.-S., The geometry of the differential equation y‴ = F(x, y, y′, y″), Sci. Rep. Nat. Tsing Hua Univ. (A), 4 (1940), 97–111.Google Scholar
[6] Duval, C. and Ovsienko, V., Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions, Lett. Math. Phys., 57 (2001), no. 1, 61–67.CrossRefGoogle Scholar
[7] Eastwood, M. G., Representations via over determined systems, Geometric analysis of PDE and several complex variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI (2005), pp. 201–210.Google Scholar
[8] Fox, D. J. F., Contact projective structures, to appear, Indiana Univ. Math. J. Google Scholar
[9] Gunning, R. C., Special coordinate coverings of Riemann surfaces, Math. Ann., 170 (1967), 67–86.Google Scholar
[10] Gunning, R. C., On projective covariant differentiation, E. B. Christoffel (Aachen/Monschau, 1979), Birkhäuser, Basel (1981), pp. 584–591.Google Scholar
[11] Lecomte, P. B. A. and Ovsienko, V., Projectively equivariant symbol calculus, Lett. Math. Phys., 49 (1999), no. 3, 173–196.Google Scholar
[12] Lecomte, P. B. A. and Ovsienko, V., Cohomology of the vector fields Lie algebra and modules of differential operators on a smooth manifold, Compositio Math., 124 (2000), no. 1, 95–110.Google Scholar
[13] Ovsienko, V. and Tabachnikov, S., Projective differential geometry old and new: from the Schwarzian derivative to cohomology of diffeomorphism groups, Cambridge University Press, Cambridge, 2004.Google Scholar
[14] Ozawa, T. and Sato, H., Contact transformations and their Schwarzian derivatives, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie (Kyoto/Nara, 1999), Adv. Stud. Pure Math., vol. 37, Math. Soc. Japan, Tokyo (2002), pp. 337–366.Google Scholar
[15] Sasaki, T. and Yoshida, M., Schwarzian derivatives and uniformization, The Kowalevski property (Leeds, 2000), CRM Proc. Lecture Notes, vol. 32, Amer. Math. Soc., Providence, RI (2002), pp. 271–286.Google Scholar
[16] Sato, H., Schwarzian derivatives of contact diffeomorphisms, Lobachevskii J. Math., 4 (1999), 89–98 (electronic), Towards 100 years after Sophus Lie (Kazan, 1998).Google Scholar
[17] Thomas, T. Y., The differential invariants of generalized spaces, Chelsea Publishing Company, New York, 1934.Google Scholar
[18] Weyl, H., The Classical Groups. Their invariants and representations, Princeton University Press, Princeton, N.J., 1939.Google Scholar