[en] Let M be an odd-dimensional Euclidean space endowed with a contact 1-form \alpha. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by \alpha. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form \alpha defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko.
Disciplines :
Mathematics
Author, co-author :
Frégier, Yaël; University of Luxembourg > Institute of Mathematics
Mathonet, Pierre ; Université de Liège - ULiège > Département de mathématique > Département de mathématique
Poncin, Norbert; Univserity of Luxembourg > Institute of Mathematics
Language :
English
Title :
Decomposition of symmetric tensor fields in the presence of a flat contact projective structure
V. I. ARNOLD. Mathematical methods of classical mechanics. Springer-Verlag, New York, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein, Graduate Texts in Mathematics, 60.
DAVID E. BLAIR. Contact manifolds in Riemannian geometry. Springer-Verlag, Berlin, 1976. Lecture Notes in Mathematics, Vol. 509.
F. BONIVER, S. HANSOUL, P. MATHONET, and N. PONCIN. Equivariant symbol calculus for differential operators acting on forms. Lett. Math. Phys., 62(3):219-232, 2002.
F. BONIVER and P. MATHONET. IFFT-equivariant quantizations. J. Geom. Phys., 56(4):712-730, 2006.
C. DUVAL, P. LECOMTE, and V. OVSIENKO. Conformally equivariant quantization: existence and uniqueness. Ann. Inst. Fourier (Grenoble), 49(6):1999-2029, 1999.
C. DUVAL and V. OVSIENKO. Projectively equivariant quantization and symbol calculus: noncommutative hypergeometric functions. Lett. Math. Phys., 57(1):61-67, 2001.
C. DUVAL and V. OVSIENKO. Space of second-order linear differential operators as a module over the Lie algebra of vector fields. Adv. Math., 132(2):316-333, 1997.
H. GARGOUBI. Sur la géométrie de l'espace des opérateurs différentiels linéaires sur R. Bull. Soc. Roy. Sci. Liège, 69(1):21-47, 2000.
H. GARGOUBI and V. YU. OVSIENKO. Space of linear differential operators on the real line as a module over the Lie algebra of vector fields. Internat. Math. Res. Notices, (5):235-251, 1996.
KH. GARGUBI and V. OVSIENKO. Modules of differential operators on the line. Punktaional. Anal, i Prilozhen., 35(1):16-22, 96, 2001.
P. B. A. LECOMTE, P. MATHONET, and E. TOUSSET. Comparison of some modules of the Lie algebra of vector fields. Indag. Math. (N.S.), 7(4):461-471, 1996.
P. B. A. LECOMTE and V. YU. OVSIENKO. Projectively equivariant symbol calculus. Lett. Math. Phys., 49(3):173-196, 1999.
PIERRE B. A. LECOMTE. On the projective classification of the modules of differential operators on ℝm. In Noncommutative differential geometry and its applications to physics (Shonan, 1999), volume 23 of Math. Phys. Stud., pages 123-129. Kluwer Acad. Publ., Dordrecht, 2001.
P. MATHONET. Intertwining operators between some spaces of differential operators on a manifold. Comm. Algebra, 27(2):755-776, 1999.
P. MATHONET. Invariant bidifferential operators on tensor densities over a contact manifold. Lett. Math. Phys., 48(3):251-261, 1999.
P. MATHONET and F. RADOUX. Natural and projectively equivariant quantizations by means of Cartan connections. Lett. Math. Phys., 72(3):183-196, 2005.
DUSA MCDUFF and DIETMAR SALAMON. Introduction to symplectic topology. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1998.
V. OVSIENKO. Vector fields in the presence of a contact structure. Enseign. Math., 52:215-229, 2006.
V. OVSIENKO and S. TABACHNIKOV. Projective differential geometry old and new, volume 165 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2005. From the Schwarzian derivative to the cohomology of diffeomorphism groups.
M. RUMIN. Formes différentielles sur les variétés de contact. J. Differential Geom., 39(2): 281-330, 1994.
H. WEYL. The classical groups. Their invariants and representations, Fifteenth printing, Princeton Paperbacks, Princeton Landmarks in Mathematics Princeton University Press, Princeton, NJ, 1997.
E. J. WILCZYNSKI. Projective differential geometry of curves and ruled surfaces. Chelsea Publishing Co., New York, 1962.