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## Banach Center Publications

2000 | 51 | 1 | 15-23
Tytuł artykułu

### Schwarzian derivative related to modules of differential operators on a locally projective manifold

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EN
Abstrakty
EN
We introduce a 1-cocycle on the group of diffeomorphisms Diff(M) of a smooth manifold M endowed with a projective connection. This cocycle represents a nontrivial cohomology class of Diff(M) related to the Diff(M)-modules of second order linear differential operators on M. In the one-dimensional case, this cocycle coincides with the Schwarzian derivative, while, in the multi-dimensional case, it represents its natural and new generalization. This work is a continuation of [3] where the same problems have been treated in the one-dimensional case.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
15-23
Opis fizyczny
Daty
wydano
2000
Twórcy
autor
• Centre de Physique Théorique, CPT-CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France
autor
• Centre de Physique Théorique, CPT-CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France
Bibliografia
• [1] L. V. Ahlfors, Cross-ratios and Schwarzian derivatives in $ℝ^n$, in: Complex Analysis, Birkhäuser, Boston, 1989.
• [2] R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. 23 (1977), 209-220.
• [3] S. Bouarroudj and V. Ovsienko, Three cocycles on $Diff(S^1)$ generalizing the Schwarzian derivative, Internat. Math. Res. Notices 1998, No.1, 25-39.
• [4] C. Duval and V. Ovsienko, Space of second order linear differential operators as a module over the Lie algebra of vector fields, Adv. in Math. 132 (1997), 316-333.
• [5] C. Duval and V. Ovsienko, Conformally equivariant quantization, Preprint CPT, 1998.
• [6] A. A. Kirillov, Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments, Lect. Notes in Math., 970 Springer-Verlag 1982, 101-123.
• [7] S. Kobayashi and C. Horst, Topics in complex differential geometry, in: Complex Differential Geometry, Birkhäuser Verlag, 1983, 4-66.
• [8] S. Kobayashi and T. Nagano, On projective connections, J. of Math. and Mech. 13 (1964), 215-235.
• [9] 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 (1996), 461-471.
• [10] P. B. A. Lecomte and V. Ovsienko, Projectively invariant symbol calculus, Lett. Math. Phys., to appear.
• [11] R. Molzon and K. P. Mortensen, The Schwarzian derivative for maps between manifolds with complex projective connections, Trans. of the AMS 348 (1996), 3015-3036.
• [12] B. Osgood and D. Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Math. J. 67 (1992), 57-99.
• [13] V. Ovsienko, Lagrange Schwarzian derivative and symplectic Sturm theory. Ann. Fac. Sci. Toulouse Math. 6 (1993), no. 1, 73-96.
• [14] V. Retakh and V. Shander, The Schwarz derivative for noncommutative differential algebras. Unconventional Lie algebras, Adv. Soviet Math. 17 (1993), 139-154.
• [15] S. Tabachnikov, Projective connections, group Vey cocycle, and deformation quantization. Internat. Math. Res. Notices 1996, No. 14, 705-722.
• [16] E. J. Wilczynski, Projective differential geometry of curves and ruled surfaces, Leipzig - Teubner - 1906.
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Bibliografia
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