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

Banach Center Publications (2000)

- Volume: 51, Issue: 1, page 15-23
- ISSN: 0137-6934

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topBouarroudj, S., and Ovsienko, V.. "Schwarzian derivative related to modules of differential operators on a locally projective manifold." Banach Center Publications 51.1 (2000): 15-23. <http://eudml.org/doc/209027>.

@article{Bouarroudj2000,

abstract = {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.},

author = {Bouarroudj, S., Ovsienko, V.},

journal = {Banach Center Publications},

keywords = {cohomology of group of diffeomorphisms; Schwarzian derivative},

language = {eng},

number = {1},

pages = {15-23},

title = {Schwarzian derivative related to modules of differential operators on a locally projective manifold},

url = {http://eudml.org/doc/209027},

volume = {51},

year = {2000},

}

TY - JOUR

AU - Bouarroudj, S.

AU - Ovsienko, V.

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

JO - Banach Center Publications

PY - 2000

VL - 51

IS - 1

SP - 15

EP - 23

AB - 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.

LA - eng

KW - cohomology of group of diffeomorphisms; Schwarzian derivative

UR - http://eudml.org/doc/209027

ER -

## References

top- [1] L. V. Ahlfors, Cross-ratios and Schwarzian derivatives in ${\mathbb{R}}^{n}$, in: Complex Analysis, Birkhäuser, Boston, 1989.
- [2] R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. 23 (1977), 209-220. Zbl0367.57004
- [3] S. Bouarroudj and V. Ovsienko, Three cocycles on $Diff\left({S}^{1}\right)$ generalizing the Schwarzian derivative, Internat. Math. Res. Notices 1998, No.1, 25-39. Zbl0919.57026
- [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. Zbl0902.17011
- [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. Zbl0498.22017
- [7] S. Kobayashi and C. Horst, Topics in complex differential geometry, in: Complex Differential Geometry, Birkhäuser Verlag, 1983, 4-66. Zbl0506.53029
- [8] S. Kobayashi and T. Nagano, On projective connections, J. of Math. and Mech. 13 (1964), 215-235. Zbl0117.39101
- [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. Zbl0892.58002
- [10] P. B. A. Lecomte and V. Ovsienko, Projectively invariant symbol calculus, Lett. Math. Phys., to appear. Zbl0989.17015
- [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. Zbl0872.53013
- [12] B. Osgood and D. Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Math. J. 67 (1992), 57-99. Zbl0766.53034
- [13] V. Ovsienko, Lagrange Schwarzian derivative and symplectic Sturm theory. Ann. Fac. Sci. Toulouse Math. 6 (1993), no. 1, 73-96. Zbl0780.34004
- [14] V. Retakh and V. Shander, The Schwarz derivative for noncommutative differential algebras. Unconventional Lie algebras, Adv. Soviet Math. 17 (1993), 139-154. Zbl0841.17003
- [15] S. Tabachnikov, Projective connections, group Vey cocycle, and deformation quantization. Internat. Math. Res. Notices 1996, No. 14, 705-722. Zbl0872.17020
- [16] E. J. Wilczynski, Projective differential geometry of curves and ruled surfaces, Leipzig - Teubner - 1906. Zbl37.0620.02

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