Geodesic
Remarks on Special Symplectic Connections
Archivum mathematicum, Tome 44 (2008) no. 5, pp. 491-510 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The notion of special symplectic connections is closely related to parabolic contact geometries due to the work of M. Cahen and L. Schwachhöfer. We remind their characterization and reinterpret the result in terms of generalized Weyl connections. The aim of this paper is to provide an alternative and more explicit construction of special symplectic connections of three types from the list. This is done by pulling back an ambient linear connection from the total space of a natural scale bundle over the homogeneous model of the corresponding parabolic contact structure.
The notion of special symplectic connections is closely related to parabolic contact geometries due to the work of M. Cahen and L. Schwachhöfer. We remind their characterization and reinterpret the result in terms of generalized Weyl connections. The aim of this paper is to provide an alternative and more explicit construction of special symplectic connections of three types from the list. This is done by pulling back an ambient linear connection from the total space of a natural scale bundle over the homogeneous model of the corresponding parabolic contact structure.
Classification : 53B15, 53C15, 53D15
Keywords: special symplectic connections; parabolic contact geometries; Weyl structures and connections 
@article{ARM_2008_44_5_a10,
     author = {Pan\'ak, Martin and \v{Z}\'adn{\'\i}k, Vojt\v{e}ch},
     title = {Remarks on {Special} {Symplectic} {Connections}},
     journal = {Archivum mathematicum},
     pages = {491--510},
     year = {2008},
     volume = {44},
     number = {5},
     mrnumber = {2501580},
     zbl = {1212.53113},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a10/}
}
TY  - JOUR
AU  - Panák, Martin
AU  - Žádník, Vojtěch
TI  - Remarks on Special Symplectic Connections
JO  - Archivum mathematicum
PY  - 2008
SP  - 491
EP  - 510
VL  - 44
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a10/
LA  - en
ID  - ARM_2008_44_5_a10
ER  - 
%0 Journal Article
%A Panák, Martin
%A Žádník, Vojtěch
%T Remarks on Special Symplectic Connections
%J Archivum mathematicum
%D 2008
%P 491-510
%V 44
%N 5
%U http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a10/
%G en
%F ARM_2008_44_5_a10
Panák, Martin; Žádník, Vojtěch. Remarks on Special Symplectic Connections. Archivum mathematicum, Tome 44 (2008) no. 5, pp. 491-510. http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a10/

[1] Bryant, R.: Bochner–Kähler metrics. J. Amer. Math. Soc. 14 (2) (2001), 623–715. | DOI | MR | Zbl

[2] Cahen, M., Gutt, S., Schwachhöfer, L.: Construction of Ricci–type connections by reduction and induction. The breadth of symplectic and Poisson geometry, 2005, pp. 41–57. | MR | Zbl

[3] Cahen, M., Schwachhöfer, L.: Special symplectic connections. eprint arXiv:math/0402221v2. | MR

[4] Čap, A.: Correspondence spaces and twistor spaces for parabolic geometries. J. Reine Angew. Math. 582 (2005), 143–172. | DOI | MR | Zbl

[5] Čap, A., Gover, A. R.: Tractor calculi for parabolic geometries. Trans. Amer. Math. Soc. 354 (2002), 1511–1548. | DOI | MR | Zbl

[6] Čap, A., Slovák, J.: Parabolic Geometries. to appear in Math. Surveys Monogr., 2008.

[7] Čap, A., Slovák, J.: Weyl Structures for Parabolic Geometries. Math. Scand. 93 (2003), 53–90. | MR | Zbl

[8] Chu, B. Y.: Symplectic homogeneous space. Trans. Amer. Math. Soc. 197 (1974), 145–159. | DOI | MR

[9] Fox, D. J.: Contact projective structures. Indiana Univ. Math. J. 54 (6) (2005), 1547–1598. | MR | Zbl

[10] Jacobowitz, H.: An introduction to CR structures. Math. Surveys Monogr. 32 (1990). | Zbl

[11] Panák, M., Schwachhöfer, L.: Bochner–Kaehler metrics and connections of Ricci–type. Proceedings of the 10th Conference on Differential Geometry and its Applications, Olomouc 2007, World Scientific, 2008, pp. 339–352. | MR | Zbl

[12] Pikulin, S. V., Tevelev, E. A.: Invariant linear connections on homogeneous symplectic varieties. Transform. Groups 6 (2) (2001), 193–198. | DOI | MR | Zbl

[13] Slovák, J.: Parabolic geometries. Tech. report, IGA preprint 97/11, Univ. of Adelaide, 1997.

[14] Takeuchi, M.: Lagrangean contact structures on projective cotangent bundles. Osaka J. Math. 31 (1994), 837–860. | MR | Zbl