The Tanaka-Webster connection for almost $\mathcal{S}$-manifolds and Cartan geometry
Lotta, Antonio ; Pastore, Anna Maria
Archivum Mathematicum, Tome 040 (2004), p. 47-61 / Harvested from Czech Digital Mathematics Library
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We prove that a CR-integrable almost $\mathcal S$-manifold admits a canonical linear connection, which is a natural generalization of the Tanaka–Webster connection of a pseudo-hermitian structure on a strongly pseudoconvex CR manifold of hypersurface type. Hence a CR-integrable almost $\mathcal S$-structure on a manifold is canonically interpreted as a reductive Cartan geometry, which is torsion free if and only if the almost $\mathcal S$-structure is normal. Contrary to the CR-codimension one case, we exhibit examples of non normal almost $\mathcal S$-manifolds with higher CR-codimension, whose Tanaka–Webster curvature vanishes.

Publié le : 2004-01-01
Classification:  32V05,  53B05,  53C10,  53C15,  53C25 
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     author = {Antonio Lotta and Anna Maria Pastore},
     title = {The Tanaka-Webster connection for almost $\mathcal{S}$-manifolds and Cartan geometry},
     journal = {Archivum Mathematicum},
     volume = {040},
     year = {2004},
     pages = {47-61},
     zbl = {1114.53022},
     mrnumber = {2054872},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107890}
}

Lotta, Antonio; Pastore, Anna Maria. The Tanaka-Webster connection for almost $\mathcal{S}$-manifolds and Cartan geometry. Archivum Mathematicum, Tome 040 (2004) pp. 47-61. http://gdmltest.u-ga.fr/item/107890/

Alekseevsky D. V.; Michor P. W. Differential Geometry of Cartan connections, Publ. Math. Debrecen 47 (1995), 349–375. (1995) | MR 1362298 | Zbl 0857.53011

Blair D. E. Contact manifolds in Riemannian Geometry, Lecture Notes in Math. 509, 1976, Springer–Verlag. (1976) | MR 0467588 | Zbl 0319.53026

Blair D. E. Geometry of manifolds with structural group ${\mathcal{U}}(n)\times {\mathcal{O}}(s)$, J. Differential Geom. 4 (1970), 155–167. (1970) | MR 0267501

Duggal K. L.; Ianus S.; Pastore A. M. Maps interchanging $f$-structures and their harmonicity, Acta Appl. Math. 67 (2001), 91–115. | MR 1847885 | Zbl 1030.53048

Kobayashi S.; Nomizu K. Foundations of Differential Geometry, Vol. I, Interscience, New-York, 1963. (1963) | MR 0152974

Kobayashi S.; Nomizu K. Foundations of Differential Geometry, Vol. II, Interscience, New-York, 1969. (1969) | MR 0238225 | Zbl 0175.48504

Lotta A. Cartan connections on $CR$ manifolds, PhD Thesis, University of Pisa, 2000. | Zbl 1053.32507

Mizner R. I. Almost CR structures, $f$-structures, almost product structures and associated connections, Rocky Mountain J. Math. 23, no. 4 (1993), 1337–1359. (1993) | MR 1256452 | Zbl 0806.53030

Sharpe R. W. Differential geometry. Cartan’s generalization of Klein’s Erlangen program, Graduate Texts in Mathematics 166, Springer-Verlag, New York, 1997. (1997) | MR 1453120 | Zbl 0876.53001

Tanaka N. On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. 20 (1976), 131–190. (1976) | MR 0589931 | Zbl 0346.32010

Tanno S. Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., Vol. 314 (1989), 349–379. (1989) | MR 1000553 | Zbl 0677.53043

Tanno S. The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J. 21 (1969), 21–38. (1969) | MR 0242094 | Zbl 0188.26705

Urakawa H. Yang-Mills connections over compact strongly pseudoconvex CR manifolds, Math. Z. 216 (1994), 541–573. (1994) | MR 1288045 | Zbl 0815.32008

Webster S. M. Pseudo-hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41. (1978) | MR 0520599 | Zbl 0379.53016

Yano K. On a structure defined by a tensor field $f$ of type $(1,1)$ satisfying $f^3+f=0$, Tensor (N.S.) 14 (1963), 99–109. (1963) | MR 0159296