Geodesic
The Tanaka-Webster connection for almost $\mathcal{S}$-manifolds and Cartan geometry
Archivum mathematicum, Tome 40 (2004) no. 1, pp. 47-61 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 32V05, 53B05, 53C10, 53C15, 53C25
Keywords: almost $\mathcal S$-structure; Tanaka–Webster connection; Cartan connection; CR manifold 
@article{ARM_2004_40_1_a5,
     author = {Lotta, Antonio and Pastore, Anna Maria},
     title = {The {Tanaka-Webster} connection for almost $\mathcal{S}$-manifolds and {Cartan} geometry},
     journal = {Archivum mathematicum},
     pages = {47--61},
     year = {2004},
     volume = {40},
     number = {1},
     mrnumber = {2054872},
     zbl = {1114.53022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2004_40_1_a5/}
}
TY  - JOUR
AU  - Lotta, Antonio
AU  - Pastore, Anna Maria
TI  - The Tanaka-Webster connection for almost $\mathcal{S}$-manifolds and Cartan geometry
JO  - Archivum mathematicum
PY  - 2004
SP  - 47
EP  - 61
VL  - 40
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ARM_2004_40_1_a5/
LA  - en
ID  - ARM_2004_40_1_a5
ER  - 
%0 Journal Article
%A Lotta, Antonio
%A Pastore, Anna Maria
%T The Tanaka-Webster connection for almost $\mathcal{S}$-manifolds and Cartan geometry
%J Archivum mathematicum
%D 2004
%P 47-61
%V 40
%N 1
%U http://geodesic.mathdoc.fr/item/ARM_2004_40_1_a5/
%G en
%F ARM_2004_40_1_a5
Lotta, Antonio; Pastore, Anna Maria. The Tanaka-Webster connection for almost $\mathcal{S}$-manifolds and Cartan geometry. Archivum mathematicum, Tome 40 (2004) no. 1, pp. 47-61. http://geodesic.mathdoc.fr/item/ARM_2004_40_1_a5/

[1] Alekseevsky D. V., Michor P. W.: Differential Geometry of Cartan connections. Publ. Math. Debrecen 47 (1995), 349–375. | MR | Zbl

[2] Blair D. E.: Contact manifolds in Riemannian Geometry. Lecture Notes in Math. 509, 1976, Springer–Verlag. | MR | Zbl

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

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

[5] Kobayashi S., Nomizu K.: Foundations of Differential Geometry, Vol. I. Interscience, New-York, 1963. | MR

[6] Kobayashi S., Nomizu K.: Foundations of Differential Geometry, Vol. II. Interscience, New-York, 1969. | MR | Zbl

[7] Lotta A.: Cartan connections on $CR$ manifolds. PhD Thesis, University of Pisa, 2000. | Zbl

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

[9] Sharpe R. W.: Differential geometry. Cartan’s generalization of Klein’s Erlangen program. Graduate Texts in Mathematics 166, Springer-Verlag, New York, 1997. | MR | Zbl

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

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

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

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

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

[15] 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. | MR