## Osaka Journal of Mathematics

### The normal holonomy of $CR$-submanifolds

#### Abstract

We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a $CR$-submanifold of a complex space form. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation of a Riemannian symmetric space. In case of a totally real submanifold we give two results about reduction of codimension. We describe explicitly the action of the normal holonomy in the case in which the totally real submanifold is contained in a totally real totally geodesic submanifold. In such a case we prove the compactness of the normal holonomy group.

#### Article information

Source
Osaka J. Math., Volume 54, Number 1 (2017), 17-35.

Dates
First available in Project Euclid: 3 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1488531782

Mathematical Reviews number (MathSciNet)
MR3619746

Zentralblatt MATH identifier
1371.53016

Subjects
Primary: 53B15: Other connections
Secondary: 53B20: Local Riemannian geometry 53B25: Local submanifolds [See also 53C40]

#### Citation

Di Scala, Antonio J.; Vittone, Francisco. The normal holonomy of $CR$-submanifolds. Osaka J. Math. 54 (2017), no. 1, 17--35. https://projecteuclid.org/euclid.ojm/1488531782

#### References

• D.V. Alekseevsky and A.J. Di Scala: The normal holonomy group of Kähler submanifolds Proc. London Math. Soc. (3) 89 (2004), 193 - 216.
• T. Adachi, S. Maeda and S. Udagawa: Simpleness and closedness of circles in compact Hermitian symmetric spaces Tsukuba J. Math. 24 (2000), 1–13.
• W. Ballman: Lectures on Kähler Manifolds, European Mathematical Society, 2006.
• A. Bejancu: $CR$-submanifolds of a Kaehler manifold. I, II. Proc. Amer. Math. Soc. 89 (1978) 135-142; Trans. Amer. Math. Soc. 250 (1979) 333-345.
• A. Bejancu, M. Kon and K. Yano: $CR$-submanifolds of a complex space form J. Differential Geometry 16 (1981) 137-145.
• J. Berndt, S. Console and C. Olmos: Submanifolds and holonomy, Chapman & Hall / CRC, Research Notes in Mathematics 434, 2003.
• B.-U-. Chen: $CR$-Submanifolds of a Kähler manifold. I J. Diff. Geom. 16, (1981), 305–322.
• B.-Y-. Chen, C.-S. Houh and H.-S. Lue: Totally real submanifolds J. Diff. Geom. 12, (1977), 251–257.
• S. Console, A.J. Di Scala and C. Olmos: A Berger type normal holonomy theorem for complex submanifolds, Math. Ann. 351 (2011), 187–214.
• A.J. Di Scala: Reducibility of complex submanifolds of the complex Euclidean space Math. Z. 235, (2000), 251–257.
• A.J. Scala and F. Vittone: Codimension reduction in symmetric spaces J. Geom. and Phys. 79 (2014), 29–33
• A.J. Di Scala and F. Vittone: Mok's characteristic varieties and the normal holonomy group http://arxiv.org/abs/1503.01941.
• M. Djoric and M. Okumura: $CR$-submanifolds of complex projective space Developments in Mathematics 19, Springer 2009.
• Y. Godoy and M. Salvai: The magnetic flow on the manifold of oriented geodesics of a three dimensional space form Osaka J. Math. 50 (2013), 749–763.
• S. Helgason: Differential Geometry and Symmetric Spaces Academic Press, 1962.
• M. Kon and K. Yano: Generic Submanifolds Ann. di Matematica Pura ed Applicata 123 (1980), 59 – 92.
• P. Libermann and C. Marle: Symplectic Geometry and Analytical Mechanics D. Reiden Publishing Company, 1987.
• S. Maeda and H. Tanabe: Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves, Math. Z. 252 (2006), 787–795.
• H. Naitoh: Parallel submanifolds of complex space forms I Nagoya Math. J. 90 (1983), 85–117.
• H. Naitoh and M. Takeuchi: Totally real submanifolds and symmetric bounded domains Osaka J. Math. 19 (1982), 717–731.
• C. Olmos: The normal holonomy group Proc. Am. Math. Soc. 110 (1990), 813–818.
• C. Olmos: Isoparametric submanifolds and their homogeneous structures J. Differ. Geom. 38 (1993), 225 – 234.
• C. Olmos: A geometric proof of the Berger holonomy theorem Ann. of Math. 161 (2005), 579–588.
• B. O'Neill: Semi-Riemannian Geometry Academic Press, 1983.
• J. Simons: On the transitivity of holonomy systems Ann. of Math. 76 (1962), 213 - 234.