Abstract
We prove a Berger type theorem for the normal holonomy \({\Phi^\perp}\) (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \({\mathbb{C} P^n}\). Namely, if \({\Phi^\perp}\) does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \({\mathbb{C}^n}\) the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \({\mathbb{C} P^n}\) case) and basic facts of complex submanifolds.
Similar content being viewed by others
References
Abe K., Magid M.A.: Indefinite Kähler submanifolds with positive index of relative nullity. Ann. Glob. Anal. Geom. 6(3), 231–258 (1988)
Alekseevsky D., Di Scala A.J.: The normal holonomy group of Kähler submanifolds. Proc. Lond. Math. Soc. (3) 89(1), 193–216 (2004)
Berger M.: Sur les groupes d’holonomie homogène des variétés à conexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83, 279–330 (1953)
Berndt, J., Console, S., Olmos, C.: Submanifolds and holonomy. In: Research Notes in Mathematics, vol. 434. Chapman & Hall/CRC, London (2003)
Borel, A., Weil, A.: Représentations linéaires et espaces homogénes kähleriens des groupes de Lie compacts. Séminaire Bourbaki, exposé no. 100 par J.-P. Serre (mai 1954)
Console S., Di Scala A.J.: Normal holonomy and submanifolds with parallel second fundamental form. Math. Z. 261(1), 1–11 (2009)
Console S., Di Scala A.J., Olmos C.: Holonomy and submanifold geometry. Enseign. Math. 48, 23–50 (2002)
Console S., Olmos C.: Submanifolds of higher rank. Quart. J. Math. Oxford (2) 48, 309–321 (1997)
Dadok J.: Polar coordinates induced by action of compact Lie groups. Trans. Am. Math. Soc. 288, 125–137 (1985)
Di Scala A.J.: Reducibility of complex submanifolds of the complex Euclidean space. Math. Z. 235, 251–257 (2000)
Dajczer M., Gromoll D.: Gauss parametrization and rigidity aspects of submanifolds. J. Differ. Geom. 22, 1–12 (1985)
Erbacher J.: Reduction of the codimension of an isometric immersion. J. Differ. Geom. 5, 333–340 (1971)
Heintze E., Olmos C., Thorbergsson G.: Submanifolds with constant principal curvatures. Int. J. Math. 2, 167–175 (1991)
Heintze E., Olmos C.: Normal holonomy groups and s-representations. Indiana Univ. Math. J. 41(3), 869–874 (1992)
Moore J.D.: Isometric immersion of Riemannian products. J. Differ. Geom. 5, 159–168 (1971)
Nakagawa H., Takagi R.: On locally symmetric Kaehler submanifolds in a complex projective space. J. Math. Soc. Jpn. 28, 638–667 (1976)
Olmos C.: The normal holonomy group. Proc. Am. Math. Soc. 110, 813–818 (1990)
Olmos C.: Isoparametric submanifolds and their homogeneous structures. J. Differ. Geom. 38, 225–234 (1993)
Olmos C.: Homogeneous submanifolds of higher rank and parallel mean curvature. J. Differ. Geom. 39, 605–627 (1994)
Olmos C.: Orbits of rank one and parallel mean curvature. Trans. Am. Math. Soc. 347, 2927–2939 (1995)
Olmos C.: A geometric proof of the Berger holonomy theorem. Ann. Math. (2) 161(1), 579–588 (2005)
Olmos C., Will A.: Normal holonomy in Lorentzian space and submanifold geometry. Indiana Univ. Math. J. 50(4), 1777–1788 (2001)
O’Neill B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)
Palais, R.S., Terng, C.L.: Critical point theory and submanifold geometry. In: Lecture Notes in Mathematics, vol. 1353. Springer-Verlag, Berlin (1988)
Simons J.: On the transitivity of holonomy systems. Ann. Math. 76, 213–234 (1962)
Terng C.L.: Isoparametric submanifolds and their Coxeter groups. J. Differ. Geom. 21, 79–107 (1985)
Terng C.L.: Submanifold with flat normal bundle. Math. Ann. 277, 95–111 (1987)
Thorbergsson G.: Isoparametric foliations and their buildings. Ann. Math. 133, 429–446 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Jorge Vargas on the occasion of his 60th birthday.
S. Console and A. J. Di Scala were partially supported by GNSAGA of INdAM, PRIN 07 “Differential Geometry and Global Analysis” and MIUR of Italy. C. Olmos was supported by FaMAF-Universidad Nacional de Córdoba, CIEM-Conicet, Secyt-UNC y ANCyT.
Rights and permissions
About this article
Cite this article
Console, S., Di Scala, A.J. & Olmos, C. A Berger type normal holonomy theorem for complex submanifolds. Math. Ann. 351, 187–214 (2011). https://doi.org/10.1007/s00208-010-0597-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-010-0597-0