The Normal Holonomy Group of Kahler Submanifolds |
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Authors: | Alekseevsky, Dmitri V. Di Scala, Antonio J. |
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Affiliation: | Department of Mathematics, The University of Hull Cottingham Road, Hull, HU6 7RX, UK. E-mail: D.V.Alekseevsky{at}hull.ac.uk Department of Mathematics, The University of Hull Cottingham Road, Hull, HU6 7RX, UK. E-mail: toni_discala{at}hotmail.com |
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Abstract: | We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a KählerEinsteinsubmanifold of CPn has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a KählerEinstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full KählerEinsteinsubmanifold of CPn is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary). |
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Keywords: | normal holonomy group Kä hler submanifolds Kä hler Einstein submanifolds holonomy algebras |
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