A Berger Type Normal Holonomy Theorem for Complex Submanifolds

We prove Berger type theorems for the normal holonomy Φ (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M both of Cn and of the complex projective space CP n. Namely, (1) for Cn, if M is irreducible, then Φ acts transitively on the unit sphere of the normal space; (2) for CP n, if Φ 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. 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 CP n case) and basic facts of complex submanifolds. Berger’s Holonomy Theorem [3] is probably the most important general (local) result of Riemannian geometry: the restricted holonomy group of an irreducible Riemannian manifold acts transitively on the unit sphere of the tangent space except in the case that the manifold is a symmetric space of rank bigger or equal to two. In submanifold geometry a prominent rôle is played by the holonomy group of the natural connection of the normal bundle, the so-called normal holonomy group. For submanifolds of R or more generally of spaces of constant curvature, a fundamental result is the Normal Holonomy Theorem [17]. It asserts roughly that the non-trivial component of the action of the normal holonomy group on any normal space is the isotropy representation of a Riemannian symmetric space (called s-representation for short). The Normal Holonomy Theorem is a very important tool for the study of submanifold geometry, especially in the context of submanifolds with “simple extrinsic geometric invariants”, like isoparametric and homogeneous submanifolds (see [4] for an introduction to this subject). In particular, in this extrinsic setting, some distinguished class of homogeneous submanifolds, the orbits of s-representations, play a similar rôle as symmetric spaces in intrinsic Riemannian geometry. Typically, requiring that a submanifold has “simple extrinsic geometric invariants” (e.g. “enough” parallel normal fields with respect to which the shape operator has constant eigenvalues) implies that the submanifold belongs to this class. Therefore, these methods based on the study of normal holonomy allowed to