Normal holonomy of CR submanifolds

TL;DR

This paper investigates the normal holonomy groups of CR submanifolds in complex space forms, providing classifications, explicit descriptions, and results on reduction of codimension, with particular focus on totally real submanifolds.

Contribution

It completes the local classification of normal holonomies for complex submanifolds and describes the action of the holonomy group in specific cases, including compactness results.

Findings

Normal holonomy of coisotropic submanifolds acts as a Riemannian symmetric space holonomy.

Explicit description of normal holonomy action when submanifold is in a totally real totally geodesic submanifold.

Proved the compactness of the normal holonomy group in certain cases.

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 complete the local classification of normal holonomies for complex submanifolds. 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.