CR submanifolds in Kaehler and nearly Kaehler manifolds

Abstract

A review of the study of CR, submanifolds within Kaehler and nearly Kaehler manifolds, and the properties of such manifolds with respect submanifold theory in differential geometry. The study in such a fashion is relatively young, most being carried out within the past thirty years. We consider CR submanifolds as a generalization of complex and real submanifolds, with the tangent bundle decomposing into real and complex parts. We demonstrate that the CR structure has strong consequences, and is heavily dependant on the properties of the ambient manifold. The integrability of the real and complex parts is examined in various spaces, and we consider the existence of CR submanifolds with product, warped product, and foliate structure. The relationship governing the curvatures of the ambient manifold, the CR submanifold and leaves of the complex and real distributions are all considered. We consider the general cases of complex, almost complex, Kaehler and nearly Kaehler manifolds. Further detail is included for the specific manifolds of flat complex space, complex hyperbolic space, complex projective space and the 6-sphere.As an example of the applications of CR structure we include some work on the index of paths, and some topological consequences. Examples of CR submanifolds are generated for the 6-sphere, and the properties of these submanifolds are considered, including the minimality and the second fundamental form. We include details of possible further study, and suggestions for how techniques used might be fruitfully employed elsewhere.