A New Approach of Structures and Connections on Submanifolds

To study the geometry of manifold, sometime it becomes more convenient to first embed it into a manifold whose geometry is known and then look for the geometry which is induced on it. The submanifolds of an almost Hermitian form an interesting geometric study as its almost complex structure transforms a vector to a vector perpendicular to it, which naturally gives rise to two types of submanifols, viz, invariant and anti-invariant submanifolds have been studied extensively. Invariant and anti-invariant submanifolds of Riemannian manifolds with different differential structures were studied by many geometers. In the notion of Cauchy-Riemann(CR-) submanifolds was introduced by A. Bejancu which generalizes both invariant and anti-invariant submanifolds in the sense that these submanifolds become the particular cases of CR-submanifolds. The differential geometry of CR-submanifolds has shown an increasing develepment and many differential geometers have contributed results on this topic.