New Riemannian and Kahlerian curvature invariants and strongly minimal submanifolds

During the last decade, B.-Y. Chen's fundamental inequalities have been investigated by many authors from various viewpoints. In Section 2 we provide an alternate proof for Chen's fundamental inequality associated with classical invariants. In Subsection 2.5, we obtain an inequality for warped product manifolds as a consequence of the previous study. Section 3 is devoted to the study of applications of Chen's fundamental inequality. It is well-known that the classical obstruction to minimal isometric immersions into Euclidean space is Ric ≥ 0. In this section, we present a method to construct examples of Riemannian manifolds with Ric < 0 which don't admit any minimal isometric immersion into Euclidean spaces for any codimension. The study of the relations between curvature invariants and the topology of the manifold yields in section 4 a Myers type theorem for almost Hermitian manifolds. Chen's fundamental inequality for Kahler submanifolds in complex space forms is discussed in Section 5. We provide an extension of the inequality and provide characterizations of strongly minimal complex surfaces in the complex three dimensional space. The last section is dedicated to the study of strong minimality through examples.