Ideal Slant Submanifolds in Complex Space Forms

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. Recently, B.-Y. Chen studied Lagrangian submanifolds in complex space forms which are ideal. He proved that such submanifolds are minimal. He also classified ideal Lagrangian submanifolds in complex space forms. In the present paper, we investigate ideal Kaehlerian slant submanifolds in a complex space form. We prove that such submanifolds are minimal. We also obtain obstructions to ideal slant immersions in complex hyperbolic space and complex projective space. 2000 Mathematics Subject Classification: 53C40, 53C25. 1. Chen invariants and Chen inequalities One of the most fundamental problems in submanifold theory is the immersibility of a Riemannian manifold in a Euclidean space (or, more generally, in a space form). According to the well-known embedding theorem of Nash, every Riemannian manifold can be isometrically embedded in some Euclidean space with sufficiently high codimension. Thus Riemannian manifolds could always be regarded as Riemannian submanifolds. This would then yield the opportunity to use extrinsic help. It is well-known that Riemannian invariants play the most fundamental role in Riemannian geometry. Riemannian invariants provide the intrinsic characteristics of Riemannian manifolds which affect the behavior in general of the Riemannian manifold.