Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Abstract

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.