Some Geometric Inequalities For Submanifolds In A Riemannian Manifold Of Quasi-constant Curvature

In the theory of submanifolds, the following problem is fundamen-tal:to establish simple relationships between the intrinsic invariants and the extrinsic invariants of the submanifolds [6]. In [1], B.Y. Chen intro-duced a new type of Riemannian invariants, known as δ-invariants, and established inequalities for submanifolds in a real space form in terms of the sectional curvature (intrinsic), the scalar curvature (intrinsic) and the squared mean curvature (extrinsic). Besides, many papers studied inequalities relating the Casorati curvature (extrinsic) and the scalar curvature (intrinsic) for submanifolds in real space forms, complex s-pace forms and quaternionic space forms [11,14,15]. As we know, a Riemannian manifold of quasi-constant curvature can be regarded as a generalization of real space forms. The aim of this paper is to establish geometric inequalities between the intrinsic curvatures and the extrin-sic curvatures for submanifolds in this kind of Riemannian manifold, which generalize some known results in real space forms. Specifically speaking:In Chapter 3, we not only provide an another proof of the in-equality concerning δ(2) for submanifolds in real space form, but also establish an inequality concerning δ(n1,…, nk) for submaifolds in a Riemannian manifold of quasi-constant curvature.In Chapter 4, we establish an inequality between the mean curva-ture and the Ricci curvature for submanifolds in a Riemannian manifold of quasi-constant curvature.In Chapter 5, by using Oprea’s optimization method, we first proof a result obtained by Decu et al. Then we establish an inequality con-cerning the normalized δ-Casorati curvature δC(n-1) for submani-folds in a Riemannian manifold of quasi-constant curvature. Besides, we characterize several kinds of submanifolds in real space form when the equality case of the inequality holds.In Chapter 6, we establish an inequality between the warping func-tion and the mean curvature for warped product submanifolds in a Riemannian manifold of quasi-constant curvature.