Some inequalities for submanifolds in a Riemannian manifold of nearly quasi-constant curvature

Abstract

In this paper, we derive a DDVV-type inequality for submianifolds in a Riemannian manifold of nearly quasi-constant curvature. Moreover, two inequalities involving the Casorati curvature and the scalar curvature are obtained.

Optimal casorati inequalities on bi-slant submanifolds of generalized sasakian space forms

“…. , 2m + 1, let us take the quadratic form φ r : (18) and the constrained extremum problem min φ r subject to the component of trace H ,…”

“…Y. Chen [5] introduced the new types of Riemannian invariants, known in the literature as Chen invariants and obtained general optimal inequalities consisting of the new intrinsic invariants and the main extrinsic invariants for any Riemannian submanifolds. It was the starting point of the theory of Chen invariants, which are one of the most interesting research topics in differential geometry [11,12,13,14,16,17,18,19,24]. Instead of concentrating on the sectional curvature with the extrinsic squared mean curvature, the Casorati curvature of a submanifold in a Riemannian manifold was considered as an extrinsic invariant defined as the normalized square of the length of the second fundamental form.…”