Recent Developments on Chen–Ricci Inequalities in Differential Geometry

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. For this respect, the first author established in 1996 a basic inequality involving the Ricci curvature and the squared mean curvature of submanifolds in real space forms, which is known today as the Chen–Ricci inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is thus to present a comprehensive survey on recent developments in this inequality done by many geometers during the last 25 years.