Recent Developments on the First Chen Inequality in Differential Geometry

Bang-Yen Chen () and Gabriel-Eduard Vîlcu
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Bang-Yen Chen: Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
Gabriel-Eduard Vîlcu: Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania

Mathematics, 2023, vol. 11, issue 19, 1-50

Abstract: 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. In this respect, the first author established, in 1993, a basic inequality involving the first δ -invariant, δ ( 2 ) , and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen’s first 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 on this inequality performed by many geometers during the last three decades.

Keywords: first Chen inequality; ? -invariant; Chen invariant; ideal immersion; statistical manifold; Riemannian submersion; submanifold; Riemannian map; contact manifold (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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