Some curvature conditions of the type $2\times 4$ on the submanifolds satisfying Chen's equality

Abstract

Submanifolds of the Euclidean spaces satisfyingequality in the basic Chen's inequality have, as is known, manyinteresting properties. In this paper, we discuss on suchsubmanifolds the curvature conditions of the form $E_2\cdotF_4=0$, where $E_2$ is the Ricci or the Einstein curvatureoperator, $F_4$ is any of the standard curvature operators $R, Z,P, K, C$, and $E_2$ acts on $F_4$ as a derivation.