Submanifolds of the Euclidean spaces satisfying equality in the basic Chen's inequality have, as is known, many interesting properties. In this paper, we discuss the curvature conditions of the form $E\cdot S=0$ on such submanifolds, where $E$ is any of the standard 4-covariant curvature operators, $S$ is the Ricci curvature operator, and $E$ acts on $S$ as a derivation.