Abstract 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.
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