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On the special class of curves on some four-dimensional
semiparallel submanifolds

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*Kaarin Riives*

**E-mail:** riives@math.ut.ee
**Abstract.** A submanifold $M^m$ in the Euclidean space $E^n$ is
said to be {\it parallel} if it satisfies the property $\bar \nabla h =
0$, where $h$ is the second fundamental form and $\bar \nabla$ denotes
the van der Waerden-Bortolotti connection. A submanifold is said to be
{\it semiparallel} if it satisfies the property $\bar R \circ h = 0$, where
$\bar R$ denotes the curvature operator of the connection. It has been
proved by Lumiste [11] that the semiparallel submanifolds are the second
order envelopes of the parallel ones. Using the Cartan moving frame method
and the exterior differential calculus the paper describes some special
classes of curves on irreducible envelopes of the reducible symmetric submanifolds
$V^2(r_1) \times S^1 (r_2) \times S^1 (r_3)$ with a Veronese component,
which is a Veronese surface in $E^5$.

**AMSclassification.** 53B25, 53C40

**Keywords.** Curves on semiparallel submanifolds, parallel submanifolds,
semiparallel submanifolds, Veronese surfaces