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

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