**
Beiträge zur Algebra und Geometrie **

Contributions to Algebra and Geometry

38(1), 125 - 134 (1997)

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On Generalized Chen and k-minimal Immersions

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Sheila Carter, Ugur Dursun

S. Carter, School of Mathematics, The University of Leeds, Leeds LS2 9JT U.K., E-mail : s.carter@leeds.ac.uk U. Dursun, After July 1997 the address will be: Istanbul Teknik Universitesi, Fen-Edebiyat Fakultesi Matematik Bolumu, Maslak, Istanbul, TURKEY

**Abstract:** The notion of an $\cal A$-submanifold (Chen submanifold) has been extended to an ${\cal A}_{k}$-submanifold by B. Rouxel [Kodai Math. J. {\bf 4} (1981), 181-188] and S.J. Li and C.S. Houh [Journal of Geometry, {\bf 48} (1993), 144-156]. \noindent In this paper we investigate the relation between their definitions. We also introduce the idea of a k-minimal submanifold for which an open subset of a k-plane subbundle of the submanifold is minimal. A k-minimal submanifold satisfies a stronger condition than that of an ${\cal A}_{k}$-submanifold. By using partial tubes we give methods for constructing k-minimal submanifolds, and hence ${\cal A}_{k}$-submanifolds, for different values of k. This generalise a construction from "Partial tubes and Chen submanifolds" by the authors [ to appear in Journal of Geometry].

**Keywords:** Chen submanifold; minimal submanifold; flat normal subbundle; mean curvature vector; parallel normal vector field; partial tube; umbilical normal vector field

**Classification (MSC91):** 53C42

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