On normal $(f,\,g,\,u,\,v,\, \lambda)$-structures on submanifolds of codimension $2$ in an even-dimensional Euclidean space

Abstract

It is well known that a hypersurface of an almost Hermitian manifold admits an almost contact metric structure naturally induced on it. The study of hypersurfaces of a Euclidean space and of a Kahlerian manifold on which the induced almost contact metric structure satisfies certain conditions has been started by one of the present authors [4, 5]. On the other hand Blair [1, 2], Goldberg [3], Ludden [1, 2], Yamaguchi [8] and the present authors [6, 9] started the study of hypersurface of an almost contact manifold and of submanifolds of codimension 2 of an almost complex manifold. These submanifolds admit, under certain conditions, what we call (/, g, u, v, λ) structure. An even-dimensional sphere of codimension 2 of an even-dimensional Euclidean space is a typical example, of a manifold which admits this kind of structure. In a previous paper [9], we have studied the (/, g, u, v, -structure and given characterizations of even-dimensional sphere. In the present paper, we study submanifolds of codimension 2 in an evendimensional Euclidean space which admit a normal (/, g, u, v, Λ)-structure. In § 1, we consider submanifolds of codimension 2 of an even-dimensional Euclidean space regarded as a flat Kahlerian manifold. In the next section, we deal with (/, g, u, v, -structure induced on a submanifold of codimension 2 of an even-dimensional Euclidean space. In § 3, we find differential equations which /, g, u, v and λ satisfy. § 4 is devoted to the study of relations between the structure equations of the submanifold and the induced (/, g, u, v, ^-structure. In § 5 we prove a series of lemmas which are valid for normal (/, g, u, v, λ}structures and in § 6 we study properties of the mean curvature vector of the submanifold with normal (/, g, u, v, -structure. In the last § 7, we study hypersurfaces of an odd-dimensional Euclidean space and determine all the hypersurfaces admitting a normal (/, g, u, v, Λ)-structure. Our main theorem appears at the end of §7.