%0 Journal Article
%T -Submanifolds of £¿ -Dimension in a Quaternionic Projective Space under Some Curvature Conditions
%A Hyang Sook Kim
%A Jin Suk Pak
%J International Journal of Mathematics and Mathematical Sciences
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/193697
%X The purpose of this paper is to study n-dimensional -submanifolds of -dimension in a quaternionic projective space and especially to determine such submanifolds under some curvature conditions. 1. Introduction Let be a connected real -dimensional submanifold of real codimension of a quaternionic K£¿hler manifold with quaternionic K£¿hler structure . If there exists an -dimensional normal distribution of the normal bundle such that at each point in , then is called a QR-submanifold of -dimension, where denotes the complementary orthogonal distribution to in (cf. [1¨C3]). Real hypersurfaces, which are typical examples of -submanifold with , have been investigated by many authors (cf. [2¨C9]) in connection with the shape operator and the induced almost contact -structure (for definition, see [10¨C13]). In their paper [2, 3], Kwon and Pak had studied -submanifolds of -dimension isometrically immersed in a quaternionic projective space and proved the following theorem as a quaternionic analogy to theorems given in [14, 15], which are natural extensions of theorems proved in [6] to the case of -submanifolds with -dimension and also extensions of theorems in [16]. Theorem K-P. Let £¿£¿be an -dimensional -submanifold of -dimension isometrically immersed in a quaternionic projective space , and let the normal vector field £¿£¿be parallel with respect to the normal connection. If the shape operator £¿£¿corresponding to £¿£¿satisfies then £¿£¿is locally a product of £¿£¿where£¿£¿ £¿£¿and£¿£¿ £¿£¿belong to some£¿£¿ -£¿£¿and -dimensional spheres ( £¿£¿is the Hopf fibration£¿£¿ ). On the other hand, when is a real hypersurface of , if is an Einstein space or a locally symmetric space, then has a parallel second fundamental form (cf. [4, 6, 7, 9]). Projecting the quantities on onto in , we can consider -submanifolds of -dimension with the conditions corresponding to or . In this paper, we will study such -submanifolds isometrically immersed in and obtain Theorem 3 and other results stated in the last Section 5 as quaternionic analogies to theorems given in [16, 17] and as the extensions of theorems given in [18] by using Theorem K-P. 2. Preliminaries Let be a real -dimensional quaternionic K£¿hler manifold. Then, by definition, there is a -dimensional vector bundle consisting of tensor fields of type over satisfying the following conditions (a), (b), and (c).(a) In any coordinate neighborhood , there is a local basis , , of such that (b) There is a Riemannian metric which is Hermite with respect to all of , , and .(c) For the Riemannian connection with respect to , £¿where , , and are local -forms
%U http://www.hindawi.com/journals/ijmms/2013/193697/