QR-Submanifolds of ( p − 1 ) QR-Dimension in a Quaternionic Projective Space QP ( n + p ) / 4 under Some Curvature Conditions

at each point x in M, then M is called a QR-submanifold of r QR-dimension, where ] denotes the complementary orthogonal distribution to ] in TM (cf. [1–3]). Real hypersurfaces, which are typical examples of QR-submanifold with r = 0, have been investigated by many authors (cf. [2–9]) in connection with the shape operator and the induced almost contact 3-structure (for definition, see [10–13]). In their paper [2, 3], Kwon and Pak had studied QR-submanifolds of (p − 1) QR-dimension isometrically immersed in a quaternionic projective space QP 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 QR-submanifolds with (p − 1) QR-dimension and also extensions of theorems in [16].


Preliminaries
Let  be a real ( + )-dimensional quaternionic Kähler manifold.Then, by definition, there is a 3-dimensional vector bundle  consisting of tensor fields of type (1, 1) over  satisfying the following conditions (a), (b), and (c).
(a) In any coordinate neighborhood U, there is a local basis {, , } of  such that (b) There is a Riemannian metric  which is Hermite with respect to all of , , and .
Next, applying  to the first equation of (9) and using ( 10), (14), and (15), we have Similarly, we have from which, taking account of the skew symmetry of  1 ,  2 , and  3 and using (11), we also have So ( 10) can be rewritten in the form ( = 2, . . ., ).Applying  and  to the first equation of ( 9) and using (3), (9), and (20), we have Similarly, the other equations of (9) yield From the first three equations of (20), we also have Equations ( 14)-( 17), (19), and ( 22)-(24) tell us that  admits the so-called almost contact 3-structure and consequently  = 4 + 3 for some integer  (cf.[12]).Now let ∇ be the Levi-Civita connection on , and let ∇ ⊥ be the normal connection induced from ∇ in the normal bundle of .Then Gauss and Weingarten formulae are given by for ,  tangent to .Here ℎ denotes the second fundamental form and   the shape operator corresponding to   .They are related by ℎ(, ) = ∑  =1 (  , )  .Furthermore, put where (  ) is the skew-symmetric matrix of connection forms of ∇ ⊥ .Differentiating the first equation of (9) covariantly and using ( 4), ( 9), ( 10), ( 14 for , , and  tangent to , where  and  denote the Riemannian curvature tensor of  and , respectively.
In the rest of this paper we assume that the distinguished normal vector field  1 :=  is parallel with respect to the normal connection ∇ ⊥ .Then it follows from (27) that  1 = 0, and consequently, ( 30

Fibrations and Immersions
From now on -dimensional QR-submanifolds of (−1) QRdimension isometrically immersed in QP (+)/4 only will be considered.Moreover, we will use the assumption and the notations as in Section 2.
Summing up, we have the following lemma.
Finally, we will prove our main theorem.
Proof.By means of (96), it follows easily from (83) that By the quite same method, we can obtain Combining with those equalities and Theorem K-P, we complete the proof.