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

Abstract

The purpose of this paper is to study n-dimensional QR-submanifolds of (𝑝−1)QR-dimension in a quaternionic projective space QP^{(𝑛+𝑝)/4} and especially to determine such submanifolds under some curvature conditions.

1. Introduction

Let 𝑀 be a connected real 𝑛-dimensional submanifold of real codimension 𝑝 of a quaternionic Kähler manifold 𝑀 with quaternionic Kähler structure {𝐹,𝐺,𝐻}. If there exists an 𝑟-dimensional normal distribution 𝜈 of the normal bundle 𝑇𝑀^⟂ such that 𝐹𝜈_𝑥⊂𝜈_𝑥,𝐺𝜈_𝑥⊂𝜈_𝑥,𝐻𝜈_𝑥⊂𝜈_𝑥,𝐹𝜈^⟂_𝑥⊂𝑇_𝑥𝑀,𝐺𝜈^⟂_𝑥⊂𝑇_𝑥𝑀,𝐻𝜈^⟂_𝑥⊂𝑇_𝑥𝑀 (1) at each point 𝑥 in 𝑀, then 𝑀 is called a QR-submanifold of 𝑟QR-dimension, where 𝜈^⟂ denotes the complementary orthogonal distribution to 𝜈 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^{(𝑛+𝑝)/4} 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].

Theorem K-P. Let 𝑀  be an 𝑛-dimensional 𝑄𝑅-submanifold of (𝑝−1)𝑄𝑅-dimension isometrically immersed in a quaternionic projective space 𝑄𝑃^{(𝑛+𝑝)/4}, and let the normal vector field 𝑁₁  be parallel with respect to the normal connection. If the shape operator 𝐴₁  corresponding to 𝑁₁  satisfies𝐴₁𝜙=𝜙𝐴₁,𝐴₁𝜓=𝜓𝐴₁,𝐴₁𝜃=𝜃𝐴₁, (2)then 𝜋⁻¹(𝑀)  is locally a product of 𝑀₁×𝑀₂  where  𝑀₁  and  𝑀₂  belong to some  (4𝑛₁+3)-  and (4𝑛₂+3)-dimensional spheres (𝜋  is the Hopf fibration  𝑆^𝑛+𝑝+3(1)→𝑄𝑃^{(𝑛+𝑝)/4}).

On the other hand, when 𝑀 is a real hypersurface of QP^{(𝑛+𝑝)/4}, if 𝜋⁻¹(𝑀) is (1) an Einstein space or (2) a locally symmetric space, then 𝜋⁻¹(𝑀) has a parallel second fundamental form (cf. [4, 6, 7, 9]). Projecting the quantities on 𝜋⁻¹(𝑀) onto 𝑀 in QP^{(𝑛+𝑝)/4}, we can consider QR-submanifolds of (𝑝−1)QR-dimension with the conditions corresponding to (1) or (2). In this paper, we will study such QR-submanifolds isometrically immersed in QP^{(𝑛+𝑝)/4} 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 𝑀 be a real (𝑛+𝑝)-dimensional quaternionic Kähler manifold. Then, by definition, there is a 3-dimensional vector bundle 𝑉 consisting of tensor fields of type (1,1) over 𝑀 satisfying the following conditions (a), (b), and (c).(a) In any coordinate neighborhood 𝒰, there is a local basis {𝐹, 𝐺, 𝐻} of 𝑉 such that 𝐹²=−𝐼,𝐺²=−𝐼,𝐻²=−𝐼,𝐹𝐺=−𝐺𝐹=𝐻,𝐺𝐻=−𝐻𝐺=𝐹,𝐻𝐹=−𝐹𝐻=𝐺. (3)(b) There is a Riemannian metric 𝑔 which is Hermite with respect to all of 𝐹, 𝐺, and 𝐻.(c) For the Riemannian connection ∇ with respect to 𝑔, ∇𝐹∇𝐺=𝐹𝐺𝐻,∇𝐻0𝑟−𝑞−𝑟0𝑝𝑞−𝑝0 (4) where 𝑝, 𝑞, and 𝑟 are local 1-forms defined in 𝒰. Such a local basis {𝐹,𝐺,𝐻} is called a canonical local basis of the bundle 𝑉 in 𝒰 (cf. [10, 19, 20]).

For canonical local bases {𝐹,𝐺,𝐻} and {^′𝐹,^′𝐺,^′𝐻} of 𝑉 in coordinate neighborhoods 𝒰 and ^′𝒰, it follows that in 𝒰∩^′𝒰^′_′𝐹_′𝐺𝐻=𝑠_𝑥𝑦𝐹𝐺𝐻𝑥,𝑦=1,2,3, (5) where 𝑠_𝑥𝑦 are local differentiable functions with (𝑠_𝑥𝑦)∈SO(3) as a consequence of (3). As is well known (cf. [19]), every quaternionic Kähler manifold is orientable.

Now let 𝑀 be an 𝑛-dimensional QR-submanifold of (𝑝−1)QR-dimension isometrically immersed in 𝑀. Then by definition, there is a unit normal vector field 𝑁 such that 𝜈^⟂_𝑥=Span{𝑁} at each point 𝑥 in 𝑀. We set 𝑈=−𝐹𝑁,𝑉=−𝐺𝑁,𝑊=−𝐻𝑁. (6) Denoting by 𝒟_𝑥 the maximal quaternionic invariant subspace 𝑇_𝑥𝑀∩𝐹𝑇_𝑥𝑀∩𝐺𝑇_𝑥𝑀∩𝐻𝑇_𝑥𝑀 of 𝑇_𝑥𝑀, we have 𝒟^⟂_𝑥=Span{𝑈,𝑉,𝑊}, where 𝒟^⟂_𝑥 means the complementary orthogonal subspace to 𝒟_𝑥 in 𝑇_𝑥𝑀 (cf. [1–3]). Thus, we have 𝑇_𝑥𝑀=𝒟_𝑥⊕Span{𝑈,𝑉,𝑊},𝑥∈𝑀, (7) which together with (3) and (6) implies 𝐹𝑇_𝑥𝑀,𝐺𝑇_𝑥𝑀,𝐻𝑇_𝑥𝑀⊂𝑇_𝑥𝑀⊕Span{𝑁}. (8) Therefore, for any tangent vector field 𝑋 and for a local orthonormal basis {𝑁_𝛼}_{𝛼=1,…,𝑝}(𝑁₁∶=𝑁) of normal vectors to 𝑀, we have (𝑋)(𝑋)(𝑋)𝐹𝑋=𝜙𝑋+𝑢𝑁,𝐺𝑋=𝜓𝑋+𝑣𝑁,𝐻𝑋=𝜃𝑋+𝑤𝑁,𝐹𝑁_𝛼=−𝑈_𝛼+𝑃₁𝑁_𝛼,𝐺𝑁_𝛼=−𝑉_𝛼+𝑃₂𝑁_𝛼,𝐻𝑁_𝛼=−𝑊_𝛼+𝑃₃𝑁_𝛼, (9)(10)(𝛼=1,…,𝑝). Then it is easily seen that {𝜙,𝜓,𝜃} and {𝑃₁,𝑃₂,𝑃₃} are skew-symmetric endomorphisms acting on 𝑇_𝑥𝑀 and 𝑇_𝑥𝑀^⟂, respectively. Moreover, the Hermitian property of {𝐹,𝐺,𝐻} implies 𝑔𝑋,𝜙𝑈_𝛼=−𝑢(𝑋)𝑔𝑁₁,𝑃₁𝑁_𝛼,𝑔𝑋,𝜓𝑉_𝛼=−𝑣(𝑋)𝑔𝑁₁,𝑃₂𝑁_𝛼𝑔,𝛼=1,…,𝑝,𝑋,𝜃𝑊_𝛼=−𝑤(𝑋)𝑔𝑁₁,𝑃₃𝑁_𝛼,𝑔𝑈_𝛼,𝑈_𝛽=𝛿_𝛼𝛽−𝑔𝑃₁𝑁_𝛼,𝑃₁𝑁_𝛽,𝑔𝑉_𝛼,𝑉_𝛽=𝛿_𝛼𝛽−𝑔𝑃₂𝑁_𝛼,𝑃₂𝑁_𝛽𝑔𝑊,𝛼,𝛽=1,…,𝑝,_𝛼,𝑊_𝛽=𝛿_𝛼𝛽−𝑔𝑃₃𝑁_𝛼,𝑃₃𝑁_𝛽. (11)(12) Also, from the hermitian properties 𝑔(𝐹𝑋,𝑁_𝛼)=−𝑔(𝑋,𝐹𝑁_𝛼), 𝑔(𝐺𝑋,𝑁_𝛼)=−𝑔(𝑋,𝐺𝑁_𝛼), and 𝑔(𝐻𝑋,𝑁_𝛼)=−𝑔(𝑋,𝐻𝑁_𝛼), it follows that 𝑔𝑋,𝑈_𝛼=𝑢(𝑋)𝛿_1𝛼,𝑔𝑋,𝑉_𝛼=𝑣(𝑋)𝛿_1𝛼,𝑔𝑋,𝑊_𝛼=𝑤(𝑋)𝛿_1𝛼, (13) and hence, 𝑔𝑈₁,𝑋=𝑢(𝑋),𝑔𝑉₁(𝑋),𝑔𝑊,𝑋=𝑣₁,𝑋=𝑤(𝑋),𝑈_𝛼𝑉=0,_𝛼=0,𝑊_𝛼=0,𝛼=2,…,𝑝. (14) On the other hand, comparing (6) and (10) with 𝛼=1, we have 𝑈₁=𝑈, 𝑉₁=𝑉, and 𝑊₁=𝑊, which together with (6) and (14) implies 𝑔()(𝑋)()(𝑋)𝑔()(𝑋)(𝑈)(𝑉)(𝑊)𝑈,𝑋=𝑢,𝑔𝑉,𝑋=𝑣𝑊,𝑋=𝑤,𝑢=1,𝑣=1,𝑤=1. (15) In the sequel, we will use the notations 𝑈, 𝑉, and 𝑊 instead of 𝑈₁, 𝑉₁, and 𝑊₁.

Next, applying 𝐹 to the first equation of (9) and using (10), (14), and (15), we have 𝜙²𝑋=−𝑋+𝑢(𝑋)𝑈,𝑢(𝑋)𝑃₁𝑁=−𝑢𝜙𝑋𝑁. (16) Similarly, we have 𝜙²𝑋=−𝑋+𝑢(𝑋)𝑈,𝜓²(𝑋)𝜃𝑋=−𝑋+𝑣𝑉,₂𝑋=−𝑋+𝑤(𝑋)𝑊,𝑢(𝑋)𝑃₁𝑁=−𝑢𝜙𝑋𝑁,𝑣(𝑋)𝑃₂𝑤(𝑋)𝑃𝑁=−𝑣𝜓𝑋𝑁,₃𝑁=−𝑤(𝜃𝑋)𝑁, (17)(18) from which, taking account of the skew symmetry of 𝑃₁, 𝑃₂, and 𝑃₃ and using (11), we also have 𝑢𝜙𝑋=0,𝑣𝜓𝑋=0,𝑤(𝜃𝑋)=0,𝜙𝑈=0,𝜓𝑉=0,𝜃𝑊=0,𝑃₁𝑃𝑁=0,₂𝑁=0,𝑃₃𝑁=0. (19) So (10) can be rewritten in the form 𝐹𝑁=−𝑈,𝐺𝑁=−𝑉,𝐻𝑁=−𝑊,𝐹𝑁_𝛼=𝑃₁𝑁_𝛼,𝐺𝑁_𝛼=𝑃₂𝑁_𝛼,𝐻𝑁_𝛼=𝑃₃𝑁_𝛼 (20)(𝛼=2,…,𝑝). Applying 𝐺 and 𝐻 to the first equation of (9) and using (3), (9), and (20), we have (𝑋)(𝑋)(𝑋)(𝑋)𝜃𝑋+𝑤𝑁=−𝜓𝜙𝑋−𝑣𝜙𝑋𝑁+𝑢𝑉,𝜓𝑋+𝑣𝑁=𝜃𝜙𝑋+𝑤𝜙𝑋𝑁−𝑢𝑊, (21) and consequently, 𝜓(𝑋)(𝑋),𝜃(𝑋)(𝑋).𝜙𝑋=−𝜃𝑋+𝑢𝑉,𝑣𝜙𝑋=−𝑤𝜙𝑋=𝜓𝑋+𝑢𝑊,𝑤𝜙𝑋=𝑣 (22) Similarly, the other equations of (9) yield 𝜙(𝑋)(𝑋),𝜃(𝑋)(𝑋),𝜙()(𝑋)()(𝑋),𝜓()(𝑋)()(𝑋).𝜓𝑋=𝜃𝑋+𝑣𝑈,𝑢𝜓𝑋=𝑤𝜓𝑋=−𝜙𝑋+𝑣𝑊,𝑤𝜓𝑋=−𝑢𝜃𝑋=−𝜓𝑋+𝑤𝑈,𝑢𝜃𝑋=−𝑣𝜃𝑋=𝜙𝑋+𝑤𝑉,𝑣𝜃𝑋=𝑢 (23) From the first three equations of (20), we also have (𝑈)𝑤(𝑈)(𝑉)(𝑉)𝑢(𝑊)(𝑊)𝜓𝑈=−𝑊,𝑣=0,𝜃𝑈=𝑉,=0,𝜙𝑉=𝑊,𝑢=0,𝜃𝑉=−𝑈,𝑤=0,𝜙𝑊=−𝑉,=0,𝜓𝑊=𝑈,𝑣=0. (24)

Equations (14)–(17), (19), and (22)–(24) tell us that 𝑀 admits the so-called almost contact 3-structure and consequently 𝑛=4𝑚+3 for some integer 𝑚 (cf. [12]).

Now let ∇ be the Levi-Civita connection on 𝑀, and let ∇^⟂ be the normal connection induced from ∇ in the normal bundle of 𝑀. Then Gauss and Weingarten formulae are given by ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),∇_𝑋𝑁_𝛼=−𝐴_𝛼𝑋+∇^⟂_𝑋𝑁_𝛼,𝛼=1,…,𝑝, (25)(26) for 𝑋, 𝑌 tangent to 𝑀. Here ℎ denotes the second fundamental form and 𝐴_𝛼 the shape operator corresponding to 𝑁_𝛼. They are related by ℎ(𝑋,𝑌)=∑^𝑝_𝛼=1𝑔(𝐴_𝛼𝑋,𝑌)𝑁_𝛼. Furthermore, put ∇^⟂_𝑋𝑁_𝛼=^𝑝_𝛽=1𝑠_𝛼𝛽(𝑋)𝑁_𝛽, (27) where (𝑠_𝛼𝛽) is the skew-symmetric matrix of connection forms of ∇^⟂.

Differentiating the first equation of (9) covariantly and using (4), (9), (10), (14) (25), and (26), we have ∇_𝑌𝜙𝑋=𝑟(𝑌)𝜓𝑋−𝑞(𝑌)𝜃𝑋+𝑢(𝑋)𝐴₁𝑌−𝑔𝐴₁∇𝑌,𝑋𝑈,_𝑌𝑢𝑋=𝑟(𝑌)𝑣(𝑋)−𝑞(𝑌)𝑤(𝑋)+𝑔𝜙𝐴₁𝑌,𝑋. (28) From the other equations of (9), we also have ∇_𝑌𝜓𝑋=−𝑟(𝑌)𝜙𝑋+𝑝(𝑌)𝜃𝑋+𝑣(𝑋)𝐴₁𝑌−𝑔𝐴₁∇𝑌,𝑋𝑉,_𝑌𝑣𝑋=−𝑟(𝑌)𝑢(𝑋)+𝑝(𝑌)𝑤(𝑋)+𝑔𝜓𝐴₁,∇𝑌,𝑋_𝑌𝜃𝑋=𝑞(𝑌)𝜙𝑋−𝑝(𝑌)𝜓𝑋+𝑤(𝑋)𝐴₁𝑌−𝑔𝐴₁∇𝑌,𝑋𝑊,_𝑌𝑤𝑋=𝑞(𝑌)𝑢(𝑋)−𝑝(𝑌)𝑣(𝑋)+𝑔𝜃𝐴₁𝑌,𝑋. (29)

Next, differentiating the first equation of (20) covariantly and comparing the tangential and normal parts, we have ∇_𝑌𝑈=𝑟(𝑌)𝑉−𝑞(𝑌)𝑊+𝜙𝐴₁𝑔𝐴𝑌,_𝛼𝑈,𝑌=−^𝑝_𝛽=2𝑠_1𝛽(𝑌)𝑃_1𝛽𝛼,𝛼=2,…,𝑝. (30) From the other equations of (20), we have similarly ∇_𝑌𝑉=−𝑟(𝑌)𝑈+𝑝(𝑌)𝑊+𝜓𝐴₁𝑔𝐴𝑌,_𝛼𝑉,𝑌=−^𝑝_𝛽=2𝑠_1𝛽(𝑌)𝑃_2𝛽𝛼∇,𝛼=2,…,𝑝,_𝑌𝑊=𝑞(𝑌)𝑈−𝑝(𝑌)𝑉+𝜃𝐴₁𝑔𝐴𝑌,_𝛼𝑊,𝑌=−^𝑝_𝛽=2𝑠_1𝛽(𝑌)𝑃_3𝛽𝛼,𝛼=2,…,𝑝. (31)

Finally the equation of Gauss is given as follows (cf. [21]): 𝑔𝑅()(𝑅())𝑋,𝑌𝑍,𝑊=𝑔𝑋,𝑌𝑍,𝑊+_𝛼𝑔𝐴_𝛼𝑋,𝑍𝑔𝐴_𝛼𝐴𝑌,𝑊−𝑔_𝛼𝑌,𝑍𝑔𝐴_𝛼𝑋,𝑊, (32) for 𝑋,𝑌, and 𝑍 tangent to 𝑀, where 𝑅 and 𝑅 denote the Riemannian curvature tensor of 𝑀 and 𝑀, respectively.

In the rest of this paper we assume that the distinguished normal vector field 𝑁₁∶=𝑁  is parallel with respect to the normal connection ∇^⟂. Then it follows from (27) that 𝑠_1𝛽=0, and consequently, (30)-(31) imply 𝐴_𝛼𝑈=0,𝐴_𝛼𝑉=0,𝐴_𝛼𝑊=0,𝛼=2,…,𝑝. (33)

On the other hand, since the curvature tensor 𝑅 of QP^{(𝑛+𝑝)/4} is of the form 𝑅𝑋,𝑌𝑍=𝑔𝑌,𝑍𝑋−𝑔𝑋,𝑍𝑌+𝑔𝐹𝑌,𝑍𝐹𝑋−𝑔𝐹𝑋,𝑍𝐹𝑌−2𝑔𝐹𝑋,𝑌𝐹𝑍+𝑔𝐺𝑌,𝑍𝐺𝑋−𝑔𝐺𝑋,𝑍𝐺𝑌−2𝑔𝐺𝑋,𝑌𝐺𝑍+𝑔𝐻𝑌,𝑍𝐻𝑋−𝑔𝐻𝑋,𝑍𝐻𝑌−2𝑔𝐻𝑋,𝑌𝐻𝑍 (34) for 𝑋,𝑌, and 𝑍 tangent to QP^{(𝑛+𝑝)/4}, (32) reduces to 𝑅()()()𝑌()()()𝑋,𝑌𝑍=𝑔𝑌,𝑍𝑋−𝑔𝑋,𝑍+𝑔𝜙𝑌,𝑍𝜙𝑋−𝑔𝜙𝑋,𝑍𝜙𝑌−2𝑔𝜙𝑋,𝑌𝜙𝑍+𝑔𝜓𝑌,𝑍𝜓𝑋−𝑔𝜓𝑋,𝑍𝜓𝑌−2𝑔𝜓𝑋,𝑌𝜓𝑍+𝑔𝜃𝑌,𝑍𝜃𝑋−𝑔𝜃𝑋,𝑍𝜃𝑌−2𝑔𝜃𝑋,𝑌𝜃𝑍+_𝛼𝑔𝐴_𝛼𝑌,𝑍𝐴_𝛼𝑋−𝑔𝐴_𝛼𝑋,𝑍𝐴_𝛼𝑌. (35)

3. Fibrations and Immersions

From now on 𝑛-dimensional QR-submanifolds of (𝑝−1)QR-dimension isometrically immersed in QP^{(𝑛+𝑝)/4} only will be considered. Moreover, we will use the assumption and the notations as in Section 2.

Let 𝑆^𝑛+𝑝+3(𝑎) be the hypersphere of radius 𝑎 (>0) in 𝑄^{(𝑛+𝑝+4)/4} the quaternionic space of quaternionic dimension (𝑛+𝑝+4)/4, which is identified with the Euclidean (𝑛+𝑝+4)-space ℝ^𝑛+𝑝+4. The unit sphere 𝑆^𝑛+𝑝+3(1) will be briefly denoted by 𝑆^𝑛+𝑝+3. Let 𝜋∶𝑆^𝑛+𝑝+3→QP^{(𝑛+𝑝)/4} be the natural projection of 𝑆^𝑛+𝑝+3 onto QP^{(𝑛+𝑝)/4} defined by the Hopf fibration 𝑆³→𝑆^𝑛+𝑝+3→QP^{(𝑛+𝑝)/4}. As is well known (cf. [10, 11, 20]), 𝑆^𝑛+𝑝+3 admits a Sasakian 3-structure whereby 𝜉̃,𝜂,and𝜁̃ are mutually orthogonal unit Killing vector fields. Thus it follows that ∇̃𝜉̃∇𝜂∇𝜁̃̃∇̃𝜁∇𝜂̃̃∇̃𝜉̃∇𝜁̃̃∇𝜂̃∇̃𝜉̃𝜉=0,𝜂=0,𝜁=0,𝜂=−𝜁=𝜉,𝜁=−𝜉=𝜂,𝜉=−𝜂=𝜁, (36) where ∇ denotes the Riemannian connection with respect to the canonical metric 𝑔 on 𝑆^𝑛+𝑝+3 (cf. [6, 9–13]). Moreover, each fibre 𝜋⁻¹(𝑥) of 𝑥 in QP^{(𝑛+𝑝)/4} is a maximal integral submanifold of the distribution spanned by 𝜉̃,𝜂, and 𝜁̃. Thus the base space QP^{(𝑛+𝑝)/4} admits the induced quaternionic Kähler structure of constant 𝑄-sectional curvature 4 (cf. [10, 11]). We have especially a fibration 𝜋∶𝜋⁻¹(𝑀)→𝑀 which is compatible with the Hopf fibration 𝜋. More precisely speaking, 𝜋∶𝜋⁻¹(𝑀)→𝑀 is a fibration with totally geodesic fibers such that the following diagram is commutative: (37) where 𝑖^′∶𝜋⁻¹(𝑀)→𝑆^𝑛+𝑝+3 and 𝑖∶𝑀→QP^{(𝑛+𝑝)/4} are isometric immersions.

Now, let 𝜉,𝜂, and 𝜁 be the unit vector fields tangent to the fibers of 𝜋⁻¹(𝑀) such that 𝑖^′_∗𝜉=𝜉̃, 𝑖^′_∗𝜂=𝜂, and 𝑖^′_∗𝜁=𝜁̃. (In what follows, we will again delete the 𝑖^′ and 𝑖^′_∗ in our notation.) Furthermore, we denote by 𝑋^∗ the horizontal lift of a vector field 𝑋 tangent to 𝑀. Then, the horizontal lifts 𝑁^∗_𝛼 (𝛼=1,…,𝑝) of the normal vectors 𝑁_𝛼 to 𝑀 form an orthonormal basis of normal vectors to 𝜋⁻¹(𝑀) in 𝑆^𝑛+𝑝+3. Let 𝐴^′_𝛼 and 𝑠^′_𝛼𝛽 be the corresponding shape operators and normal connection forms, respectively. Then, as shown in [3, 9, 22], the fundamental equations for the submersion 𝜋 are given by ^′∇_𝑋^∗𝑌^∗=∇_𝑋𝑌^∗+𝑔^′𝜙𝑋^∗,𝑌^∗𝜉+𝑔^′𝜓𝑋^∗,𝑌^∗𝜂+𝑔^′(𝜃𝑋)^∗,𝑌^∗𝜁,𝑋^∗,𝑌^∗=[𝑋,𝑌]^∗+2𝑔^′𝜙𝑋^∗,𝑌^∗𝜉+2𝑔^′𝜓𝑋^∗,𝑌^∗𝜂+2𝑔^′(𝜃𝑋)^∗,𝑌^∗𝜁,_′∇_𝑋^∗𝜉=^′∇_𝜉𝑋^∗=−𝜙𝑋^∗,_′∇_𝑋^∗𝜂=^′∇_𝜂𝑋^∗=−𝜓𝑋^∗,_′∇_𝑋^∗𝜁=^′∇_𝜁𝑋^∗=−(𝜃𝑋)^∗,𝑋^∗,𝜉=0,𝑋^∗,𝜂=0,𝑋^∗,𝜁=0, (38)(39)(40)(41) where 𝑔^′ denotes the Riemannian metric of 𝜋⁻¹(𝑀) induced from 𝑔 in 𝑆^𝑛+𝑝+3 and ^′∇ the Levi-Civita connection with respect to 𝑔^′. The same equations are valid for the submersion 𝜋 by replacing 𝜙, 𝜓, and 𝜃 (resp., 𝜉, 𝜂, and 𝜁) with 𝐹, 𝐺, and 𝐻 (resp., 𝜉̃, 𝜂, and 𝜁̃), respectively. We denote by ^′∇^⟂ the normal connection of 𝜋⁻¹(𝑀) induced from ∇. Since the diagram is commutative, ∇_𝑋^∗𝑁^∗_𝛼 implies ^′∇^⟂_𝑋^∗𝑁^∗_𝛼−𝐴^′_𝛼𝑋^∗=∇_𝑋𝑁_𝛼^∗+𝑔(𝐹𝑋)^∗,𝑁^∗_𝛼𝜉̃+𝑔(𝐺𝑋)^∗,𝑁^∗_𝛼𝜂+𝑔(𝐻𝑋)^∗,𝑁^∗_𝛼𝜁̃=−𝐴_𝛼𝑋^∗+𝑔𝑈_𝛼,𝑋^∗𝜉+𝑔𝑉_𝛼,𝑋^∗𝜂+𝑔𝑊_𝛼,𝑋^∗𝜁+∇^⟂_𝑋𝑁_𝛼^∗ (42) because of (10), (26), and (38), from which, comparing the tangential part, we have 𝐴^′_𝛼𝑋^∗=𝐴_𝛼𝑋^∗−𝑔𝑈_𝛼,𝑋^∗𝜉−𝑔𝑉_𝛼,𝑋^∗𝜂−𝑔𝑊_𝛼,𝑋^∗𝜁. (43) Next, calculating ∇_𝜉𝑁^∗_𝛼 and using (10), (26), and (40), we have ^′∇^⟂_𝜉𝑁^∗_𝛼−𝐴^′_𝛼𝜉=−𝐹𝑁_𝛼^∗=𝑈^∗_𝛼−𝑃₁𝑁_𝛼^∗, (44) which yields 𝐴^′_𝛼𝜉=−𝑈^∗_𝛼 (45) and similarly 𝐴^′_𝛼𝜉=−𝑈^∗_𝛼,𝐴^′_𝛼𝜂=−𝑉^∗_𝛼,𝐴^′_𝛼𝜁=−𝑊^∗_𝛼. (46) Hence, (43) and (46) with 𝛼=1 imply 𝐴^′₁𝑋^∗=𝐴₁𝑋^∗−𝑔(𝑈,𝑋)^∗𝜉−𝑔(𝑉,𝑋)^∗𝜂−𝑔(𝑊,𝑋)^∗𝐴𝜁,_′1𝜉=−𝑈^∗,𝐴^′₁𝜂=−𝑉^∗,𝐴^′₁𝜁=−𝑊^∗. (47)

4. Co-Gauss Equations for the Submersion 𝜋∶𝜋⁻¹(𝑀)→𝑀

In this section, we derive the co-Gauss and co-Codazzi equations of the submersion 𝜋∶𝜋⁻¹(𝑀)→𝑀 for later use.

Differentiating (38) with 𝑌=𝑈 covariantly along 𝜋⁻¹(𝑀) and using (24), (38), and (39), we have ^′∇_𝑌^∗′∇_𝑋^∗𝑈^∗=∇_𝑌∇_𝑋𝑈^∗+𝑣(𝑋)𝜃𝑌−𝑤(𝑋)𝜓𝑌^∗+𝑔𝜙𝑌,∇_𝑋𝑈^∗𝜉+𝑔𝜓𝑌,∇_𝑋𝑈+𝑔∇_𝑌𝑋,𝑊+𝑔𝑋,∇_𝑌𝑊^∗𝜂+𝑔𝜃𝑌,∇_𝑋𝑈−𝑔∇_𝑌𝑋,𝑉−𝑔𝑋,∇_𝑌𝑉^∗𝜁. (48) Similarly (38) with 𝑌=𝑉 and (38) with 𝑌=𝑊 give ^′∇_𝑌^∗′∇_𝑋^∗𝑉^∗=∇_𝑌∇_𝑋𝑉^∗+𝑤(𝑋)𝜙𝑌−𝑢(𝑋)𝜃𝑌^∗+𝑔𝜙𝑌,∇_𝑋𝑉−𝑔∇_𝑌𝑋,𝑊−𝑔𝑋,∇_𝑌𝑊^∗𝜉+𝑔𝜓𝑌,∇_𝑋𝑉^∗𝜂+𝑔𝜃𝑌,∇_𝑋𝑉+𝑔∇_𝑌𝑋,𝑈+𝑔𝑋,∇_𝑌𝑈^∗𝜁,_′∇_𝑌^∗′∇_𝑋^∗𝑊^∗=∇_𝑌∇_𝑋𝑊^∗−𝑣(𝑋)𝜙𝑌−𝑢(𝑋)𝜓𝑌^∗+𝑔𝜙𝑌,∇_𝑋𝑊+𝑔∇_𝑌𝑋,𝑉+𝑔𝑋,∇_𝑌𝑉^∗𝜉+𝑔𝜓𝑌,∇_𝑋𝑊−𝑔∇_𝑌𝑋,𝑈−𝑔𝑋,∇_𝑌𝑈^∗𝜂+𝑔(𝜃𝑌,∇_𝑋𝑊)^∗𝜁, (49)(50) respectively. On the other hand, it follows from (19), (24), (38), and (39) that ^′∇_[𝑌^∗_,𝑋^∗_]𝑈^∗=∇_[𝑌,𝑋]𝑈^∗+2𝑔𝜓𝑌,𝑋^∗𝑊^∗−2𝑔(𝜃𝑌,𝑋)^∗𝑉^∗+𝑔([𝑌,𝑋],𝑊)^∗𝜂−𝑔([𝑌,𝑋],𝑉)^∗𝜁,_′∇_[𝑌^∗_,𝑋^∗_]𝑉^∗=∇_[𝑌,𝑋]𝑉^∗−2𝑔𝜙𝑌,𝑋^∗𝑊^∗+2𝑔(𝜃𝑌,𝑋)^∗𝑈^∗−𝑔([𝑌,𝑋],𝑊)^∗𝜉+𝑔([𝑌,𝑋],𝑈)^∗𝜁,_′∇_[𝑌^∗_,𝑋^∗_]𝑊^∗=∇_[𝑌,𝑋]𝑊^∗+2𝑔𝜙𝑌,𝑋^∗𝑉^∗−2𝑔𝜓𝑌,𝑋^∗𝑈^∗+𝑔([𝑌,𝑋],𝑉)^∗𝜉−𝑔([𝑌,𝑋],𝑈)^∗𝜂. (51)(52)(53) By means of (48) and (51), we have ^′𝑅𝑌^∗,𝑋^∗𝑈^∗={𝑅(𝑌,𝑋)𝑈}^∗+𝑤(𝑌)(𝑋)(𝑌)(𝑋)()𝑉𝑊𝜓𝑋−𝑤𝜓𝑌−𝑣𝜃𝑋+𝑣𝜃𝑌+2𝑔𝜃𝑌,𝑋−2𝑔𝜓𝑌,𝑋^∗+𝑔𝜙𝑌,∇_𝑋𝑈−𝑔𝜙𝑋,∇_𝑌𝑈^∗𝜉+𝑔𝜓𝑌,∇_𝑋𝑈−𝑔𝜓𝑋,∇_𝑌𝑈+𝑔𝑋,∇_𝑌𝑊−𝑔𝑌,∇_𝑋𝑊^∗𝜂+𝑔𝜃𝑌,∇_𝑋𝑈−𝑔𝜃𝑋,∇_𝑌𝑈−𝑔(𝑋,∇_𝑌𝑉)+𝑔(𝑌,∇_𝑋𝑉)^∗𝜁, (54) where ^′𝑅 denotes the curvature tensor of 𝜋⁻¹(𝑀) with respect to the connection ^′∇. Using (30), (31), (33), and (35), we can easily see that ^′𝑅𝑌^∗,𝑋^∗𝑈^∗=𝑢(𝑋)(𝑌)𝑋𝐴𝑌−𝑢+𝑢₁𝑋𝐴₁𝑌−𝑢𝐴₁𝑌𝐴₁𝑋^∗+𝑟(𝑌)𝑤(𝑋)(𝑋)𝑤(𝑌)(𝑌)𝑣(𝑋)(𝑋)𝑣(𝑌)(𝑋)𝑢𝐴−𝑟+𝑞−𝑞+𝑢₁𝑌−𝑢(𝑌)𝑢𝐴₁𝑋^∗𝜉+𝑝(𝑋)𝑣(𝑌)(𝑌)𝑣(𝑋)(𝑋)𝑢𝐴−𝑝+𝑣₁𝑌−𝑣(𝑌)𝑢𝐴₁𝑋^∗𝜂+𝑝(𝑋)𝑤(𝑌)−𝑝(𝑌)𝑤(𝑋)+𝑤(𝑋)𝑢(𝐴₁𝑌)−𝑤(𝑌)𝑢(𝐴₁𝑋)^∗𝜁. (55) By the same method, we can easily verify that (49), (50), (52), and (53) yield ^′𝑅𝑌^∗,𝑋^∗𝑉^∗=𝑣(𝑋)𝑌−𝑣(𝑌)𝑋+𝑣𝐴₁𝑋𝐴₁𝑌−𝑣𝐴₁𝑌𝐴₁𝑋^∗+𝑞(𝑋)𝑢(𝑌)(𝑌)𝑢(𝑋)(𝑌)𝑣𝐴−𝑞−𝑢₁𝑋+𝑢(𝑋)𝑣𝐴₁𝑌^∗𝜉+𝑟(𝑌)𝑤(𝑋)(𝑋)𝑤(𝑌)(𝑌)𝑢(𝑋)(𝑋)𝑢(𝑌)(𝑋)𝑣𝐴−𝑟+𝑝−𝑝+𝑣₁𝑌−𝑣(𝑌)𝑣𝐴₁𝑋^∗𝜂+𝑞(𝑋)𝑤(𝑌)(𝑌)𝑤(𝑋)(𝑌)𝑣𝐴−𝑞−𝑤₁𝑋+𝑤(𝑋)𝑣𝐴₁𝑌^∗𝜁,_′𝑅𝑌^∗,𝑋^∗𝑊^∗=𝑤(𝑋)𝑌−𝑤(𝑌)𝑋+𝑤𝐴₁𝑋𝐴₁𝑌−𝑤𝐴₁𝑌𝐴₁𝑋^∗+𝑟(𝑋)𝑢(𝑌)(𝑌)𝑢(𝑋)(𝑌)𝑤𝐴−𝑟−𝑢₁𝑋+𝑢(𝑋)𝑤𝐴₁𝑌^∗𝜉+𝑟(𝑋)𝑣(𝑌)(𝑌)𝑣(𝑋)(𝑌)𝑤𝐴−𝑟−𝑣₁𝑋+𝑣(𝑋)𝑤𝐴₁𝑌^∗𝜂+𝑞(𝑌)𝑣(𝑋)(𝑋)𝑣(𝑌)(𝑌)𝑢(𝑋)(𝑋)𝑢(𝑌)−𝑞+𝑝−𝑝+𝑤(𝑋)𝑤(𝐴₁𝑌)−𝑤(𝑌)𝑤(𝐴₁𝑋)^∗𝜁. (56)

Differentiating (38) with 𝑋=𝑈 covariantly along 𝜋⁻¹(𝑀) and using (24), we have ^′∇_𝑌^∗′∇_𝑈^∗𝑋^∗=∇_𝑌∇_𝑈𝑋^∗+𝑤(𝑋)𝜓𝑌−𝑣(𝑋)𝜃𝑌^∗+𝑔𝜙𝑌,∇_𝑈𝑋^∗𝜉+𝑔𝜓𝑌,∇_𝑈𝑋−𝑔∇_𝑌𝑊,𝑋−𝑔𝑊,∇_𝑌𝑋^∗𝜂+𝑔𝜃𝑌,∇_𝑈𝑋+𝑔∇_𝑌𝑉,𝑋+𝑔𝑉,∇_𝑌𝑋^∗𝜁. (57) Similarly, (38) with 𝑋=𝑉 and (38) with 𝑋=𝑊, respectively, give ^′∇_𝑌^∗′∇_𝑉^∗𝑋^∗=∇_𝑌∇_𝑉𝑋^∗−𝑤(𝑋)𝜙𝑌−𝑢(𝑋)𝜃𝑌^∗+𝑔(𝜓𝑌,∇_𝑉𝑋)^∗𝜂+{𝑔(𝜙𝑌,∇_𝑉𝑋)+𝑔(∇_𝑌𝑊,𝑋)+𝑔(𝑊,∇_𝑌𝑋)}^∗𝜉+{𝑔(𝜃𝑌,∇_𝑉𝑋)−𝑔(∇_𝑌𝑈,𝑋)−𝑔(𝑈,∇_𝑌𝑋)}^∗𝜁,_′∇_𝑌^∗′∇_𝑊^∗𝑋^∗=(∇_𝑌∇_𝑊𝑋)^∗+𝑣(𝑋)𝜙𝑌−𝑢(𝑋)𝜓𝑌^∗+𝑔𝜃𝑌,∇_𝑊𝑋^∗𝜁+𝑔𝜙𝑌,∇_𝑊𝑋−𝑔∇_𝑌𝑉,𝑋−𝑔𝑉,∇_𝑌𝑋^∗𝜉+𝑔𝜓𝑌,∇_𝑊𝑋+𝑔∇_𝑌𝑈,𝑋+𝑔𝑈,∇_𝑌𝑋^∗𝜂. (58)(59) Differentiating (38) also covariantly in the direction of 𝑈^∗ and using (24), we have ^′∇_𝑈^∗′∇_𝑌^∗𝑋^∗=∇_𝑈∇_𝑌𝑋^∗+𝑔𝜓𝑌,𝑋^∗𝑊^∗−𝑔(𝜃𝑌,𝑋)^∗𝑉^∗+𝑔∇_𝑈𝜙𝑌,𝑋+𝑔𝜙𝑌,∇_𝑈𝑋^∗𝜉+𝑔∇_𝑈𝜓𝑌,𝑋+𝑔𝜓𝑌,∇_𝑈𝑋−𝑔𝑊,∇_𝑌𝑋^∗𝜂+𝑔∇_𝑈(𝜃𝑌),𝑋+𝑔𝜃𝑌,∇_𝑈𝑋+𝑔𝑉,∇_𝑌𝑋^∗𝜁. (60) Similarly, differentiating (38) covariantly in the direction of 𝑉^∗ and 𝑊^∗, respectively, we have ^′∇_𝑉^∗′∇_𝑌^∗𝑋^∗=∇_𝑉∇_𝑌𝑋^∗−𝑔𝜙𝑌,𝑋^∗𝑊^∗−𝑔(𝜃𝑌,𝑋)^∗𝑈^∗+𝑔∇_𝑉𝜙𝑌,𝑋+𝑔𝜙𝑌,∇_𝑉𝑋+𝑔𝑊,∇_𝑌𝑋^∗𝜉+𝑔∇_𝑉𝜓𝑌,𝑋+𝑔𝜓𝑌,∇_𝑉𝑋^∗𝜂+𝑔∇_𝑉(𝜃𝑌),𝑋+𝑔𝜃𝑌,∇_𝑉𝑋−𝑔𝑈,∇_𝑌𝑋^∗𝜁,_′∇_𝑊^∗′∇_𝑌^∗𝑋^∗=∇_𝑊∇_𝑌𝑋^∗−𝑔𝜓𝑌,𝑋^∗𝑈^∗+𝑔𝜙𝑌,𝑋^∗𝑉^∗+𝑔∇_𝑊𝜙𝑌,𝑋+𝑔𝜙𝑌,∇_𝑊𝑋−𝑔𝑉,∇_𝑌𝑋^∗𝜉+𝑔∇_𝑊𝜓𝑌,𝑋+𝑔𝜓𝑌,∇_𝑊𝑋+𝑔𝑈,∇_𝑌𝑋^∗𝜂+𝑔∇_𝑊(𝜃𝑌),𝑋+𝑔𝜃𝑌,∇_𝑊𝑋^∗𝜁. (61)(62)

On the other hand, (38) and (39) with 𝑋=𝑈 imply ^′∇_[𝑌^∗_,𝑈^∗_]𝑋^∗=∇_[𝑌,𝑈]𝑋^∗−2𝑤(𝑌)𝜓𝑋−𝑣(𝑌)𝜃𝑋^∗+𝑔𝜙[𝑌,𝑈],𝑋^∗𝜉+𝑔𝜓[𝑌,𝑈],𝑋^∗𝜂+𝑔(𝜃[𝑌,𝑈],𝑋)^∗𝜁. (63) Similarly, from (39) with 𝑋=𝑉 and (39) with 𝑋=𝑊, respectively, we find that ^′∇_[𝑌^∗_,𝑉^∗_]𝑋^∗=∇_[𝑌,𝑉]𝑋^∗+2𝑤(𝑌)𝜙𝑋−𝑢(𝑌)𝜃𝑋^∗+𝑔𝜙[𝑌,𝑉],𝑋^∗𝜉+𝑔𝜓[𝑌,𝑉],𝑋^∗𝜂+𝑔(𝜃[𝑌,𝑉],𝑋)^∗𝜁,_′∇_[𝑌^∗_,𝑊^∗_]𝑋^∗=∇_[𝑌,𝑊]𝑋^∗−2𝑣(𝑌)𝜙𝑋−𝑢(𝑌)𝜓𝑋^∗+𝑔𝜙[𝑌,𝑊],𝑋^∗𝜉+g𝜓[𝑌,𝑊],𝑋^∗𝜂+𝑔(𝜃[𝑌,𝑊],𝑋)^∗𝜁. (64)(65)

Using (28)–(31), it follows from (57), (60), and (63) that ^′𝑅𝑌^∗,𝑈^∗𝑋^∗={𝑅(𝑌,𝑈)𝑋}^∗+𝑤(𝑋)(𝑋)()𝑉(𝑌)(𝑌)𝜓𝑌−𝑣𝜃𝑌−𝑔𝜓𝑌,𝑋𝑊+𝑔𝜃𝑌,𝑋+2𝑤𝜓𝑋−2𝑣𝜃𝑋^∗−𝑟(𝑈)𝑔(𝑈)𝑔()(𝑌)𝑢𝐴𝜓𝑌,𝑋−𝑞𝜃𝑌,𝑋+𝑢₁𝑋(𝑌)𝑤(𝑋)(𝑌)𝑣(𝑋)𝐴+𝑟+𝑞−𝑔₁𝑌,𝑋^∗𝜉−(𝑌)𝑣(𝑋)(𝑈)𝑔(𝑈)𝑔()(𝑌)𝑢𝐴−𝑝−𝑟𝜙𝑌,𝑋+𝑝𝜃𝑌,𝑋+𝑣₁𝑋^∗𝜂−−𝑝(𝑌)𝑤(𝑋)+𝑞(𝑈)𝑔𝜙𝑌,𝑋−𝑝(𝑈)𝑔(𝜓𝑌,𝑋)+𝑤(𝑌)𝑢(𝐴₁𝑋)^∗𝜁, (66) from which, taking account of (35) and using (24) and (33), we obtain ^′𝑅𝑌^∗,𝑈^∗𝑋^∗=𝑢(𝑋)()𝑈𝐴𝑌−𝑔𝑌,𝑋+𝑢₁𝑋𝐴₁𝑌−𝑔𝐴₁𝑌,𝑋𝐴₁𝑈^∗−𝑟(𝑈)𝑔(𝑈)𝑔()(𝑌)𝑢𝐴𝜓𝑌,𝑋−𝑞𝜃𝑌,𝑋+𝑢₁𝑋(𝑌)𝑤(𝑋)(𝑌)𝑣(𝑋)𝐴+𝑟+𝑞−𝑔₁𝑌,𝑋^∗𝜉−(𝑌)𝑣(𝑋)(𝑈)𝑔(𝑈)𝑔()(𝑌)𝑢𝐴−𝑝−𝑟𝜙𝑌,𝑋+𝑝𝜃𝑌,𝑋+𝑣₁𝑋^∗𝜂−−𝑝(𝑌)𝑤(𝑋)+𝑞(𝑈)𝑔𝜙𝑌,𝑋−𝑝(𝑈)𝑔(𝜓𝑌,𝑋)+𝑤(𝑌)𝑢(𝐴₁𝑋)^∗𝜁. (67) Similarly, by using (58), (59), (61), (62), (64), and (65), we can easily obtain ^′𝑅𝑌^∗,𝑉^∗𝑋^∗=𝑣(𝑋)()𝑉𝐴𝑌−𝑔𝑌,𝑋+𝑣₁𝑋𝐴₁𝑌−𝑔𝐴₁𝑌,𝑋𝐴₁𝑉^∗−(𝑉)𝑔(𝑉)𝑔()(𝑌)𝑣𝐴−𝑟𝜙𝑌,𝑋+𝑝𝜃𝑌,𝑋+𝑣₁𝑋(𝑌)𝑤(𝑋)(𝑌)𝑢(𝑋)𝐴+𝑟+𝑝−𝑔₁𝑌,𝑋^∗𝜂+𝑞(𝑌)𝑢(𝑋)(𝑉)𝑔(𝑉)𝑔()(𝑌)𝑣𝐴−𝑟𝜓𝑌,𝑋+𝑞𝜃𝑌,𝑋−𝑢₁𝑋^∗𝜉−−𝑞(𝑌)𝑤(𝑋)+𝑞(𝑉)𝑔𝜙𝑌,𝑋−𝑝(𝑉)𝑔(𝜓𝑌,𝑋)+𝑤(𝑌)𝑣(𝐴₁𝑋)^∗𝜁,_′𝑅𝑌^∗,𝑊^∗𝑋^∗=𝑤(𝑋)()𝑊𝐴𝑌−𝑔𝑌,𝑋+𝑤₁𝑋𝐴₁𝑌−𝑔𝐴₁𝑌,𝑋𝐴₁𝑊^∗−𝑞(𝑊)𝑔(𝑊)𝑔(𝑌)𝑤𝐴𝜙𝑌,𝑋−𝑝𝜓𝑌,𝑋+𝑤₁𝑋(𝑌)𝑣(𝑋)(𝑌)𝑢(𝑋)𝐴+𝑞+𝑝−𝑔₁𝑌,𝑋^∗𝜁−(𝑌)𝑢(𝑋)(𝑊)𝑔(𝑊)𝑔()(𝑌)𝑤𝐴−𝑟+𝑟𝜓𝑌,𝑋−𝑞𝜃𝑌,𝑋+𝑢₁𝑋^∗𝜉+𝑟(𝑌)𝑣(𝑋)+𝑟(𝑊)𝑔𝜙𝑌,𝑋−𝑝(𝑊)𝑔(𝜃𝑌,𝑋)−𝑣(𝑌)𝑤(𝐴₁𝑋)^∗𝜂. (68)

5. Main Results

It is well known [3] that if 𝜋⁻¹(𝑀) is locally symmetric then ^′∇𝐴₁=0 which implies identities (2) in Theorem K-P. In this point of view, we consider the following assumptions in (69) which are weaker conditions than the locally symmetry of 𝜋⁻¹(𝑀).

In order to obtain our main results, let 𝑀 be 𝑛-dimensional QR-submanifolds of (𝑝−1)QR-dimension in QP^{(𝑛+𝑝)/4} with the assumptions ^′∇_𝜉^′𝑅𝑌^∗,𝑋^∗𝑈^∗=0,^′∇_𝜂^′𝑅𝑌^∗,𝑋^∗𝑉^∗=0,_′∇_𝜁^′𝑅𝑌^∗,𝑋^∗𝑊^∗=0,_′∇_𝜉^′𝑅𝑌^∗,𝑈^∗𝑋^∗=0,^′∇_𝜂^′𝑅𝑌^∗,𝑉^∗𝑋^∗=0,_′∇_𝜁^′𝑅𝑌^∗,𝑊^∗𝑋^∗=0. (69)

We notice here that the above curvature conditions in (69) are different from those in [18] due to Pak and Sohn.

We first consider the assumption ^′∇_𝜉^′𝑅𝑌^∗,𝑋^∗𝑈^∗=0. (70)

Differentiating (55) covariantly in the direction of 𝜉 and using (19), (36), and (40), and the assumption (^′∇_𝜉^′𝑅)(𝑌^∗,𝑋^∗)𝑈^∗=0, we have −^′𝑅𝜙𝑌^∗,𝑋^∗𝑈^∗−^′𝑅𝑌^∗,𝜙𝑋^∗𝑈^∗=(𝑋)(𝑌)𝐴−𝑢𝜙𝑌+𝑢𝜙𝑋−𝑢₁𝑋𝜙𝐴₁𝑌+𝑢𝐴₁𝑌𝜙𝐴₁𝑋^∗+𝑝(𝑋)𝑤(𝑌)(𝑌)𝑤(𝑋)(𝑋)𝑢𝐴−𝑝+𝑤₁𝑌−𝑤(𝑌)𝑢𝐴₁𝑋^∗𝜂−𝑝(𝑋)𝑣(𝑌)−𝑝(𝑌)𝑣(𝑋)+𝑣(𝑋)𝑢(𝐴₁𝑌)−𝑣(𝑌)𝑢(𝐴₁𝑋)^∗𝜁, (71) from which, taking the vertical component of 𝜉,𝜂, and 𝜁, respectively, and using (22)–(24) and (55) itself, we can get 𝑟(𝑋)𝑣(𝑌)(𝑌)𝑣(𝑋)(𝑋)𝑤(𝑌)(𝑌)𝑤(𝑋)𝑤(𝑋)𝑤(𝑌)𝑣(𝑋)𝑣(𝑌)(𝑋)𝑢𝐴−𝑟−𝑞+𝑞−𝑟𝜙𝑌+𝑟𝜙𝑋−𝑞𝜙𝑌+𝑞𝜙𝑋−𝑢₁𝜙𝑌+𝑢(𝑌)𝑢𝐴₁𝜙𝑋=0,−𝑢𝐴₁𝜙𝑌𝑣(𝑋)+𝑢𝐴₁𝑣(𝑌)𝑣(𝑋)𝑣(𝑌)𝜙𝑋+𝑝𝜙𝑌−𝑝𝜙𝑋=0,−𝑢𝐴₁𝜙𝑌𝑤(𝑋)+𝑢𝐴₁𝑤(𝑌)𝑤(𝑋)𝑤(𝑌)𝜙𝑋+𝑝𝜙𝑌−𝑝𝜙𝑋=0. (72)(73)(74)

Putting 𝑌=𝑈 in (72) and using (19) and (24), we have 𝜙𝐴₁𝑈+𝑟(𝑈)𝑉−𝑞(𝑈)𝑊=0, (75) and consequently, 𝑟(𝑈)=𝑤𝐴₁𝑈=𝑢𝐴₁𝑊,𝑞(𝑈)=𝑣𝐴₁𝑈=𝑢𝐴₁𝑉. (76) Putting 𝑌=𝑊 and 𝑋=𝑉 in (73) and using (15) and (24) yield 𝑝(𝑉)=𝑣𝐴₁𝑈=𝑢𝐴₁𝑉. (77) Also, putting 𝑌=𝑉 and 𝑋=𝑊 in (74) and using (15) and (24), we have 𝑝(𝑊)=𝑤𝐴₁𝑈=𝑢𝐴₁𝑊. (78)

Summing up, we have 𝐴₁𝑈=𝑢𝐴₁𝑈(𝑉)(𝑊)𝑝(𝑉)𝐴𝑈+𝑝𝑉+𝑝𝑊,=𝑣₁𝑈=𝑢𝐴₁𝑉(𝑈),𝑝(𝑊)𝐴=𝑞=𝑤₁𝑈=𝑢𝐴₁𝑊=𝑟(𝑈). (79)

Thus we get the following lemma.

Lemma 1. Let 𝑀 be an 𝑛-dimensional 𝑄𝑅-submanifold of (𝑝−1)𝑄𝑅-dimension in a quaternionic projective space 𝑄𝑃^{(𝑛+𝑝)/4}, and let the normal vector field 𝑁₁ be parallel with respect to the normal connection. If the equalities in (69) are established, then 𝐴₁𝑈=𝑢𝐴₁𝑈(𝑉)(𝑊)𝐴𝑈+𝑝𝑉+𝑝𝑊,₁𝑉=𝑞(𝑈)𝑈+𝑣𝐴₁𝑉(𝑊)𝐴𝑉+𝑞𝑊,₁𝑊=𝑟(𝑈)𝑈+𝑟(𝑉)𝑉+𝑤𝐴₁𝑊𝑝(𝑉)𝐴𝑊,=𝑣₁𝑈=𝑢𝐴₁𝑉(𝑈),𝑝(𝑊)𝐴=𝑞=𝑤₁𝑈=𝑢𝐴₁𝑊(𝑈),𝑞(𝑊)𝐴=𝑟=𝑤₁𝑉=𝑣𝐴₁𝑊=𝑟(𝑉). (80)

Next, we assume the additional condition ^′∇_𝜉^′𝑅𝑌^∗,𝑈^∗𝑋^∗=0. (81)

Differentiating (67) covariantly in the direction of 𝜉 and using (36), (40) and the assumption (^′∇_𝜉^′𝑅)(𝑌^∗,𝑈^∗)𝑋^∗=0, we have −^′𝑅𝜙𝑌^∗,𝑈^∗𝑋^∗−^′𝑅𝑌^∗,𝑈^∗𝜙𝑋^∗=−𝑢(𝑋)𝜙𝑌−𝑢𝐴₁𝑋𝜙𝐴₁𝑌+𝑔(𝐴₁𝑌,𝑋)𝜙𝐴₁𝑈^∗−(𝑌)𝑤(𝑋)(𝑈)𝑔(𝑈)𝑔(𝑌)𝑢𝐴−𝑝+𝑞𝜙𝑌,𝑋−𝑝𝜓𝑌,𝑋+𝑤₁𝑋^∗𝜂+−𝑝(𝑌)𝑣(𝑋)−𝑟(𝑈)𝑔𝜙𝑌,𝑋+𝑝(𝑈)𝑔(𝜃𝑌,𝑋)+𝑣(𝑌)𝑢(𝐴₁𝑋)^∗𝜁, (82) from which, taking the vertical component of 𝜉,𝜂, and 𝜁, respectively, and using (22)–(24) and (67) itself, we can find 𝑟(𝑈()(𝑌)𝑣(𝑋)(𝑌)𝑢(𝑋(𝑈(𝑌)𝑤(𝑋)(𝑌)𝑢(𝑋(𝑌)𝑢𝐴)−2𝑔𝜃𝑌,𝑋+𝑢−𝑣)−𝑞)2𝑔𝜓𝑌,𝑋+𝑢−𝑤)+𝑢₁𝑤(𝑋)(𝑌)𝑣(𝑋)𝑣(𝑋)(𝑌)𝑤(𝑋)𝜙𝑋+𝑟𝜙𝑌+𝑟+𝑞𝜙𝑌−𝑞+𝑔𝜙𝐴₁𝑌−𝐴₁𝜙𝑌,𝑋=0,−𝑞(𝑈)𝑔𝜙𝑌,𝑋+𝑝𝜙𝑌𝑣(𝑋)−𝑣(𝑌)𝑢𝐴₁(𝑈𝑔(𝑌)𝑤(𝑋)(𝑌)𝑢(𝑋𝜙𝑋−𝑝)𝜓𝑌,𝑋+𝑢−𝑤)}=0,−𝑟(𝑈)𝑔𝜙𝑌,𝑋+𝑝𝜙𝑌𝑤(𝑋)−𝑤(𝑌)𝑢𝐴₁(𝑈𝑔()(𝑌)𝑢(𝑋)(𝑌)𝑣(𝑋𝜙𝑋−𝑝)𝜃𝑌,𝑋+𝑣−𝑢)=0. (83)(84)(85)

Taking the skew-symmetric part of (83) with respect to 𝑋 and 𝑌, we have (𝑈()(𝑌)𝑣(𝑋)(𝑌)𝑢(𝑋(𝑈(𝑌)𝑤(𝑋)(𝑌)𝑢(𝑋(𝑌)𝑢𝐴2𝑟)−2𝑔𝜃𝑌,𝑋+𝑢−𝑣)−2𝑞)2𝑔𝜓𝑌,𝑋+𝑢−𝑤)+𝑢₁𝜙𝑋−𝑢(𝑋)𝑢𝐴₁𝑤(𝑋)𝑤(𝑌)(𝑌)𝑣(𝑋)(𝑋)𝑣(𝑌)𝑣(𝑋)𝑣(𝑌)(𝑌)𝑤(𝑋)(𝑋)𝑤(𝑌)𝜙𝑌+𝑟𝜙𝑌−𝑟𝜙𝑋+𝑟−𝑟+𝑞𝜙𝑌−𝑞𝜙𝑋−𝑞+𝑞=0. (86) Replacing 𝑌 with 𝜃𝑌 in (86) and using (19) and (22)–(24), we have 𝑟(𝑈()(𝑌)𝑤(𝑋)(𝑌)𝑢(𝑋)(𝑌)𝑣(𝑋(𝑈(𝑌)𝑣(𝑋)(𝑋)𝑣(𝑌()𝑤(𝑋)𝑣(𝑋)𝑢(𝑌)()𝑣(𝑋)(𝑋)𝑢(𝑌)𝑤(𝑋)(𝑌)𝑢𝐴)4𝑔𝑌,𝑋−3𝑤−2𝑢−2𝑣)}−𝑞)4𝑔𝜙𝑌,𝑋+3𝑤−2𝑤)−𝑞𝜃𝑌−𝑞𝜓𝑌−𝑞𝜙𝑋+𝑟𝜃𝑌−𝑟−𝑟𝜓𝑌−𝑣₁𝜙𝑋+𝑢(𝑋)𝑢𝐴₁𝐴𝜓𝑌−𝑢₁𝑈𝑢(𝑋)𝑤(𝑌)=0. (87) Now we consider the following orthonormal basis: 𝑈,𝑉,𝑊,𝑒₁,…,𝑒_𝑚,𝜙𝑒₁,…,𝜙𝑒_𝑚,𝜓𝑒₁,…,𝜓𝑒_𝑚,𝜃𝑒₁,…,𝜃𝑒_𝑚, (88) which will be called 𝑄-basis, where 4𝑚+3=dim𝑀. Taking the trace of the above equation with respect to the 𝑄-basis and using (76), we can easily see 4𝑚𝑟(𝑈)=0; that is, 𝑟(𝑈)=0. (89)

Replacing also 𝑌 with 𝜓𝑌 in (86) and using (19) and (22)–(24), we have 𝑞(𝑈)=0. (90) Substituting (89) and (90) into (75), we have 𝜙𝐴₁𝑈=0 and hence 𝐴₁𝑈=𝑢𝐴₁𝑈𝑈. (91)

On the other hand, replacing 𝑌 with 𝜓𝑌 in (84) and using (19), (22), (23), and (90), we obtain 𝑝(𝑈𝑔()(𝑌)𝑢(𝑋)(𝑌)𝑤(𝑋()𝑣(𝑋))𝑌,𝑋−𝑢−𝑤)+𝑝𝜃𝑌=0, (92) from which, taking the trace with respect to the 𝑄-basis and using (15) and (24), we find 4𝑚𝑝(𝑈)=0; that is, 𝑝(𝑈)=0 (93) which together with (84) and (90) implies 𝑝𝜙𝑌=0. (94) Replacing 𝑌 with 𝜙𝑌 in the above equation and using (17) and (93), we can easily see that 𝑝(𝑌)=0 (95) for any vector 𝑌 tangent to 𝑀.

Summing up, we have the following lemma.

Lemma 2. Let 𝑀 be as in Lemma 1, and let the normal vector field 𝑁₁ be parallel with respect to the normal connection. If the equalities in (69) and (5.2) are established, then 𝐴₁𝑈=𝑢𝐴₁𝑈𝑈,𝐴₁𝑉=𝑣𝐴₁𝑉𝐴𝑉,₁𝑊=𝑤𝐴₁𝑊𝑊,𝑝=𝑞=𝑟=0. (96)

Finally, we will prove our main theorem.

Theorem 3. Let 𝑀 be an 𝑛-dimensional 𝑄𝑅-submanifold of (𝑝−1)𝑄𝑅-dimension in a quaternionic projective space 𝑄𝑃^{(𝑛+𝑝)/4}, and let the normal vector field 𝑁₁ be parallel with respect to the normal connection. If the equalities in (69) and (5.2) are established, then 𝜋⁻¹(𝑀) is locally a product of 𝑀₁×𝑀₂ where 𝑀₁ and 𝑀₂ belong to some (4𝑛₁+3)- and (4𝑛₂+3)-dimensional spheres (𝜋 is the Hopf fibration 𝑆^𝑛+𝑝+3(1)→𝑄𝑃^{(𝑛+𝑝)/4}).

Proof. By means of (96), it follows easily from (83) that 𝜙𝐴₁=𝐴₁𝜙. (97) By the quite same method, we can obtain 𝜓𝐴₁=𝐴₁𝜓,𝜃𝐴₁=𝐴₁𝜃. (98) Combining with those equalities and Theorem K-P, we complete the proof.

Corollary 4. Let 𝑀 be an 𝑛-dimensional 𝑄𝑅-submanifold of (𝑝−1)𝑄𝑅-dimension in a quaternionic projective space 𝑄𝑃^{(𝑛+𝑝)/4}, and let the normal vector field 𝑁₁ be parallel with respect to the normal connection. If the following equalities: ^′∇_𝜉^′𝑅=0,^′∇_𝜂^′𝑅=0,^′∇_𝜁^′𝑅=0 (99) are established, then 𝜋⁻¹(𝑀) is locally a product of 𝑀₁×𝑀₂ where 𝑀₁ and 𝑀₂ belong to some (4𝑛₁+3)- and (4𝑛₂+3)-dimensional spheres.

Acknowledgment

This work was supported by the 2013 Inje University research grant.