research article generic submanifolds of nearly kaehler...

Research ArticleGeneric Submanifolds of Nearly Kaehler Manifolds withCertain Parallel Canonical Structure

Qingqing Zhu and Biaogui Yang

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350108, China

Correspondence should be addressed to Biaogui Yang; [email protected]

Received 30 June 2014; Accepted 17 August 2014; Published 29 October 2014

Academic Editor: Francesco Tornabene

Copyright © 2014 Q. Zhu and B. Yang.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds, and CR-submanifolds. In this paper we initiate the study of generic submanifolds in a nearly Kaehler manifold from differential geometricpoint of view. Some fundamental results in this paper will be obtained.

1. Introduction

Nearly Kaehler manifolds have been studied intensively inthe 1970’s by Gray [1]. These nearly Kaehler manifolds arealmost Hermitianmanifolds with almost complex structure 𝐽for which the tensor field∇𝐽 is skew-symmetric. In particular,the complex structure is nonintegrable if the manifold isnon-Kaehler. As we all know, there are two natual types ofsubmanifolds of nearly Kaehler (or more generally, almostHermitian)manifold, namely, almost complex and totally realsubmanifolds. Almost complex submanifolds are submani-folds whose tangent spaces are invariant under 𝐽 and totallyreal submanifolds are opposite. A well known example is thenearly Kaehler 6-dimensional sphere which has been studiedby many authors (see, e.g., [2–7]).

In 1981, Chen introduced preliminary the differentialgeometry of real submanifolds in a Kaehler manifold ([8])and gave some basic formulas and definitions. Inspired bythat paper, we will generalize some important formulasand proprerties in a Kaehler manifold to a nearly Kaehlermanifold. The paper is organized as follows: the basic onnearly Kaehler manifolds and submanifold theory will berecapitulated in Section 2. In Section 3, we give the inte-grability conditions of the two natural distributions H andH⊥ associated with a generic submanifold of nearly Kaehlermanifold. Finally, we consider generic submanifolds with oneof its canonical structures to be parallel. These results enableus to prove the following theorem.

Theorem 1. Let 𝑀 be a generic submanifold in a nearlyKaehler manifold 𝑀. If 𝑃 (or 𝐹) is parallel, then the holomor-phic distributionH is intergrable.

The operator 𝑃 (or 𝐹) is a canonical structure as thefollowing paper introduced.

2. Preliminaries

An almost Hermitian manifold (𝑀, 𝑔, 𝐽) is a manifoldendowedwith an almost complex structure 𝐽, that is, compat-ible with the metric 𝑔, that is, an endomorphism 𝐽 : 𝑇𝑀 →

𝑇𝑀 such that 𝐽2𝑃= −𝐼𝑑 for every 𝑝 ∈ 𝑀 and 𝑔(𝐽𝑋, 𝐽𝑌) =

𝑔(𝑋, 𝑌). A nearly Kaehler manifold is an almost Hermitianmanifold with the extra condition that the (1,2)-tensor field𝐺 = ∇𝐽 is skew-symmetric:

(∇𝑋𝐽) 𝑌 + (∇

𝑌𝐽)𝑋 = 0, (1)

for every 𝑋,𝑌 ∈ 𝑇𝑀. Here ∇ stands for the Levi-Civitaconnection of the metric 𝑔. The tensor field 𝐺 on𝑀 satisfiesthe following properties ([1, 2]):

𝐺 (𝑋, 𝑌) = − 𝐺 (𝑌,𝑋) , (2)

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014, Article ID 363429, 5 pageshttp://dx.doi.org/10.1155/2014/363429

2 International Scholarly Research Notices

𝐺 (𝑋, 𝐽𝑌) = − 𝐽𝐺 (𝑋, 𝑌) , (3)

𝑔 (𝐺 (𝑋, 𝑌) , 𝑍) = − 𝑔 (𝐺 (𝑋, 𝑍) , 𝑌) , (4)

𝑔 (𝐺 (𝑋, 𝑌) , 𝑍) = 𝑔 (𝐺 (𝑌, 𝑍) , 𝑋) = 𝑔 (𝐺 (𝑍,𝑋) , 𝑌) , (5)

where𝑋,𝑌, and 𝑍 are arbitrary vector fields on𝑀.We denote themetrics of𝑀 and its submanifold𝑀 by the

same letter 𝑔, 𝑇𝑀 is the tangent bundle of 𝑀, and 𝑇⊥𝑀 is

the normal bundle of𝑀. If ∇ and ∇⊥ denote the Riemannianconnection induced on𝑀 and the connection in the normalbundle 𝑇

⊥𝑀, respectively, then the Gauss and Weingarten

formulas are

∇𝑋𝑌 = ∇

𝑋𝑌 + 𝜎 (𝑋, 𝑌) , (6)

∇𝑋𝜉 = −𝐴

𝜉𝑋 + ∇

⊥

𝑋𝜉, (7)

where 𝑋,𝑌 ∈ 𝑇𝑀 and 𝜉 ∈ 𝑇⊥𝑀. The second fundamental

form 𝜎 and the shape operator𝐴𝜉are related to each other by

𝑔 (𝜎 (𝑋, 𝑌) , 𝜉) = 𝑔 (𝐴𝜉𝑋,𝑌) . (8)

For any vector field𝑋 tangent to𝑀, we put

𝐽𝑋 = 𝑃𝑋 + 𝐹𝑋, (9)

where 𝑃𝑋 and 𝐹𝑋 are the tangential and normal componentsof 𝐽𝑋, respectively.Then𝑃 is an endomorphismof the tangentbundle 𝑇𝑀 and 𝐹 is a normal-bundle-valued 1-form on 𝑇𝑀.For any vector field 𝜉 normal to𝑀, we put

𝐽𝜉 = 𝑡𝜉 + 𝑓𝜉, (10)

where 𝑡𝜉 and𝑓𝜉 are the tangential and normal components of𝐽𝜉, respectively. Then 𝑓𝜉 is an endomorphism of the normalbundle 𝑇

⊥𝑀 and 𝑡 is a tangent-bundle-valued 1-form on

𝑇⊥𝑀.For a submanifold𝑀 in a nearly Kaehler manifold𝑀 we

define

H𝑥= 𝑇𝑥𝑀∩ 𝐽𝑇

𝑥𝑀, (11)

the holomorphic tangent space of𝑀 at 𝑥.H𝑥is the maximal

complex subspace of 𝑇𝑥𝑀 which is contained in 𝑇

𝑥𝑀.

Similar to [8], we will give several definitions as follows.

Definition 2. A submanifold 𝑀 in a nearly Kahler manifold(or in an almost complex manifold in general) is called ageneric submanifold if dim H

𝑥is constant along 𝑀 and

H𝑥defines a differentiable distribution on 𝑀, called the

holomorphic distribution.

Definition 3. A generic submanifold 𝑀 in a nearly Kaehlermanifold is a totally real (resp., complex) submanifold if𝐽𝑇𝑀 ⊆ 𝑇

⊥𝑀 (resp., 𝐽𝑇𝑀 = 𝑇𝑀).

For a generic submanifold𝑀 in a nearlyKaehlermanifold𝑀, the orthogonal complementary distribution H⊥, calledthe purely real distribution, satisfies

H𝑥⊥ H⊥

𝑥, 𝑃H

⊥

𝑥⊆ H⊥

𝑥,

H⊥

𝑥∩ 𝐽H

⊥

𝑥= {0} .

(12)

From (9) it is clear that the normal-bundle-valued 1-form 𝐹

induces an isomorphism from H⊥𝑥onto 𝐹H⊥

𝑥. Let V

𝑥be the

vector space of holomorphic normal vectors to 𝑀 at 𝑥, orsimply the holomorphic normal space of𝑀 at 𝑥; that is,

V𝑥= 𝑇⊥

𝑥𝑀∩ 𝐽𝑇

⊥

𝑥𝑀. (13)

Then V𝑥defines a differentiable vector subbundle of𝑇⊥𝑀. we

have that𝑇⊥𝑀 = 𝐹H

⊥⊕ V, 𝑡 (𝑇

⊥𝑀) = H

⊥, (14)

𝑔 (𝐹H⊥, V) = 0. (15)

3. Integrability

In this section we study the integrability of the holomorphicdistributionH and the purely real distributionH⊥. First wegive the following.

Lemma 4. Let𝑀 be a generic submanifold in a nearly Kaehlermanifold𝑀. Then

𝑔 (𝜎 (𝑋, 𝐽𝑌) − 𝜎 (𝐽𝑋, 𝑌) , 𝜂) = 𝑔 (2𝐺(𝑋, 𝑌)⊥, 𝜂) , (16)

for any vector𝑋,𝑌 ∈ H and 𝜂 ∈ V.

Proof. From (2) and (6), we obtain

𝑔 (𝐽𝜎 (𝑋,𝑈) , 𝜂) = 𝑔 (𝐽∇𝑈𝑋, 𝜂)

= 𝑔 (∇𝑈𝐽𝑋 − 𝐺(𝑈,𝑋)

⊥, 𝜂)

= 𝑔 (𝜎 (𝐽𝑋,𝑈) + 𝐺(𝑋,𝑈)⊥, 𝜂) ,

(17)

where𝑋 ∈ H, 𝑈 ∈ 𝑇𝑀, and 𝜂 ∈ V. This implies that𝑔 (𝐽𝜎 (𝑋, 𝑌) , 𝜂) = 𝑔 (𝜎 (𝐽𝑋, 𝑌) + 𝐺(𝑋, 𝑌)

⊥, 𝜂) , (18)

where 𝑋,𝑌 ∈ H and 𝜂 ∈ V. Since the second fundamentalform 𝜎 is symmetric, we have𝑔 (𝜎 (𝐽𝑋, 𝑌) + 𝐺(𝑋, 𝑌)

⊥, 𝜂) = 𝑔 (𝜎 (𝐽𝑌,𝑋) + 𝐺(𝑌,𝑋)

⊥, 𝜂) .

(19)From the equations above, we prove the lemma.

Proposition 5. Let 𝑀 be a generic submanifold in a nearlyKaehler manifold 𝑀. Then the holomorphic distribution H isintegrable if and only if

𝑔 (𝜎 (𝑋, 𝐽𝑌) − 𝜎 (𝐽𝑋, 𝑌) , 𝐹𝑍) = 𝑔 (2𝐺(𝑋, 𝑌)⊥, 𝐹𝑍) , (20)

for any vector fields𝑋, 𝑌 ∈ H, and 𝑍 ∈ H⊥.

Proof. Since 𝑀 is nearly Kaehlerian, using formulas (2) and(6), we have

𝜎 (𝑋, 𝐽𝑌) − 𝜎 (𝐽𝑋, 𝑌)

= ∇𝑋𝐽𝑌 − ∇

𝑋𝐽𝑌 − ∇

𝑌𝐽𝑋 + ∇

𝑌𝐽𝑋

= 𝐺 (𝑋, 𝑌) + 𝐽∇𝑋𝑌 − 𝐺 (𝑌,𝑋)

− 𝐽∇𝑌𝑋 + ∇

𝑌𝐽𝑋 − ∇

𝑋𝐽𝑌

= 2𝐺 (𝑋, 𝑌) + 𝐽 [𝑋, 𝑌] + ∇𝑌𝐽𝑋 − ∇

𝑋𝐽𝑌.

(21)

International Scholarly Research Notices 3

So we get

𝜎 (𝑋, 𝐽𝑌) − 𝜎 (𝐽𝑋, 𝑌) − 2𝐺 (𝑋, 𝑌)

= 𝐽 [𝑋, 𝑌] + ∇𝑌𝐽𝑋 − ∇

𝑋𝐽𝑌,

(22)

for any vector field𝑋,𝑌 inH. If the holomorphic distributionH is integrable, the right-hand-side of (22) lies in 𝑇𝑀; thuswe obtain 𝜎(𝑋, 𝐽𝑌) − 𝜎(𝐽𝑋, 𝑌) = 2𝐺(𝑋, 𝑌)

⊥. In particular,we have (20). Conversely, if (20) holds, then by Lemma 4 and(14) we have 𝜎(𝑋, 𝐽𝑌)−𝜎(𝐽𝑋, 𝑌) = 2𝐺(𝑋, 𝑌)

⊥ for any vectors𝑋,𝑌 in H. Thus by (22) we obtain 𝐽[𝑋, 𝑌] = ∇

𝑋𝐽𝑌 − ∇

𝑌𝐽𝑋.

Since ∇𝑋𝐽𝑌 − ∇

𝑌𝐽𝑋 is tangent to 𝑀, this implies that [𝑋, 𝑌]

lies inH.Thuswe proved the proposition from the Frobeniustheorem.

Proposition 6. Let 𝑀 be a generic submanifold in a nearlykaehler manifold 𝑀. Then the purely real distribution H⊥ isintegrable if and only if

𝑃 {𝐴𝐹𝑊

𝑍 − 𝐴𝐹𝑍𝑊+ ∇

𝑊𝑃𝑍 − ∇

𝑍𝑃𝑊} − 2𝐽𝐺 (𝑊,𝑍) (23)

lies inH⊥, for any vector fields 𝑍,𝑊 ∈ H⊥.

Proof. For any vector fields 𝑍,𝑊 ∈ H⊥, (6) gives

𝐽 (∇𝑍𝑊) = 𝐽∇

𝑍𝑊+ 𝐽𝜎 (𝑍,𝑊) , (24)

and it follows immediately from (6), (7), and (9) that

𝐽 (∇𝑍𝑊) = ∇

𝑍𝐽𝑊 − 𝐺 (𝑍,𝑊)

= ∇𝑍𝑃𝑊 + ∇

𝑍𝐹𝑊 − 𝐺 (𝑍,𝑊)

= ∇𝑍𝑃𝑊 + 𝜎 (𝑍, 𝑃𝑊) − 𝐴

𝐹𝑊𝑍

+ 𝐷𝑍𝐹𝑊 − 𝐺 (𝑍,𝑊) .

(25)

From which we obtain

− ∇𝑍𝑊− 𝜎 (𝑍,𝑊) = 𝐽∇

𝑍𝑃𝑊 + 𝐽𝜎 (𝑍, 𝑃𝑊)

− 𝐽𝐴𝐹𝑊

𝑍 + 𝐽𝐷𝑍𝐹𝑊 − 𝐽𝐺 (𝑍,𝑊) .

(26)

Comparing the tangential parts, we have

∇𝑍𝑊 = −𝑃∇

𝑍𝑃𝑊 − 𝑡𝜎 (𝑍, 𝑃𝑊)

+ 𝑃𝐴𝐹𝑊

𝑍 − 𝑡𝐷𝑍𝐹𝑊 − 𝐺(𝑍, 𝐽𝑊)

⊤.

(27)

Thus we get

[𝑍,𝑊] = 𝑃 {𝐴𝐹𝑊

𝑍 − 𝐴𝐹𝑍𝑊+ ∇

𝑊𝑃𝑍 − ∇

𝑍𝑃𝑊}

+ 𝑡 {𝜎 (𝑊, 𝑃𝑍) − 𝜎 (𝑍, 𝑃𝑊) + 𝐷𝑊𝐹𝑍 − 𝐷

𝑍𝐹𝑊}

− 2𝐽𝐺 (𝑊,𝑍) .

(28)

Since 𝑡(𝑇⊥𝑀) = H⊥, this implies that [𝑍,𝑊] lies in H⊥ ifand only if (23) lies inH⊥. The proposition is proved.

Theorem 7. Let 𝑀 be a generic submanifold in a nearlyKaehler manifold𝑀. IfH is integrable and its leaves are totallygeodesic in𝑀, then

𝑔 ((𝐺 + 𝐽𝜎) (H,H) ,H⊥) = 0. (29)

Proof. SinceH is integrable and its leaves are totally geodesicin𝑀, we have ∇

𝑋𝑌 ∈ H, for any 𝑋,𝑌 ∈ H. So 𝑔(∇

𝑋𝑍,𝑌) =

𝑔(∇𝑋𝑌,𝑍) = 0, and we can get ∇

𝑋𝑍 ∈ H⊥ for any vector

𝑋 ∈ H and 𝑍 ∈ H⊥. From (2), (6), (7), and (9), we get

𝑔 (∇𝑋𝑍, 𝐽𝑌)

= −𝑔 (𝐽∇𝑋𝑍, 𝑌)

= 𝑔 (𝐺 (𝑋, 𝑍) − ∇𝑋𝐽𝑍, 𝑌)

= 𝑔 (𝐺 (𝑋, 𝑍) , 𝑌) − 𝑔 (∇𝑋𝑃𝑍, 𝑌) − 𝑔 (∇

𝑋𝐹𝑍, 𝑌)

= 𝑔 (𝐺 (𝑋, 𝑍) , 𝑌) − 𝑔 (∇𝑋𝑃𝑍, 𝑌) + 𝑔 (𝐴

𝐹𝑍𝑋,𝑌) ,

(30)

which implies that

− 𝑔 (𝐺 (𝑋, 𝑌) , 𝑍) + 𝑔 (𝜎 (𝑋, 𝑌) , 𝐽𝑍)

= −𝑔 (𝐺 (𝑋, 𝑌) + 𝐽𝜎 (𝑋, 𝑌) , 𝑍) = 0.(31)

This proves the theorem.

Theorem 8. Let 𝑀 be a generic submanifold in a nearlyKaehler manifold 𝑀. If H⊥ is integrable and its leaves aretotally geodesic in𝑀, then

𝑔 ((𝐺 − 𝐽𝜎) (H,H⊥) ,H⊥) = 0. (32)

Proof. Under the hypothesis, for any vector fields𝑋 inH and𝑍,𝑊 inH⊥, it follows from (3), (6), (7), and (9) that

𝑔 (∇𝑍𝑋,𝑊) = 𝑔 (𝐽∇

𝑍𝑋, 𝐽𝑊)

= 𝑔 (−𝐺 (𝑍,𝑋) + ∇𝑍𝐽𝑋, 𝑃𝑊 + 𝐹𝑊)

= 𝑔 (𝐺 (𝑋, 𝑍) , 𝐽𝑊) + 𝑔 (∇𝑍𝐽𝑋, 𝑃𝑊)

+ 𝑔 (∇𝑍𝐽𝑋, 𝐹𝑊)

= 𝑔 (𝐺 (𝑋, 𝑍) , 𝐽𝑊) + 𝑔 (𝜎 (𝐽𝑋, 𝑍) , 𝐹𝑊)

= −𝑔 (𝐽𝐺 (𝑋, 𝑍) + 𝐽𝜎 (𝐽𝑋, 𝑍) ,𝑊)

= 𝑔 (𝐺 (𝐽𝑋, 𝑍) − 𝐽𝜎 (𝐽𝑋, 𝑍) ,𝑊) .

(33)

That is,

𝑔 (𝐺 (𝐽𝑋, 𝑍) − 𝐽𝜎 (𝐽𝑋, 𝑍) ,𝑊) = 0. (34)

From this we obtain the theorem.

4. Generic Submanifolds with ParallelCanonical Structure

For the endomorphism 𝑃 : 𝑇𝑀 → 𝑇𝑀, we put

(∇𝑋𝑃)𝑌 = ∇

𝑋𝑃𝑌 − 𝑃∇

𝑋𝑌, (35)

4 International Scholarly Research Notices

for any vector fields𝑋,𝑌 ∈ 𝑇𝑀.The endomorphism 𝑃 is saidto be parallel if ∇

𝑋𝑃 = 0 for any vector 𝑋 ∈ 𝑇𝑀. From (6),

(7), and (9) we can obtain the following:

𝐽∇𝑋𝑌 + 𝐽𝜎 (𝑋, 𝑌) = 𝐽∇

𝑋𝑌

= ∇𝑋𝐽𝑌 − 𝐺 (𝑋, 𝑌)

= ∇𝑋𝑃𝑌 + ∇

𝑋𝐹𝑌 − 𝐺 (𝑋, 𝑌)

= ∇𝑋𝑃𝑌 + 𝜎 (𝑋, 𝑃𝑌)

− 𝐴𝐹𝑌𝑋 + 𝐷

𝑋𝐹𝑌 − 𝐺 (𝑋, 𝑌) .

(36)

That is,

𝑃∇𝑋𝑌 + 𝐹∇

𝑋𝑌 + 𝑡𝜎 (𝑋, 𝑌) + 𝑓𝜎 (𝑋, 𝑌)

= ∇𝑋𝑃𝑌 + 𝜎 (𝑋, 𝑃𝑌) − 𝐴

𝐹𝑌𝑋 + 𝐷

𝑋𝐹𝑌 − 𝐺 (𝑋, 𝑌) .

(37)

By comparing the tagential parts, we have the following:

(∇𝑋𝑃)𝑌 = 𝑡𝜎 (𝑋, 𝑌) + 𝐴

𝐹𝑌𝑋 + 𝐺(𝑋, 𝑌)

⊤. (38)

Therefore, for any vector fields𝑋,𝑌, 𝑍 ∈ 𝑇𝑀, we have

𝑔 ((∇𝑋𝑃)𝑌, 𝑍) = 𝑔 (𝑡𝜎 (𝑋, 𝑌) + 𝐴

𝐹𝑌𝑋 + 𝐺(𝑋, 𝑌)

⊤, 𝑍)

= −𝑔 (𝜎 (𝑋, 𝑌) , 𝐽𝑍) + 𝑔 (𝐴𝐹𝑌𝑋,𝑍)

+ 𝑔 (𝐺(𝑋, 𝑌)⊤, 𝑍)

= 𝑔 (𝐴𝐹𝑌𝑍 − 𝐴

𝐹𝑍𝑌 + 𝐺(𝑌, 𝑍)

⊤, 𝑋) .

(39)

So, we obtain the Lemma as follows.

Lemma 9. Let𝑀 be a generic submanifold in a nearly Kaehlermanifold𝑀. The P is parallel, that is, ∇𝑃 = 0, if and only if

𝐺(𝑈,𝑉)⊤= 𝐴𝐹𝑉𝑈 − 𝐴

𝐹𝑈𝑉, (40)

for any vectors 𝑈,𝑉 ∈ 𝑇𝑀.

Theorem 10. Let 𝑀 be a generic submanifold in a nearlyKaehler manifold𝑀. If 𝑃 is parallel, then

(i) 𝐺(𝑈,𝑋)⊤= −𝐴

𝐹𝑈𝑋, for any vector fields 𝑋 ∈ H and

𝑈 ∈ 𝑇𝑀

(ii) the holomorphic distributionH is intergrable.

Proof. From Lemma 9, for any vector fields 𝑋 ∈ H and 𝑈 ∈

𝑇𝑀, we know 𝐹𝑋 = 0; this implies that 𝐺(𝑈,𝑋)⊤= −𝐴

𝐹𝑈𝑋.

On the other hand, for any vector fields𝑋,𝑌 ∈ H,𝐺(𝑋, 𝑌)⊤ =−𝐴𝐹𝑋𝑌 = 0, then 𝐺(𝑋, 𝑌) is normal to𝑀. By (i), we can get

𝑔 (𝐺 (𝑈,𝑋) , 𝑌) = 𝑔 (−𝐴𝐹𝑈

𝑋,𝑌) ; (41)

that is,

𝑔 (𝐺 (𝑋, 𝑌) − 𝐽𝜎 (𝑋, 𝑌) , 𝑈) = 0, (42)

for any vector fields 𝑋,𝑌 ∈ H and 𝑈 ∈ 𝑇𝑀. The equationsabove imply that

𝑔 (𝜎 (𝑋, 𝑌) , 𝐹𝑍) = 0, (43)

for any vector fields𝑋,𝑌 ∈ H and 𝑍 ∈ H⊥. These give

𝑔 (𝜎 (𝑋, 𝐽𝑌) − 𝜎 (𝐽𝑋, 𝑌) , 𝐹𝑍) = 𝑔 (2𝐺(𝑋, 𝑌)⊥, 𝐹𝑍) . (44)

From Proposition 5, the theorem holds.

For the normal bundle-valued 1-form 𝐹, we put

(∇𝑋𝐹)𝑌 = ∇

𝑋𝐹𝑌 − 𝐹∇

𝑋𝑌. (45)

For any vector fields 𝑋,𝑌 ∈ 𝑇𝑀. The endomorphism 𝐹 issaid to be parallel if ∇

𝑋𝐹 = 0 for any vector 𝑋 ∈ 𝑇𝑀. By

comparing the normal parts of (37), we have the following:

(∇𝑋𝐹)𝑌 = 𝑓𝜎 (𝑋, 𝑌) − 𝜎 (𝑋, 𝑃𝑌) + 𝐺(𝑋, 𝑌)

⊥, (46)

for any vectors 𝑋,𝑌 ∈ 𝑇𝑀. Hence, for any vector field 𝜉 ∈

𝑇⊥𝑀, it follows from (4), (8), and (10) that

𝑔 ((∇𝑋𝐹)𝑌, 𝜉)

= 𝑔 (𝑓𝜎 (𝑋, 𝑌) − 𝜎 (𝑋, 𝑃𝑌) + 𝐺(𝑋, 𝑌)⊥, 𝜉)

= 𝑔 (𝐽𝜎 (𝑋, 𝑌) , 𝜉) − 𝑔 (𝜎 (𝑋, 𝑃𝑌) , 𝜉) + 𝑔 (𝐺 (𝑋, 𝑌) , 𝜉)

= −𝑔 (𝜎 (𝑋, 𝑌) , 𝑓𝜉) − 𝑔 (𝜎 (𝑋, 𝑃𝑌) , 𝜉) + 𝑔 (𝐺 (𝑋, 𝑌) , 𝜉)

= −𝑔 (𝐴𝑓𝜉𝑌 + 𝐴

𝜉𝑃𝑌 − 𝐺(𝑌, 𝜉)

⊤

, 𝑋) .

(47)

From which we obtain the Lemma as follows.

Lemma 11. Let𝑀 be a generic submanifold in a nearly Kaehlermanifold𝑀. The 𝐹 is parallel, that is, ∇𝐹 = 0, if and only if

𝐴𝑓𝜉𝑋 + 𝐴

𝜉𝑃𝑋 = 𝐺(𝑋, 𝜉)

⊤

, (48)

for any vectors𝑋 ∈ 𝑇𝑀 and 𝜉 ∈ 𝑇⊥𝑀.

Theorem 12. Let 𝑀 be a generic submanifold in a nearlyKaehler manifold 𝑀. If 𝐹 is parallel, then the holomorphicdistributionH is intergrable.

Proof. From (46) we have,

𝑔 ((∇𝑋𝐹)𝑌, 𝜉)

= 𝑔 (𝑓𝜎 (𝑋, 𝑌) − 𝜎 (𝑋, 𝑃𝑌) + 𝐺(𝑋, 𝑌)⊥, 𝜉)

= 𝑔 (𝐽𝜎 (𝑋, 𝑌) , 𝜉) − 𝑔 (𝜎 (𝑋, 𝑃𝑌) , 𝜉)

+ 𝑔 (𝐺(𝑋, 𝑌)⊥, 𝜉) ,

(49)

for any vectors 𝑋,𝑌 ∈ H and 𝜉 ∈ 𝑇⊥𝑀. Since 𝐹 is parallel,

then

𝑔 (𝐽𝜎 (𝑋, 𝑌) , 𝜉) − 𝑔 (𝜎 (𝑋, 𝑃𝑌) , 𝜉) + 𝑔 (𝐺(𝑋, 𝑌)⊥, 𝜉) = 0.

(50)

International Scholarly Research Notices 5

That is,

𝐽𝜎 (𝑋, 𝑌) = 𝜎 (𝑋, 𝐽𝑌) − 𝐺(𝑋, 𝑌)⊥. (51)

This implies that

𝑔 (𝜎 (𝑋, 𝐽𝑌) − 𝜎 (𝐽𝑋, 𝑌) , 𝐹𝑍) = 𝑔 (2𝐺(𝑋, 𝑌)⊥, 𝐹𝑍) , (52)

for any vectors 𝑋,𝑌 ∈ H and 𝑍 ∈ H⊥. From Proposition 5,the theorem holds.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the Grant no. 2011J05001 ofNSF of Fujian Province, China, and the Grant no. JA11052 ofthe Fund of the Education Department of Fujian Province,China, and was partially supported by Grant no. 11171139 ofNSFC.

References

[1] A. Gray, “Nearly Kahler manifold,” Journal of DifferentialGeometry, vol. 4, pp. 283–309, 1970.

[2] F. Belgun and A. Moroianu, “Nearly Kahler 6-manifolds withreduced holonomy,” Annals of Global Analysis and Geometry,vol. 19, no. 4, pp. 307–319, 2001.

[3] B. Chen and T. Nagano, “Totally geodesic submanifolds ofsymmetric spaces, I,” Duke Mathematical Journal, vol. 44, no.4, pp. 745–755, 1977.

[4] F.Dillen, L. Verstraelen, and L.Vrancken, “Almost complex sub-manifolds of a 6-dimensional sphere II,” Kodai MathematicalJournal, vol. 10, pp. 161–171, 1987.

[5] N. Ejiri, “Totally real submanifolds in a 6-sphere,” Proceedings ofthe American Mathematical Society, vol. 83, no. 4, pp. 759–763,1981.

[6] A. Gray, “Minimal varieties and almost Hermitian submani-folds,”MichiganMathematical Journal, vol. 12, pp. 273–287, 1965.

[7] A. Gray, “Almost complex submanifolds of the six sphere,”Proceedings of the American Mathematical Society, vol. 20, pp.277–279, 1969.

[8] B.-Y. Chen, “Differential geometry of real submanifolds in aKahlermanifold,”Monatshefte furMathematik, vol. 91, no. 4, pp.257–274, 1981.

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