On Submersion of CR-Submanifolds of lcqK Manifold

International Scholarly Research Network ISRN Geometry Volume 2012, Article ID 309145, 13 pages doi:10.5402/2012/309145

Research Article On Submersion of CR-Submanifolds of l.c.q.K. Manifold Majid Ali Choudhary, Mahmood Jaafari Matehkolaee, and Mohd. Jamali Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India Correspondence should be addressed to Majid Ali Choudhary, majid [email protected] Received 27 September 2012; Accepted 15 October 2012 Academic Editors: A. Ferrandez and T. Friedrich Copyright q 2012 Majid Ali Choudhary et al. 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. We study submersion of CR-submanifolds of an l.c.q.K. manifold. We have shown that if an almost Hermitian manifold B admits a Riemannian submersion π : M → B of a CR-submanifold M of a locally conformal quaternion Kaehler manifold M, then B is a locally conformal quaternion Kaehler manifold.

1. Introduction The concept of locally conformal Kaehler manifolds was introduced by Vaisman in 1. Since then many papers appeared on these manifolds and their submanifolds see 2 for details. However, the geometry of locally conformal quaternion Kaehler manifolds has been studied in 2–4 and their QR-submanifolds have been studied in 5. A locally conformal quaternion Kaehler manifold shortly, l.c.q.K. manifold is a quaternion Hermitian manifold whose metric is conformal to a quaternion Kaehler metric in some neighborhood of each point. The main difference between locally conformal Kaehler manifolds and l.c.q.K. manifolds is that the Lee form of a compact l.c.q.K. manifold can be chosen as parallel form without any restrictions 2. The study of the Riemannian submersion π : M → B of a Riemannian manifold M onto a Riemannian manifold B was initiated by O’Neill 6. A submersion naturally gives rise to two distributions on M called the horizontal and vertical distributions, respectively, of which the vertical distribution is always integrable giving rise to the fibers of the submersion which are closed submanifold of M. The notion of Cauchy-Riemann CR submanifold was introduced by Bejancu 7 as a natural generalization of complex submanifolds and totally real submanifolds. A CR-submanifolds M of a l.c.q.K. manifold M requires a differentiable

2

ISRN Geometry

holomorphic distribution D, that is, Jx Dx Dx for all x ∈ M, whose orthogonal complement D⊥ is totally real distribution on M, that is, Jx Dx⊥ ⊂ T M⊥ for all x ∈ M. A CR-submanifold is called holomorphic submanifold if dim Dx⊥ 0, totally real if dim Dx 0 and proper if it is neither holomorphic nor totally real. A CR-submanifold of a l.c.q.K. manifold M is called a CR-product if it is Riemannian product of a holomorphic submanifold N and a totally real submanifold N ⊥ of M. Kobayashi 8 has proved that if an almost Hermitian manifold B admits a Riemannian submersion π : M → B of a CR-submanifold M of a Kaehler Manifold M, then B is a Kaehler manifold. However, Deshmukh et al. 9 studied similar type of results for CR-submanifolds of manifolds in different classes of almost Hermitian manifolds, namely, Hermitian manifolds, quasi-Kaehler manifolds, and nearly Kaehler manifolds. In the present paper, we investigate submersion of CR-submanifold of a l.c.q.K. manifold M and prove that if an almost Hermitian manifold B admits a Riemannian submersion π : M → B of a CR-submanifold M of a l.c.q.K. manifold M, then B is an l.c.q.K. manifold.

2. Preliminaries Let M, g, H be a quaternion Hermitian manifold, where H is a subbundle of end T M of rank 3 which is spanned by almost complex structures J1 , J2 , and J3 . The quaternion Hermitian metric g is said to be a quaternion Kaehler metric if its Levi-Civita connection ∇ satisfies ∇H ⊂ H. A quaternion Hermitian manifold with metric g is called a locally conformal quaternion Kaehler l.c.q.K. manifold if over neighborhoods {Ui } covering M, g|Ui efi gi where gi is a quaternion Kaehler metric on Ui . In this case, the Lee form ω is locally defined by ω|Ui dfi and satisfies 3 dθ w ∧ θ,

dw 0.

2.1

Let M be l.c.q.K. manifold and ∇ denotes the Levi-Civita connection of M. Let B be the Lee vector field given by gX, B wX.

2.2

Then for l.c.q.K. manifold, we have 3

X Ja Y 1 θY X − wY Ja X − gX, Y A − ΩX, Y B ∇ 2

2.3

Qab XJb Y Qac XJc Y for any X, Y ∈ T M, where Qab is skew-symmetric matrix of local forms θ w ◦ Ja and A −Ja B.

ISRN Geometry

3

We also have θX gJa X, B,

ΩX, Y gX, Ja Y .

2.4

Let M be a Riemannian manifold isometrically immersed in M. Let T M be the Lie algebra of vector fields in M and T M⊥ , the set of all vector fields normal to M. Denote by ∇ the Levi connection of M. Then the Gauss and Weingarten formulas are given by ∇X Y ∇X Y hX, Y ,

2.5

∇X N −AN X ∇⊥X N

2.6

for any X, Y ∈ T M, and N ∈ T M⊥ , where ∇⊥ is the connection in the normal bundle T M⊥ , h is the second fundamental form, and AN is the Weingarten endomorphism associated with N. The second fundamental form and shape operator are related by g AN X, Y ghX, Y , N.

2.7

The curvature tensor R of the submanifold M is related to the curvature tensor R of M by the following Gauss formula: RX, Y ; Z, W RX, Y ; Z, W ghX, Z, hY, W − ghX, W, hY, Z,

2.8

for any X, Y, Z, W ∈ T M. For submersion of a l.c.q.K. manifold onto an almost Hermitian manifold, we have the following. Definition 2.1. Let M be a CR-submanifold of a locally conformal quaternion Kaehler manifold M. By a submersion π : M → B of M onto an almost Hermitian manifold B, we mean a Riemannian submersion π : M → B together with the following conditions: i D⊥ is the kernel of π∗ , that is, π∗ D⊥ {0}, ∗ ∗ ii π∗ : Dp → Dπp is a complex isometry of the subspace Dp onto Dπp for every ∗ p ∈ M, where Dπp denotes the tangent space of B at πp,

iii J interchanges D⊥ and ν, that is, JD⊥ T M⊥ . For a vector field X on M, we set 8 X HX V X, where H and V denoted the horizontal and vertical part of X.

2.9

4

ISRN Geometry

We recall that a vector field X on M for submersion π : M → B is said to be a basic vector field if X ∈ D and X is π related to a vector field on B, that is, there is a vector field X∗ on B such that π∗ Xp X∗ πp

for each p ∈ M.

2.10

If J and J are the almost complex structures on M and B, respectively, then from Definition 2.1ii we have π∗ ◦ J J ◦ π∗ on D. We have the following lemma for basic vector fields 6. Lemma 2.2. Let X and Y be basic vector fields on M. Then i gX, Y g∗ X∗ , Y∗ ◦ π, g is the metric on M, and g∗ is the Riemannian metric on B; ii the horizontal part HX, Y of X, Y is a basic vector field and corresponds to X∗ , Y∗ , that is, π∗ HX, Y X∗ , Y∗ ◦ π;

2.11

iii H∇X Y is a basic vector field corresponding to ∇∗X∗ Y∗ , where ∇∗ is a Riemannian connection on B; iv X, W ∈ D⊥ for W ∈ D⊥ . ∗ for For a covariant differentiation operator ∇∗ , we define a corresponding operator ∇ basic vector fields of M by ∗ Y H∇X Y , ∇ X

X, Y ∈ D,

2.12

∗ Y is a basic vector field, and from the above lemma we have then ∇ X ∗ Y ∇∗ Y∗ ◦ π. π∗ ∇ X X∗

2.13

Now, we define a tensor field C on M by setting ∇X Y H∇X Y CX, Y ,

X, Y ∈ D,

2.14

that is, CX, Y is the vertical component of ∇X Y . In particular, if X and Y are basic vector fields, then we have ∗ Y CX, Y . ∇X Y ∇ X

2.15

The tensor field C is skew-symmetric and it satisfies CX, Y

1 V X, Y , 2

X, Y ∈ D.

2.16

ISRN Geometry

5

For X ∈ D and V ∈ D⊥ define an operator A on M by setting ∇X V ν∇X V AX V , that is, AX V is the horizontal component of ∇X V . Using iv of Lemma 2.2 we have H∇X V H∇V X AX V.

2.17

The operator C and A are related by gAX V, Y −gV, CX, Y ,

X, Y ∈ D, V ∈ D⊥ .

2.18

For a CR-submanifold M in a locally conformal quaternion Kaehler manifold M, we denote by ν the orthogonal complement of JD⊥ in T M⊥ . Hence, we have the following orthogonal decomposition of the normal bundle: T M⊥ JD⊥ ⊕ ν,

JD⊥ ⊥ ν.

2.19

Set P X tanJX, tN tanJN,

FX norJX, fN norJN,

for X ∈ T M, for N ∈ T M⊥ .

2.20

Here, tanx and norx are the natural projections associated with the orthogonal direct sum decomposition Tx M Tx M ⊕ T Mx⊥

2.21

for any x ∈ M.

Then the following identities hold: P 2 −I − tF,

FP tF 0,

P t tf 0,

f 2 −I − Ft,

2.22

where I is the identity transformation. We have following results. Lemma 2.3. Let M be a CR-submanifold in a l.c.q.K. manifold M. Then i holomorphic distribution D is integrable iff hX, Ja Y − hJa X, Y ΩX, Y nor B 0,

∀X, Y ∈ D

2.23

or equivalently, g hX, Ja Y − hJa X, Y ΩX, Y B, Ja Z 0,

∀X, Y ∈ D, Z ∈ D⊥ ;

2.24

6

ISRN Geometry ii anti-invariant distribution D⊥ of M is integrable iff AJa W T AJa T W,

∀W, T ∈ D⊥ .

2.25

Proof. i Using 2.3, we have ∇X Ja Y Pa ∇X Y ta hX, Y

1 θY X − wY Ja X − gX, Y tanA − ΩX, Y tanB 2

Qab XJb Y Qac XJc Y, hX, Ja Y Fa ∇X Y fa hX, Y −

1 gX, Y norA ΩX, Y norB . 2

2.26

From the second of these equations, we have hX, Ja Y − hY, Ja X ΩX, Y norB Fa X, Y ,

∀X, Y ∈ D.

2.27

If we need D to be integrable, we have hX, Ja Y − hY, Ja X ΩX, Y norB 0

2.28

or ghX, Ja Y − hY, Ja X ΩX, Y B, Ja Z 0,

∀Z ∈ D⊥ .

2.29

ν-part of hX, Ja Y − hJa X, Y ΩX, Y B vanishes for all X, Y ∈ D. ii We have ∇X Ja Y Ja ∇X Y

1 θY X − ωY Ja X − ΩX, Y B gX, Y Ja B 2

2.30

Qab X Jb Y Qac XJc Y. Then for any T , W ∈ D⊥ , and X ∈ D, we have 1 1 g ∇T Ja W, X g Ja ∇T W, X θWgT, X − ωWgJa T, X 2 2 1 1 − ΩT, WgB, X gT, WgJa B, X 2 2

2.31

Qab T gJb W, X Qac T gJc W, X

1 ⇒ −AJa W T, X − ∇T , Ja X − gT, WgB, Ja X. 2

2.32

ISRN Geometry

7

So, we have

1 AJa W T, X ∇T W, Ja X gT, WgB, Ja X, 2

1 AJa T W, X ∇W T, Ja X gW, T gB, Ja X. 2

2.33

From these two equations, we have

AJa W T − AJa T W, X ∇T W − ∇W T, Ja X

⇒ AJa W T − AJa T W, X W, T , Ja X.

2.34

So, we conclude that if AJa W T AJa T W then D⊥ is integrable. Converse is obvious. Lemma 2.4. Let M be a CR-submanifold of l.c.q.K. manifold. Then ∇X Ja Y ∇Y Ja X

2.35

iff Lee vector field B is orthogonal to anti-invariant distribution D⊥ . Proof. Since ∇ is metric connection, for X, Y ∈ D, and Z ∈ D⊥ , using 2.3, we have

1 1 ∇X Ja Y, Z Ja ∇X Y, Z θY X, Z − ΩX, Y gB, Z 2 2 1 1 − ωY gJa X, Z gX, Y gJa B, Z 2 2 Qab XgJb Y, Z Qac XgJcY, Z

2.36

1 1 − ∇X Y, Ja Z − ΩX, Y ωZ − gX, Y gB, Ja Z 2 2

1 1 Y, ∇X Ja Z − ΩX, Y ωZ − gX, Y gB, Ja Z 2 2 or ∇X Ja Y hX, Ja Y Z

⊥ 1 Y, −AJa Z X ∇X Ja Z − ΩX, Y ωZ 2 1 − gX, Y gB, Ja Z. 2

2.37

8

ISRN Geometry

This gives 1 1 ∇X Ja Y, Z − Y, AJa Z X − ΩX, Y ωZ − gX, Y gB, Ja Z, 2 2

1 1 ∇Y Ja X, Z − X, AJa Z Y − ΩY, XωZ − gY, XgB, Ja Z. 2 2

2.38

The above two equations give ∇X Ja Y − ∇Y Ja X, Z 1 1 − AJa Z X, Y X, AJa Z Y − ΩX, Y ωZ ΩY, XωZ 2 2

2.39

ΩY, XωZ or ∇X Ja Y − ∇Y Ja X, Z ΩY, XgB, Z.

2.40

This gives ∇X Ja Y ∇Y Ja X iff ωZ 0.

3. Submersions of CR-Submanifolds On a Riemannian manifold M, a distribution S is said to be parallel if ∇X Y ∈ S, X, Y ∈ S, where ∇ is a Riemannian connection on M. It is proved earlier that horizontal distribution D is integrable. If, in addition, D⊥ is parallel, then we prove the following. Proposition 3.1. Let π : M → B be a submersion of a CR-submanifold M of a locally conformal quaternion Kaehler manifold M onto an almost Hermitian manifold B. If (horizontal distribution) D is integrable and (vertical distribution) D⊥ is parallel, then M is a CR-product (Rienannian product M1 × M2 , where M1 is an invariant submanifold and M2 is a totally real submanifold of M). Proof. Since the horizontal distribution D is integrable for X, Y ∈ D, we have X, Y ∈ D. Therefore, V X, Y 0. Then from 2.16, we have CX, Y 0,

∀X, Y ∈ D.

3.1

Thus, from the definition of C, we have ∗ Y ∈ D, ∇X Y ∇ X

that is, D is parallel.

3.2

Since D and D⊥ are both parallel, using de Rham’s theorem, it follows that M is the product M1 × M2 , where M1 is invariant submanifold of M and M2 is totally real submanifold of M. Hence, M is a CR-product.

ISRN Geometry

9

In 10, Simons defined a connection and an invariant inner product on HT, V HomT M, V M, where V M is vector bundle over M and T M be tangent bundle of M. In fact, if r, s ∈ HT, V m , we set

r, s

p

rei , sei ,

where {ei } is a frame in T Mm .

3.3

i1

Define Qab X DJa , Jb , which implies Qab XJa Y DJa , Jb Ja Y . Let D be 4n dimensional distribution whose basis is given by {e1 , . . . , en , ea1 , . . . , ean , eb1 , . . . , ebn , ec1 , . . . , ecn } where eai Ja ei , ebi Jb ei , eci Jc ei and Ja ◦ Jb Jc , Jb ◦ J c J a , J c ◦ J a J b . Now, component of Qab X is defined as follows: n n

DX Ja ei , Jb ei DX Ja eai , Jb eai Qab X DX Ja , Jb i1

n

i1

DX Ja ebi , Jb ebi

i1

q

n

DX Ja eci , Jb eci

3.4

i1

D X Ja ej , Jb ej .

j1

So, we have

Qab X

n

n

DX Ja ei , Jb ei DX Ja Ja ei , Jb Ja ei

i1

i1

n

n

DX Ja Jb ei , Jb Jb ei DX Ja Jc ei , Jb Jc ei

i1

q

i1

D X Ja ej , Jb ej

j1

n

DX Ja ei , Jb ei

i1

−

n

n

n

i1

DX ei , Jc ei

i1

i1

n

DX Ja ei , Jb ei

i1

n n

DX Jc ei , ei − DX Ja ei , Jb ei − DX Jb ei , Ja ei i1

i1

q

DX Ja ei , Jb ei D X Ja ej , Jb ej i1

3.5

10

ISRN Geometry

or

Qab XJb Y

n

n

DX Ja ei , Jb ei Jb Y DX ei , Jc ei Jb Y

i1

−

i1

n

n

D X Ja ei , Jb ei Jb Y −

i1

i1

DX Jc ei , ei Jb Y

3.6

q n

D X Jb ei , Ja ei Jb Y DX Ja ej , Jb ej Jb Y. i1

i1

Applying π ∗ and using Lemma 2.2, we get

π ∗ Qab XJb Y

n

n

∗ ∗ DX∗ Ja ei∗ , Jb ei∗ Jb Y∗ DX∗ ei∗ , Jc ei∗ Jb Y∗

i1

i1

n

n

∗ ∗ DX∗ Ja ei∗ , Jb ei∗ Jb Y∗ − DX∗ Jc ei∗ , ei∗ Jb Y∗

i1

−

i1

n

q

∗

∗ ∗ DX∗ Jb ei∗ , Ja ei∗ Jb Y∗ DX∗ Ja ej∗ , Jb ej∗ Jb Y∗ DX∗ Ja , Jb Jb Y∗

i1

i1

3.7

or ∗ π ∗ Qab XJb Y Qab X∗ Jb∗ Y∗ .

3.8

Now, we prove the main result of this paper. Theorem 3.2. Let M be an l.c.q.K. manifold and M be a CR-submanifold of M. Let B be an almost Hermitian manifold and π : M → B be a submersion. Then B is an l.c.q.K. manifold. Proof. Let X, Y ∈ D be basic vector fields. Then from 2.5 and 2.15, we have ∗ Y CX, Y hX, Y . ∇X Y ∇ X

3.9

Replacing Y by Ja Y in 3.9, we have ∗ Ja Y CX, Ja Y hX, Ja Y . ∇X Ja Y ∇ X

3.10

ISRN Geometry

11

Using 2.3, we get ∗ Ja Y CX, Ja Y hX, Ja Y ∇ X ∗ Y 1 θY X − ωY Ja X − ΩX, Y B gX, Y Ja B Ja ∇ X 2

3.11

Qab XJb Y Qac XJc Y or ∗ Ja Y CX, Ja Y hX, Ja Y Ja ∇ ∗ Y Ja CX, Y Ja hX, Y ∇ X X


3.12

Qab XJb Y Qac XJc Y.

Thus, we have

∗ Ja Y CX, Ja Y hX, Ja Y − Ja CX, Y − Ja hX, Y ∇ X −


3.13

− Qab XJb Y − Qac XJc Y 0.

Comparing horizontal, vertical, and normal components in the above equation to get

∗ Ja Y − 1 θY X − ωY Ja X − ΩX, Y B gX, Y Ja B ∇ X 2

3.14

− Qab XJa Y − Qac XJc Y 0, CX, Ja Y Ja hX, Y ,

3.15

hX, Ja Y Ja CX, Y

3.16

from 3.14, we have ∗ Ja Y − Ja ∇ ∗ Y − 1 gJa Y, BX − gB, Y Ja X − gX, Ja Y B gX, Y Ja B ∇ X X 2 − Qab XJa Y − Qac XJc Y 0.

3.17

12

ISRN Geometry

Then for any X∗ , Y∗ ∈ χB, and J being almost complex structure on B, we have after operating π ∗ on the above equation ∇∗X∗ Ja Y∗ −Ja ∇∗X∗ Y∗ −

1 g∗ Ja Y∗ , B∗ X∗ −g∗ B∗ , Y∗ Ja X∗ −g∗ X∗ , Ja Y∗ B∗ g∗ X∗ , Y∗ Ja B∗ 2

∗ ∗ − Qab X∗ Ja∗ Y∗ − Qac X∗ Jc∗ Y∗ 0.

3.18

This gives

1 ∇∗X∗ Ja Y∗ − θ Y∗ X∗ − ω Y∗ Ja X∗ − Ω X∗ , Y∗ B∗ g∗ X∗ , Y∗ Ja B∗ 2 −

Qab X∗ Ja Y∗

−

Qac X∗ Jc Y∗

3.19

0.

This shows that B is l.c.q.K. manifold. Now, using 2.17 and 2.18, we obtain a relation between curvature tensor R on M and curvature tensor R∗ of B as follows: RX, Y, Z, W R∗ X∗ , Y∗ , Z∗ , W∗ − gCY, Z, CX, W gCX, Z, CY, W 2gCX, Y , CZ, W,

3.20

where π∗ X X∗ , π∗ Y Y∗ , π∗ Z Z∗ , and π∗ W W∗ ∈ B. Now, using the above equation together with 2.8 and using the fact that C is skewsymmetric, we obtain HX RX, Ja X, Ja X, X H ∗ X∗ hX, Ja X2 − ghJa X, Ja X, hX, X

3.21

− 3CX, Ja X2 , where HX and H ∗ X∗ are the holomorphic sectional curvature tensors of M and B, respectively. If we assume that D is integrable then using Lemma 2.3i, we have hJa X, Ja X −hX, X.

3.22

Also from 3.15, we have CX, Ja X 0. Then, 3.21 reduces to HX H ∗ X∗ hX, Ja X2 hX, X2 , This gives HX ≥ H ∗ X∗ . Thus, we have the following result.

∀X ∈ D.

3.23

ISRN Geometry

13

Theorem 3.3. Let M be a CR-submanifold of a l.c.q.K. manifold M with integrable D. Let B be an almost Hermitian manifold and π : M → B be a submersion. Then holomorphic sectional curvatures H and H ∗ of M and B, respectively, satisfy HX ≥ H ∗ X∗ ,

for all unit vectors X ∈ D.

3.24

Note. The above result was obtained in 9 by taking M to be quasi-Kaehler manifold. Later, similar type of relation was derived in 11, considering M to be l.c.K manifold.

Acknowledgments The first author is thankful to the Department of Science and Technology, Government of India, for its financial assistance provided through INSPIRE fellowship no. DST/INSPIRE Fellowship/2009/XXV to carry out this research work.

References 1 I. Vaisman, “On locally conformal almost kähler manifolds,” Israel Journal of Mathematics, vol. 24, no. 3-4, pp. 338–351, 1976. 2 S. Dragomir and L. Ornea, Locally Conformal Kähler Geometry, vol. 155 of Progress in Mathematics, Birkhäauser, Boston, Mass, USA, 1998. 3 L. Ornea and P. Piccinni, “Locally conformal kähler structures in quaternionic geometry,” Transactions of the American Mathematical Society, vol. 349, no. 2, pp. 641–655, 1997. 4 L. Ornea, “Weyl structure on quaternioric manifolds, a state of the art,” http://arxiv.org/abs/ math/0105041. 5 B. Sahin and R. Gunes, “QR-submanifolds of a locally conformal quaternion kaehler manifold,” ¨ Publicationes Mathematicae Debrecen, vol. 63, no. 1-2, pp. 157–174, 2003. 6 B. O’Neill, “The fundamental equations of a submersion,” The Michigan Mathematical Journal, vol. 13, pp. 459–469, 1966. 7 A. Bejancu, “CR submanifolds of a kaehler manifold. I,” Proceedings of the American Mathematical Society, vol. 69, no. 1, pp. 135–142, 1978. 8 S. Kobayashi, “Submersions of CR submanifolds,” The Tohoku Mathematical Journal, vol. 39, no. 1, pp. 95–100, 1987. 9 S. Deshmukh, T. Ghazal, and H. Hashem, “Submersions of CR-submanifolds on an almost hermitian manifold,” Yokohama Mathematical Journal, vol. 40, no. 1, pp. 45–57, 1992. 10 J. Simons, “Minimal varieties in riemannian manifolds,” Annals of Mathematics, vol. 88, no. 1, pp. 62– 105, 1968. 11 R. Al-Ghefari, M. H. Shahid, and F. R. Al-Solamy, “Submersion of CR-submanifolds of locally conformal kaehler manifold,” Contributions to Algebra and Geometry, vol. 47, no. 1, pp. 147–159, 2006.

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