International Scholarly Research Network ISRN Geometry Volume 01, Article ID 309145, 13 pages doi:10.540/01/309145 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 11005, India Correspondence should be addressed to Majid Ali Choudhary, majid alichoudhary@yahoo.co.in Received 7 September 01; Accepted 15 October 01 Academic Editors: A. Ferrandez and T. Friedrich Copyright q 01 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 for details. However, the geometry of locally conformal quaternion Kaehler manifolds has been studied in 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. 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
ISRN Geometry holomorphic distribution D,thatis,J x D x D x for all x M, whose orthogonal complement D is totally real distribution on M, thatis,j x D x TM for all x M. A CR-submanifold is called holomorphic submanifold if dim D x 0, totally real if dim D x 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.. Preliminaries Let M, g, H be a quaternion Hermitian manifold, where H is a subbundle of end TM of rank 3 which is spanned by almost complex structures J 1, J,andJ 3. 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 {U i } covering M, g Ui e f i g i where g i is a quaternion Kaehler metric on U i. In this case, the Lee form ω is locally defined by ω Ui df i and satisfies 3 dθ w θ, dw 0..1 Let M be l.c.q.k. manifold and denotes the Levi-Civita connection of M. LetB be the Lee vector field given by g X, B w X.. Then for l.c.q.k. manifold, we have 3 ( ) X J a Y 1 { θ Y X w Y Ja X g X, Y A Ω X, Y B } Q ab X J b Y Q ac X J c Y.3 for any X, Y TM, where Q ab is skew-symmetric matrix of local forms θ w J a and A J a B.
ISRN Geometry 3 We also have θ X g J a X, B, Ω X, Y g X, J a Y..4 Let M be a Riemannian manifold isometrically immersed in M. LetT M be the Lie algebra of vector fields in M and TM, 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 h X, Y, X N A N X X N.5.6 for any X, Y T M, andn TM, where is the connection in the normal bundle TM, h is the second fundamental form, and A N is the Weingarten endomorphism associated with N. The second fundamental form and shape operator are related by ( ) g A N X, Y g h X, Y,N..7 The curvature tensor R of the submanifold M is related to the curvature tensor R of M by the following Gauss formula: R X, Y ; Z, W R X, Y ; Z, W g h X, Z,h Y, W g h X, W,h Y, Z,.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.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,thatis,π D {0}, ii π : D p D π p is a complex isometry of the subspace D p onto D for every π p p M, where D denotes the tangent space of B at π p, π p iii J interchanges D and ν, thatis,jd TM. For a vector field X on M,weset 8 X HX VX,.9 where H and V denoted the horizontal and vertical part of X.
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 X p X π p for each p M..10 If J and J are the almost complex structures on M and B, respectively, then from Definition.1 ii we have π J J π on D. We have the following lemma for basic vector fields 6. Lemma.. Let X and Y be basic vector fields on M.Then i g X, Y g X,Y π, g is the metric on M, and g is the Riemannian metric on B; ii the horizontal part H X, Y of X, Y is a basic vector field and corresponds to X,Y, that is, π H X, Y X,Y π;.11 iii H X Y is a basic vector field corresponding to X Y,where connection on B; iv X, W D for W D. is a Riemannian For a covariant differentiation operator, we define a corresponding operator for basic vector fields of M by X Y H XY, X,Y D,.1 then XY is a basic vector field, and from the above lemma we have π ( X Y ) X Y π..13 Now, we define a tensor field C on M by setting X Y H X Y C X, Y, X,Y D,.14 that is, C X, Y is the vertical component of X Y. In particular, if X and Y are basic vector fields, then we have X Y XY C X, Y..15 The tensor field C is skew-symmetric and it satisfies C X, Y 1 V X, Y, X,Y D..16
ISRN Geometry 5 For X D and V D define an operator A on M by setting X V ν X V A X V,thatis, A X V is the horizontal component of X V.Using iv of Lemma. we have H X V H V X A X V..17 The operator C and A are related by g A X V, Y g V, C X, Y, X,Y D, V D..18 For a CR-submanifold M in a locally conformal quaternion Kaehler manifold M, we denote by ν the orthogonal complement of JD in TM. Hence, we have the following orthogonal decomposition of the normal bundle: TM JD ν, JD ν..19 Set PX tan JX, FX nor JX, for X TM, tn tan JN, fn nor JN, for N TM..0 Here, tan x and nor x are the natural projections associated with the orthogonal direct sum decomposition T x M T x M TM x for any x M..1 Then the following identities hold: P I tf, FP tf 0, Pt tf 0, f I Ft,. where I is the identity transformation. We have following results. Lemma.3. Let M be a CR-submanifold in a l.c.q.k. manifold M. Then i holomorphic distribution D is integrable iff h X, J a Y h J a X, Y Ω X, Y nor B 0, X, Y D.3 or equivalently, g h X, J a Y h J a X, Y Ω X, Y B, J a Z 0, X, Y D, Z D ;.4
6 ISRN Geometry ii anti-invariant distribution D of M is integrable iff A Ja WT A Ja TW, W, T D..5 Proof. i Using.3, we have X J a Y P a X Y t a h X, Y 1 { θ Y X w Y Ja X g X, Y tan A Ω X, Y tan B } Q ab X J b Y Q ac X J c Y, h X, J a Y F a X Y f a h X, Y 1 { g X, Y nor A Ω X, Y nor B }..6 From the second of these equations, we have h X, J a Y h Y, J a X Ω X, Y nor B F a X, Y, X, Y D..7 If we need D to be integrable, we have h X, J a Y h Y, J a X Ω X, Y nor B 0.8 or g h X, J a Y h Y, J a X Ω X, Y B,J a Z 0, Z D..9 ν-part of h X, J a Y h J a X, Y Ω X, Y B vanishes for all X, Y D. ii We have X J a Y J a X Y 1 { θ Y X ω Y Ja X Ω X, Y B g X, Y J a B } Q ab X J b Y Q ac X J c Y..30 Then for any T, W D,andX D, we have ( ) ( ) g T J a W, X g J a T W, X 1 θ W g T, X 1 ω W g J at, X A Ja WT, X 1 Ω T, W g B, X 1 g T, W g J ab, X Q ab T g J b W, X Q ac T g J c W, X T,J a X 1 g T, W g B, J ax..31.3
ISRN Geometry 7 So, we have AJa WT, X T W, J a X 1 g T, W g B, J ax, AJa TW, X W T, J a X 1 g W, T g B, J ax..33 From these two equations, we have AJa WT A Ja TW, X T W W T, J a X A Ja WT A Ja TW, X W, T,J a X..34 So, we conclude that if A Ja WT A Ja TW then D is integrable. Converse is obvious. Lemma.4. Let M be a CR-submanifold of l.c.q.k. manifold. Then X J a Y Y J a X.35 iff Lee vector field B is orthogonal to anti-invariant distribution D. Proof. Since is metric connection, for X, Y D,andZ D,using.3, we have X J a Y, Z J a X Y, Z 1 θ Y X, Z 1 Ω X, Y g B, Z 1 ω Y g J ax, Z 1 g X, Y g J ab, Z Q ab X g J b Y, Z Q ac X g JcY,Z X Y, J a Z 1 Ω X, Y ω Z 1 g X, Y g B, J az Y, X J a Z 1 Ω X, Y ω Z 1 g X, Y g B, J az.36 or X J a Y h X, J a Y Z Y, A Ja ZX XJ a Z 1 Ω X, Y ω Z 1 g X, Y g B, J az..37
8 ISRN Geometry This gives X J a Y, Z Y, A Ja ZX 1 Ω X, Y ω Z 1 g X, Y g B, J az, Y J a X, Z X, A Ja ZY 1 Ω Y, X ω Z 1 g Y, X g B, J az..38 The above two equations give X J a Y Y J a X, Z A Ja ZX, Y X, A Ja ZY 1 Ω X, Y ω Z 1 Ω Y, X ω Z.39 Ω Y, X ω Z or X J a Y Y J a X, Z Ω Y, X g B, Z..40 This gives X J a Y Y J a X iff ω Z 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 M 1 M,whereM 1 is an invariant submanifold and M 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.16, we have C X, Y 0, X, Y D. 3.1 Thus, from the definition of C, we have X Y XY D, that is,d is parallel. 3. Since D and D are both parallel, using de Rham s theorem, it follows that M is the product M 1 M, where M 1 is invariant submanifold of M and M 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 H T, V Hom T M,V M, where V M is vector bundle over M and T M be tangent bundle of M. In fact, if r, s H T, V m,weset r, s p r e i,s e i, where {e i } is a frame in T M m. 3.3 Define Q ab X DJ a,j b, which implies Q ab X J a Y DJ a,j b J a Y. Let D be 4n dimensional distribution whose basis is given by {e 1,...,e n,e a1,..., e an,e b1,...,e bn,e c1,...,e cn } where e ai J a e i, e bi J b e i, e ci J c e i and J a J b J c, J b J c J a, J c J a J b. Now, component of Q ab X is defined as follows: Q ab X D X J a,j b D X J a e i,j b e i D X J a e ai,j b e ai D X J a e bi,j b e bi D X J a e ci,j b e ci q ( ) ( ) D X J a ej,jb ej. j 1 3.4 So, we have Q ab X n ) D X J a e i,j b e i (D X J a J a e i,j b J a e i D X J a J b e i,j b J b e i D X J a J c e i,j b J c e i q ( ) ( ) D X J a ej,jb ej j 1 D X J a e i,j b e i D X e i,j c e i D X J a e i,j b e i n D X J c e i,e i D X J a e i,j b e i D X J b e i,j a e i q ( ) ( ) D X J a e i,j b e i D X J a ej,jb ej 3.5
10 ISRN Geometry or Q ab X J b Y D X J a e i,j b e i J b Y D X e i,j c e i J b Y n n D X J a e i,j b e i J b Y D X J c e i,e i J b Y n q ( ) ( ) D X J b e i,j a e i J b Y D X J a ej,jb ej J b Y. 3.6 Applying π and using Lemma., weget π Q ab X J b Y D X J ae i,j b e i J b Y D X e i,j ce i D X J ae i,j b e i J b Y D X J b e i,j ae i J b Y D X J ce i,e i J b Y q D X J ae j,j b e j J b Y J b Y D X J a,j b J b Y 3.7 or π Q ab X J b Y Q ab X J b Y. 3.8 Now, we prove the main result of this paper. Theorem 3.. Let M be an l.c.q.k. manifold and M be a CR-submanifold of M. LetB 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.5 and.15, we have X Y XY C X, Y h X, Y. 3.9 Replacing Y by J a Y in 3.9, we have X J a Y X J ay C X, J a Y h X, J a Y. 3.10
ISRN Geometry 11 Using.3, weget X J ay C X, J a Y h X, J a Y J a X Y 1 { θ Y X ω Y Ja X Ω X, Y B g X, Y J a B } 3.11 Q ab X J b Y Q ac X J c Y or X J ay C X, J a Y h X, J a Y J a X Y J ac X, Y J a h X, Y 1 { θ Y X ω Y Ja X Ω X, Y B g X, Y J a B } 3.1 Q ab X J b Y Q ac X J c Y. Thus, we have ( X J a) Y C X, J a Y h X, J a Y J a C X, Y J a h X, Y 1 { θ Y X ω Y Ja X Ω X, Y B g X, Y J a B } 3.13 Q ab X J b Y Q ac X J c Y 0. Comparing horizontal, vertical, and normal components in the above equation to get ( X a) J Y 1 { θ Y X ω Y Ja X Ω X, Y B g X, Y J a B } Q ab X J a Y Q ac X J c Y 0, C X, J a Y J a h X, Y, h X, J a Y J a C X, Y 3.14 3.15 3.16 from 3.14, we have X J ay J a X Y 1 { g Ja Y, B X g B, Y J a X g X, J a Y B g X, Y J a B } Q ab X J a Y Q ac X J c Y 0. 3.17
1 ISRN Geometry Then for any X,Y χ B, andj being almost complex structure on B, we have after operating π on the above equation X J ay J a X Y 1 { ( ) g J a Y,B X g B,Y J ax ( g X,J ay ) B g X,Y J ab } Q ab X J a Y Q ac X J c Y 0. 3.18 This gives ( X a) J Y 1 { θ Y X ω Y J ax Ω X,Y B g X,Y J ab } Q ab X J ay Q ac X J cy 0. 3.19 This shows that B is l.c.q.k. manifold. Now, using.17 and.18, we obtain a relation between curvature tensor R on M and curvature tensor R of B as follows: R X, Y, Z, W R X,Y,Z,W g C Y, Z,C X, W g C X, Z,C Y, W g C X, Y,C Z, W, 3.0 where π X X, π Y Y, π Z Z,andπ W W B. Now, using the above equation together with.8 and using the fact that C is skewsymmetric, we obtain H X R X, J a X, J a X, X H X h X, J a X g h J a X, J a X,h X, X 3.1 3 C X, J a X, where H X and H X are the holomorphic sectional curvature tensors of M and B, respectively. If we assume that D is integrable then using Lemma.3 i, we have h J a X, J a X h X, X. 3. Also from 3.15, we have C X, J a X 0. Then, 3.1 reduces to H X H X h X, J a X h X, X, X D. 3.3 This gives H X H X. Thus, we have the following result.
ISRN Geometry 13 Theorem 3.3. Let M be a CR-submanifold of a l.c.q.k. manifold M with integrable D. LetB 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 H X H X, for all unit vectors X D. 3.4 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/009/ XXV to carry out this research work. References 1 I. Vaisman, On locally conformal almost kähler manifolds, Israel Mathematics, vol. 4, no. 3-4, pp. 338 351, 1976. 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., 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. Günes, QR-submanifolds of a locally conformal quaternion kaehler manifold, Publicationes Mathematicae Debrecen, vol. 63, no. 1-, pp. 157 174, 003. 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 14, 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, 199. 10 J. Simons, Minimal varieties in riemannian manifolds, Annals of Mathematics, vol. 88, no. 1, pp. 6 105, 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, 006.
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