We study lightlike submersions from a totally umbilical semi-transversal lightlike sub- manifold of an indefinite Kaehler manifold onto an indefinite almost Hermitian manifold

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Vol. 19 (2018), No. 2, pp. 953–968 DOI: 10.18514/MMN.2018.2483

LIGHTLIKE SUBMERSIONS FROM TOTALLY UMBILICAL SEMI-TRANSVERSAL LIGHTLIKE SUBMANIFOLDS

RUPALI KAUSHAL, RAKESH KUMAR, AND RAKESH KUMAR NAGAICH Received 21 December, 2017

Abstract. We study lightlike submersions from a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifold onto an indefinite almost Hermitian manifold. We show that if an indefinite almost Hermitian manifoldBadmits a lightlike submersionWM!B from a totally umbilical semi-transversal lightlike submanifoldMof an indefinite Kaehler man- ifoldMN thenBis necessarily an indefinite Kaehler manifold. We investigate the condition for a totally umbilical semi-transversal lightlike submanifoldMto becomes a product manifold and its fibers become geodesic. Finally, we obtain some characterization theorems related to the sectional curvature of an indefinite Kaehler manifold.

2010Mathematics Subject Classification: 53C20; 53C50

Keywords: indefinite Kaehler manifold, semi-transversal lightlike submanifolds, lightlike submersions

1. INTRODUCTION

The study of Riemannian submersionsWM !B, from a Riemannian manifold M onto a Riemannian manifold B was initiated by O’Neill [10]. A Riemannian submersion naturally yields a vertical distribution, which is always integrable and a horizontal distribution. On the other hand, for aCR-submanifold M of a Kaehler manifoldMN there are two orthogonal complementary distributionsDandD^?, such thatDisJN-invariant andD^?is totally real and always integrable (cf. Bejancu [2]), where JN is almost complex structure of MN . Kobayashi [9] observed the similar- ity between the total space of a Riemannian submersion and aCR-submanifold of a Kaehler manifold in terms of distributions. Then Kobayashi [9] introduced a submer- sionWM !B, from aCR-submanifoldM of a Kaehler manifoldMN onto an almost Hermitian manifoldBsuch that the distributionsDandD^? of theCR-submanifold become the horizontal and the vertical distributions respectively, as required by the submersions andrestricted toDbecomes a complex isometry.

Later, semi-Riemannian submersions were introduced by O’Neill in [11]. As it is known that whenM andB are Riemannian manifolds then the fibers are always

c 2018 Miskolc University Press

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Riemannian manifolds. However, when the manifolds are semi-Riemannian manifolds then the fibers may not be Riemannian (hence semi-Riemannian) manifolds, (see [15]). Therefore in [13], Sahin introduced a screen lightlike submersion from a lightlike manifold onto a semi-Riemannian manifold and in [15], Sahin and Gun- duzalp introduced a lightlike submersion from a semi-Riemannian manifold onto a lightlike manifold. It is well-known that semi-Riemannian submersions are of in- terest in mathematical physics, owing to their applications in the Yang-Mills theory, Kaluza-Klein theory, supergravity and superstring theories [3,4,8,16]. Moreover, the geometry of lightlike submanifolds has potential for applications in mathematical physics, particularly in general relativity (for detail, see [5]) therefore in present pa- per, we study lightlike submersions from a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifold onto an almost Hermitian manifold.

2. LIGHTLIKE SUBMANIFOLDS

Let.M ;N g/N be a real.mCn/-dimensional semi-Riemannian manifold of constant indexqsuch thatm; n1,1qmCn 1and.M; g/be anm-dimensional submanifold ofMN andgbe the induced metric ofgN onM. IfgN is degenerate on the tangent bundleTM ofM thenM is called a lightlike submanifold ofMN , (see [5]). For a degenerate metricgonM,TM^?is a degeneraten-dimensional subspace ofTxMN. Thus bothTxM andTxM^?are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace Rad.TxM /DTxM \TxM^? which is known as radical (null) subspace. If the mappingRad.TM /Wx2M ! Rad.TxM /, defines a smooth distribution onM of rankr > 0then the submanifold M ofMN is called an r-lightlike submanifold and Rad.TM / is called the radical distribution onM.

Screen distributionS.TM /is a semi-Riemannian complementary distribution of Rad.TM / in TM, that is, TM DRad.TM /?S.TM / and S.TM^?/ is a complementary vector subbundle to Rad.TM / in TM^?. Let t r.TM / and lt r.TM / be complementary (but not orthogonal) vector bundles to TM in TMN j^M and to Rad.TM / in S.TM^?/^? respectively. Then TMN j^MD TM ˚t r.TM / D .Rad TM˚lt r.TM //?S.TM /?S.TM^?/:

Theorem 1 ([5]). Let .M; g; S.TM /; S.TM^?// be an r-lightlike submanifold of a semi-Riemannian manifold.M ;N g/. Then there exists a complementary vectorN bundlelt r.TM /ofRad.TM /inS.TM^?/^?and a basis oflt r.TM /jUconsisting of smooth sectionfNigofS.TM^?/^?jU, whereUis a coordinate neighborhood of M such that

N

g.Ni; j/Dıij; g.NN i; Nj/D0;for any i; j 2 f1; 2; ::; rg; (2.1) wheref1; :::; rgis a lightlike basis ofRad.TM /.

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LetrN be the Levi-Civita connection onMN then for anyX; Y 2 .TM /andU 2 .t r.TM //, the Gauss and Weingarten formulas are given by

rN^XY D r^XY Ch.X; Y /; rN^XU D AUXC rX^?U; (2.2) where frXY; A_UXgandfh.X; Y /;r_X^?Ugbelong to .TM /and .t r.TM //, respectively. Herer is a torsion-free linear connection onM,his a symmetric bilinear form on .TM /which is called the second fundamental form, A_U is a linear operator onM and known as a shape operator.

Considering the projection morphisms L and S of t r.TM / on lt r.TM / and S.TM^?/, respectively, then (2.2) becomes

rN^XY D r^XY Ch^l.X; Y /Ch^s.X; Y /; rN^XU D AUXCD_X^l UCD_X^sU; (2.3) whereh^l.X; Y /DL.h.X; Y //; h^s.X; Y /DS.h.X; Y //; D_X^l U DL.rX^?U /,D_X^sU D S.r_X^?U /. As h^l and h^s are lt r.TM /-valued and S.TM^?/-valued respectively, therefore they are called as the lightlike second fundamental form and the screen second fundamental form onM. In particular

rN^XN D ANXC rX^l NCD^s.X; N /; rN^XW D AWXC rX^sWCD^l.X; W /;

(2.4) whereX2 .TM /; N2 .lt r.TM //andW 2 .S.TM^?//. Using (2.3) and (2.4), we obtain

N

g.h^s.X; Y /; W /C Ng.Y; D^l.X; W //Dg.AWX; Y /: (2.5) LetRN andRbe the curvature tensors ofrN andr, respectively then by straightforward calculations (see [5]), we have

R.X; Y /ZN DR.X; Y /ZCA_hl.X;Z/Y A_hl.Y;Z/XCA_h^s_.X;Z/Y

A_h^s_.Y;Z/XC.r^Xh^l/.Y; Z/ .r^Yh^l/.X; Z/

CD^l.X; h^s.Y; Z// D^l.Y; h^s.X; Z//C.r^Xh^s/.Y; Z/

.rYh^s/.X; Z/CD^s.X; h^l.Y; Z// D^s.Y; h^l.X; Z//: (2.6) 3. SEMI-TRANSVERSAL LIGHTLIKE SUBMANIFOLDS

Let.M ;N J ;N g/N be an indefinite almost Hermitian manifold andrN be the Levi-Civita connection onMN with respect to the indefinite metricg. ThenN MN is called an indefinite Kaehler manifold [1] if the almost complex structureJN is parallel with respect torN, that is.rN^XJ /YN D0, for anyX; Y 2 .TM /.N

Definition 1 ([12]). Let M be a lightlike submanifold of an indefinite Kaehler manifoldMN then M is called a semi-transversal lightlike submanifold ofMN if the following conditions are satisfied:

(i) Rad.TM /is transversal with respect toJN.

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(ii) There exists a real non-null distribution DS.TM / such that S.TM / = D˚D^?,J .D/N DD,J DN ^?S.TM^?/, whereD^? is orthogonal complementary toDinS.TM /.

Then tangent bundle of a semi-transversal lightlike submanifold is decomposed as TM DD?D⁰, whereD⁰DD^??Rad.TM /. We sayM is a proper semi-transversal lightlike submanifold ifD¤ f0gandD^?¤ f0g. Therefored i m.Rad.TM //2 and for a properM, d i m.D/2s; s > 1,d i m.D^?/1andd i m.Rad.TM //D d i m.lt r.TM //. Thusd i m.M /5andd i m.M /N 8. Next, we give example of semi-transversal lightlike submanifolds.

Example 1. LetM be a 5-dimensional submanifold of .R₂¹⁰;g/N given by x1D u1cosh,x2Du2cosh,x3Du1si nh,x4Du2si nh,x5Du3,x6D

q 1 u²₃, x7 D u4, x8 D u8, x9 D u2, x10 D u1, where gN is of signature . ; ;C;C;C;C;C;C;C;C/ with respect to the canonical basis f@x1; @x2; @x3; @x4; @x5; @x6; @x7; @x8; @x9; @x10g. Then TM is spanned by Z1D cosh @x1Csi nh @x3C@x10; Z2Dcosh @x2Csi nh @x4C@x9; Z3Dx6@x5

x5@x6; Z4D@x7; Z5D@x8:ClearlyM is a2-lightlike submanifold withRad.TM / DspanfZ1; Z2gand the lightlike transversal bundle is spanned by

N1D1

2. cosh @x1 si nh @x3C@x10/; N2D 1

2.cosh @x2Csi nh @x4 @x9/;

and J ZN 1 D 2N2 and J ZN 2 D2N1. Hence J .Rad.TM //N Dlt r.TM /. Since J ZN 4 DZ5 thenD DspanfZ4; Z5gwhich is an invariant distribution on M. By direct calculations, the transversal screen bundleS.TM^?//is spanned by

W₁Dsi nh @x₁Ccosh @x₃; W₂Dsi nh @x₂Ccosh @x₄; W₃Dx₆@x₆Cx₅@x₅: Thus J WN 3 D Z3. Hence D^? DspanfZ3g is an anti-invariant distribution on M and spanfW1; W2g is invariant and spanfW3g is anti-invariant subbundles of S.TM^?/ respectively. Thus it enables us to chooseS.TM /DspanfZ3; Z4; Z5g. HenceM is a proper semi-transversal lightlike submanifold.

LetM be a semi-transversal lightlike submanifold of an indefinite Kaehler manifold MN . Let Q, P1, P2 and P be the projection morphisms from TM on D, Rad.TM /,D^?andD⁰respectively. Then for anyX2 .TM /, we put

XDQXCP1XCP2X: (3.1)

ApplyingJN to (3.1), we obtainJ XN D NJ QXC NJ P1XC NJ P2X, can be written as N

J XDTQXCwP1XCwP2X:PutwP1Dw1andwP2Dw2, then we have

J XN DTXCw1XCw2X; (3.2)

whereTX 2 .D/; w1X 2 .lt r.TM //andw2X2 .J DN ^?/S.TM^?/. Simil- arly, for anyV 2 .S.TM^?//, we can write

J VN DEVCF V; (3.3)

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whereEV 2 .D^?/andF V 2 ./, whereis a complementary bundle ofJ DN ^? in S.TM^?/. Differentiating (3.2) and using (2.3), (2.4) and (3.3), for any X 2

.TM /, we have the following lemma.

Lemma 1. Let M be a semi-transversal lightlike submanifold of an indefinite Kaehler manifoldMN. Then we have

.r^XT /Y DAw1YXCAw2YXC NJ h^l.X; Y /CEh^s.X; Y /; (3.4) .rXw1/Y D h^l.X; T Y / D^l.X; w2Y /; (3.5) .rXw2/Y DF h^s.X; Y / h^s.X; T Y / D^s.X; w1Y /;where (3.6) .rXT /Y D rXT Y TrXY; .rXw1/Y D rX^l w1Y w1rXY; (3.7) .rXw₂/Y D rX^sw₂Y w₂rXY: (3.8) Definition 2([6]). A lightlike submanifold.M; g/of a semi-Riemannian manifold .M ;N g/N is said to be a totally umbilical inMN if there is a smooth transversal vector field H 2 .t r.TM // on M, called the transversal curvature vector field of M, such that h.X; Y /DHg.X; Y /, forN X; Y 2 .TM /. Using (2.3), clearly M is a totally umbilical, if and only if, forX; Y 2 .TM /andW 2 .S.TM^?//, on each coordinate neighborhoodUthere exist smooth vector fieldsH^l 2 .lt r.TM //and H^s2 .S.TM^?//such that

h^l.X; Y /DH^lg.X; Y /; h^s.X; Y /DH^sg.X; Y /; D^l.X; W /D0: (3.9) Lemma 2. LetM be a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifoldMN then the distributionD⁰defines a totally geodesic foliation inM.

Proof. Let X; Y 2 .D⁰/ then using (3.4) and (3.7), we obtain Tr^XY D Aw1YX Aw2YX J hN ^l.X; Y / Eh^s.X; Y /:On taking inner product both sides withZ2 .D/, we further obtain

g.Tr^XY; Z/D Ng.rN^Xw1Y ; Z/C Ng.rN^Xw2Y ; Z/D g.N J Y;N rN^XZ/

D Ng.Y;rN^XJ Z/N Dg.Y;r^XZ⁰/; (3.10) where Z⁰ D NJ Z 2 .D/. Since M is a totally umbilical lightlike submanifold then for anyX 2 .D⁰/ andZ2 .D/, with (3.5) and (3.7), we havew1r^XZD h^l.X; T Z/ DH^lg.X; T Z/D0 and using (3.6) and (3.8), we have w2r^XZ D F h^s.X; Z/Ch^s.X; T Z/D FH^sg.X; Z/CH^sg.X; T Z/D0, these facts im- ply thatr^XZ2 .D/, for anyX 2 .D⁰/andZ2 .D/. Therefore (3.10) implies thatg.Tr^XY; Z/D0, then the non degeneracy of the distribution D implies that

TrXY D0. Hence the result follows.

Theorem 2 ([12]). Let M be a semi-transversal lightlike submanifold of an indefinite Kaehler manifoldMN. Then the distributionD⁰is integrable, if and only if AwZV DAwVZ, for anyZ; V 2 .D⁰/.

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Theorem 3. LetM be a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifoldMN then the distributionD⁰is integrable.

Proof. LetX; Y 2 .D⁰/ then using (3.4) and (3.7) with the Lemma 2, we get AwYXD J hN ^l.X; Y / Eh^s.X; Y /this implies thatAwYX2 .D⁰/and moreover the symmetric property of the second fundamental formhgives thatA_wYXDA_wXY. Hence by virtue of the Theorem2, the result follows.

4. SEMI-TRANSVERSAL LIGHTLIKE SUBMERSIONS

LetWM !Bbe a mapping from a Riemannian manifoldM onto a Riemannian manifoldBthen it is said to be a Riemannian submersion if it satisfies the following axioms:

A1. has maximal rank. This implies that for eachb2B, ¹.b/is a submanifold ofM, known as fiber, of dimensiond i mM d i mB. A vector field tangent to the fibers is called vertical vector field and orthogonal to fibers is called horizontal vector field.

A2. preserves the lengths of horizontal vectors.

The Riemannian submersions were introduced by O’Neill in [10] and since then plenty of work on this subject matter has been done (for detail, see [7,14] and many references therein). In the study of submersions, the vertical distribution V ofM is defined byVp Dker dp; p2M, which is always integrable and the orthogonal complementary distribution to V is defined by Hp D.ker dp/^?, denoted by H and called a horizontal distribution. Therefore the tangent bundleTM ofM has the following decompositionTM DV˚H:

Since the vertical distribution of the Riemannian submersionWM !B and the totally real distribution D^? of theCR-submanifold M of a Kaehler manifold are always integrable. Therefore Kobayashi [9] introduced the submersionWM !B from a CR-submanifold M of a Kaehler manifold onto an almost Hermitian manifold B such that the distributions D andD^? of the CR-submanifold become the horizontal and the vertical distributions respectively, required by the submersion and restricted toDbecomes a complex isometry.

We have seen that for a Riemannian submersion, the tangent bundle of the source manifold splits into horizontal and vertical part. On the other hand, the tangent bundle of a lightlike submanifold splits into screen and radical part and these natural splitting of the tangent bundle plays an important role in the study of lightlike submanifolds.

Therefore Sahin [13] introduced screen lightlike submersion between a lightlike manifold and a semi-Riemannian manifold. Further in [15], Sahin and Gunduzalp introduced the idea of a lightlike submersion from a semi-Riemannian manifold onto a lightlike manifold.

From Theorem3, we know that for a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifold the distributionD⁰is integrable. Then

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a totally umbilical semi-transversal lightlike submanifold meets our requirements to define a submersion on it analogous to a submersion of aCR-submanifold. Signific- ant applications of semi-Riemannian submersions in physics and the growing import- ance of lightlike submanifolds and hypersurfaces in mathematical physics, especially in relativity (see [5]), motivated us to work on this subject matter.

Definition 3. Let.M; gM; D/be a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifold MN and.B; g_B/ be an indefinite almost Hermitian manifold. Then we say that a smooth mappingW.M; gM; D/!.B; gB/ is a lightlike submersion if

(a) at everyp2M;VpDker.d/pDD⁰.

(b) at each pointp2M, the differentialdp restricts to an isometry of the horizontal spaceHpDDpontoT_.p/B, that is,gD.X; Y /DgB.d.X /; d.Y //, for every vector fieldsX; Y 2 .D/.

Obviously from the definition, the restriction of the differentialdp to the distri- butionHp DDp maps that space isomorphically ontoT_.p/B. Then for any tangent vectorXe2T.p/B, we say that the tangent vectorX 2Dp is a horizontal lift ofXe as for submersions. IfXe is a vector field on an open subsetU ofB then the horizontal lift ofXe is the vector fieldX 2 .D/on ¹.U /such thatd.X /DX oe and the vector fieldXis called abasic vector field. Now, we give example of lightlike submersions.

Example2. LetM be a5-dimensional semi-transversal lightlike submanifold of R₂¹⁰ as in Example (1) and B DR²₁ be an indefinite almost Hermitian manifold.

Let the metrics be defined asgM D .dx1/² .dx2/²C.dx3/²C.dx4/²C.dx5/² andgB D .dy1/²C.dy2/², wherex1; x2; x3; x4; x5; x6; x7; x8; x9; x10 andy1; y2

be the canonical co-ordinates of R¹⁰₂ and R², respectively. We define a map W .x1; x2; x3; x4; x5; x6; x7; x8; x9; x10/2R₂¹⁰7!.x7; x8/2R₁²:Then the kernel ofd is

ker.d/DD⁰DspanfZ1Dcosh @x1Csi nh @x3C@x10;

Z2Dcosh @x2Csi nh @x4C@x9; Z3Dx6@x5 x5@x6g; whered.Z1/D0; d.Z2/D0andd.Z3/D0. By direct computation, we obtain DDspanfZ4D@x7; Z5D@x8g, whered.Z4/D@y1; d.Z5/D@y2. Then it follows that g_M.Z4; Z4/ D g_B.d.Z4/; d.Z4// D 1 and g_M.Z5; Z5/ D gB.d.Z5/; d.Z5//D 1:Hence is a semi-transversal lightlike submersion.

Theorem 4. Let WM !B be a lightlike submersion from a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldB. IfX andY are basic vectors-related to X ;e Yerespectively, then

(i) gM.X; Y /DgB.eX ;eY /o.

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(ii) ŒX; Y ^H is the basic vector field and-related toŒeX ;eY .

(iii) .r_X^MY /^H is the basic vector field and-related to.r^B eXY /.e (iv) For any vertical vector fieldV,ŒX; V is vertical.

Proof. Let X and Y be basic vector fields of M then .i / follows immediately from part (b) of the Definition3. SinceP andQ be the projections fromTM on the distributionsD⁰andD of a semi-transversal lightlike submanifold of indefinite Kaehler manifold respectively, thenŒX; Y DP ŒX; Y CQŒX; Y :Therefore the horizontal partQŒX; Y ofŒX; Y is a basic vector field and corresponds toŒeX ;eY , that is,d.QŒX; Y /DŒd.X /; d.Y /. Next, from the Koszul’s formula, we have

2gM.r^XY; Z/DX.gM.Y; Z//CY .gM.Z; X // Z.gM.X; Y //

gM.X; ŒY; Z/CgM.Y; ŒZ; X /CgM.Z; ŒX; Y / (4.1) for anyX; Y; Z2 .D/. Consider X; Y andZare the horizontal lifts of the vector fields X ;e eY and eZ respectively, then X.gM.Y; Z// D X .ge B.eY ;eZ//o and gM.Z; ŒX; Y /DgB.eZ; ŒX ;ee Y /othen from (4.1), we have

2g_M.rX^MY; Z/DX .ge _B.eY ;eZ//oCeY .g_B.eZ;X //oe Z.ge _B.eX ;eY //o gB.eX ; ŒeY ;eZ/oCgB.eY ; ŒeZ;X /oe CgB.eZ; ŒX ;ee Y /o D2gB.r^B

eXeY ;eZ/: (4.2)

Thus from (4.2), (iii) follows, since is surjective andeZis arbitrarily chosen. Fi- nally, letV 2 .D⁰/thenŒX; V is-related toŒeX ; 0, hence.iv/follows and this

completes the proof of the theorem.

Let r^B be the covariant differentiation on B then we define the corresponding operatorre^B for basic vector fields of B by assuming er_X^BY D.r_X^MY /^H; for any basic vector fieldsX andY. Thus from (iii) the Theorem4,er_X^BY is a basic vector field and d.r_X^MY /^H Dd.er_X^BY /D r^B

eXY :e Thus we define the tensor fields C1

andC2, using (3.1) as

rX^MY DerX^BY CC1.X; Y /CC2.X; Y /; (4.3) for any X; Y 2 .D/, where C1.X; Y / and C2.X; Y / denote the vertical parts of rX^MY. It is easy to check thatC1andC2are bilinear maps fromDD!Rad.TM / andDD!D^?respectively.

Theorem 5. LetWM !Bbe a lightlike submersion of a totally umbilical semi- transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldB then for any basic vector fieldsX andY, we have

(i) the tensor fields C1 and C2 are skew-symmetric, that is, C1.X; Y / D C1.Y; X /andC2.X; Y /D C2.Y; X /;

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(ii) P1ŒX; Y D2C1.X; Y /andP2ŒX; Y D2C2.X; Y /,

Proof. (i) LetZ2 .D^?/be any vertical vector field then for any basic vector fieldX 2 .D/, we have

0DZ.g.X; X //D2g.N rNZX; X /D2g.rX^MZ ŒX; Z; X /D 2g.Z;N rNXX / D 2g.Z;erX^BXCC1.X; X /CC2.X; X //D 2g.Z; C2.X; X //;

then the non degeneracy of the distributionD^? implies that C2.X; X /D0, that is C2is skew-symmetric. Similarly, letJ NN 2 .Rad.TM //be a vertical vector field whereN 2 .lt r.TM //, we have

0D NJ N.g.X; X //D 2g.N rN^NX; X /D 2g.rX^MN ŒX; N ; X / D2g.N;erX^BXCC1.X; X /CC2.X; X //D2g.N; C1.X; X //;

then using (2.1), we obtainC1.X; X /D0, that isC1is skew-symmetric.

(ii) For basic vector fields X; Y 2 .D/, we have ŒX; Y D r_X^MY r_Y^MX, using (3.1), (4.3) and skew-symmetric property ofC1andC2, result follows.

Next for a basic vector fieldX and a vertical vector fieldZ, using (3.1), we define the tensor fieldT as

rX^MZD.rX^MZ/^HC.rX^MZ/^V DTXZC.rX^MZ/^V; (4.4) whereT is a bilinear map fromDD⁰!D. SinceŒX; ZD r_X^MZ r_Z^MX and ŒX; Zis vertical therefore

Q.rX^MZ/DQ.rZ^MX /DTXZ; .rX^MZ/^V D.rZ^MX /^V: (4.5) LetX andY be basic vector fields and Z be a vertical vector field such that Z2

.D^?/then using (4.3), the tensor fieldsT andC2are related by

g.TXZ; Y /D Ng.rN^XZ; Y /D g.Z;r^XY /D g.Z; C2.X; Y //; (4.6) and ifZ2 .Rad.TM //then

g.TXZ; Y /D g.Z; hN ^l.X; Y //: (4.7) Theorem 6. LetWM !Bbe a lightlike submersion of a totally umbilical semi- transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldB thenBis also an indefinite Kaehler manifold.

Moreover ifHN andH^B denote the holomorphic sectional curvatures of MN andB, respectively then for any unit basic vectorX 2 .H/ofM, we have

RN^M^N.X;J X; X;N J X /N DR^B.eX ;JNX ;e X ;e JNX /e C4kH^sk²:

Proof. LetX; Y 2 .D/be basic vector fields then using (2.3) and (4.3), we have rNXY DerX^BY CC1.X; Y /CC2.X; Y /Ch^l.X; Y /Ch^s.X; Y /: (4.8)

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On applyingJNon both sides of (4.8), we obtain

JNrN^XY D NJerX^BY C NJ C1.X; Y /C NJ C2.X; Y /C NJ h^l.X; Y /

CEh^s.X; Y /CF h^s.X; Y /; (4.9) on replacingY byJ YN in (4.8), we have

rNXJ YN DerX^BJ YN CC1.X;J Y /N CC2.X;J Y /N Ch^l.X;J Y /N Ch^s.X;J Y /:N (4.10) SinceMN is a Kaehler manifold thereforerN^XJ YN D NJrN^XY, then equating (4.9) and (4.10), we obtain

erX^BJ YN D NJerX^BY 2 .H/; (4.11) C1.X;J Y /N D NJ h^l.X; Y /2 .Rad.TM //; (4.12) C2.X;J Y /N DEh^s.X; Y /2 .D^?/; (4.13) h^s.X;J Y /N D NJ C2.X; Y /CF h^s.X; Y /2 .S.TM^?//; (4.14) h^l.X;J Y /N D NJ C1.X; Y /2 .lt r.TM //: (4.15) From (4.11), we see that almost complex structureJN ofBis parallel and henceBis also an indefinite Kaehler manifold.

From (3.3), it is clear that U 2 .J DN ^?/S.TM^?/, if and only if, F U D0 then J UN DEU andU 2 .D.J DN ^?/^?/S.TM^?/, if and only if, EU D0 then J UN DF U. Therefore from (4.13), (4.14) and skew-symmetric property of C2, we obtainC2.X;J Y /N DC2.Y;J X /,N C2.J X; Y /N DC2.J Y; X /,N C2.J X;N J Y /N D C2.X; Y /andh^s.X;J Y /N Ch^s.Y;J X /N D2F h^s.X; Y /. On the other hand, sinceM is a totally umbilical semi-transversal lightlike submanifold then we haveh^s.X;J Y /N C h^s.Y;J X /N Dg.X;J Y /HN ^sCg.Y;J X /HN ^sD0:ThereforeF h^s.X; Y /D0and this implies thath^s.X; Y /2 .J DN ^?/, for anyX; Y 2 .D/. By virtue of totally umbilical property ofM, we also haveh^s.J X;N J Y /N Dh^s.X; Y /. Similarly using (4.12) and (4.15), we obtainC1.X;J Y /N DC1.Y;J X /,N C1.J X; Y /N DC1.J Y; X /,N C1.J X;N J Y /N DC1.X; Y /andh^l.J X;N J Y /N Dh^l.X; Y /, h^l.J X; Y /N Ch^l.X;J Y /N D0. Now, for anyX; Y; Z2 .D/, using (4.3) and (4.4), we have

rXrYZDerX^Ber^BYZCTXC1.Y; Z/CTXC2.Y; Z/Cvert i cal; (4.16) rYrXZDer^BYerX^BZCT_YC1.X; Z/CT_YC2.X; Z/Cvert i cal; (4.17) rŒX;Y ZDer^B_QŒX;YZC2TZC1.X; Y /C2TZC2.X; Y /Cvert i cal: (4.18) Further using (4.16)-(4.18), we obtain

R^M.X; Y /ZD.R^B.eX ;eY /eZ/CTXC1.Y; Z/CTXC2.Y; Z/ TYC1.X; Z/

T_YC2.X; Z/ 2T_ZC1.X; Y / 2T_ZC2.X; Y /

Cvert i cal; (4.19)

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where .R^B.eX ;Y /ee Z/ denotes the basic vector field of M corresponding to R^B.eX ;Y /ee Z. Using (4.19) in (2.6), we obtain

RN^M^N.X; Y /ZD.R^B.X ;ee Y /eZ/CTXC1.Y; Z/CTXC2.Y; Z/ TYC1.X; Z/

T_YC2.X; Z/ 2T_ZC1.X; Y / 2T_ZC2.X; Y /CA_hl.X;Z/Y A_hl.Y;Z/XCA_h^s_.X;Z/Y A_h^s_.Y;Z/XC.r^Xh^l/.Y; Z/

.r^Yh^l/.X; Z/CD^l.X; h^s.Y; Z// D^l.Y; h^s.X; Z//

C.r^Xh^s/.Y; Z/ .r^Yh^s/.X; Z/CD^s.X; h^l.Y; Z//

D^s.Y; h^l.X; Z//Cvert i cal:

Now, for basic vector fieldW 2 .D/with (2.4), (2.5), (4.4)-(4.7), we obtain RN^M^N.X; Y; Z; W /DR^B.eX ;Y ;e Z;e W /e g.CN 1.Y; Z/; h^l.X; W //

g.C2.Y; Z/; C2.X; W //C Ng.C1.X; Z/; h^l.Y; W //

Cg.C2.X; Z/; C2.Y; W //C2g.CN 1.X; Y /; h^l.Z; W //

C2g.C2.X; Y /; C2.Z; W //Cg.A_h^l_.X;Z/Y; W / g.A_hl.Y;Z/X; W /C Ng.h^s.X; Z/; h^s.Y; W //

N

g.h^s.Y; Z/; h^s.X; W //: (4.20) Now, using (2.4) and (4.3), we have g.A_hl.X;Z/Y; W / = g.hN ^l.X; Z/;rN^YW / =

N

g.h^l.X; Z/; C1.Y; W // and similarly g.A_hl.Y;Z/X; W /D Ng.h^l.Y; Z/; C1.X; W //.

Using these expressions with (4.15) in (4.20), we obtain

RN^M^N.X; Y; Z; W /DR^B.eX ;Y ;e Z;e W /e C Ng.J hN ^l.Y;J Z/; hN ^l.X; W //

g.C2.Y; Z/; C2.X; W // g.N J hN ^l.X;J Z/; hN ^l.Y; W //

Cg.C2.X; Z/; C2.Y; W // 2g.N J hN ^l.X;J Y /; hN ^l.Z; W //

C2g.C2.X; Y /; C2.Z; W // g.N J hN ^l.Y;J W /; hN ^l.X; Z//

C Ng.J hN ^l.X;J W /; hN ^l.Y; Z//C Ng.h^s.X; Z/; h^s.Y; W //

N

g.h^s.Y; Z/; h^s.X; W //: (4.21) To compare holomorphic sectional curvature of MN with that of B, set Y D NJ X, ZDX and W D NJ X in (4.21) and then using the hypothesis that M is a totally umbilical semi-transversal lightlike submanifold, we obtainRN^M^N.X;J X; X;N J X /N D R^B.eX ;JNX ;e X ;e JNX /e C kC2.X; X /k² C 3kC2.X;J X /N k² C kh^s.X; X /k²: Since F h^s.X; Y /D0therefore (4.14) implieskh^s.X; X /k²D kC2.X;J X /N k²and by virtue of the totally umbilical property ofM, (4.14) implies thatC2.X; X /D J hN ^s.X;J X /N D J .HN ^Sg.X;J X //N D0. Thus the holomorphic sectional curvature ofMN is given

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as

RN^M^N.X;J X; X;N J X /N DR^B.eX ;JNX ;e X ;e JNX /e C4kC2.X;J X /N k² DR^B.eX ;JNX ;e X ;e JNX /e C4kh^s.X; X /k² DR^B.eX ;JNX ;e X ;e JNX /e C4kH^sk²:

This completes the proof.

Theorem 7. LetWM !Bbe a lightlike submersion of a totally umbilical semi- transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldB. If the distributionDis integrable, thenM is a lightlike product manifold.

Proof. Let the distributionDbe an integrable thereforeP1ŒX; Y D0andP2ŒX; Y D0, for anyX; Y 2 .D/, where P1 andP2 are the projection morphisms from TM toRad.TM /andD^?, respectively. Therefore using the Theorem5, we have C1.X; Y /D0andC2.X; Y /D0. Hence using (4.3), we obtain thatrX^MY 2 .D/, for anyX; Y 2 .D/, consequently the distributionDdefines a totally geodesic foliation inM. Moreover, from the Lemma2, the distributionD⁰also defines a totally geodesic foliation inM. Thus using the De Rham’s theorem,M is a product mani- foldM1M2, whereM1andM2are the leaves of the distributions ofDandD⁰. Theorem 8. LetWM !Bbe a lightlike submersion of a totally umbilical semi- transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldBsuch thatJ .DN ^?/DS.TM^?/. Then the fibers are totally geodesic submanifolds ofM.

Proof. LetU; V 2 .D⁰/and then define

rU^MV D Or^UVCL.U; V /; (4.22) where rO^UV D.rU^MV /^V and L.U; V /D.rU^MV /^H. Since the distribution D⁰ is integrable always, thenL.U; V /DL.V; U /. Now, using the Kaehlerian property of

N

M, we haverNUJ VN D NJrNUV, sinceJ .DN ^?/DS.TM^?/, then A_{J V}_N UC rU^t J VN D NJrO^UV C NJ L.U; V /C NJ h.U; V /:

On comparing the horizontal and vertical components both sides, we get

H.A_{J V}_N U /D J L.U; V /;N V.A_{J V}_N U /D J h.U; V /:N (4.23) From (4.22), it is clear that the fibers are totally geodesic submanifolds ofM, if and only if,L.U; V /D0or using (4.23)1, if and only if,A_{J V}_N U 2 .D⁰/, for anyU; V 2 .D⁰/. Now, particularly chooseV 2D^?then using the hypothesis of this theorem J VN 2 .S.TM^?//. LetY 2 .D/then using (2.5) with the fact that M is a totally

umbilical lightlike submanifold, we obtaing.A_{J V}_N U; Y /D Ng.h^s.U; Y /;J V /N Dg.U; Y /g.HN ^s;J V /N D 0:Similarly, letV2 .Rad.TM //theng.A_{J V}_N U; Y /D Ng.J V;N rNUY /D Ng.V; h^l.U;J Y //N D

g.U;J Y /N g.V; HN ^l/D0:ThusA_{J V}_N U 2 .D⁰/and the assertion follows.

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Theorem 9. LetWM !Bbe a lightlike submersion of a totally umbilical semi- transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldB. Then the sectional curvature ofMN and of the fiber are related by

K.UN ^V /D OK.U^V /Cg.A_hl.U;U /V; V / g.A_hl.V;U /U; V / Cg.ŒA_{J V}_N ; A_{J U}_N U; V /;

for any orthonormal vector fieldsU; V 2 .D^?/.

Proof. Letr andrO be the connections of semi-transversal lightlike submanifold M and its fiber, respectively. Let R and RO be the curvature tensors of r andrO, respectively then for anyU; V 2 .D^?/, using (4.22) we have

R.U; V /U D r^U.rO^VUCL.V; U // r^V.rO^UUCL.U; U //

.rOŒU;V UCL.ŒU; V ; U //;

this further implies that

R.U; V; U; V /Dg.rUrOVU; V /Cg.rUL.V; U /; V / g.rVrOUU; V / g.r^VL.U; U /; V / g.rOŒU;V U; V /:

Again using (4.22), it leads to

R.U; V; U; V /D OR.U; V; U; V /Cg.r^UL.V; U /; V / g.r^VL.U; U /; V /: (4.24) Now, using the fact thatM is totally umbilical lightlike submanifold, we get

g.r^UL.V; W /; F /Dg.rN^UL.V; W / g.h^l.U; L.V; W //; F /

D g.L.V; W /;rUF /D g.L.V; W /; L.U; F //;

for anyU; V; W; F 2 .D^?/therefore (4.24) becomes

R.U; V; U; V /D OR.U; V; U; V / g.L.U; V /; L.U; V //Cg.L.U; U /; L.V; V //:

(4.25) Using (2.5), (2.6) andM is totally umbilical lightlike submanifold, we have

N

R.U; V; U; V /DR.U; V; U; V /Cg.A_hl.U;U /V; V / g.A_hl.V;U /U; V / C Ng.h^s.V; V /; h^s.U; U // g.hN ^s.U; V /; h^s.V; U //:

Further using (4.23), (4.25) and the factL.U; V /DL.V; U /, we obtain R.U; V; U; V /N D OR.U; V; U; V / g.H.A_{J U}_N V /;H.A_{J U}_N V //

Cg.H.A_{J U}_N U /;H.A_{J V}_N V //Cg.A_h^l_{.U;U /}V; V / g.A_hl.V;U /U; V /Cg.V.A_{J V}_N V /;V.A_{J U}_N U //

g.V.A_{J V}_N U /;V.A_{J V}_N U //:

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SinceU; V 2 .D^?/and letX 2 .D/then using (2.3), we getg.A_{J U}_N V; X /D0;

which further implies thatA_{J U}_N V 2 .D^?/andA_{J U}_N V DA_{J V}_N U, then R.U; V; U; V /N D OR.U; V; U; V / g.A_{J U}_N V; A_{J U}_N V /Cg.A_{J U}_N U; A_{J V}_N V /

Cg.A_hl.U;U /V; V / g.A_hl.V;U /U; V /: (4.26) Now, letW 2 .S.TM^?//then forU; V 2 .D^?/, using (2.5), we haveg.AWU; V / Dg.U; AWV /. Using this fact withA_{J U}_N V 2 .D^?/, we get

g.A_{J U}_N V; A_{J U}_N V / g.A_{J U}_N U; A_{J V}_N V /Dg.A_{J V}_N U; A_{J U}_N V / g.A_{J U}_N U; A_{J V}_N V / Dg.A_{J U}_N A_{J V}_N U; V / g.A_{J V}_N A_{J U}_N U; V / D g.ŒA_{J V}_N ; A_{J U}_N U; V /: (4.27)

On using (4.27) in (4.26), the assertion follows.

Now we define O’Neill’s tensors [10] for a lightlike submersion. Letr be a connection ofM then tensorsT andAof type.1; 2/are given by

TXY DHrVXVY CVrVXHY; AXY DHrHXVY CVrHXHY: (4.28) Using (4.28), we have the following lemma.

Lemma 3. LetWM !Bbe a lightlike submersion of a totally umbilical semi- transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldB. Then we have the following:

(i) r^UV DTUVCVr^UV. (ii) rVXDHrVXCTVX. (iii) r^XV DAXVCVr^XV. (iv) rXY DHrXY CA_XY, for anyX; Y 2H andU; V 2V.

Theorem 10. Let WM !B be a lightlike submersion of a totally umbilical semi-transversal lightlike submanifold of an indefinite Kaehler manifoldMN onto an indefinite almost Hermitian manifoldBsuch thatJ .DN ^?/DS.TM^?/. ThenK.XN ^ V /D kH^sk² kTXVk², for any unit vector fieldsX 2 .D/andV 2 .D^?/.

Proof. LetX 2 .D/andV 2 .D^?/then using the Theorem5and Lemma3 with (4.3), we obtain

g.R.V; X /X; V /Dg.r^VH.r^XX /; V / g.r^XH.r^VX /; V / g.rXTVX; V /Cg.TŒX;V X; V /:

It should be noted that g.rVH.rXX /; V /D g.H.rXX /;rVV /, and similarly g.r^XH.r^VX /; V /D g.H.r^VX /;r^XV /:Therefore we have

g.R.V; X /X; V /D g.H.rXX /;rVV /Cg.H.rVX /;rXV /

g.r^XTVX; V /Cg.T_ŒX;VX; V /: (4.29)

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Since J .DN ^?/DS.TM^?/ then using the Theorem 8, we have L.U; V / D0, for U; V 2 .D^?/. Hence using the definition ofT with (2.3) and (4.22), we get

g.T_VX; U /D g.T_VU; X /D g.L.V; U /; X /D0: (4.30) Now, using (4.22), we have

g.H.rXX /;rVV /Dg.H.rXX /; L.V; V //D0: (4.31) SinceM is a totally umbilical then using (4.30), we obtain

g.rXT_VX; V /D g.T_VX;rNXV /D g.T_VX;V.rXV //

Dg.L.V;V.r^XV //; X /D0: (4.32) Since for a vertical vector field V, ŒX; V is always vertical therefore again using (4.30), we have

g.T_ŒX;VX; V /D g.L.ŒX; V ; V /; X /D0: (4.33) Using (4.6) and (4.31)-(4.33) in (4.29), we obtain

g.R.V; X /X; V /Dg.TXV; TXV /: (4.34) SinceM is a totally umbilical then using (2.6) and (4.34), we get

R.X; V; X; V /N D g.TXV; TXV /Cg.h^l.X; X /;r^VV /

Cg.h^s.X; X /; h^s.V; V //: (4.35) Now, using Kaehlerian property ofMN, we haverN^VJ N D NJrN^V;forV 2 .D^?/and 2 .Rad.TM //. Using the Lemma3with (2.4) and then comparing the horizontal components of resulting equation, we obtain

A_J_N V D JNTV: (4.36)

SinceM is semi-transversal lightlike submanifold then for2 .Rad.TM //,J N 2 .lt r.TM // and using (4.28) for anyU; V 2V, TUV DHrVUVV 2H. There- fore (4.36) implies thatA_J_N V 2H orANV 2H. Then forV 2 .D^?/andN 2 .lt r.TM //, we haveg.rVV; N /D g.V;rNVN /Dg.V; A_NV /D0:This implies thatr^VV has no component inRad.TM /. Using this fact in (4.35) with (3.9), the

assertion follows.

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Authors’ addresses

Rupali Kaushal

Punjabi University, Department of Mathematics, Patiala, Punjab, India.

E-mail address:rupalimaths@pbi.ac.in

Rakesh Kumar

Punjabi University, Department of Basic & Applied Sciences, Patiala, Punjab, India.

E-mail address:dr rk37c@yahoo.co.in

Rakesh Kumar Nagaich

Punjabi University, Department of Mathematics, Patiala, Punjab, India.

E-mail address:nagaich58rakesh@gmail.com