68, 2 (2016), 119–129 June 2016

June 2016

RADICAL TRANSVERSAL SCREEN SEMI-SLANT LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KAEHLER MANIFOLDS

S. S. Shukla and Akhilesh Yadav

Abstract. In this paper, we introduce the notion of radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributionsD1,D2 and RadT Mon radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler man- ifolds have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic.

1. Introduction

The theory of lightlike submanifolds of a semi-Riemannian manifold was intro- duced by Duggal and Bejancu [2]. A submanifoldM of a semi-Riemannian manifold M is said to be lightlike submanifold if the induced metricgonM is degenerate, i.e., there exists a non-zeroX∈Γ(T M) such that g(X, Y) = 0, ∀Y ∈Γ(T M). Various classes of lightlike submanifolds of indefinite Kaehler manifolds have been defined according to the behaviour of distributions on these submanifolds with respect to the action of (1,1) tensor field J in Kaehler structure of the ambient manifolds.

Such submanifolds have been studied in [3, 7].

The geometry of slant submanifolds of Kaehler manifolds was studied by B.

Y. Chen in [1] and the geometry of semi-slant submanifolds of Kaehler manifolds was studied by N. Papaghuic in [5]. In [6], Sahin studied screen-slant lightlike submanifolds of an indefinite Hermitian manifold. The theory of radical transver- sal, transversal, semi-transversal lightlike submanifolds has been studied in [8]. In [9–11], the authors studied lightlike submanifolds, radical transversal lightlike sub- manifolds and radical transversal screen semi-slant lightlike submanifolds. In this paper, we introduce the notion of radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds. This new class of lightlike submani- folds of an indefinite Kaehler manifold includes radical transversal and transversal lightlike submanifolds as its sub-cases.

2010 Mathematics Subject Classification: 53C15, 53C40, 53C50

Keywords and phrases: Semi-Riemannian manifold; degenerate metric; radical distribution;

screen distribution; screen transversal vector bundle; lightlike transversal vector bundle; Gauss and Weingarten formulae.

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The paper is arranged as follows. There are some basic results in Section 2. In Section 3, we introduce radical transversal screen semi-slant lightlike submanifolds of an indefinite Kaehler manifold, giving some examples. Section 4 is devoted to the study of foliations determined by distributions on radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds.

2. Preliminaries

A submanifold (M^m, g) immersed in a semi-Riemannian manifold (M^m+n, g) is called a lightlike submanifold [2] if the metric g induced from g is degenerate and the radical distribution RadT M is of rank r, where 1≤r ≤m. Let S(T M) be a screen distribution which is a semi-Riemannian complementary distribution ofRadT M in TM, that is

T M =RadT M⊕orthS(T M).

Now consider a screen transversal vector bundle S(T M^⊥), which is a semi- Riemannian complementary vector bundle ofRadT M inT M^⊥. Since for any local basis{ξi} of RadT M, there exists a local null frame{Ni}of sections with values in the orthogonal complement of S(T M^⊥) in [S(T M)]^⊥ such thatg(ξi, Nj) =δij

and g(Ni, Nj) = 0, it follows that there exists a lightlike transversal vector bun- dle ltr(T M) locally spanned by {Ni}. Let tr(T M) be complementary (but not orthogonal) vector bundle toT M inT M|M. Then

tr(T M) =ltr(T M)⊕orthS(T M^⊥), T M|M =T M⊕tr(T M),

T M|M =S(T M)⊕orth[RadT M⊕ltr(T M)]⊕orthS(T M^⊥).

Following are four cases of a lightlike submanifold¡

M, g, S(T M), S(T M^⊥)¢ : Case 1. r-lightlike ifr <min (m, n),

Case 2. co-isotropic ifr=n < m,S¡ T M^⊥¢

={0}, Case 3. isotropic ifr=m < n, S(T M) ={0},

Case 4. totally lightlike ifr=m=n,S(T M) =S(T M^⊥) ={0}.

The Gauss and Weingarten formulae are given as

∇XY =∇XY +h(X, Y), (2.1)

∇XV =−AVX+∇^t_XV, (2.2)

for all X, Y ∈ Γ(T M) andV ∈Γ(tr(T M)), where ∇XY, AVX belong to Γ(T M) and h(X, Y),∇^t_XV belong to Γ(tr(T M)). ∇and ∇^tare linear connections on M and on the vector bundle tr(T M) respectively. The second fundamental form h is a symmetric F(M)-bilinear form on Γ(T M) with values in Γ(tr(T M)) and the shape operatorAV is a linear endomorphism of Γ(T M). From (2.1) and (2.2), for

anyX, Y ∈Γ(T M),N ∈Γ(ltr(T M)) andW ∈Γ(S(T M^⊥)), we have

∇XY =∇XY +h^l(X, Y) +h^s(X, Y), (2.3)

∇XN =−ANX+∇^l_XN+D^s(X, N), (2.4)

∇XW =−AWX+∇^s_XW+D^l(X, W), (2.5) where h^l(X, Y) = L(h(X, Y)), h^s(X, Y) = S(h(X, Y)), D^l(X, W) = L(∇^t_XW), D^s(X, N) = S(∇^t_XN). L and S are the projection morphisms of tr(T M) on ltr(T M) andS(T M^⊥) respectively. ∇^land∇^sare linear connections onltr(T M) and S(T M^⊥) called the lightlike connection and screen transversal connection on M respectively.

Now by using (2.1), (2.3)–(2.5) and metric connection∇, we obtain g(h^s(X, Y), W) +g(Y, D^l(X, W)) =g(AWX, Y),

g(D^s(X, N), W) =g(N, AWX).

Denote the projection of T M on S(T M) by P. Then from the decomposition of the tangent bundle of a lightlike submanifold, for any X, Y ∈ Γ(T M) and ξ ∈ Γ(RadT M), we have

∇XP Y =∇^∗_XP Y +h^∗(X, P Y),

∇Xξ=−A^∗_ξX+∇^∗t_Xξ, By using the above equations, we obtain

g(h^l(X, P Y), ξ) =g(A^∗_ξX, P Y), g(h^∗(X, P Y), N) =g(ANX, P Y),

g(h^l(X, ξ), ξ) = 0, A^∗_ξξ= 0.

It is important to note that in general ∇ is not a metric connection. Since ∇ is metric connection, by using (2.3), we get

(∇Xg)(Y, Z) =g(h^l(X, Y), Z) +g(h^l(X, Z), Y).

An indefinite almost Hermitian manifold (M , g, J) is a 2m-dimensional semi- Riemannian manifold M with semi-Riemannian metric g of constant index q, 0 < q < 2m and a (1, 1) tensor field J on M such that following conditions are satisfied:

J²X=−X,

g(JX, JY) =g(X, Y), (2.6) for allX, Y ∈Γ(T M).

An indefinite almost Hermitian manifold (M , g, J) is called an indefinite Kaehler manifold ifJ is parallel with respect to∇, i.e.,

(∇XJ)Y = 0, (2.7)

for allX, Y ∈Γ(T M), where∇ is Levi-Civita connection with respect tog.

3. Radical transversal screen semi-slant lightlike submanifolds In this section, we introduce the notion of radical transversal screen semi- slant lightlike submanifolds of indefinite Kaehler manifolds. At first, we state the following lemma for later use:

Lemma 3.1. Let M be a 2q-lightlike submanifold of an indefinite Kaehler manifold M, of index 2q such that 2q < dim(M). Then the screen distribution S(T M)on lightlike submanifold M is Riemannian.

The proof of above Lemma follows as in Lemma 3.1 of [6], so we omit it.

Definition 3.1. LetM be a 2q-lightlike submanifold of an indefinite Kaehler manifoldM of index 2qsuch that 2q < dim(M). Then we say thatM is a radical transversal screen semi-slant lightlike submanifold ofM if the following conditions are satisfied:

(i) J(RadT M) =ltr(T M),

(ii) there exist non-degenerate orthogonal distributionsD1andD2onM such that S(T M) =D1⊕orthD2,

(iii) the distributionD1 is an invariant, i.e. JD1=D1,

(iv) the distributionD2 is slant with angle θ(6= 0), i.e. for eachx∈M and each non-zero vectorX ∈(D2)x, the angle θbetweenJX and the vector subspace (D2)xis a non-zero constant, which is independent of the choice ofx∈M and X ∈(D2)x.

This constant angle θ is called the slant angle of distributionD2. A radical transversal screen semi-slant lightlike submanifold is said to be proper ifD16={0}, D26={0}andθ6= ^π₂.

From the above definition, we have the following decomposition T M =RadT M⊕orthD1⊕orthD2.

Let (R^2m_2q , g, J) denote the manifoldR^2m_2q with its usual Kaehler structure given by g= ¹₄(−P_q

i=1dxⁱ⊗dxⁱ+dyⁱ⊗dyⁱ+P_m

i=q+1dxⁱ⊗dxⁱ+dyⁱ⊗dyⁱ), J(P_m

i=1(Xi∂xi+Yi∂yi)) =P_m

i=1(Yi∂xi−Xi∂yi),

where (xⁱ, yⁱ) are the cartesian coordinates onR^2m_2q . Now we construct some exam- ples of radical transversal screen semi-slant lightlike submanifolds of an indefinite Kaehler manifold.

Example 1. Let (R¹²₂ , g, J) be an indefinite Kaehler manifold, where g is of signature (−,+,+,+,+,+,−,+,+,+,+,+) with respect to the canonical basis {∂x1, ∂x2, ∂x3, ∂x4, ∂x5, ∂x6, ∂y1, ∂y2, ∂y3, ∂y4, ∂y5, ∂y6}.

SupposeM is a submanifold of R¹²₂ given byx¹=−y²=u1,x²=−y¹=u2, x³ = u3cosβ, y³ = −u4cosβ, x⁴ =u4sinβ, y⁴ = u3sinβ, x⁵ = u5sinu6, y⁵ = u5cosu6,x⁶= sinu5,y⁶= cosu5.

The local frame ofT M is given by{Z1, Z2, Z3, Z4, Z5, Z6}, where Z1= 2(∂x1−∂y2),Z2= 2(∂x2−∂y1),

Z3= 2(cosβ∂x3+ sinβ∂y4),Z4= 2(sinβ∂x4−cosβ∂y3), Z5= 2(sinu6∂x5+ cosu6∂y5+ cosu5∂x6−sinu5∂y6), Z6= 2(u5cosu6∂x5−u5sinu6∂y5).

HenceRadT M =span{Z1, Z2} andS(T M) =span{Z3, Z4, Z5, Z6}.

Nowltr(T M) is spanned byN1=−∂x1−∂y2,N2=−∂x2−∂y1andS(T M^⊥) is spanned by

W₁= 2(sinβ∂x₃−cosβ∂y₄),W₂= 2(cosβ∂x₄+ sinβ∂y₃), W3= 2(sinu6∂x5+ cosu6∂y5−cosu5∂x6+ sinu5∂y6), W4= 2(u5sinu5∂x6+u5cosu5∂y6).

It follows that JZ1 = 2N2 and JZ2 = 2N1, which implies that JRadT M = ltr(T M). On the other hand, we can see that D1 = span{Z3, Z4} such that JZ₃ =Z₄ and JZ₄ =−Z₃, which implies that D₁ is invariant with respect to J andD2=span{Z5, Z6}is a slant distribution with slant angleπ/4. HenceM is a radical transversal screen semi-slant 2-lightlike submanifold ofR¹²₂ .

Example 2. Let (R¹²₂ , g, J) be an indefinite Kaehler manifold, where g is of signature (−,+,+,+,+,+,−,+,+,+,+,+) with respect to the canonical basis {∂x1, ∂x2, ∂x3, ∂x4, ∂x5, ∂x6, ∂y1, ∂y2, ∂y3, ∂y4, ∂y5, ∂y6}.

Suppose M is a submanifold of R¹²₂ given by x¹ = u1, y¹ = −u2, x² = u1cosα−u2sinα, y² = u1sinα+u2cosα, x³ = y⁴ = u3, x⁴ = −y³ = u4, x⁵=u₅cosθ,y⁵=u₆cosθ,x⁶=u₆sinθ, y⁶=u₅sinθ.

The local frame ofT M is given by{Z1, Z2, Z3, Z4, Z5, Z6}, where Z1= 2(∂x1+ cosα∂x2+ sinα∂y2),Z2= 2(−∂y1−sinα∂x2+ cosα∂y2), Z3= 2(∂x3+∂y4),Z4= 2(∂x4−∂y3),

Z5= 2(cosθ∂x5+ sinθ∂y6),Z6= 2(sinθ∂x6+ cosθ∂y5).

HenceRadT M =span{Z1, Z2} andS(T M) =span{Z3, Z4, Z5, Z6}.

Now ltr(T M) is spanned by N1 =−∂x1+ cosα∂x2+ sinα∂y2, N2 =∂y1− sinα∂x2+ cosα∂y2 andS(T M^⊥) is spanned by

W1= 2(∂x3−∂y4),W2= 2(∂x4+∂y3),

W3= 2(sinθ∂x5−cosθ∂y6),W4= 2(cosθ∂x6−sinθ∂y5).

It follows thatJZ1=−2N2,JZ2= 2N1, which implies thatJRadT M=ltr(T M).

On the other hand, we can see thatD1=span{Z3, Z4}such thatJZ3=Z4,JZ4=

−Z3, which implies thatD1is invariant with respect toJ andD2=span{Z5, Z6} is a slant distribution with slant angle 2θ. HenceM is a radical transversal screen semi-slant 2-lightlike submanifold ofR¹²₂ .

Now, for any vector fieldX tangent toM, we putJX=P X+F X, whereP X and F X are tangential and transversal parts of JX respectively. We denote the

projections on RadT M, D1 and D2 in T M by P1, P2 and P3 respectively. Then for anyX ∈Γ(T M), we get

X =P1X+P2X+P3X. (3.1)

Now applyingJ to (3.1), we have

JX=JP1X+JP2X+JP3X, which gives

JX =JP1X+JP2X+f P3X+F P3X, (3.2) where f P3X (resp.F P3X) denotes the tangential (resp. transversal) component of JP₃X. Thus we get JP₁X ∈ Γ(ltr(T M)), JP₂X ∈Γ(D₁), f P₃X ∈Γ(D₂) and F P3X ∈Γ(S(T M^⊥)).

Similarly, we denote the projections oftr(T M) on ltr(T M) andS(T M^⊥) by Q₁and Q₂ respectively. Then for anyW ∈Γ(tr(T M)), we have

W =Q1W+Q2W. (3.3)

ApplyingJ to (3.3), we obtain

JW =JQ1W +JQ2W, which gives

JW =JQ₁W+BQ₂W+CQ₂W, (3.4)

whereBQ2W (resp. CQ2W) denotes the tangential (resp. transversal) component of JQ2W. Thus we get JQ1W ∈ Γ(RadT M), BQ2W ∈ Γ(D2) and CQ2W ∈ Γ(S(T M^⊥)).

Now, by using (2.7), (3.2), (3.4) and (2.3)–(2.5) and identifying the components onRadT M,D1,D2,ltr(T M) andS(T M^⊥), we obtain

P1(∇XJP2Y) +P1(∇Xf P3Y) =P1(A_JP

1YX) +P1(AF P3YX) +Jh^l(X, Y), P2(∇XJP2Y) +P2(∇Xf P3Y) =P2(AF P3YX) +P2(A_JP

1YX) +JP2∇XY, (3.5) P3(∇XJP2Y) +P3(∇Xf P3Y)

=P3(AF P3YX) +P3(A_JP1YX) +f P3∇XY +Bh^s(X, Y), (3.6)

∇^l_XJP1Y +h^l(X, JP2Y) +h^l(X, f P3Y) =JP1∇XY −D^l(X, F P3Y), (3.7) D^s(X, JP1Y) +h^s(X, JP2Y) +h^s(X, f P3Y)

=Ch^s(X, Y)− ∇^s_XF P3Y +F P3∇XY. (3.8) Theorem 3.2. Let M be a 2q-lightlike submanifold of an indefinite Kaehler manifoldM. ThenM is a radical transversal screen semi-slant lightlike submanifold if and only if

(i) Jltr(T M)is a distribution onM such thatJltr(T M) =RadT M, (ii) distributionD1 is invariant with respect toJ, i.e. JD1=D1, (iii) there exists a constant λ∈[0,1)such that P²X =−λX.

Moreover, there also exists a constant µ∈ (0,1] such that BF X =−µX, for all X ∈ Γ(D2), whereD1 andD2 are non-degenerate orthogonal distributions on M such that S(T M) =D1⊕orthD2 andλ= cos²θ,θ is slant angle of D2.

Proof. LetM be a radical transversal screen semi-slant lightlike submanifold of an indefinite Kaehler manifoldM. Then distributionD1is invariant with respect to J andJRadT M =ltr(T M). ThusJX∈ltr(T M), for allX ∈Γ(RadT M). Hence J(JX)∈J(ltr(T M)), which implies−X ∈J(ltr(T M)), for allX ∈Γ(RadT M), which proves (i) and (ii).

Now for anyX ∈Γ(D2), we have|P X|=|JX|cosθ, which implies cosθ= |P X|

|JX|. (3.9)

In view of (3.9), we get cos²θ=^{|P X|}_|JX|2² = ^{g(P X,P X)}_g(JX,JX) =^g(X,P2X)

g(X,J²X), which gives g(X, P²X) = cos²θ g(X, J²X). (3.10) Since M is radical transversal screen semi-slant lightlike submanifold, cos²θ = λ(constant)∈[0,1) and therefore from (3.14), we getg(X, P²X) =λg(X, J²X) = g(X, λJ²X), which implies

g(X,(P²−λJ²)X) = 0.

Now for anyX ∈Γ(D2), we haveJ²(X) =P²X+F P X+BF X+CF X. Taking the tangential component, we get P²X = −X −BF X ∈ Γ(D2), for any X ∈ Γ(D₂). Thus (P²−λJ²)X ∈ Γ(D₂). Since the induced metric g = g|_D2×D2 is non-degenerate(positive definite), by the facts above, we have (P²−λJ²)X = 0, which implies

P²X =λJ²X=−λX. (3.11)

Now, for any vector fieldX∈Γ(D2), we have

JX=P X+F X, (3.12)

whereP X andF X are tangential and transversal parts ofJX respectively.

ApplyingJ to (3.12) and taking tangential component, we get

−X =P²X+BF X. (3.13)

From (3.11) and (3.13), we getBF X =−µX, where 1−λ=µ(constant)∈(0,1].

This proves (iii).

Conversely suppose that conditions (i), (ii) and (iii) are satisfied. From (i), we have JN ∈RadT M, for all N ∈Γ(ltr(T M)). HenceJ(JN)∈J(RadT M), which implies−N ∈ J(RadT M), for all N ∈ Γ(ltr(T M)). ThusJRadT M =ltr(T M).

From (3.18), for anyX ∈Γ(D2), we get−X =P²X−µX, which impliesP²X =

−λX, where 1−µ=λ(constant)∈[0,1).

Now cosθ= ^{g(JX,P X)}_{|JX||P X|} =−^{g(X,JP X)}_{|JX||P X|} =−^g(X,P_{|JX||P X|}^2X) =−λ^g(X,J_{|JX||P X|}^2X) =λ^g(JX,JX)_{|JX||P X|}. From the above equation, we get

cosθ=λ|JX|

|P X|. (3.14)

Therefore (3.9) and (3.14) give cos²θ=λ(constant).

HenceM is a radical transversal screen semi-slant lightlike submanifold.

Corollary 3.1. Let M be a radical transversal screen semi-slant lightlike submanifold of an indefinite Kaehler manifold M with slant angleθ, then for any X, Y ∈Γ(D2), we have

(i) g(P X, P Y) = cos²θ g(X, Y), (ii) g(F X, F Y) = sin²θ g(X, Y).

The proof of above Corollary follows by using similar steps as in proof of Corollary 3.2 of [6].

Theorem 3.3. Let M be a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifoldM with structure vector field tangent to M. Then RadT M is integrable if and only if

(i) P2(A_JP

1YX) =P2(A_JP

1XY)andP3(A_JP

1YX) =P3(A_JP

1XY), (ii) D^s(Y, JP1X) =D^s(X, JP1Y), for allX, Y ∈Γ(RadT M).

Proof. Let M be a radical transversal screen semi-slant lightlike submani- fold of an indefinite Kaehler manifold M. Let X, Y ∈ Γ(RadT M). From (3.8), we have D^s(X, JP1Y) = Ch^s(X, Y) +F P3∇XY, which gives D^s(X, JP1Y)− D^s(Y, JP1X) =F P3[X, Y]. In view of (3.5), we haveP2(A_JP1YX)+JP2∇XY = 0, which implies P2(A_JP

1XY)−P2(A_JP

1YX) = JP2[X, Y]. Also from (3.6), we have P3(A_JP

1YX) + Bh^s(X, Y) +f P3∇XY = 0, which gives P3(A_JP

1XY) − P3(A_JP

1YX) =f P3[X, Y]. This concludes the theorem.

Theorem 3.4. Let M be a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifold M. Then D1 is integrable if and only if

(i) h^l(Y, JP2X) =h^l(X, JP2Y)andh^s(Y, JP2X) =h^s(X, JP2Y), (ii) P3(∇XJP2Y) =P3(∇YJP2X), for allX, Y ∈Γ(D1).

Proof. Let M be a radical transversal screen semi-slant lightlike subman- ifold of an indefinite Kaehler manifold M. Let X, Y ∈ Γ(D1). From (3.8),

we have h^s(X, JP2Y) = Ch^s(X, Y) +F P3∇XY, which implies h^s(X, JP2Y)− h^s(Y, JP2X) = F P3[X, Y]. In view of (3.7), we have h^l(X, JP2Y) = JP1∇XY, which gives h^l(X, JP₂Y)−h^l(Y, JP₂X) = JP₁[X, Y]. From (3.6), we obtain P3(∇XJP2Y) = f P3∇XY + Bh^s(X, Y), which implies P3(∇XJP2Y) − P3(∇YJP2X) =f P3[X, Y]. This proves the theorem.

Theorem 3.5. Let M be a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifold M. Then D2 is integrable if and only if

(i) h^l(X, f P3Y) +D^l(X, F P3Y) =h^l(Y, f P3X) +D^l(Y, F P3X), (ii) P2(∇Xf P3Y − ∇Yf P3X) =P2(AF P3YX−AF P3XY), for allX, Y ∈Γ(D2).

Proof. Let M be a radical transversal screen semi-slant lightlike subman- ifold of an indefinite Kaehler manifold M. Let X, Y ∈ Γ(D2). From (3.7), we have h^l(X, f P₃Y) +D^l(X, F P₃Y) = JP₁∇_XY, which gives h^l(X, f P₃Y) + D^l(X, F P3Y)−h^l(Y, f P3X)−D^l(Y, F P3X) = JP1[X, Y]. Also from (35), we obtain P2(∇Xf P3Y) = P2(AF P3YX) +JP2∇XY, which implies P2(∇Xf P3Y −

∇Yf P3X) =P2(AF P3YX−AF P3XY) +JP2[X, Y]. Thus, we obtain the required results.

Theorem 3.6. Let M be a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifold M. Then the induced connection∇ is a metric connection if and only if

(i) BD^s(X, N) =f P3ANX,

(ii) JP2ANX = 0, for all X∈Γ(T M) andN∈Γ(ltr(T M)).

Proof. LetM be a radical transversal screen semi-slant lightlike submanifold of an indefinite Kaehler manifoldM. Then the induced connection∇onM is a metric connection if and only ifRadT Mis parallel distribution with respect to∇[2]. From (2.3), (2.4) and (2.7), we obtain ∇XJN +h^l(X, JN) +h^s(X, JN) =−JANX + J∇^l_XN+JD^s(X, N). Now, on comparing tangential components of both sides of above equation, we get∇XJN=−JP2ANX−f P3ANX+J∇^l_XN+BD^s(X, N), which completes the proof.

4. Foliations determined by distributions

In this section, we obtain necessary and sufficient conditions for foliations determined by distributions on a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifold to be totally geodesic.

Definition 4.1. An equivalence relation on ann-dimensional semi-Rieman- nian manifold (M , g) in which the equivalence classes are connected, immersed submanifolds (called the leaves of the foliation) of a common dimensionk, 0< k≤n is called a foliation on M. If each leaf of a foliation F on a semi-Riemannian

manifold (M , g) is totally geodesic submanifold of M, we say that F is a totally geodesic foliation.

Theorem 4.1. Let M be a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifold M. Then RadT M defines a totally geodesic foliation if and only ifg(AF P3ZX, JY) =g(∇XJP2Z+∇Xf P3Z, JY), for allX, Y ∈Γ(RadT M)andZ ∈Γ(S(T M)).

Proof. Let M be a radical transversal screen semi-slant lightlike subman- ifold of an indefinite Kaehler manifold M. To prove RadT M defines a total- ly geodesic foliation, it is sufficient to show that ∇XY ∈ Γ(RadT M), for all X, Y ∈ Γ(RadT M). Since ∇ is metric connection, using (2.3), (2.6), (2.7) and (3.2), for any X, Y ∈ Γ(RadT M) and Z ∈ Γ(S(T M)), we obtain g(∇XY, Z) =

−g(∇X(JP2Z+f P3Z+F P3Z), JY), which impliesg(∇XY, Z) =g(P1AF P3ZX− P1∇XJP2Z−P1∇Xf P3Z, JY). This proves the theorem.

Theorem 4.2. Let M be a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifoldM. Then D1 defines a totally geodesic foliation if and only if

(i) g(AF ZX, JY) =g(∇Xf Z, JY), for allX, Y ∈Γ(D1)andZ ∈Γ(D2), (ii) A^∗_JN vanishes onD1, for all N∈Γ(ltr(T M)).

Proof. LetM be a radical transversal screen semi-slant lightlike submanifold of an indefinite Kaehler manifoldM. The distribution D1 defines a totally geodesic foliation if and only if ∇XY ∈ Γ(D1), for all X, Y ∈ Γ(D1). Since ∇ is metric connection, from (2.3), (2.6) and (2.7), for any X, Y ∈ Γ(D1) and Z ∈ Γ(D2), we obtain g(∇_XY, Z) =−g(∇_XJZ, JY), which implies g(∇_XY, Z) = g(A_{F Z}X −

∇Xf Z, JY). Also, from (2.3), (2.6) and (2.7), for any X, Y ∈ Γ(D1) and N ∈ Γ(ltr(T M)), we have g(∇XY, N) = −g(JY,∇XJN), which gives g(∇XY, N) =

−g(JY,∇XJN) =g(JY, A^∗_JNX). This concludes the theorem.

Theorem 4.3. Let M be a radical transversal screen semi-slant lightlike sub- manifold of an indefinite Kaehler manifoldM. Then D2 defines a totally geodesic foliation if and only if

(i) g(f Y,∇_XJZ) =−g(F Y, h^s(X, JZ)), (ii) g(f Y,∇XJN) =−g(F Y, h^s(X, JN)),

for allX, Y ∈Γ(D2),Z∈Γ(D1)andN ∈Γ(ltr(T M)).

Proof. LetM be a radical transversal screen semi-slant lightlike submanifold of an indefinite Kaehler manifoldM. The distribution D2 defines a totally geodesic foliation if and only if ∇XY ∈ Γ(D2), for all X, Y ∈ Γ(D2). Since ∇ is metric connection, using (2.3), (2.6) and (2.7), for any X, Y ∈Γ(D2) andZ ∈Γ(D1), we get g(∇XY, Z) = −g(JY,∇XJZ), which implies g(∇XY, Z) = −g(f Y,∇XJZ)− g(F Y, h^s(X, JZ)). Now, from (2.3), (2.6) and (2.7), for anyX, Y ∈Γ(D2) andN∈

Γ(ltr(T M)), we have g(∇XY, N) = −g(JY,∇XJN), which gives g(∇XY, N) =

−g(f Y,∇XJN)−g(F Y, h^s(X, JN)). Thus, we obtain the required results.

Acknowledgement: Akhilesh Yadav gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (C.S.I.R.), India.

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(received 13.05.2015; in revised form 17.01.2016; available online 03.02.2016) Department of Mathematics, University of Allahabad, Allahabad-211002, India E-mail:ssshukla [email protected], akhilesh [email protected]