Radical transversal screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds

(1)

Radical transversal screen pseudo-slant lightlike

submanifolds of indefinite Kaehler manifolds

S.S. Shukla and Akhilesh Yadav

(Received May 20, 2014; Revised September 25, 2014)

Abstract. In this paper, we introduce the notion of radical transversal screen

pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds giving char-acterization theorem with some non-trivial examples of such submanifolds. Inte-grability conditions of distributions D1, D2and RadT M on radical transversal

screen pseudo-slant lightlike submanifolds of an indefinite Kaehler manifold have been obtained. Further, we obtain necessary and suﬃcient conditions for folia-tions determined by above distribufolia-tions to be totally geodesic. We also study mixed geodesic radical transversal screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds.

AMS 2010 Mathematics Subject Classification. 53C15; 53C40; 53C50.

Key words and phrases. Semi-Riemannian manifold, degenerate metric, radi-cal distribution, screen distribution, screen transversal vector bundle, lightlike transversal vector bundle, Gauss and Weingarten formulae.

§1. Introduction

In 1990, B.Y. Chen defined slant immersions in complex geometry as a natural generalization of both holomorphic and totally real immersions ([3]). Further, A. Lotta introduced the concept of slant immersions of a Riemannian manifold into an almost contact metric manifold ([9]). A. Carriazo defined and studied bi-slant submanifolds of almost Hermitian and almost contact metric manifolds and gave the notion of pseudo-slant submanifolds ([2]). The theory of lightlike submanifolds of a semi-Riemannian manifold was introduced by Duggal and Bejancu ([6]). A submanifold M of a semi-Riemannian manifold M is said to be lightlike submanifold if the induced metric g on M is degenerate, i.e. there exists a non-zero X∈ Γ(T M) such that g(X, Y ) = 0, ∀Y ∈ Γ(T M).

The theory of radical transversal, transversal, semi-transversal lightlike sub-manifolds has been studied in ([14]). In this article, we introduce the notion

(2)

of radical transversal screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds. This new class of lightlike submanifolds of an indefinite Kaehler manifold includes radical transversal and transversal lightlike sub-manifolds as its sub-cases. The paper is arranged as follows. There are some basic results in section 2 . In section 3, we introduce radical transversal screen pseudo-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 pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds.

§2. Preliminaries

A submanifold (Mm, g) immersed in a semi-Riemannian manifold (Mm+n, g) is called a lightlike submanifold ([6]) 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 of RadT M in TM, that is

(2.1) 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 of RadT M in T M⊥. Since for any local basis {ξi} of RadT M, there exists a local null frame {Ni} of

sec-tions with values in the orthogonal complement of S(T M⊥) in [S(T M )]⊥such that g(ξi, Nj) = δij and g(Ni, Nj) = 0, it follows that there exists a lightlike

transversal vector bundle ltr(T M ) locally spanned by {Ni}. Let tr(T M) be

complementary (but not orthogonal) vector bundle to T M in T M|M. Then

(2.2) tr(T M ) = ltr(T M )⊕orthS(T M⊥),

(2.3) T M|M = T M⊕ tr(T M),

(2.4) 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 if r < min (m, n),

Case.2 co-isotropic if r = n < m, S(T M⊥)={0}, Case.3 isotropic if r = m < n, S (T M ) ={0},

Case.4 totally lightlike if r = m = n, S(T M ) = S(T M⊥) ={0}. The Gauss and Weingarten formulae are given as

(3)

(2.6) ∇XV =−AVX +∇tXV, ∀ V ∈ Γ(tr(T M)),

where{∇XY, AVX} and

{

h(X, Y ),∇t_XV} belong to Γ(T M ) and Γ(tr(T M )) respectively. ∇ 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 operator AV

is a linear endomorphism of Γ(T M ). From (2.5) and (2.6), for any X, Y ∈ Γ(T M ), N ∈ Γ(ltr(T M)) and W ∈ Γ(S(T M⊥)), we have

(2.7) ∇XY =∇XY + hl(X, Y ) + hs(X, Y ) ,

(2.8) ∇XN =−ANX +∇lXN + Ds(X, N ) ,

(2.9) ∇XW =−AWX +∇sXW + Dl(X, W ) ,

where hl(X, Y ) = L (h(X, Y )), hs(X, Y ) = S (h(X, Y )), Dl(X, W ) = L(∇t_XW ), Ds(X, N ) = S(∇t_XN ). L and S are the projection morphisms of tr(T M ) on ltr(T M ) and S(T M⊥) respectively. ∇l _and _∇s _{are linear connections on} ltr(T M ) and S(T M⊥) called the lightlike connection and screen transversal connection on M respectively.

Now by using (2.5), (2.7)-(2.9) and metric connection∇, we obtain (2.10) g(hs(X, Y ), W ) + g(Y, Dl(X, W )) = g(AWX, Y ),

(2.11) g(Ds(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 (2.12) ∇XP Y =∇∗XP Y + h∗(X, P Y ), (2.13) ∇Xξ =−A∗ξX +∇∗tXξ, where { ∇∗ XP Y, A∗ξX }

and {h∗(X, P Y ),∇∗t_Xξ} belong to Γ(S(T M )) and Γ (Rad(T M )) respectively. By using above equations, we obtain

(2.14) g(hl(X, P Y ), ξ) = g(A∗_ξX, P Y ),

(4)

(2.16) g(hl(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 (2.7), for any X, Y, Z∈ Γ(T M), we get

(2.17) (∇Xg)(Y, Z) = g(hl(X, Y ), Z) + g(hl(X, Z), Y ).

An indefinite almost Hermitian manifold (M , g, J ) is a 2m-dimensional semi-Riemannian manifold M with semi-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:

(2.18) J2X =−X,

(2.19) g(J X, J Y ) = g(X, Y ),

for all X, Y ∈ Γ(T M).

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

(2.20) (∇XJ )Y = 0,

for all X, Y ∈ Γ(T M), where ∇ is Levi-Civita connection with respect to g. For any vector field X tangent to M , we put

(2.21) J X = P X + F X,

where P X and F X are tangential and transversal parts of J X respectively.

§3. Radical Transversal Screen Pseudo-Slant Lightlike

Submanifolds

In this section, we introduce the notion of radical transversal screen pseudo-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 [12], so we omit it. Definition 3.1. Let M be a 2q-lightlike submanifold of an indefinite Kaehler manifold M of index 2q such that 2q < dim(M ). Then we say that M is

(5)

a radical transversal screen pseudo-slant lightlike submanifold of M if the following conditions are satisfied:

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

(ii) there exists non-degenerate orthogonal distributions D1and D2on M such

that S(T M ) = D1⊕orthD2,

(iii) the distribution D1 is anti-invariant, i.e. J D1 ⊂ S(T M⊥),

(iv) the distribution D2 is slant with angle θ ∈ [0, π/2), i.e. there exists

θ∈ [0, π/2) such that |P X| = |JX| cos θ, for any X ∈ Γ(D2).

This constant angle θ is called the slant angle of distribution D2. A radical

transversal screen pseudo-slant lightlike submanifold is said to be proper if D1 ̸= {0}, D2 ̸= {0} and θ ̸= 0.

From the above definition, we have the following decomposition

(3.1) T M = RadT M⊕orthD1⊕orthD2.

Let (R2m_2q , g, J ) denote the manifoldR2m_2q with its usual Kaehler structure given by g = 1₄ ( −∑q i=1(dxi⊗ dxi+ dyi⊗ dyi) + ∑m i=q+1(dxi⊗ dxi+ dyi⊗ dyi) ) , J (∑m_i=1(Xi∂xi+ Yi∂yi)) = ∑m i=1(Yi∂xi− Xi∂yi),

where (xi, yi) are the cartesian coordinates onR2m_2q . Now, we construct some examples of radical transversal screen pseudo-slant lightlike submanifolds of an indefinite Kaehler manifold.

Example 1. Let (R12

2 , 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 = y2 = u1, x2 = y1 = u2,

x3 = u3cos β, y3 = u3sin β, x4 = u4sin β, y4 = u4cos β, x5 = u5, y5 = u6,

x6 = k cos u6, y6= k sin u6, where k is any constant.

The local frame of T M is given by {Z1, Z2, Z3, Z4, Z5, Z6}, where

Z1 = 2(∂x1+ ∂y2), Z2= 2(∂x2+ ∂y1),

Z3 = 2(cos β∂x3+ sin β∂y3),

Z4 = 2(sin β∂x4+ cos β∂y4),

Z5 = 2(∂x5),

Z6 = 2(∂y5− k sin u6∂x6+ k cos u6∂y6).

Hence RadT M = span{Z1, Z2} and S(T M) = span {Z3, Z4, Z5, Z6}.

Now ltr(T M ) is spanned by N1 = −∂x1 + ∂y2, N2 = −∂x2 + ∂y1 and

S(T M⊥) is spanned by

W1= 2(sin β∂x3− cos β∂y3),

W2= 2(cos β∂x4− sin β∂y4),

W3= 2(k cos u6∂x6+ k sin u6∂y6),

(6)

It follows that J Z1 = −2N2, J Z2 = −2N1, which implies that J RadT M =

ltr(T M ). On the other hand, we can see that D1 = span{Z3, Z4} such that

J Z3 = W1, J Z4 = W2, which implies that D1 is anti-invariant with respect

to J and D2 = span{Z5, Z6} is a slant distribution with slant angle θ =

arccos(1/√1 + k2_{). Hence M is a radical transversal screen pseudo-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 ofR12₂ given by x1= u1, y1 = u2, x2=−u1cos α+

u2sin α, y2 = u1sin α + u2cos α, x3 = y4 = u3, x4 = y3 = u4, x5 = u5cos u6,

y5 = u5sin u6, x6 = cos u5, y6= sin u5.

The local frame of T M is given by {Z1, Z2, Z3, Z4, Z5, Z6}, where

Z1 = 2(∂x1− cos α∂x2+ sin α∂y2),

Z2 = 2(∂y1+ sin α∂x2+ cos α∂y2),

Z3 = 2(∂x3+ ∂y4), Z4= 2(∂x4+ ∂y3),

Z5 = 2(cos u6∂x5+ sin u6∂y5− sin u5∂x6+ cos u5∂y6),

Z6 = 2(−u5sin u6∂x5+ u5cos u6∂y5).

Hence RadT M = span{Z1, Z2} and S(T M) = span {Z3, Z4, Z5, Z6}.

Now ltr(T M ) is spanned by N1 =−∂x1− cos α∂x2+ sin α∂y2, N2 =−∂y1+

sin α∂x2+ cos α∂y2 and S(T M⊥) is spanned by

W1= 2(∂x3− ∂y4), W2 = 2(∂x4− ∂y3),

W3= 2(cos u6∂x5+ sin u6∂y5+ sin u5∂x6− cos u5∂y6),

W4= 2(u5cos u5∂x6+ u5sin u5∂y6).

It follows that J Z1 = 2N2, J Z2 = −2N1, which implies that J RadT M =

ltr(T M ). On the other hand, we can see that D1 = span{Z3, Z4} such that

J Z3= W2, J Z4= W1, which implies that D1 is anti-invariant with respect to

J and D2 = span{Z5, Z6} is a slant distribution with slant angle π/4. Hence

M is a radical transversal screen pseudo-slant 2-lightlike submanifold ofR12₂ . Now, We denote the projections on RadT M , D1 and D2 in T M by P1, P2 and

P3 respectively. Similarly, we denote the projections of tr(T M ) on ltr(T M ),

J (D1) and D′ by Q1, Q2 and Q3 respectively, where D′ is non-degenerate

orthogonal complementary subbundle of J (D1) in S(T M⊥). Then, for any

X∈ Γ(T M), we get

(3.2) X = P1X + P2X + P3X.

Now applying J to (3.2), we have

(3.3) J X = J P1X + J P2X + J P3X,

which gives

(7)

where f P3X(resp. F P3X) denotes the tangential (resp. transversal)

com-ponent of J P3X. Thus we get J P1X ∈ Γ(ltr(T M)), JP2X ∈ Γ(J(D1)) ⊂

Γ(S(T M⊥)), f P3X∈ Γ(D2) and F P3X∈ Γ(D′). Also, for any W∈ Γ(tr(T M)),

we have (3.5) W = Q1W + Q2W + Q3W. Applying J to (3.5), we obtain (3.6) J W = J Q1W + J Q2W + J Q3W, which gives (3.7) J W = J Q1W + J Q2W + BQ3W + CQ3W,

where BQ3W (resp. CQ3W ) denotes the tangential (resp. transversal)

com-ponent of J Q3W . Thus we get J Q1W ∈ Γ(RadT M), JQ2W ∈ Γ(D1),

BQ3W ∈ Γ(D2) and CQ3W ∈ Γ(D′).

Now, by using (2.20), (3.4), (3.7) and (2.7)-(2.9) and identifying the compo-nents on RadT M , D1, D2, ltr(T M ), J (D1) and D′, we obtain

P1(A_{J P}₂_YX) + P1(A_{J P}₁_YX) + P1(AF P3YX) = P1(∇Xf P3Y ) − Jhl_{(X, Y ),} (3.8) P2(A_{J P}₂_YX) + P2(A_{J P}₁_YX) + P2(AF P3YX) = P2(∇Xf P3Y ) − JQ2hs(X, Y ), (3.9) P3(AJ P2YX) + P3(AJ P1YX) + P3(AF P3YX) = P3(∇Xf P3Y ) − BQ3hs(X, Y )− fP3∇XY, (3.10) (3.11) ∇l_XJ P1Y + Dl(X, J P2Y ) + hl(X, f P3Y ) + Dl(X, F P3Y ) = J P1∇XY, Q2∇sXJ P2Y + Q2∇sXF P3Y =J P2∇XY − Q2Ds(X, J P1Y ) − Q2hs(X, f P3Y ), (3.12) Q3∇sXJ P2Y +Q3∇sXF P3Y − F P3∇XY = CQ3hs(X, Y ) − Q3hs(X, f P3Y )− Q3Ds(X, J P1Y ). (3.13)

Theorem 3.2. Let M be a 2q-lightlike submanifold of an indefinite Kaehler manifold M . Then M is a radical transversal screen pseudo-slant lightlike submanifold of M if and only if

(i) J ltr(T M ) is a distribution on M such that J ltr(T M ) = RadT M , (ii) distribution D1is anti-invariant with respect to J , i.e. J D1⊂ S(T M⊥),

(iii) there exists a constant λ ∈ (0, 1] such that P2X = −λX, for all X∈ Γ(D2), where D1 and D2 are non-degenerate orthogonal distributions on

(8)

Proof. Let M be a radical transversal screen pseudo-slant lightlike subman-ifold of an indefinite Kaehler mansubman-ifold M . Then distribution D1 is

anti-invariant with respect to J and J RadT M = ltr(T M ). Thus for any X ∈ Γ(RadT M ), J X ∈ ltr(T M). Hence J(JX) ∈ J(ltr(T M)), which implies −X ∈ J(ltr(T M)), for all X ∈ Γ(RadT M), which proves (i) and (ii).

Now for any X∈ Γ(D2), we have|P X| = |JX| cos θ, which implies

(3.14) cos θ = |P X|

|JX|. In view of (3.14), we get cos2θ = |P X|_|JX|22 =

g(P X,P X) g(J X,J X) =

g(X,P2X)

g(X,J2X), which gives

(3.15) g(X, P2X) = cos2θ g(X, J2X).

Since M is radical transversal screen pseudo-slant lightlike submanifold, cos2θ = λ(constant) ∈ (0, 1] and therefore from (3.15), we get g(X, P2X) = λg(X, J2X) = g(X, λJ2X), which implies

(3.16) g(X, (P2− λJ2)X) = 0.

Now for any X ∈ Γ(D2), we have J 2

(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 − λJ 2

)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

(3.17) P2X = λJ2X =−λX, ∀X ∈ Γ(D2).

This proves (iii).

Conversely suppose that conditions (i), (ii) and (iii) are satisfied. From (i), we have J N ∈ RadT M, for all N ∈ Γ(ltr(T M)). Hence J(JN) ∈ J(RadT M), which implies−N ∈ J(RadT M), for all N ∈ Γ(ltr(T M)). Thus JRadT M = ltr(T M ). From (iii), we have P2X = λJ2X, for all X ∈ Γ(D2) , where

λ(constant)∈ (0, 1] . Now cos θ = g(J X,P X) |JX||P X| =− g(X,J P X) |JX||P X| =− g(X,P2_X) |JX||P X| =−λ g(X,J2X) |JX||P X| = λ g(J X,J X) |JX||P X|.

From above equation, we get

(3.18) cos θ = λ|JX|

|P X|.

Therefore (3.14) and (3.18) give cos2θ = λ(constant).

(9)

Theorem 3.3. Let M be a 2q-lightlike submanifold of an indefinite Kaehler manifold M . Then M is a radical transversal screen pseudo-slant lightlike submanifold of M if and only if

(i) J ltr(T M ) is a distribution on M such that J ltr(T M ) = RadT M , (ii) distribution D1is anti-invariant with respect to J , i.e. J D1⊂ S(T M⊥),

(iii) there exists a constant µ ∈ [0, 1) such that BF X = −µX, for all X∈ Γ(D2), where D1 and D2 are non-degenerate orthogonal distributions on

M such that S(T M ) = D1⊕orthD2 and µ = sin2θ, θ is slant angle of D2.

Proof. Let M be a radical transversal screen pseudo-slant lightlike subman-ifold of an indefinite Kaehler mansubman-ifold M . Then distribution D1 is

anti-invariant with respect to J and J RadT M = ltr(T M ). Thus for any X ∈ Γ(RadT M ) J X ∈ ltr(T M). Hence J(JX) ∈ J(ltr(T M)), which implies −X ∈ J(ltr(T M)), for all X ∈ Γ(RadT M), which proves (i) and (ii). Now, for any vector field X ∈ Γ(D2) , we have

(3.19) J X = P X + F X,

where P X and F X are tangential and transversal parts of J X respectively. Applying J to (3.19) and taking tangential component, we get

(3.20) −X = P2X + BF X.

Since M is a radical transversal screen pseudo-slant lightlike submanifold, P2X =− cos2θX, for all X ∈ Γ(D2), where cos2θ = λ(constant)∈ (0, 1] and

therefore from (3.20), for any X∈ Γ(D2), we get

(3.21) BF X =− sin2θ X,

where sin2θ = 1− λ = µ(constant) ∈ [0, 1). This proves (iii).

Conversely suppose that conditions (i), (ii) and (iii) are satisfied. From (i), we have J N ∈ RadT M, for all N ∈ Γ(ltr(T M)). Hence J(JN) ∈ J(RadT M), which implies−N ∈ J(RadT M), for all N ∈ Γ(ltr(T M)). Thus JRadT M = ltr(T M ). From (3.20), for any X ∈ Γ(D2), we get

(3.22) −X = P2X− µX,

which implies

(3.23) P2X =− cos2θ X,

where cos2θ = 1− µ = λ(constant) ∈ (0, 1]. Now the proof follows from theorem (3.2).

(10)

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

(3.24) g(P X, P Y ) = cos2θ g(X, Y ),

(3.25) 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 [12].

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

(i) Q2Ds(Y, J P1X) = Q2Ds(X, J P1Y ),

(ii) Q3Ds(Y, J P1X) = Q3Ds(X, J P1Y ),

(iii) P3A_{J P}₁_XY = P3A_{J P}₁_YX, for all X, Y ∈ Γ(RadT M).

Proof. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . From (3.12), for any X, Y ∈ Γ(RadT M), we have

(3.26) Q2Ds(X, J P1Y ) = J P2∇XY.

On interchanging X and Y in (3.26), we get

(3.27) Q2Ds(Y, J P1X) = J P2∇YX.

From (3.26) and (3.27), we obtain

Q2Ds(X, J P1Y )− Q2Ds(Y, J P1X) = J P2[X, Y ].

(3.28)

From (3.13), for any X, Y ∈ Γ(RadT M), we have

Q3Ds(X, J P1Y ) = CQ3hs(X, Y ) + F P3∇XY.

(3.29)

Interchanging X and Y in (3.29), we get

Q3Ds(Y, J P1X) = CQ3hs(Y, X) + F P3∇YX.

(3.30)

In view of (3.29) and (3.30), we obtain

Q3Ds(X, J P1Y )− Q3Ds(Y, J P1X) = F P3[X, Y ].

(11)

From (3.10), for any X, Y ∈ Γ(RadT M), we have

P3A_{J P}₁_YX + BQ3hs(X, Y ) =−fP3∇XY.

(3.32)

P3A_{J P}₁_XY + BQ3hs(Y, X) =−fP3∇YX.

(3.33)

P3AJ P1XY − P3AJ P1YX = f P3[X, Y ]. (3.34)

Now the proof follows from (3.28), (3.31) and 3.34.

Theorem 3.5. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . Then D1 is integrable if

and only if

(i) Q3(∇s_YJ P2X) = Q3(∇s_XJ P2Y ) and P3A_{J P}₂_XY = P3A_{J P}₂_YX,

(ii) Dl(X, J P2Y ) = Dl(Y, J P2X), for all X, Y ∈ Γ(D1).

Proof. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . From (3.11), for any X, Y ∈ Γ(D1), we

have

(3.35) Dl(X, J P2Y ) = J P1∇XY.

(3.36) Dl(Y, J P2X) = J P1∇YX.

(3.37) Dl(X, J P2Y )− Dl(Y, J P2X) = J P1[X, Y ].

From (3.10), for any X, Y ∈ Γ(D1), we have

(3.38) P3A_{J P}₂_YX + BQ3hs(X, Y ) =−fP3∇XY.

(3.39) P3A_{J P}₂_XY + BQ3hs(Y, X) =−fP3∇YX.

(12)

(3.41) Q3(∇sXJ P2Y )− CQ3hs(X, Y ) = F P3∇XY.

(3.42) Q3(∇sYJ P2X)− CQ3hs(Y, X) = F P3∇YX.

From (3.41) and (3.42), we get

(3.43) Q3(∇sXJ P2Y )− Q3(∇sYJ P2X) = F P3[X, Y ].

The proof follows from (3.37), (3.40) and (3.43).

Theorem 3.6. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . Then D2 is integrable if

and only if

(i) Dl(X, F P3Y )− hl(Y, f P3X) = Dl(Y, F P3X)− hl(X, f P3Y ),

(ii) Q2(∇s_XF P3Y − hs(Y, f P3X)) = Q2(∇s_YF P3X− hs(X, f P3Y )),

for all X, Y ∈ Γ(D2).

Proof. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . From (3.11), for any X, Y ∈ Γ(D2), we

have

(3.44) hl(X, f P3Y ) + Dl(X, F P3Y ) = J P1∇XY.

(3.45) hl(Y, f P3X) + Dl(Y, F P3X) = J P1∇YX.

hl(X, f P3Y )− hl(Y, f P3X) + Dl(X, F P3Y )

− Dl_{(Y, F P}

3X) = J P1[X, Y ].

(3.46)

(3.47) Q2∇sXF P3Y + Q2hs(X, f P3Y ) = J P2∇XY.

(3.48) Q2∇sYF P3X + Q2hs(Y, f P3X) = J P2∇YX.

Q2∇sXF P3Y − Q2∇sYF P3X + Q2hs(X, f P3Y )

− Q2hs(Y, f P3X) = J P2[X, Y ].

(3.49)

(13)

Theorem 3.7. Let M be a radical transversal screen pseudo-slant lightlike sub-manifold of an indefinite Kaehler sub-manifold M . Then the induced connection∇ is a metric connection if and only if J Q2Ds(X, N ) = 0 and BQ3Ds(X, N ) =

f P3ANX, for all X∈ Γ(T M) and N ∈ Γ(ltr(T M)).

Proof. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . Then the induced connection∇ on M is a metric connection if and only if RadT M is parallel distribution with respect to∇ ([6]). From (2.20), for any X ∈ Γ(T M) and N ∈ Γ(ltr(T M)), we have

∇XJ N = J∇XN.

(3.50)

From (2.7), (2.8) and (3.50), we obtain

∇XJ N =−JANX + J∇lXN + J Q2Ds(X, N ) + J Q3Ds(X, N ).

(3.51)

On comparing tangential components of both sides of above equation, we get

∇XJ N =−fP3ANX + J∇Xl N + J Q2Ds(X, N ) + BQ3Ds(X, N ),

(3.52)

which completes the proof.

§4. Foliations Determined by Distributions

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

Theorem 4.1. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . Then RadT M defines a totally geodesic foliation if and only if P1A_{J P}₂_ZX + P1AF P3ZX = P1∇Xf P3Z,

for all X ∈ Γ(RadT M) and Z ∈ Γ(S(T M)).

Proof. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . It is easy to see that RadT M defines a totally geodesic foliation if and only if ∇XY ∈ Γ(RadT M), for all X, Y ∈

Γ(RadT M ). Since ∇ is metric connection, using (2.7) and (2.19), for any X, Y ∈ Γ(RadT M) and Z ∈ Γ(S(T M)), we get

g(∇XY, Z) = g((∇XJ )Z− ∇XJ Z, J Y ).

(4.1)

In view of (2.20), (3.4) and (4.1), we obtain

g(∇XY, Z) =−g(∇X(J P2Z + f P3Z + F P3Z), J Y ).

(14)

From (2.7), (2.9) and (4.2), we get g(∇XY, Z) = g(A_{J P}₂_ZX + AF P3ZX −

∇Xf P3Z, J Y ), which gives

g(∇XY, Z) = g(P1A_{J P}₂_ZX + P1AF P3ZX− P1∇Xf P3Z, J Y ). (4.3)

This completes the proof.

Theorem 4.2. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . Then D1 defines a totally

geodesic foliation if and only if

(i) g(hs(X, f Z), J Y ) =−g(∇s_XF Z, J Y ), (ii) g(hs(X, J N ), J Y ) = 0 ,

for all X, Y ∈ Γ(D1), Z ∈ Γ(D2) and N ∈ Γ(ltr(T M)).

Proof. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . The distribution D1 defines a totally

geodesic foliation if and only if∇XY ∈ D1, for all X, Y ∈ Γ(D1). From (2.7),

(2.19) and (2.20), for any X, Y ∈ Γ(D1) and Z ∈ Γ(D2), we obtain

g(∇XY, Z) = g(∇XJ Y, J Z).

(4.4)

Since ∇ is metric connection, using (4.4), we get g(∇XY, Z) =−g(∇XJ Z, J Y ).

(4.5)

g(∇XY, Z) =−g(hs(X, f Z) +∇sXF Z, J Y ).

(4.6)

Now from (2.7), (2.19) and (2.20), for any X, Y ∈ Γ(D1) and N ∈ Γ(ltr(T M)),

we have g(∇XY, N ) = g(∇XJ Y, J N ). (4.7) From (4.7), we get g(∇XY, N ) =−g(JY, ∇XJ N ). (4.8)

Also, from (2.7) and (4.8), we obtain

g(∇XY, N ) =−g(JY, hs(X, J N )).

(4.9)

Thus the proof is completed.

Definition 4.1. A radical transversal screen pseudo-slant lightlike submani-fold M of an indefinite Kaehler manisubmani-fold M is said to be mixed geodesic screen pseudo-slant lightlike submanifold if its second fundamental form h satisfies h(X, Y ) = 0, for all X ∈ Γ(D1) and Y ∈ Γ(D2). Thus M is mixed geodesic

radical transversal screen pseudo-slant lightlike submanifold if hl(X, Y ) = 0 and hs(X, Y ) = 0, for all X ∈ Γ(D1) and Y ∈ Γ(D2).

(15)

Corollary 4.1. Let M be a mixed geodesic radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . Then D1

defines a totally geodesic foliation if and only if (i) g(∇s

XF Z, J Y ) = 0, (ii) g(hs(X, J N ), J Y ) = 0,

Proof. Since M is a mixed geodesic radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M , we have hs(X, Z) = 0 for all X ∈ Γ(D1) and Z ∈ Γ(D2). Now the proof follows from theorem 4.2.

Theorem 4.3. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . Then D2 defines a totally

geodesic foliation if and only if (i) g(f Y, A_{J Z}X) = g(F Y,∇s

XJ Z), (ii) g(f Y, A∗_{J N}X) = g(F Y, hs(X, J N )),

Proof. Let M be a radical transversal screen pseudo-slant lightlike submanifold of an indefinite Kaehler manifold M . The distribution D2 defines a totally

geodesic foliation if and only if∇XY ∈ D2, for all X, Y ∈ Γ(D2). From (2.7),

(2.19) and (2.20), for any X, Y ∈ Γ(D2) and Z ∈ Γ(D1), we obtain

g(∇XY, Z) = g(∇XJ Y, J Z).

(4.10)

Since ∇ is metric connection, using (4.10), we get g(∇XY, Z) =−g(JY, ∇XJ Z).

(4.11)

g(∇XY, Z) = g(f Y, AJ ZX)− g(F Y, ∇ s XJ Z).

(4.12)

Now, from (2.7), (2.19) and (2.20), for all X, Y ∈ Γ(D2) and N ∈ Γ(ltr(T M)),

we have g(∇XY, N ) = g(∇XJ Y, J N ). (4.13) From (4.13), we get g(∇XY, N ) =−g(JY, ∇XJ N ). (4.14)

g(∇XY, N ) = g(f Y, A∗_{J N}X)− g(F Y, hs(X, J N )),

(4.15)

which completes the proof.

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

(16)

References

[1] Bejan, C. L. and Duggal, K. L., Global Lightlike Manifolds and Harmonicity, Kodai Math. J., Vol. 28, 131-145(2005).

[2] Carriazo, A., New Developments in Slant Submanifolds Theory, Narosa Publishing House, New Delhi, India, 2002.

[3] Chen, B. Y., Geometry of Slant Submanifolds, Katholieke Universiteit, Leuven, 1990.

[4] Chen, B. Y., Slant immersions, Bull. Austral. Math. Soc. Vol. 41, 135- 147(1990). [5] Chen, B. Y. and Tazawa, Y., Slant submanifolds in complex Euclidean spaces,

Tokyo J. Math. Vol. 14, 101-120(1991).

[6] Duggal, K.L. and Bejancu, A., Lightlike Submanifolds of Semi-Riemannian

Man-ifolds and Applications, Vol. 364 of Mathematics and its applications, Kluwer

Aca-demic Publishers, Dordrecht, The Netherlands, 1996.

[7] Duggal, K.L. and Sahin, B., Diﬀerential Geomety of Lightlike Submanifolds, Birkhauser Verlag AG, Basel, Boston, Berlin, 2010.

[8] Johnson, D.L. and Whitt, L.B., Totally Geodesic Foliations, J. Diﬀerential Geom-etry, Vol. 15, 225-235(1980).

[9] Lotta, A., Slant Submanifolds in Contact geometry, Bull. Math. Soc. Roumanie, Vol. 39, 183-198(1996).

[10] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Aca-demic Press New York, 1983.

[11] Papaghiuc, N., Semi-slant submanifolds of a Kaehlerian manifold, An. Stiint. Al.I.Cuza. Univ. Iasi, Vol. 40, 55-61(1994).

[12] Sahin, B., Screen Slant Lightlike Submanifolds, Int. Electronic J. of Geometry, Vol. 2, 41-54(2009).

[13] Sahin, B., Slant lightlike submanifolds of indefinite Hermitian manifolds, Balkan Journal of Geometry and Its Appl., Vol. 13(1), 107-119(2008).

[14] Sahin, B., Transversal lightlike submanifolds of indefinite Kaehler manifolds, Analele. Universitatii din Timisoara, Vol. 44(1), 119-145(2006).

[15] Sahin, B and Gunes, R., Geodesic CR-lightlike submanifolds, Beitrage Algebra and Geometry, Vol. 42(2), 583-594(2001).

[16] Shukla, S.S. and Akhilesh Yadav, lightlike submanifolds of indefinite

para-Sasakian manifolds, Matematicki Vesnik,Vol. 66(4), 371-386(2014).

[17] Yano, K. and Kon, M., Structures on Manifolds, Vol. 3 of Series in Pure Mathe-matics, World Scientfic, Singapore, 1984.

(17)

S.S. Shukla,

Department of Mathematics,

University of Allahabad, Allahabad-211002, India E-mail : ssshukla [email protected]

Akhilesh Yadav

Department of Mathematics,

University of Allahabad, Allahabad-211002, India E-mail : akhilesh [email protected];