indefinite nearly Kaehler manifolds

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indefinite nearly Kaehler manifolds

Megha Pruthi and Sangeet Kumar

Abstract.The aim of present paper is to study normal semi-transversal lightlike submanifolds of indefinite nearly Kaehler manifolds. We find some necessary and suﬃcient conditions for an isometrically immersed semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifold to be a normal semi-transversal lightlike submanifold.

M.S.C. 2010: 53C15, 53C40, 53C50.

Key words: Indefinite nearly Kaehler manifolds; semi-transversal lightlike subman- ifolds; normal semi-transversal lightlike submanifolds.

1 Introduction

The concept of CR-submanifolds of Kaehler manifolds was introduced by Bejancu [1], as a generalization of totally real and complex submanifolds and has been further developed by many others (for details, see [2, 3, 4]). The premise ofCR-submanifolds has perceived several important contributions in complex and contact Riemannian (or pseudo-Riemannian) geometries and have been successfully applied in diﬀerential geometry and mathematical physics, particularly in, theory of general relativity. From last two decades, finding an interplay between Riemannian and semi-Riemannian geometries is a topic of chief interest. In the process of generalization of submanifold theory from Riemannian manifolds to semi-Riemannian manifolds, the lightlike submanifolds arise naturally in the semi-Riemannian category. In case of lightlike submanifolds, the normal bundle intersects with the tangent bundle and this characteris- tic feature makes the study of lightlike submanifolds more complicated and strikingly diﬀerent from the study of non-degenerate submanifolds. As a result, one fails to use the results of non-degenerate submanifolds in case of lightlike submanifolds. Thus to generalize the concept ofCR-submanifolds in lightlike geometry, Duggal and Bejancu [5] introduced the notion ofCR-lightlike submanifolds of indefinite Kaehler manifolds and proved that this class of lightlike submanifolds has direct relation with physically important asymptotically flat space time, which further leads to Twistor theory of

Balkan Journal of Geometry and Its Applications, Vol.24, No.2, 2019, pp. 42-52.∗

⃝c Balkan Society of Geometers, Geometry Balkan Press 2019.

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Penrose and Heaven theory of Newman. But CR-lightlike submanifolds do not in- clude complex and totally real lightlike submanifolds. Then Duggal and Sahin [6] in- troducedSCR-lightlike submanifolds of indefinite Kaehler manifolds, which contains complex and totally real subcases. But there was no inclusion relation betweenCR andSCR cases, therefore, Duggal and Sahin [7], introducedGCR-lightlike submanifolds of indefinite Kaehler manifolds, which behaves as an umbrella of complex, totally real, screen real andCR-lightlike submanifolds. Later on, Sahin [11] introduced the notion of semi-transversal lightlike submanifolds of indefinite Kaehler manifolds. Re- cently, Kumar [10] proved the existence of semi-transversal lightlike submanifolds in indefinite nearly Kaehler manifolds and proved various characterization results for semi-transversal lightlike submanifolds to be semi-transversal lightlike warped prod- ucts.

In [2], Bejancu initiated the study of normalCR-submanifolds of Kaehler manifolds and proved several characterization theorems for aCR-submanifold of a Kaehler manifold to be a normal CR-submanifold. Haihua et. al. [9] investigated CR- submanifolds of nearly Kaehler manifolds and derived some results for aCR-submanifold to be normal. The available literature on lightlike submanifolds demonstrate that several classes of lightlike submanifolds have been introduced as a generalization of non-degenerate CR-submanifolds in lightlike geometry but no attempts have been made to generalize the idea of normalCR-submanifolds in lightlike geometry. More- over, it is quite interesting to seek conditions under which a lightlike submanifold becomes a normal lightlike submanifold. Therefore, in this paper, we find some necessary and suﬃcient conditions for an isometrically immersed semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifold to be a normal semi-transversal lightlike submanifold.

2 Preliminaries

2.1 Lightlike submanifolds

Let ( ¯M ,¯g) be a real (m+n)-dimensional semi-Riemannian manifold of constant index qsuch thatm, n≥1, 1≤q≤m+n−1 and (M, g) be anm-dimensional submanifold of ¯M andgbe the induced metric of ¯gonM. If ¯gis degenerate on the tangent bundle T M ofM, thenM is called a lightlike submanifold of ¯M, (see [5]). For a degenerate metricg on M, T M^⊥ is a degenerate n-dimensional subspace of TxM¯. Thus both T_xM andT_xM^⊥are degenerate orthogonal subspaces, but no longer complementary.

In this case, there exists a subspaceRad(T_xM) =T_xM ∩T_xM^⊥, which is known as radical (null) subspace. If the mappingRad(T M) :x∈M −→Rad(T_xM), defines a smooth distribution onM of rankr >0, then the submanifoldM of ¯M is called an r-lightlike submanifold andRad(T M) is called the radical distribution onM.

Screen distribution S(T M) is a semi-Riemannian complementary distribution of Rad(T M) in T M, that is

(2.1) T M =Rad(T M)⊥S(T M)

and S(T M^⊥) is a complementary vector subbundle to Rad(T M) in T M^⊥. Let tr(T M) andltr(T M) be complementary (but not orthogonal) vector bundles toT M

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inTM¯ |M and to Rad(T M) inS(T M^⊥)^⊥ respectively. Then we have (2.2) tr(T M) =ltr(T M)⊥S(T M^⊥).

(2.3) TM¯ |M=T M ⊕tr(T M) = (Rad(T M)⊕ltr(T M))⊥S(T M)⊥S(T M^⊥).

For a quasi-orthonormal fields of frames onT M, we have

Theorem 2.1. ([5]). Let(M, g, S(T M), S(T M^⊥))be anr-lightlike submanifold of a semi-Riemannian manifold( ¯M ,g). Then there exists a complementary vector bundle¯ ltr(T M)ofRad(T M)inS(T M^⊥)^⊥and a basis ofΓ(ltr(T M)|u)consisting of smooth section{Ni} ofS(T M^⊥)^⊥|u, whereuis a coordinate neighborhood ofM such that (2.4) g(N¯ i, ξj) =δij, g(N¯ i, Nj) = 0,for any i, j∈ {1,2, .., r},

where{ξ1, ..., ξr} is a lightlike basis ofΓ(Rad(T M)).

Let ¯∇ be the Levi-Civita connection on ¯M, then according to the decomposition (2.3), the Gauss and Weingarten formulae are given by

(2.5) ∇¯XY =∇XY +h(X, Y), ∇¯XU =−AUX+∇^⊥XU,

for anyX, Y ∈Γ(T M) andU ∈Γ(tr(T M)), where{∇XY, AUX}and{h(X, Y),∇^⊥XU} belong to Γ(T M) and Γ(tr(T M)), respectively. Here∇ is a torsion-free linear connection on M, h is a symmetric bilinear form on Γ(T M) which is called second fundamental form,A_U is a linear operator onM and is known as shape operator.

According to (2.2), considering the projection morphismsL andS oftr(T M) on ltr(T M) andS(T M^⊥) respectively, then Gauss and Weingarten formulae become (2.6) ∇¯XY =∇XY +h^l(X, Y) +h^s(X, Y), ∇¯XU =−AUX+D^l_XU+D^s_XU, where we put h^l(X, Y) = L(h(X, Y)), h^s(X, Y) = S(h(X, Y)), D_X^l U = L(∇^⊥XU), D^s_XU = S(∇^⊥XU). As h^l and h^s are Γ(ltr(T M))-valued and Γ(S(T M^⊥))-valued respectively, therefore they are called the lightlike second fundamental form and the screen second fundamental form onM. In particular,

(2.7) ¯∇XN=−A_NX+∇^lXN+D^s(X, N), ∇¯XW =−A_WX+∇^sXW+D^l(X, W), whereX ∈ Γ(T M), N ∈ Γ(ltr(T M)) and W ∈Γ(S(T M^⊥)). Using (2.6) and (2.7), we obtain

(2.8) ¯g(h^s(X, Y), W) + ¯g(Y, D^l(X, W)) =g(AWX, Y), (2.9) g(D¯ ^s(X, N), W) = ¯g(AWX, N),

for anyX, Y ∈Γ(T M),W ∈Γ(S(T M^⊥)) andN ∈Γ(ltr(T M)).

Let P be the projection morphism of T M onS(T M), then using (2.1), we can induce some new geometric objects on the screen distributionS(T M) onM as (2.10) ∇XP Y =∇^∗XP Y +h^∗(X, Y), ∇Xξ=−A^∗_ξX+∇^∗X^tξ,

for anyX, Y ∈Γ(T M) andξ∈Γ(Rad(T M)), where {∇^∗XP Y, A^∗_ξX} and{h^∗(X, Y),

∇^∗X^tξ} belong to Γ(S(T M)) and Γ(Rad(T M)), respectively. Using (2.6) and (2.10), we obtain

(2.11) g(h¯ ^l(X, P Y), ξ) =g(A^∗_ξX, P Y), g(h¯ ^∗(X, P Y), N) =g(A_NX, P Y), for anyX, Y ∈Γ(T M), ξ∈Γ(Rad(T M)) andN ∈Γ(ltr(T M)).

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2.2 Indefinite nearly Kaehler manifolds

Let ¯M be an indefinite almost Hermitian manifold with an almost complex structure J¯of type (1,1) and Hermitian metric ¯gsuch that for allX, Y ∈Γ(TM¯) (see [12]), we have

J¯²=−I, g( ¯¯ J X,J Y¯ ) = ¯g(X, Y).

Let ¯∇ be the Levi-Civita connection of ¯M with respect to ¯g, then the covariant derivative of ¯J is defined by

(2.12) ( ¯∇XJ¯)Y = ¯∇XJ Y¯ −J¯∇¯XY, for allX, Y ∈Γ(TM¯).

An indefinite almost Hermitian manifold ¯M is called an indefinite nearly Kaehler manifold (see [8]), if

(2.13) ( ¯∇XJ¯)Y + ( ¯∇YJ¯)X = 0, ∀X, Y ∈Γ(T M), which is equivalent to

(2.14) ( ¯∇XJ¯)X= 0, ∀X ∈Γ(T M).

It is well known that every Kaehler manifold is a nearly Kaehler manifold but converse is not true. S⁶ with its canonical almost complex structure is a nearly Kaehler manifold but not a Kaehler manifold. Due to rich geometric and topological properties, the study of nearly Kaehler manifolds is as important as that of Kaehler manifolds.

3 Semi-transversal lightlike submanifolds

Definition 3.1. ([10]). Let M be a lightlike submanifold of an indefinite nearly Kaehler manifold ¯M, thenM is called a semi-transversal lightlike submanifold of ¯M, if the following conditions are satisfied:

(A) Rad(T M) is transversal with respect to ¯J, that is, ¯J Rad(T M) =ltr(T M).

(B) There exists a real non-null distributionD⊂S(T M) such that S(T M) =D⊕D^⊥, J D¯ ^⊥⊂S(T M^⊥), J¯(D) =D, whereD^⊥ is orthogonal complementary toD in S(T M).

Thus we obtain that the tangent bundle T M of a semi-transversal lightlike submanifold is decomposed asT M =D⊥D^′, whereD^′=D^⊥⊥Rad(T M).

Let M be a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifold ¯M. LetQ, P1, P2 andP be the projections onD, Rad(T M), D^⊥ andD^′, respectively. Then for anyX∈Γ(T M), we have

(3.1) X=QX+P₁X+P₂X.

Applying ¯J to (3.1), we obtain

(3.2) J X¯ =f X+ω₁X+ω₂X,

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and we can rewrite (3.2) as

(3.3) J X¯ =f X+ωX,

wheref XandωX are the tangential and transversal components of ¯J X, respectively.

Similarly,

(3.4) J V¯ =BV +CV,

for any V ∈ Γ(tr(T M)), where BV and CV are the sections of T M and tr(T M), respectively.

According to definition of semi-transversal lightlike submanifold, considering the de- compositionTM¯ =D⊕D^′⊕J D¯ ^′⊕µ, using (3.3) and (3.4), we havef X ∈Γ(D), ωX∈ Γ( ¯J D^′), BV ∈Γ(D^′) andCV ∈Γ(µ), for anyX ∈Γ(T M) andV ∈Γ( ¯J D^′⊕µ).

Moreover, the covariant derivatives off andω are respectively, given by (3.5) (∇Xf)Y =∇Xf Y −f∇XY, (∇^tXω)Y =∇^tXωY −ω∇XY, for anyX, Y ∈Γ(T M).

Lemma 3.1. ([12]). IfM¯ is a nearly Kaehler manifold, then (3.6) ( ¯∇XJ¯)Y + ( ¯∇J X^¯ J¯) ¯J Y = 0, ( ¯∇XJ¯)Y = 1

4

J¯[ ¯J ,J¯](X, Y),

for any X, Y ∈ Γ(TM¯), where [ ¯J ,J¯](X, Y) is the torsion tensor or the Nijenhuis tensor ofJ¯given by

(3.7) [ ¯J ,J¯](X, Y) = [ ¯J X,J Y¯ ]−J¯[X,J Y¯ ]−J¯[ ¯J X, Y]−[X, Y].

Firstly, we will prove a basic lemma for later use.

Lemma 3.2. LetM be a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifoldM¯. Then we have

(∇Xf)Y =AωYX+Bh(X, Y) +1

4( ¯J[ ¯J ,J¯](X, Y))^T, (3.8)

(∇^tXω)Y =Ch^s(X, Y)−h(X, f Y) +1

4( ¯J[ ¯J ,J¯](X, Y))^⊥, (3.9)

for anyX, Y ∈Γ(T M).

Proof. For anyX, Y ∈Γ(T M), using (2.5), (3.3) and (3.4) in (2.12), we obtain ( ¯∇XJ¯)Y =∇X(f Y) +h(X, f Y)−AωYX+∇^tX(ωY)

−f(∇XY)−ω(∇XY)−Bh(X, Y)−Ch^s(X, Y).

(3.10)

Then from (3.5) and (3.10), we get

(3.11) ( ¯∇XJ¯)Y = (∇Xf)Y+(∇^tXω)Y+h(X, f Y)−A_ωYX−Bh(X, Y)−Ch^s(X, Y).

Further using (3.6) in (3.11), we have

(∇Xf)Y+ (∇^tXω)Y+h(X, f Y)−AωYX−Bh(X, Y)−Ch^s(X, Y) = 1 4

J¯[ ¯J ,J¯](X, Y).

Thus on comparing the tangential and transversal components, the assertion follows.

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4 Normal semi-transversal lightlike submanifolds

Define a tensor fieldS as

(4.1) S(X, Y) = [f, f](X, Y)−2Bdω(X, Y), ∀ X, Y ∈Γ(T M), where

(4.2) [f, f](X, Y) = [f X, f Y] +f²[X, Y]−f([f X, Y] + [X, f Y]) and

(4.3) dω(X, Y) =1

2{∇^tX(ωY)− ∇^tY(ωX)−ω[X, Y]}. Since∇and∇^t are torsion free, therefore from (4.2) and (4.3), we obtain (4.4) [f, f](X, Y) = (∇f Xf)Y −(∇f Yf)X−f((∇Xf)Y −(∇Yf)X) and

(4.5) dω(X, Y) =1

2{(∇^tXω)Y −(∇^tYω)X}. Then using (4.4) and (4.5) in (4.1), we derive

S(X, Y) =(∇f Xf)Y −(∇f Yf)X−f((∇Xf)Y −(∇Yf)X)

−B{(∇^tXω)Y −(∇^tYω)X}. (4.6)

Further using (3.8) and (3.9) in (4.6), we have

S(X, Y) =AωYf X−f AωYX−AωXf Y +f AωXY +1

4( ¯J[ ¯J ,J¯](f X, Y))^T

−1

4( ¯J[ ¯J ,J¯](f Y, X))^T −1

2f( ¯J[ ¯J ,J](X, Y¯ ))^T −1

2B( ¯J[ ¯J ,J¯](X, Y))^⊥. (4.7)

Now, we define a normal semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifold as follows:

Definition 4.1. A semi-transversal lightlike submanifold M of an indefinite nearly Kaehler manifold ¯M is said to be normal, if the tensor fieldS vanishes identically on M, that is, if

S(X, Y) = 0, ∀ X, Y ∈Γ(T M).

Theorem 4.1. Let M be a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifoldM¯ with the totally real distributionD^′ being integrable. Then M is normal, if and only if

0 =AωYf X−f AωYX−AωXf Y +f AωXY +1

4( ¯J[ ¯J ,J](f X, Y¯ ))^T

−1

4( ¯J[ ¯J ,J¯](f Y, X))^T −1

2f( ¯J[ ¯J ,J¯](X, Y))^T −1

2B( ¯J[ ¯J ,J¯](X, Y))^⊥, for anyX, Y ∈Γ(T M).

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Proof. The proof of assertion follows directly from Definition 4.1 and (4.7).

Theorem 4.2. Let M be a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifoldM¯ and

(4.8) [ ¯J ,J](X, Y¯ )∈Γ(µ),

for anyX, Y ∈Γ(T M). ThenM is normal, if and only if

(4.9) AωYf X=f AωYX,

for anyX ∈Γ(D)andY ∈Γ(D^′).

Proof. Using (4.8) in (4.7), we derive

(4.10) S(X, Y) =AωYf X−f AωYX−AωXf Y +f AωXY, for anyX, Y ∈Γ(T M).

Assume thatM be a normal semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifold, then for any X ∈ Γ(D) and Y ∈ Γ(D^′) from (4.10), we obtain (4.9).

Conversely, letM be a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifold satisfying (4.9). ForX, Y ∈Γ(D), using (4.10), we getS(X, Y) = 0.

Now for X ∈ Γ(D) and Y ∈ Γ(D^′), from (4.10), we obtain S(X, Y) = A_ωYf X − f A_ωYX, which on using (4.9) reduces toS(X, Y) = 0. Similarly, forX ∈Γ(D^′) and Y ∈Γ(D), from (4.10), we haveS(X, Y) = 0.

Finally forX, Y ∈Γ(D^′), using (4.10), we get

(4.11) S(X, Y) =f(A_ωXY −A_ωYX).

Next for anyX, Y ∈Γ(D^′) from (2.5) and (2.12), we obtain ( ¯∇XJ¯)Y =−A_ωYX +

∇^t_X(ωY)−J¯( ¯∇XY), then considering inner product withZ∈Γ(D), we obtain (4.12) g(A_ωYX, Z) =−g(( ¯¯ ∇XJ¯)Y, Z)−g( ¯¯ J( ¯∇XY), Z).

On interchanging the role ofX andY in (4.12), we get

(4.13) g(A_ωXY, Z) =−¯g(( ¯∇YJ)X, Z)¯ −¯g( ¯J( ¯∇YX), Z).

Now subtracting (4.12) from (4.13), we derive

(4.14) g(AωXY −AωYX, Z) = 2¯g(( ¯∇XJ¯)Y, Z) + ¯g( ¯J[X, Y], Z).

Then for anyX, Y ∈Γ(D^′),(4.8) gives thatD^′is integrable. Further using (3.6) and (4.8) in (4.14), we obtaing(A_ωXY−A_ωYX, Z) = 0, then non-degeneracy ofDyields that A_ωXY =A_ωYX, thus from (4.11), we have S(X, Y) = 0, which completes the

proof.

Corollary 4.3. A semi-transversal lightlike submanifold M of an indefinite nearly Kaehler manifoldM¯ satisfying (4.8) is normal, if and only if

(i) ¯g(h^s(X, f Y), ωZ) + ¯g(h^s(f X, Y), ωZ) = 0,

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(ii) ¯g(h^s(f X, W), ωZ) = 0,

for anyX, Y ∈Γ(D)andZ, W ∈Γ(D^⊥).

Suppose{E1, E2, E3, ..., Eq} is a local field of orthogonal frames forD^⊥. Denote Ai, the fundamental tensor of Weingarten with respect toVi= ¯J Ei, then from above theorem, we have

Corollary 4.4. Let M be a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifoldM¯ and

(4.15) [ ¯J ,J](X, Y¯ )∈Γ(µ),

for anyX, Y ∈Γ(T M). Then M is normal if and only if the fundamental tensors of WeingartenAi commute withf on invariant distribution, that is, if and only if

(4.16) Ai◦f =f◦Ai.

Next using (2.5), we derive

(4.17) ∇XEi =f AJ E¯ iX−B∇^tXJ E¯ i−1

4( ¯J[ ¯J ,J¯](X, Ei))^T and

(4.18) ∇^tXJ E¯ _i=w∇XE_i+Ch^s(X, E_i) +1

4( ¯J[ ¯J ,J¯](X, E_i))^⊥. Definition 4.2. A vector fieldX is said to be aD-Killing vector field, if

g(∇ZX, Y) +g(Z,∇YX) = 0, for anyY, Z∈Γ(D).

Now we are ready to give a necessary and suﬃcient condition for a semi-transversal lightlike submanifold to be normal. Thus we have

Theorem 4.5. A necessary and suﬃcient condition for a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifold and [ ¯J ,J](X, Y¯ ) ∈ Γ(µ) for X, Y ∈Γ(T M)to be normal is that E_i,(i= 1,2,3, ..., q)beD-Killing vector fields.

Proof. ForY, Z ∈Γ(D), using (4.17), we obtain

(4.19) g(∇ZEi, Y) +g(Z,∇YEi) =g(f AJ E¯ _iZ, Y) +g(Z, f AJ E¯ _iY), Now using (2.8), we have

g(Z, f AJ E¯ _iY) =−g(f Z, AJ E¯ _iY) =−g(h¯ ^s(Y, f Z),J E¯ i)

=−¯g( ¯∇f ZY,J E¯ i) = ¯g(Y,∇¯f ZJ E¯ i)

=−g(Y, AJ E¯ _if Z).

(4.20)

Thus from (4.19) and (4.20), we derive

(4.21) g(∇ZE_i, Y) +g(Z,∇YE_i) =g(f AJ E¯ iZ−AJ E¯ if Z, Y).

Hence, the result follows from Theorem 4.2 and (4.21).

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The Lie derivative off with respect toY ∈Γ(T M) is given by (4.22) (LYf)X= [Y, f X]−f[Y, X],

for anyX∈Γ(T M).

The normal semi-transversal lightlike submanifold can be characterized by another tensor fieldS^∗ defined by

(4.23) S^∗(Y, X) = (LYf)X,

for anyX, Y ∈Γ(T M).

Theorem 4.6. Let M be a semi-transversal lightlike submanifold of an indefinite nearly Kaehler manifoldM¯ satisfying

(i) P(∇XY) = 0, f or any X ∈Γ(D), Y ∈Γ(D^′), (ii) [ ¯J ,J¯](X, Y)∈Γ(µ), f or any X, Y ∈Γ(T M).

ThenM is normal if and only if we have

(4.24) S^∗(Y, X) = 0,

for allX ∈Γ(D)andY ∈Γ(D^′).

Proof. From Theorem 4.2, it follows that M is a normal semi-transversal lightlike submanifold, if and only if,S(X, Y) = 0, for allX ∈Γ(D) and Y ∈Γ(D^′). For any X∈Γ(D) andY ∈Γ(D^′), using (4.1) and (4.2), we derive

(4.25) S(X, Y) =f([Y, f X]−f[Y, X])−2Bdω(X, Y).

Taking into account∇^t is torsion free and using (3.9), (4.5) becomes dω(X, Y) = 1

2{(∇^tXω)Y −(∇^tYω)X}

= 1

2h(Y, f X) +1

4( ¯J[ ¯J ,J¯](X, Y))^⊥, which further gives

(4.26) 2Bdω(X, Y) =Bh(Y, f X).

Now using (4.22), (4.23) and (4.26) in (4.25), we have S(X, Y) =f(S^∗(Y, X))−Bh(f X, Y).

(4.27)

Now for anyX∈Γ(D) andY ∈Γ(D^′), using (3.9), we obtain h(Y, f X) =ω∇YX+Ch^s(X, Y)−1

4( ¯J[ ¯J ,J](X, Y¯ ))^⊥. Applying ¯J on both sides, we get

Bh(Y, f X) +Ch^s(Y, f X) =−P∇YX+ ¯J Ch^s(X, Y) +1

4([ ¯J ,J¯](X, Y))^⊥,

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then comparing the tangential components, we obtain Bh(Y, f X) =−P(∇YX).

Thus, (4.27) reduces to

(4.28) S(X, Y) =f(S^∗(Y, X)) +P(∇YX).

SinceM is normal, therefore we must have

f(S^∗(Y, X)) = 0, P(∇YX) = 0, which implies that

(4.29) QS^∗(X, Y) = 0, P(∇YX) = 0.

Again from (4.22) and (4.23), we get

(4.30) P(S^∗(Y, X)) =P(∇Yf X− ∇f XY),

which on using hypothesis alongwith second part of (4.29), yields thatP(S^∗(Y, X)) = 0 and henceS^∗(Y, X) = 0.

Conversely, suppose thatM is a semi-transversal lightlike submanifold of indefinite nearly Kaehler manifold ¯M satisfying (4.24). Then using hypothesis and (4.24), from (4.30), we get

(4.31) P(∇Yf X) = 0.

Thus using (4.24) and (4.31) in (4.28), we obtainS(X, Y) = 0, which completes the

proof.

Acknowledgment

Sangeet Kumar is grateful to SERB-DST, New Delhi for the financial funding under grant no. ECR/2017/000786.

References

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Authors’ address:

Megha Pruthi and Sangeet Kumar (corresponding author) Department of Mathematics,

Sri Guru Teg Bahadur Khalsa College, Sri Anandpur Sahib-140118, India.

E-mail: [email protected] , [email protected]