lightlike submanifolds of indefinite kenmotsu...

Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 10, 475 - 496

Lightlike Submanifolds of

Indefinite Kenmotsu Manifolds

Ram Shankar Gupta

University School of Basic and Applied Sciences

Guru Gobind Singh Indraprastha University Kashmere Gate, Delhi-110006, India

[email protected]

A. Sharfuddin

Department of Mathematics, Faculty of Natural Science Jamia Millia Islamia (Central University), New Delhi-110025, India

[email protected]

Abstract. In this paper we introduce the notion of lightlike submanifolds of an indefinite Kenmotsu manifold. We have studied the invariant, contact CR, contact screen Cauchy-Riemann (contact SCR) lightlike submanifolds of an indefinite Kenmotsu manifold. We give the condition under which lightlike submanifold of an indefinite Kenmotsu manifold is minimal. We have also studied totally contact umbilical lightlike submanifolds. Examples of lightlike submanifold of an indefinite Kenmotsu manifold have also been given.

Mathematics Subject Classification: 53C15, 53C40, 53C50, 53D15 Keywords: Degenerate metric, Kenmotsu manifold, CR-submanifold Introduction In the theory of submanifolds of semi-Riemannian manifolds it is interesting to study the geometry of lightlike submanifolds due to the fact that the intersection of normal vector bundle and the tangent bundle is non-trivial making it more interesting and remarkably different from the study of non-degenerate submanifolds. The geometry of lightlike submanifolds of indefinite Kaehler manifolds was studied by Duggal and Bejancu [6] and a general notion of lightlike submanifolds of indefinite Sasakian manifolds was introduced by Duggal and Sahin [8]. However, a general notion of lightlike submanifolds of indefinite Kenmotsu manifolds has not been introduced as yet. This research is partly supported by the UNIVERSITY GRANTS COMMISSION (UGC), India under a Major Research Project No. SR. 36-321/2008. The first author would like to thank the UGC for providing the financial support to pursue this research work.

476 R. Shankar Gupta and A. Sharfuddin In section 1, we have collected the formulae and information which are useful in subsequent sections. In section 2, we have studied the invariant lightlike submanifolds of indefinite Kenmotsu manifold. In Section 3, we introduced the notion of contact CR- lightlike submanifolds of indefinite Kenmotsu manifold and have given an example of contact CR-lightlike submanifold of 9

2R . In section 4, we have studied contact Screen Cauchy-Riemann lightlike submanifolds of indefinite Kenmotsu manifold and have given an example of contact SCR lightlike submanifold in 9

2R . In section 5, we have studied the minimal lightlike submanifolds of invariant, contact CR, contact SCR lightlike submanifold of indefinite Kenmotsu manifold and have given an example of minimal lightlike submanifold in 11

4R . 1. Preliminaries

An odd-dimensional semi-Riemannian manifold M is said to be an indefinite almost contact metric manifold if there exist structure tensors (φ , V, η, g ), where φ is a (1, 1) tensor field, V a vector field, η a 1-form and g is the semi-Riemannian metric on M satisfying 2φ X = -X + η(X) V, η o φ = 0, φ V= 0 η(V) =1 g (φ X, φ Y) = g (X, Y) - η(X) η(Y), g (X,V ) = η(X) (1.1) for any X,Y∈ MT , where MT denotes the Lie algebra of vector fields on . An indefinite almost contact metric manifold M is called an indefinite Kenmotsu manifold if [5], ( Xφ∇ )Y = - g (φ X,Y)V + η(Y)φ X , and ∇ XV = - X + η(X) V (1.2) for any X,Y∈ MT , where ∇ denotes the Levi-Civita connection on . A submanifold Mm immersed in a semi-Riemannian manifold ( , )m kM g+ is called a lightlike submanifold if it admits a degenerate metric g induced from g whose radical distribution Rad(TM) is of rank r, where 1 ≤r ≤m. Now, Rad(TM) =TM ∩TM⊥, where

{ }: ( , ) 0,x xx M

TM u T M g u v v T M⊥

∈

= ∈ = ∀ ∈U (1.3)

Let S(TM) be a screen distribution which is a semi-Riemannian complementary distribution of Rad(TM) in TM, that is, TM =Rad(TM)⊥S(TM). We consider a screen transversal vector bundle S(TM⊥), which is a semi-Riemannian complementary vector bundle of Rad(TM) in TM⊥. Since, for any local basis {ξi} of Rad(TM), there exists a local frame {Ni} of sections with values in the orthogonal complement of S(TM⊥) in [S(TM)]⊥ such that g (ξi, Nj) =δij and g (Ni, Nj) = 0, it follows that there exists a lightlike transversal vector bundle ltr(TM) locally spanned by {Ni} [ ]( ). 6 , 144cf page . Let tr(TM) be complementary

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

M

M

Lightlike submanifolds of indefinite Kenmotsu manifolds 477

[ ]

( ) ( ) ( )

( ) ( ) ( ) ( )M

tr TM l tr TM S TM

TM S TM Rad TM l tr TM S TM

⊥

⊥

⎫= ⊥ ⎪⎬

= ⊥ ⊕ ⊥ ⎪⎭ (1.4)

A submanifold (M, g, S(TM), S(TM⊥)) of M is said to be

(1) r-lightlike if r < min {m, k};

(2) Coisotropic if r = k <m, S(TM⊥) = {0};

(3) Isotropic if r = m<k, S(TM) = {0};

(4) Totally lightlike if r = m = k, S(TM) = {0} = S(TM⊥).

Let∇ ,∇ and ∇ t denote the linear connections on M , M and vector bundle tr(TM), respectively. Then the Gauss and Weingarten formulae are given by

( , ), , ( ),X XY Y h X Y X Y TM∇ = ∇ + ∀ ∈Γ (1.5)

, ( ( )),tX U XU A X U U tr TM∇ = − +∇ ∀ ∈Γ (1.6)

where { },X UY A X∇ and { }( , ), tXh X Y U∇ belong to ( )TMΓ and (tr( ))TMΓ ,

respectively and UA is the shape operator of M with respect to U. Moreover, according to decomposition (1.4), hl, hs are

( ( ) valued and ( ( ) valuedltr TM S TM ⊥Γ − Γ − lightlike second fundamental form and screen second fundamental form of M respectively, then

( , ) ( , ), , ( ) , (1.7)

( ) ( , ), ( ( ) ) , (1.8)

( ) ( , ), ( ( ) ), (1.9)

l sX X

l sX N X

s lX W X

Y Y h X Y h X Y X Y TM

N A X N D X N N l tr TM

W A X W D X W W S TM ⊥

∇ = ∇ + + ∀ ∈Γ

∇ = − +∇ + ∈Γ

∇ = − +∇ + ∈Γ

where Dl(X,W), Ds(X,N) are the projections of ∇ t on ( ( ) and ( ( )ltr TM S TM ⊥Γ Γ , respectively and ∇ l, ∇ s are linear connections on ( ( ) and ( ( )ltr TM S TM ⊥Γ Γ , respectively. We call ∇ l, ∇ s the lightlike and screen transversal connections on M, and AN, AW are shape operators on M with respect to N and W, respectively.

Using (1.5) and (1.7)~(1.9), we obtain

( ( , ), ) ( , ( , ) ) ( , ), (1.10)

( ( , ), ) ( , ) (1.11)

s lW

sW

g h X Y W g Y D X W g A X Y

g D X N W g N A X

+ =

=

Let P denote the projection of TM on S(TM) and , t∗ ∗∇ ∇ denote the linear connections on S(TM) and Rad(TM), respectively. Then from the decomposition of tangent bundle of lightlike submanifold, we have

478 R. Shankar Gupta and A. Sharfuddin

for X, Y∈ ( )TMΓ and (Rad )TMξ ∈Γ , where ,h A∗ ∗ are the second fundamental forms of distributions S(TM) and Rad(TM).

From (1.12) and (1.13), we get

( ( , ), ) ( , ), (1.14)

( ( , ), ) ( , ), (1.15)

( ( , ), ) 0, 0 (1.16)

l

Nl

g h X PY g A X PY

g h X PY N g A X PY

g h X A

ξ

ξ

ξ

ξ ξ ξ

∗

∗

∗

=

=

= =

In general the induced connection ∇ on M is not a metric connection. Since∇ is a metric connection, by using (1.7), we obtain

( )( , ) ( ( , ), ) ( ( , ), ) (1.17)l lX g Y Z g h X Y Z g h X Z Y∇ = +

However, it is important to note that , t∗ ∗∇ ∇ are metric connections on S(TM) and Rad(TM), respectively.

A plane section ∏ in Tx M of a Kenmotsu manifold M is called a φ -section if it is spanned by a unit vector X orthogonal to V and φ X, where X is a non-null vector field on M . The sectional curvature K(∏ ) with respect to ∏ determined by X is called a φ -sectional curvature. If M has a φ -sectional curvature c which does not depend on the φ -section at each point, then c is constant in M . Then, M is called an indefinite Kenmotsu space form and is denoted by M (c). The curvature tensor R of M (c) is given by [5]

{ }

( )

( 3)( , ) ( , ) ( , )4

( ) ( ) ( ) ( ) ( , ) ( )1(1.18)

( , ) ( ) ( , ) ( , ) 2 ( , )4

cR X Y Z g Y Z X g X Z Y

X Z Y Y Z X g X Z Y Vcg Y Z X V g Y Z X g Z X Y g X Y Z

η η η η ηη φ φ φ φ φ φ

−= −

− ++ ⎧ ⎫+ ⎨ ⎬− + + −⎩ ⎭

for any X, Y, and Z vector fields on M .

Definition 1.1. [9]. A lightlike submanifold (M, g) of a semi-Riemannian manifold ( M , g ) is totally umbilical in M if there is a smooth transversal vector field H∈ Γ(tr(TM)) on M, called the transversal curvature vector field of M, such that for all X,Y ∈ Γ(TM),

( , ) ( , ) (1.19)h X Y Hg X Y=Using (1.7) and (1.19), it is easy to see that M is totally umbilical if and only if on each coordinate neighborhood U , there exist smooth vector fields l ∈HΓ(ltr(TM)) and s ∈H Γ(S(TM⊥)) such that

( , ), (1.12)

, (1.13)X X

tX X

PY PY h X PY

A Xξξ ξ

∗ ∗

∗ ∗

∇ = ∇ +

∇ = − +∇


( , ) ( , ), ( , ) 0(1.20)

( , ) ( , ), , ( ), ( ( ) )

l l l

s s

h X Y g X Y D X Wh X Y g X Y X Y TM W S TM ⊥

⎫= = ⎪⎬

= ∀ ∈Γ ∈Γ ⎪⎭

HH

Similar to the concept of contact totally umbilical submanifold of Sasakian manifold introduced in the book of Yano and Kon [cf. [10], page 374], we define:

Definition 1.2. If the second fundamental form h of a submanifold M, tangent to the structure vector field V, of an indefinite Kenmotsu manifold M , is of the form

[ ]( , ) ( , ) ( ) ( ) (1.21)h X Y g X Y X Yη η α= −for any X,Y ∈ Γ(TM), where α is a vector field transversal to M, then M is called contact totally umbilical and totally geodesic if α = 0.

The above definition also holds for a lightlike submanifold M. For a contact totally umbilical submanifold M, we have

A general notion of minimal lightlike submanifold M, introduced by Bejancu and Duggal [3], is as follows:

Definition 1.3. A lightlike submanifold (M, g, S(TM)) isometrically immersed in a semi-Riemannian manifold ( , )M g is minimal if

(i) 0sh = on Rad(TM);

(ii) traceh = 0, where trace is written with respect to g restricted to S(TM).

Definition 1.4. [4]. A lightlike submanifold M of a semi-Riemannian manifold is said to be an irrotational submanifold if Xξ∇ ∈ Γ(TM), for all X ∈ Γ(TM) and ξ ∈Γ(RadTM). From (1.7), we conclude that M is an irrotational lightlike submanifold if and only if the following are satisfied:

( , ) 0, ( , ) 0 (1.23)s lh X h Xξ ξ= =

Definition 1.5. A lightlike submanifold M, of an indefinite Kenmotsu manifold

M , is screen real submanifold if Rad(TM) and S(TM) are, respectively, invariant and anti-invariant with respect toφ .

The above definition is the lightlike version (cf. [7]) of totally real submanifolds of an almost Hermitian (or contact) manifold [10].

The following result is important for our work.

[ ][ ]

( , ) ( , ) ( ) ( ) ,(1.22)

( , ) ( , ) ( ) ( )

where ( ( )) and ( ( )).

lL

sS

S L

h X Y g X Y X Y

h X Y g X Y X Y

S TM ltr TM

η η α

η η α

α α⊥

⎫= − ⎪⎬

= − ⎪⎭∈Γ ∈Γ


Proposition 1.1 [6]. The lightlike second fundamental forms of a lightlike submanifold M do not depend on S(TM), S(TM⊥), and ltr(TM).

2. Invariant submanifolds

Let (M, g, S(TM), S(TM⊥)) be a lightlike submanifold of indefinite Kenmotsu manifold ( M , g ). For any vector field X tangent to M, we write

(2.1)X PX FXφ = +where PX and FX are the tangential and transversal parts of φ X, respectively. It is known that if M is tangent to the structure vector field V, then V belongs to S(TM). Using this fact, we say that M is invariant in M if M is tangent to the structure vector field V and

, that is, ( ), ( ) (2.2)X PX X TM X TMφ φ= ∈Γ ∀ ∈ΓFrom (1.2) and (1.7), we have

( )( , ) 0, ( , ) 0, , (2.3)l sXh X V h X V V X X Vη= = ∇ =− +

Moreover, for invariant submanifolds, using (1.2), (1.7) and (2.2), we get

( , ) ( , ) ( , ), , ( ) (2.4)h X Y h X Y h X Y X Y TMφ φ φ= = ∀ ∈Γ

Now, we have

Proposition 2.1. Let (M, g, S(TM), S(TM⊥)) be a lightlike submanifold of an indefinite Kenmotsu manifold M . If the second fundamental forms hl and hs are parallel, then M is totally geodesic.

Proof: Let us assume that hl is parallel. Then we have

( )( , ) ( , ) ( , ) ( , ) 0 (2.5)t l l l lX X X Xh Y V h Y V h Y V h Y V∇ = ∇ − ∇ − ∇ =

Thus using (2.3), we get hl(Y, X) = 0. Similarly, we have hs(Y, X) = 0, which completes the proof.

Theorem 2.2. Let (M, g, S(TM), S(TM⊥)) be an invariant lightlike submanifold of codimension two of an indefinite Kenmotsu manifold M . Then, RadTM defines a totally geodesic foliation on M. Moreover, M =M1 ×M2 is a lightlike product manifold if and only if h∗ = 0, where M1 is a leaf of the radical distribution and M2 is a semi-Riemannian manifold.

Proof: Since rank (RadTM) = 2, we can write X, Y ∈ Γ(RadTM) as a linear combination of ξ and φξ , that is, X = A1 ξ +B1φξ , Y = A2 ξ +B2φξ . Thus using (1.7), we find


2 1 2 1

2 1 2 1

( , ) ( , ( , ) ) ( ( , ) , )

( , ( , ) ) ( ( , ), ) (2.6)

l lX

l l

g Y PZ A A g h PZ A B g h PZ

B A g h PZ B B g h PZ

ξ ξ φξ ξ

φξ ξ φξ φξ

∇ = − −

− −

Now, by using (1.16) and (2.4), we obtain g( XY∇ , P Z) = 0. This shows that RadTM defines a totally geodesic foliation. Then, the proof of the theorem follows from Theorem 2.6, page 162[ ]6 .

Theorem 2.3. Let (M, g, S(TM), S(TM⊥)) be an invariant lightlike submanifold of codimension two of an indefinite Kenmotsu manifold M . Suppose ( M ′ , g) is a nondegenerate submanifold of M such that M ′ is a leaf of integrable S(TM). Then M is totally geodesic, with an induced metric connection if M ′ is being so immersed as a submanifold of M .

Proof. Since dim (RadTM) = dim (ltr(TM)) = 2, hl(X,Y) = A1N + B1 Nφ , where A1 and B1 are functions on M. Thus hl(X, ξ) = 0 if and only if g (hl(X, ξ), ξ) = 0 and g (hl (X, ξ),φξ ) = 0, for all X ∈ Γ(TM), and ξ ∈ Γ(RadTM). From (1.16), we have g (hl( X, ξ), ξ) = 0. Using (2.4), we get g (hl(X, ξ),φξ ) = - g (hl ( Xφ , ξ), ξ) = 0. Similarly, hl(X,φξ ) = 0. On M ′ , we have

( , ), , ( ), (2.7)X XY Y h X Y X Y TM′ ′ ′∇ =∇ + ∀ ∈Γwhere ′∇ is a metric connection on M ′ and h′ is the second fundamental form ofM ′ . Thus ( , ) ( , ) ( , ),lh X Y h X Y h X Y for∗′ = + all X, Y ( )TM ′∈Γ . Also, g(X,Y) =

( , )g X Y′ ,for all X,Y ( )TM∈Γ ,which completes the proof.

Theorem 2.4. Let (M, g, S(TM), S(TM⊥)) be a lightlike submanifold, tangent to structure vector field V, of an indefinite Kenmotsu manifold M . If M is totally umbilical, then M is totally geodesic.

Proof. Using (1.2), (1.7), (2.1), and considering the transversal parts, we get

( , ) ( , ) 0, ( ) (2.8)s lh X V h X V X TM+ = ∀ ∈ΓThus from (2.8), we have hl(X, V) = 0 and hs(X, V) = 0 which imply that hl(V,V) = 0 and hs(V,V) = 0. As M is totally umbilical and V is nonnull, using (1.20), we have hl = 0 and hs = 0. Hence M is totally geodesic.

3. Contact CR-lightlike submanifolds

In this section, we state the following definition for a contact CR-lightlike submanifold:

Definition 3.1. Let (M, g, S(TM), S(TM⊥)) be a lightlike submanifold, tangent to structure vector field V, immersed in an indefinite Kenmotsu manifold ( M , g ).


We say that M is a contact CR-lightlike submanifold of M if the following conditions are satisfied:

(A) RadTM is a distribution on M such that RadTM ∩φ (RadTM) = {0}; (B) there exist vector bundles D0 and D′ over M such that

{ } { }0

0 0 1

( ) ( ) ,(3.1)

, ( ) ( )S TM RadTM D D V

D D D L ltr TMφ

φ φ

′ ⎫= ⊕ ⊥ ⊥ ⎪⎬

′= = ⊥ ⎪⎭

where D0 is nondegenerate and L1 is vector subbundle of S(TM⊥).

Thus we have the following decomposition:

{ } 0, ( ) (3.2)TM D D V D RadTM RadTM Dφ′= ⊕ ⊥ = ⊥ ⊥ A contact CR-lightlike submanifold is proper if Do ≠ {0} and L1 ≠ {0}.

It is easy to see that any contact CR-lightlike three-dimensional submanifold is 1-lightlike.

Hereafter, ( 2 1mqR + , 0φ , V, η, g) will denote the manifold 2 1m

qR + with its usual Kenmotsu structure given by

/22

1 1

01 1

, ,

( ), (3.3)

( ( ) ) ( )

q mz i i i i i i i i

i i q

m mi i

i i i i i ii i

dz V z

g e dx dx dy dy dx dx dy dy

X x Y y Z z Y x X y

η

η η

φ

= = +

= =

⎫⎪= = ∂ ⎪⎪⎪= ⊗ + − ⊗ + ⊗ + ⊗ + ⊗ ⎬⎪⎪

∂ + ∂ + ∂ = ∂ − ∂ ⎪⎪⎭

∑ ∑

∑ ∑ where ( , , )i ix y z are the Cartesian coordinates.

Example 3.2. Let M = ( 92R , g ) be a semi-Euclidean space, where g is of

signature (−,+,+, +,−,+,+,+,+) with respect to canonical basis

{ }1 2 3 4 1 2 3 4

92

1 4 2 2 2 2

1 1 4 2 4 12

3 3 4 3 5 22

, , , , , , , , (3.4)

Consider a submanifold of ,defined by

, 1 ( ) , 1 (3.5)Then a local frameof isgiven by

( ), ( ),

, , (

z z

z z z

x x x x y y y y z

M R

x y x y yTM

Z e x y Z e x y

yZ e x Z e y Z e x yx

− −

− − −

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= = − ≠ ±

= ∂ + ∂ = ∂ − ∂

= ∂ = ∂ = − ∂ + ∂ 2

6 4 1 7

), (3.6)

( ),zZ e x y Z V z−

⎫⎪⎪⎬⎪⎪= ∂ + ∂ = = ∂ ⎭

Hence, RadTM = span{Z1}, 0φ RadTM = span{Z2}, and RadTM ∩ 0φ RadTM =


{0}. Hence, (A) holds. Next, 0φ (Z3) =−Z4 implies that D0 = {Z3, Z4} is invariant with respect to 0φ . By direct calculations, we get

2

2 2 0 52( ) ( ) ( ) (3.7)z yS TM span W e x y such that W Zx

φ⊥ −⎧ ⎫= = ∂ + ∂ = −⎨ ⎬

⎩ ⎭

and ltr(TM) is spanned by N = 1 4( )2

ze x y−

= −∂ + ∂ such that 0φ (N) = 12

Z6. Hence,

M is a contact CR-lightlike submanifold.

Proposition 3.3. There exist no isotropic or totally lightlike contact CR-lightlike submanifolds in an indefinite Kenmotsu manifold.

Proof. If M is isotropic or totally lightlike, then S(TM) = {0}. Hence, conditions (A) and (B) of Definition 3.1 are not satisfied.

Proposition 3.4. Let M be a contact CR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then,

(i) D is integrable if and only if ( , ) ( , ) , , ( )s sB h X Y B h Y X X Y Dφ φ= ∀ ∈Γ

(ii) and D′ ⊕D is integrable. Proof. Suppose D is integrable, then g([X,Y],V) = 0, g([X,Y],W)= 0, for X,Y ∈ Γ(D) and ( )W D′∈ Γ . Also from (1.7), we derive g([X,Y],V) = g (∇ XY,V) − g (∇ YX,V). Then, using (1.2) and the fact that ∇ is a metric connection, we have

g([X,Y],V) = −g(Y, - X + η(X) V) + g(- Y + η(Y) V,X). Hence, g([X,Y],V) = 0 implies that D is integrable.

Similarly, we have

g([X,Y],W)= g ( ( , ) ( , ),s sh X Y h Y X Wφ φ φ− ),

Hence, g([X,Y],W)=0 if and only if ( , ) ( , ) , , ( )s sB h X Y B h Y X X Y Dφ φ= ∀ ∈Γ .

It is easy to see that D′ ⊕D is also integrable.

Denote the orthogonal complement subbundle to the vector subbundle L1 in S(TM⊥) by 1L⊥ . For a contact CR-lightlike submanifold M, we put

1

, ( ), (3.8)where ( ) nd ( ( ) ). Similarly, wehave

X fX X X TMfX D a X L l tr TM

φ ωω

= + ∀ ∈Γ∈Γ ∈Γ ⊥

1 1

, ( ( ) ), (3.9)

where ( ) and ( ).

W BW CW W S TM

BW L CW L

φ

φ⊥

⊥= + ∀ ∈Γ

∈Γ ∈Γ


Proposition 3.5. Let M be a contact CR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then, D⊥{V} is integrable if and only if

h(X, φ Y) = h(φ X,Y) (3.10)

Proof. From (1.2), (1.7), (3.8), (3.9), and using the transversal parts, we obtain ω(∇XY)=−Chs(X,Y) + h(X,φ Y), for all X,Y ∈ Γ(D⊥{V}).

Consequently, ω[X,Y] = h(X,φ Y) − h(φ X,Y), for all X,Y ∈ Γ(D⊥{V}). Therefore, { }D V⊥ is integrable if and only if ( , ) ( , )h X Y h X Yφ φ= .

Proposition 3.6. Let M be a contact CR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then D⊥{V} is a totally geodesic foliation if and only if

1( , ) 0, ( , ) has no component in . (3.11)l sh X Y and h X Y Lφ = Proof. In view of conditions (A) and (B) of Definition 3.1, D⊥{V} defines a totally geodesic foliation if and only if ( , ) ( , ) 0X Xg Y g Y Wφξ∇ = ∇ = for

{ } 1, ( ) ( ).X Y D V and W Lφ∈Γ ⊥ ∈Γ Then, from (1.7), we have

g(∇ XY, φ ξ)=− g (φ ∇ XY, ξ).

Using (1.2) and (1.7), we get

( , ) ( ( , ), ) (3.12)In a similar way, weobtain

( , ) ( ( , ), ) (3.13)

lX

sX

g Y g h X Y

g Y W g h X Y W

φξ φ ξ

φ φ

∇ = −

∇ = −Thus, from (3.12) and (3.13) and as D⊥{V} is invariant with respect to φ , we obtain (3.11), which completes the proof.

Proposition 3.7. Let M be a contact CR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then, Dўdoes not define a totally geodesic foliation.

Proof. In view of conditions (A) and (B) of Definition 3.1, Dўdefines a totally geodesic foliation if and only if

0

( , ) ( , ) ( , ) ( , ) 0, (3.14), ( ), ( tr( )), and ( ).

From (1.2) and (1.7), wefind( , ) ( , ) (3.15)

which is not zero and contradicts the equation (3.14), proving our ass

Z Z Z Z

Z

g W N g W N g W X g W Vwhere Z W D N l TM X D

g W V g W Z

φ∇ = ∇ = ∇ = ∇ =′∈Γ ∈Γ ∈Γ

∇ = −ertion.

Now we prove a Theorem for irrotational lightlike submanifolds of indefinite Kenmotsu manifold:


Theorem 3.8. Let M be an irrotational contact CR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then M does not define contact CR-lightlike product.

Proof: By classical definition of product manifolds, a submanifold M is a contact CR-lightlike product if D⊥{V}and Dўdefines totally geodesic foliations in M. From Proposition 3.7., Dўdoes not define a totally geodesic foliation, which completes the proof.

We have the following:

Lemma 3.9. Let M be a totally contact umbilical proper contact CR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then Lα = 0.

Proof. Let M be a totally contact umbilical proper contact CR-lightlike submanifold. Then, by direct calculations, using (1.2), (1.7), (1.9), and taking the tangential parts, we have

1

( , ) ( , ) 0 (3.16)

for ( ). Hence, we obtain ( , ) ( ( , ), ) 0.

Using (1.10), we find( ( , ), ) ( ( , ), ) 0.

Thus from (1.22), weget( , ) ( , ) 0

l sz Z

lz

s l

L

A Z f Z h Z Z Bh Z Z

Z L g A Z g h Z Z

g h Z Z g h Z Z

g Z Z g

φ

φ

φ

φ φξ ξ

φξ φ ξ

α ξ

+ ∇ + + =

∈Γ + =

+ =

− =Since φ L1 is nondegenerate, we get Lα = 0, which completes the proof.

Theorem 3.10. Let M be a totally contact umbilical proper contact CR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then either M is totally geodesic or dim(φ L1) = 1.

Proof. Assume M proper is totally contact umbilical. From (1.2), (1.7), (3.9), and Lemma 3.9., we get

ω(∇XX) + Chs(X, X) = 0 for X ∈ Γ(Do).

Hence,

1( , ) ( ) (3.17)sh X X L∈ΓNow from (3.16) and (1.10), we have

1( ( , ), ) ( ( , ), ) for , ( )s sg h Z Z W g h Z W Z Z W Lφ φ φ= ∈Γ

Since M is totally contact umbilical, we obtain


( , ) ( , ) ( , ) ( , ) (3.18)s sg Z Z g W g Z W g Zα φ α φ=Interchanging Z and W and subtracting the two, we obtain

2( , )( , ) ( , ) (3.19)( , ) ( , )s s

g Z Wg Z g Zg Z Z g W W

α φ α φ=

Considering (3.17), (3.19) has solutions if either (a) dim(L1) = 1, or (b) sα = 0. Thus the proof follows from Lemma 3.9.

It is known that CR-submanifolds of Riemannian manifold are generalization of both invariant and totally real submanifolds[ ]1 . Therefore, it is important to know whether contact CR-lightlike submanifolds admit invariant submanifolds and, also, are there any real submanifolds. From Definition 1.4., we have

Proposition 3.11. Contact CR-lightlike submanifolds are nontrivial.

Proof. Suppose M is an invariant lightlike submanifold of an indefinite Kenmotsu manifold. Then we can easily see that the radical distribution is invariant. Thus condition (A) of Definition 3.1 is not satisfied. Similarly, one can prove that the screen real lightlike case is not possible.

4. Contact SCR-lightlike submanifolds

We know from Proposition 3.11 that contact CR-lightlike submanifolds exclude the invariant and the screen real subclasses, and therefore, do not serve the central purpose of introducing a CR-structure. To include these two subclasses, we introduce a new class, called contact screen Cauchy-Riemann (SCR)-lightlike submanifold, as follows:

Definition 4.1. Let (M, g, S(TM), S(TM⊥)) be a lightlike submanifold tangent to the structure vector field V of an indefinite Kenmotsu manifold M . Then M is a contact SCR-lightlike submanifold of M if the following conditions are satisfied:

(1) There exist real nonnull distributions D and D⊥ such that

{ } { }( ) , ( ) ( ), 0 , (4.1)S TM D D V D S TM D Dφ⊥ ⊥ ⊥ ⊥= ⊕ ⊥ ⊂ ∩ =

where D⊥ is orthogonally complementary to D⊥{V} in S(TM).

(2) The distributions D and Rad(TM) are invariant with respect toφ .

It follows that ltr(TM) is also invariant with respect toφ . Hence we have

{ }, ( ) (4.2)TM D D V D D Rad TM⊥= ⊕ ⊥ = ⊥

Denote the orthogonal complement of φ (D⊥) in S(TM⊥) by μ. We say that M is a


proper contact SCR-lightlike submanifold of M if D ≠ {0} and D⊥≠ {0}. For a contact SCR-lightlike submanifold, we have:

(i) Condition (2) implies that dim(RadTM) = r = 2p ≥ 2;

(ii) For proper M, (2) implies that dim(D) = 2s ≥ 2, dim(D⊥) ≥ 1. Thus, dim(M) ≥ 5, dim( M ) ≥ 9.

For any X ∈Γ(TM) and W ∈ Γ(S(TM⊥)), we put

, , (4.3)where ( ), ( ) , ( ) , ( ).

X P X F X W B W C WP X D F X D B W D C V

φ φ

φ μ⊥ ⊥

′ ′ ′ ′= + = +

′ ′ ′ ′∈Γ ∈Γ ∈Γ ∈Γ

We have the following:

Example 4.2. Let M be a submanifold of M = ( 92R , g ) defined by

1 2 1 2 4 4 2 4

1 1 2 2 1 2

3 3 4 34 4

5 4 4

, , 1 ( ) , 1 (4.4)It is easy tosee that a local frameof is given by,

( ), ( ),

, , (4.5)

( ), ,

z z

z z

z

x x y y x y yTM

Z e x x Z e y y

Z e x Z e y

Z e y x x y V z

− −

− −

−

= = = − ≠ ±

⎫= ∂ + ∂ = ∂ + ∂⎪

= ∂ = ∂ ⎬⎪= − ∂ + ∂ = ∂ ⎭

Then, RadTM = Span {Z1, Z2} and 0φ (Z1)=−Z2. Thus, RadTM is invariant with respect to 0φ . Also, 0φ (Z3)=−Z4 implies that D = Span{Z3,Z4}. By direct calculations, we get that S(TM⊥) = span{W = ze- (x4∂x4 + y4∂y4 )} such that

0φ (W)=−Z5, and the lightlike transversal bundle ltr(TM) is spanned by

1 1 2 2 1 2( ), ( ) ( 4.6)2 2

z ze eN x x N y y− −

= −∂ + ∂ = −∂ + ∂

It follows that 0φ (N2) = N1. Thus, ltr(TM) is also invariant. Hence, M is a contact SCR lightlike submanifold.

The following results can be easily proved by using Definition 4.1:

a) A contact SCR-lightlike submanifold of M is invariant (resp., screen real) and only if D⊥ = {0} (resp., D = {0}).

b) Any contact SCR-coisotropic or isotropic and or totally lightlike submanifold of M is an invariant lightlike submanifold. Consequently, there exist no proper contact SCR or screen real coisotropic or isotropic or totally lightlike submanifold of M .

We now prove:


Theorem 4.3. Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then the induced connection ∇ is a metric connection if and only if the following two conditions hold:

1. hs (X, ξ) has no components in φ (D⊥) 2. Ax

* X has no components in D, for all X ∈ Γ(TM), ξ ∈ Γ(RadTM).

Proof. Equation (1.2) implies that X Xφξ φ ξ∇ = ∇ and from (1.7), (1.13), (4.2), we get

( , ) (4.7)s tX XB h X P A Xξφξ ξ φ ξ∗ ∗′ ′∇ = + ∇ −

We know that the induced connection is a metric connection if and only if RadTM is parallel with respect to ∇. Suppose that RadTM is parallel. Then from (4.7), we have Bўhs(X,ξ) = 0 and PўAx

* X = 0. Hence hs(X,ξ) has no components inφ (D⊥)

and Ax* X has no components in D. Conversely, assume that (1) and (2) are

satisfied. Then from (4.7), we get ∇X φ ξ ∈Γ(RadTM). Thus, RadTM is parallel and ∇ is a metric connection, which completes the proof.

Proposition 4.4. There exists a Levi-Civita connection on an irrotational screen real lightlike submanifold of an indefinite Kenmotsu manifold.

Proof. From (1.7), we have

( ( , ), ) ( , ) ( , ) ( ( ) , ) ( 4.8)lX X X Xg h X Y g Y g Y g Y Yξ ξ φ φξ φ φ φξ= ∇ = ∇ = − ∇ +∇

for all X,Y ∈ Γ(TM).

From (1.2), we obtain

g (hl(X,Y),ξ) = g (С X φ Y, φ ξ)

SinceС is a metric connection, we have

g (hl(X,Y),ξ) = - g (φ Y, С X φ ξ).

Using (1.7), (1.13) and (1.14) in the above equation, we obtain

g (hl(X,Y),ξ) = - g (φ Y, hs(X,φ ξ)).

M being irrotational implies that g (hl(X,Y), ξ)= 0, that is, hl = 0. Then the proof follows from (1.17).

From (1.2), (1.7), and (4.3), we have the following:

( ) ( , ) ( , ) ( ) , (4.9)

( ) ( , ) ( , ) ( ) , (4.10)

( , ) ( , ) ( , ), , ( ) (4.11)

sX F Y

s sXl l l

P Y A X B h X Y g X Y V Y P X

F Y C h X Y h X P Y Y F X

h X Y h X P Y D X F Y X Y TM

φ η

η

φ

′′ ′ ′∇ = + − +

′ ′ ′ ′∇ = − +

′ ′= + ∀ ∈Γ


The following results are similar to those proved in Propositions 3.5 and 3.6.

Proposition 4.5. Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then

(1) the distribution D⊥ is integrable if and only if

Aφ XY = Aφ YX, ∀X, Y ∈ Γ(D⊥); (4.12)

(2) the distribution D and D ⊥{V} is integrable if and only if

( , ) ( , ), , ( ) ; (4.13)s sh X P Y h P X Y X Y D′ ′= ∀ ∈ΓTheorem 4.6. Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then D ⊥{V} defines a totally geodesic foliation in M if and only if hs(X, φ Y) has no components in φ ( D⊥), for X, Y ∈ Γ( D ⊥{V}).

Proof. From (1.7), we have

g(∇XY,Z) = g (С XY, Z), for X,Y ∈ Γ( D ⊥{V}) and Z ∈Γ(D⊥).

Using (1.2), we get

g(∇XY,Z) = g (С X φ Y, φ Z).

Hence,

g(∇XY,Z) = g(hs(X, φ Y), φ Z), which proves our assertion.

Theorem 4.7. Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then D⊥ and D do not define a totally geodesic foliation on M.

Proof: By definition of contact SCR-lightlike submanifold, D⊥ defines a totally geodesic foliation in M if and only if

g(∇XY, Z) = g(∇XY, V) = g (∇XY, N) = 0, (4.14)

for X,Y ∈Γ(D⊥), Z ∈ Γ( D ), and N ∈ Γ(ltr(TM)).

From (1.7) and (1.2), we obtain

g(∇XY, V) = g(Y, X) (4.15)

for X,Y ∈Γ(D⊥). Since Γ(D⊥) is non degenerate, we can choose non null vectors X, Y ∈Γ(D⊥) such that g(Y, X) ≠ 0, which imply that D⊥ does not define a totally geodesic foliation on M. Similarly we can show that D does not define a totally geodesic foliation on M.


Lemma 4.8. Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then

( , ) 0, ( , ) 0, ( ) , ( ), (4.16), ( ), (4.17)

, ( ), (4.18)

l sX

X

X

h X V h X V V X X V X TMV X X D

V X X D

η

⊥

= = ∇ = − + ∀ ∈Γ

∇ = ∀ ∈Γ

∇ = ∀ ∈ΓProof. Using (1.2) and (1.7), we get

( , ) ( , ) ( ) , ( ).l sXV h X V h X V X X V for X TMη∇ + + = − + ∈Γ

Then, (4.16)~(4.18) follow directly.

Theorem 4.9. Any totally contact umbilical proper contact SCR-lightlike submanifold M of M admits a metric connection.

Proof. From (4.11), we obtain hl(X, φ Y) = hl(φ Y, X), for all X, Y ∈ Γ( D ). Using this and (1.22), we get g(X,φ Y) Lα = g(φ X,Y) Lα . Thus, g(X,φ Y) Lα = 0, and since D is nondegenerate, it follows that Lα = 0. Thus, hl(X, Y) = 0 for X, Y ∈ Γ( D ). Again, from (4.11), we have ( , ) ( , ), , ( )l lh X Y D X F Y X Y Dφ ⊥′= ∀ ∈Γ and using (1.22), we find that φ hl(X, Y) = 0.Thus, hl(X, Y) = 0 for X, Y ∈ Γ(D⊥). From Lemma 4.8, if X ∈ Γ(TM) and Y = V, then from (4.16), we get that hl(X, V) = 0. Thus hl = 0 on M. Finally, our assertion follows from (1.15).

Theorem 4.10. Let M be a totally contact umbilical contact SCR-lightlike submanifold of M . If dim (D⊥) > 1, then M is totally geodesic.

Proof. The proof is similar to the proof of Theorem 3.10.

Theorem 4.11. There exist no totally contact umbilical proper contact SCR-lightlike submanifolds in M (c) with c № -1.

Proof. Suppose M is totally umbilical proper SCR-lightlike submanifold of M . From equation (1.18) and Gauss equation (3.9) (cf. [6], page171), we get


1 (1 ) ( , ) ( , ) (( )( , ), ) (( )( , ), ), (4.19)2

( ), ( ), where( )( , ) ( , ) ( , ) ( , )

Since is totally contact umbilical, we have ( , ) 0,and from (1.22), weget

s sX X

s s s s sX X X X

s

c g X X g Z Z g h X Z Z g h X Z Z

X D Z Dh X Z h X Z h X Z h X Z

M h X Z

φφ φ φ

φ φ φ φ

φ

⊥

− + = ∇ − ∇

∀ ∈Γ ∈Γ

∇ =∇ − ∇ − ∇

=

( , ) ( , ) ,(4.20)

( , ) ( , )Thus from(4.20), we obtain

( )( , ) ( , ) ( , ) (4.21)On theother hand,since ( , ) 0, taking thecovariant derivative with respect to

sX X s

sX X s

sX X s X s

h X Z g X Z

h X Z g X Z

h X Z g X Z g X Zg X Z X

φ φ α

φ φ α

φ φ α φ αφ

⎫− ∇ = − ∇ ⎪⎬

− ∇ = − ∇ ⎪⎭

∇ = − ∇ − ∇

= ,weobtain ( , ) ( , ).Hence, weget

( )( , ) 0 (4.22)X X

sX

g X Z g X Z

h X Z

φ φ

φ

∇ = − ∇

∇ =In a similar way, we have

( )( , ) 0 (4.23)Thus from (4.19), (4.22) and (4.23) , weobtain

1 (1 ) ( , ) ( , ) 0 (4.24)2

sX h X Z

c g X X g Z Z

φ∇ =

− + =

This implies (c +1) g(X, X)g(Z, Z) = 0. Since D and D⊥are nondegenerate, we can choose non-null vector fields X and Z, so c=−1, which proves the result.

5. Minimal lightlike submanifolds

In this section, we study minimal lightlike submanifold of an indefinite Kenmotsu manifold. As in the semi-Riemannian case, any lightlike totally geodesic submanifold M is minimal. Thus, from Theorem 2.4, any totally umbilical lightlike submanifold of indefinite Kenmotsu manifold, with structure vector field tangent to submanifold, is minimal. Furthermore, from Theorems 3.10 and 4.10 of this paper, it follows that totally contact umbilical contact CR-lightlike submanifold with (dim(φL1) > 1) and totally contact umbilical contact SCR-lightlike submanifolds with (dim(D⊥ > 1)) are minimal.

Example 5.1. Let M = ( 114R , g ) be a semi-Euclidean space, where g is of

signature (−,−, +,+,+,−,−,+,+,+,+) with respect to the canonical basis


1 2 3 4 5 1 2 3 4 5114

1 1 1 5

2 2 3 2 2 3

3 2 3 3 2 3

4 4 4 6

5 1 5 5 1 5

( , , , , , , , , , , ) (5.1)

Suppose is a submanifold of given by

,cosh cosh , cosh sinhs h cosh , s h s h

,cos s , s c

x x x x x y y y y y z

M R

x u y ux u u y u ux in u u y in u in ux u y ux u u in y u in uθ θ θ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= = −

= =

= =

= = −

= + = −7

1 1 5 5

2 5 1 52 3 2 3

3 2 32 3 2 3

2 3

4

(5.2)

os

Then it is easy to see that a local frame of is given by

( cos sin )

(sin cos )

(sinh cosh cosh cosh

sinh sinh cosh sinh )

(

z

z

z

z

z uTM

Z e x x y

Z e x y y

Z e u u x u u x

u u y u u y

Z e

θ

θ θ

θ θ

−

−

−

−

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪= ⎭

= ∂ + ∂ + ∂

= ∂ −∂ − ∂

= ∂ + ∂

+ ∂ + ∂

= 2 3 2 32 3

2 3 2 32 3

45 4 6 7

(5.3)

cosh s h sinh sinh

cosh sh sinh cosh )

, , .z z

u in u x u u x

u co u y u u y

Z e x Z e y Z z− −

⎫⎪⎪⎪⎪⎪⎬⎪∂ + ∂ ⎪⎪+ ∂ + ∂⎪⎪= ∂ = − ∂ = ∂ ⎭

We see that M is a 2-lightlike submanifold with { }1 2,RadTM span Z Z= and

{ }0 1 2 0 0 5 6 5 6.Thus is invariant with respect to . Since ( ) , , is alsoinvariant.

Z Z RadTM Z Z D Z Zφ φ φ= = =

{ }

0 3 0 4

0 3 0 4

1 1 5 5

2 5

Moreover, since and are perpendicular to and they are nonnull , wecan choose

( ) ,Furthere more, the lightlike transversal bundle ( ) spanned by

( cos sin )2

(sin2

z

z

Z Z TM

S TM Span Z Zltr TM

eN x x y

eN x y

φ φ

φ φ

θ θ

θ

⊥

−

−

=

= −∂ + ∂ + ∂

= ∂ + ∂ 1 5

114

(5.4)cos )

is also invariant. Thus we conclude that is a contact lightlike submanifoldof .Then a quasiorthonormal basis of along is given by

y

M SCRR M M

θ

⎫⎪⎪⎬⎪− ∂ ⎪⎭

−


1 1 2 2

1 3 2 42 3 2 3 2 3 2 3

3 5 4 6 7

1 0 3 2 0 42 3 2 3 2 3 2 3

1 2

1 1 1 2 2 2

,1 1,

cosh sinh cosh sinh, , (5.5)

1 1,cosh sinh cosh sinh

, .where ( , ) 1, ( , ) 1, and is the degenerate metricon

Z Z

e Z e Zu u u u

e Z e Z Z Z

W Z W Zu u u u

N Ng e e g e e g

ξ ξ

φ φ

ε ε

= = ⎫⎪⎪= =⎪+ +⎪= = = ⎬⎪⎪= =⎪+ +⎪⎭

= = = = −

1 2 3 4

1 1 2 2 2 22 3 2 3 3/2 2 3 2 3 3/2

.By direct calculation and using Gauss formula (1.7) , we get

( , ) ( , ) ( , ) ( , ) 0, 0, ( )(5.6)

( , ) , ( , )(cosh sinh ) (cosh sinh )

Therefore,

s s s s l

z zs s

M

h X h X h X e h X e h X TM

e eh e e W h e e Wu u u u

trace h

ξ ξ− −

⎫= = = = = ∀ ∈Γ⎪⎬

= = ⎪+ + ⎭

( ) 1 1 1 2 2 2 1 1 2 2

114

( , ) ( , ) ( , ) ( , ) 0 (5.7)

Thus is a minimalcontact lightlike submanifold of .S TM

s s s sg h e e h e e h e e h e e

M SCR R

ε ε= + = − =

−

Now we prove characterisation results for minimal lightlike submanifold of all thecasesdiscussed in the paper.

Theorem 5.2. Let M be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold M . Then M is minimal if and only if

( ) 0, 0 , ( , ) 0 (5.8)

for ( ) and ( ( ) ).Proof . Since 0, from (1.7), we get ( , ) ( , ) 0.

kj

lW S TM

l sV

trace A A on D D X W

X RadTM W S TMV h V V h V V

ξ∗ ⊥

⊥

= = =

∈Γ ∈Γ

∇ = = =

Now, take an orthonormal frame { } { }1 1 2,......., such that ,......., are tangent tom r ae e e e D−


{ }[ ]

2 1 2and ,......., are tangent to .

First from 3 , we know that 0 on ( ). Now from (4.11), for , ( ), we have

( , ) ( , ) (5.9)

a m r

l

l l

e e D

h Rad TM Y Z D

h Y Z h Y Zφ φ

⊥+ −

= ∈Γ

=

2

12

( )1

2 2

1 2 1

Hence, we obtain ( , ) ( , ) .Thus

( , ) 0.

Since ( ( , ) ( , ) ), then is minimal if and only if

( , ) ( ( , ) ( , ) ) 0 (5.10)

On the other

l l

al

i ii

m rl s

i i i i iS TMi

a m rs l s

i i i i i i i ii a

h Z Y h Y Z

h e e

trace h h e e h e e M

h e e h e e h e e

φ φ

ε

ε ε

=

−

=

−

= +

= −

=

= +

+ + =

∑

∑

∑ ∑hand, we have

2 2

( )1 1

2 2

1 2 1

2 2

1 2 1

( )1

1 ( ( , ), )2

1 ( ( , ), ) (5.11)21 ( ( , ), ) .2

Using (1.10) and (1.14), we get

12

a n rs

S TM i i i j ji j

n r m rs

i i i j jj i a

r m rl

i i i k kk i a

S TM ij

trace h g h e e W Wn r

g h e e W Wn r

g h e e Nr

trace hn r

ε

ε

ε ξ

ε

−

= =

− −

= = +

−

= = +

=

⎫= ⎪− ⎪

⎪⎪+ ⎬− ⎪⎪

+ ⎪⎪⎭

=−

∑ ∑

∑ ∑

∑ ∑

2 2

1

2 2

1 2 1

2 2

1 2 1

( , )

1 ( , ) (5.12)21 ( , ) .2

j

j

k

a n r

W i i ji

n r m r

i W i i jj i a

r m r

i i i kk i a

g A e e W

g A e e Wn r

g A e e Nr ξ

ε

ε

−

=

− −

= = +

−∗

= = +

⎫⎪⎪⎪⎪+ ⎬− ⎪⎪

+ ⎪⎪⎭

∑ ∑

∑ ∑

∑ ∑

On the other hand, from (1.10), we obtain

( ( , ), ) ( , ( , )), (5.13), ( ), and ( ( ) ).

s lg h X Y W g Y D X WX Y RadTM W S TM ⊥

= −

∀ ∈Γ ∀ ∈ΓThus our assertion follows from (5.12) and (5.13).

Theorem 5.3. Let M be an irrotational screen real lightlike submanifold of an indefinite Kenmotsu manifold M . Then M is minimal if and only if

0 ( ) (5.14)aWtrace A on S TM=

Proof. Proposition 4.4, implies that 0lh = . Thus M is minimal if and only if


0sh = on RadTM and ( ) 0sS TMtrace h = . Thus the proof follows from Theorem

5.2.

Theorem 5.4. Let M be an invariant lightlike submanifold of an indefinite . ( , ) 0

( ) ( ( ) ).Proof . If is invariant , then ( ) ( ) and ( ) ( ).Hence, ( ( )) ( ) and ( ( )) ( ).

lKenmotsu manifold M Then M is minimal in M if and only if D X Wfor X RadTM and W S TM

M Rad TM Rad TM S TM S TMltr TM ltr TM S TM S TM

φ φ

φ φ

⊥

⊥ ⊥

=

∈Γ ∈Γ= =

= =Then using (2.4), we obtain

{ } { }2 2

( )1 1

( , ) ( , ) (5.15)for , ( ), ( , ) ( , ).Thus

( , ) ( , ) ( , ) ( , ) 0 (5.16)m r m r

S TM i i i i i i i i i ii i

h X Y h X YX Y TM and consequently h X Y h X Y

trace h h e e h e e h e e h e e

φ φφ φ

ε φ φ ε− −

= =

=∈Γ = −

= + = − =∑ ∑From (1.10), we get

( ( , ), ) ( ( , ), ) for , ( ) ( ( ) ).

The proof follows from Definition1.3., and the fact that 0on .

s l

l

g h X Y W g D X W Y X Y RadTM W S TMh RadTM

⊥= − ∈Γ ∈Γ

=

Theorem 5.5. Let M be an irrotational contact CR- lightlike submanifold of an indefinite Kenmotsu manifold M . Then M is minimal in M if and only if

0 1

1

1

(1) ,

(2) ( , ) ,(3) 0,

( ( )) ( ), ( ( )) ( ).a

N

s

W D L

A and A N have no components in D

D N N has no components in Ltrace A

for N ltr TM and RadTM where D ltr TM L

ξ

φ

φξ φ

φ

ξ φ φ

∗

⊥

⊥

′

=

′∈Γ ∈Γ = ⊥

Proof. Suppose M is irrotational. From (1.2) and (1.7), we have

1 1

1 1 1

( ( , ), ) ( , ).Then using (1.7) and (1.13), we obtain

( ( , ), ) ( , ), , ( ). (5.17)

l

l

g h g

g h g A RadTM

φξ

ξ

φξ φξ ξ ξ φξ

φξ φξ ξ φξ φξ ξ ξ∗

= − ∇

= ∀ ∈Γ

In a similar way, from (1.2), (1.7), (1.13), (3.8) and (3.9), we get

( ( , ), ) ( , ), ( ), ( ( )). (5.18)sg h W g A BW RadTM W S TMξφξ φξ φξ ξ∗ ⊥= ∀ ∈Γ ∈ΓNow using (1.2), (1.7), (1.9), (3.8) and (3.9), we derive

( , ) ( , ), ( ( )). (5.19)sNh N N A N CD N N N ltr TMφ φ ω φ φ= − + ∀ ∈Γ

Then the proof follows from (5.17)~(5.19) and Theorem 5.2.


References

[1] A. Bejancu, Geometry of CR-Submanifolds, vol. 23 of Mathematics and Its Applications (East European Series), D. Reidel, Dordrecht, the Netherlands, 1986. [2] C. Calin, Contributions to geometry of CR-submanifold, Thesis, University of Iasi, Romania, 1998. [3] C. L. Bejan and K. L. Duggal, Global lightlike manifolds and harmonicity, Kodai Mathematical Journal, vol. 28 (2005), no. 1, pp. 131–145. [4] D. N. Kupeli, Singular Semi-Riemannian Geometry, vol. 366 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996. [5] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math J., 21 (1972), 93-103. [6] K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi- Riemannian Manifolds and Applications, vol. 364 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands,1996. [7] K. L. Duggal and B. Sahin, Screen cauchy Riemann lightlike submanifolds, Acta Mathematica Hungarica, vol. 106 (2005), no. 1-2, pp. 137-165. [8] K. L. Duggal and B. Sahin, Lightlike submanifolds of indefinite Sasakian manifolds, International Journal of Mathematics and Mathematical Sciences,Volume 2007 (2007), Article ID 57585, 21 pages. [9] K. L. Duggal and D. H. Jin, Totally umbilical lightlike submanifolds, Kodai Mathematical Journal, vol. 26 (2003), no.1, pp. 49–68. [10] K. Yano and M. Kon, Structures on Manifolds, vol. 3 of Series in Pure Mathematics, World Scientific, Singapore, 1984. [11] N. Aktan, On non existence of lightlike hypersurfaces of indefinite Kenmotsu space form, Turk. J. Math., 32 (2008), 1-13.

Received: June, 2009

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