Lightlike Submanifolds of Indefinite Kenmotsu Manifolds - hikari

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 R29 . 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 R29 . 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 R411 .

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 semiRiemannian metric on M satisfying φ 2 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∈ TM , where TM denotes the Lie algebra of vector fields on M . An indefinite almost contact metric manifold M is called an indefinite Kenmotsu manifold if [5], ( ∇ X φ )Y = - g ( φ X,Y)V + η(Y) φ X , and ∇ XV = - X + η(X) V

(1.2)

for any X,Y∈ TM , where ∇ denotes the Levi-Civita connection on M . A submanifold Mm immersed in a semi-Riemannian manifold ( M m + k , 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 (1.3) TM ⊥ = U {u ∈ Tx M : g (u , v ) = 0, ∀v ∈ Tx M } x∈M

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 semiRiemannian 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} ( cf .[ 6] , page144 ) . Let tr(TM) be complementary

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

Lightlike submanifolds of indefinite Kenmotsu manifolds

477

⎫⎪ ⎬ TM M = S (TM ) ⊥ [ Rad (TM ) ⊕ l tr (TM )] ⊥ S (TM ) ⎪⎭ A submanifold (M, g, S(TM), S(TM⊥)) of M is said to be tr (TM ) = l tr (TM ) ⊥ S (TM ⊥ )

⊥

(1) r-lightlike if

r < min {m, k};

(2) Coisotropic if

r = k 1) and totally contact umbilical contact SCRlightlike submanifolds with (dim(D⊥ > 1)) are minimal.

Example 5.1. Let M = ( R411 , g ) be a semi-Euclidean space, where g is of signature (−,−, +,+,+,−,−,+,+,+,+) with respect to the canonical basis


492

(∂x1 , ∂x2 , ∂x3 , ∂x4 , ∂x5 , ∂y1 , ∂y2 , ∂y3 , ∂y4 , ∂y5 , ∂z )

(5.1)

Suppose M is a submanifold of R411 given by ⎫ ⎪ x 2 = cosh u 2 cosh u 3 , y 2 = cosh u 2 sinh u 3 ⎪ x3 = s in h u 2 cosh u 3 , y 3 = s in h u 2 s in h u 3 ⎪⎪ ⎬ x4 = u 4 , y 4 = −u 6 ⎪ 5 1 5 5 1 5 x = u cos θ + u s inθ , y = u s inθ − u cos θ ⎪ ⎪ 7 ⎪⎭ z =u Then it is easy to see that a local frame of TM is given by x1 = u1 ,

y1 = −u 5

(5.2)

Z1 = e − z (∂x1 + cos θ ∂x5 + sin θ ∂y5 )

⎫ ⎪ Z 2 = e − z (sin θ∂x5 − ∂y1 − cos θ ∂y5 ) ⎪ 2 3 2 3 −z Z 3 = e (sinh u cosh u ∂x2 + cosh u cosh u ∂x3 ⎪⎪ ⎪ (5.3) + sinh u 2 sinh u 3∂y2 + cosh u 2 sinh u 3∂y3 ) ⎬ ⎪ Z 4 = e − z (cosh u 2 s in h u 3∂x2 + sinh u 2 sinh u 3∂x3 ⎪ + coshu 2 co sh u 3∂y2 + sinh u 2 cosh u 3∂y3 ) ⎪ ⎪ ⎪⎭ Z 5 = e − z ∂x4 , Z 6 = −e− z ∂y 4 , Z 7 = ∂z. We see that M is a 2-lightlike submanifold with RadTM = span {Z1 , Z 2 } and

φ0Z1 = Z2 .Thus RadTM is invariant with respect to φ0 . Since φ0 (Z5 ) = Z6 , D ={Z5 , Z6 } is also invariant. Moreover, since φ0 Z 3 and φ0 Z 4 are perpendicular to TM and they are nonnull , we

can choose S (TM ⊥ ) = Span { φ0 Z 3 , φ0 Z 4 } Furthere more, the lightlike transversal bundle ltr (TM ) spanned by ⎫ e− z (−∂x1 + cos θ ∂x5 + sin θ ∂y5 ) ⎪ ⎪ 2 (5.4) ⎬ −z e ⎪ N2 = (sin θ ∂x5 + ∂y1 − cos θ ∂y5 ) ⎪⎭ 2 is also invariant. Thus we conclude that M is a contact SCR − lightlike submanifold N1 =

of R411.Then a quasiorthonormal basis of M along M is given by


ξ1 = Z1 ,

493

ξ2 = Z2

⎫ ⎪ 1 1 e1 = Z3 , e2 = Z4 ⎪ 2 3 2 3 2 3 2 3 ⎪ cosh u + sinh u cosh u + sinh u ⎪ (5.5) e3 = Z 5 , e4 = Z 6 , Z = Z7 ⎬ ⎪ 1 1 φ0 Z 3 , φ0 Z 4 ⎪ W1 = W2 = ⎪ cosh 2 u 3 + sinh 2 u 3 cosh 2 u 3 + sinh 2 u 3 ⎪ N1 , N2. ⎭ ε 2 = g (e2 , e2 ) = −1, and g is the degenerate metric on M . where ε1 = g (e1 , e1 ) = 1, By direct calculation and using Gauss formula (1.7) , we get h s ( X , ξ1 ) = h s ( X , ξ 2 ) = h s ( X , e3 ) = h s ( X , e4 ) = 0,

⎫ ⎪ ⎬ (5.6) e e s s h (e1 , e1 ) = W h e e W = , ( , ) 2 2 2 2 ⎪ (cosh 2 u 3 + sinh 2 u 3 )3/2 (cosh 2 u 3 + sinh 2 u 3 )3/2 ⎭ Therefore, hl = 0, ∀X ∈ Γ(TM )

−z

−z

trace hg S ( TM ) = ε1h s (e1 , e1 ) + ε 2 h s (e2 , e2 ) = h s (e1 , e1 ) − h s (e2 , e2 ) = 0

(5.7)

Thus M is a minimal contact SCR − lightlike submanifold of R411 .

Now we prove characterisation results for minimal lightlike submanifold of all the cases discussed 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 trace AW j S (TM ) = 0,

Aξ∗k = 0 on D ⊥ , D l ( X , W ) = 0

for X ∈ Γ( RadTM ) and W ∈ Γ( S (TM ⊥ ) ). Proof . Since ∇V V = 0, from (1.7), we get hl (V , V ) = h s (V , V ) = 0.

Now, take an orthonormal frame {e1 ,......., em−r } such that {e1 ,......., e2a } are tangent to D

(5.8)


494

and {e2 a +1 ,......., em − 2 r } are tangent to D ⊥ . First from [3] , we know that hl = 0 on Rad (TM ). Now from (4.11), for Y , Z ∈ Γ( D), we have hl (φY , Z ) = φ hl (Y , Z )

(5.9)

Hence , we obtain h (φ Z , φY ) = −h (Y , Z ) . l

l

Thus 2a

∑ h (e , e ) = 0. l

i =1

Since trace h S (TM ) =

i

i

m−2r

∑ ε (h (e , e ) + h (e , e ) ), then M i =1

l

i

2a

∑ ε i h s (ei , ei ) + i =1

s

i

i

i

m−2r

i

is minimal if and only if

∑ ε ( h (e , e ) + h ( e , e ) ) = 0

2 a +1

l

i

s

i

i

i

i

(5.10)

On the other hand, we have

⎫ ⎪ i =1 ⎪ ⎪⎪ 1 n−2r m−2r s + ( ( , ), ) ε g h e e W W ∑ ∑ i i i j j ⎬ n − 2r j =1 i = 2 a +1 ⎪ 2r m−2r ⎪ 1 l +∑ ⎪ ( ( , ), ) . ε g h e e ξ N ∑ i i i k k ⎪⎭ k =1 2r i = 2 a +1 Using (1.10) and (1.14), we get trace h

2a

S (TM )

=∑

⎫ ⎪ i =1 ⎪ ⎪⎪ 1 n−2r m−2r ε i g ( AW j ei , ei )W j ⎬ + ∑ ∑ n − 2r j =1 i = 2 a +1 ⎪ 2r m−2r ⎪ 1 ∗ ε g ( A e , e ) N . +∑ ∑ i ξk i i k ⎪⎪⎭ k =1 2r i = 2 a +1 2a

trace h

S (TM )

1 n−2 r ε i g (h s (ei , ei ),W j )W j ∑ n − 2r j =1

=∑

(5.11)

1 n−2 r ∑ ε i g ( AW j ei , ei )W j n − 2r j =1

(5.12)

On the other hand, from (1.10), we obtain g ( h s ( X , Y ), W ) = − g (Y , D l ( X , W )),

(5.13)

⊥

∀X , Y ∈ Γ ( RadTM ), and ∀W ∈ Γ ( S (TM ) ). Thus 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 trace AWa = 0 on S (TM )

(5.14)

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


h s = 0 on RadTM and trace h s

S (TM )

495

= 0 . Thus the proof follows from Theorem

5.2. Theorem 5.4. Let M be an invariant lightlike submanifold of an indefinite Kenmotsu manifold M . Then M is minimal in M if and only if Dl ( X , W ) = 0

for X ∈ Γ( RadTM ) and W ∈ Γ( S (TM ⊥ ) ). Proof. If M is invariant , then φ Rad (TM ) = Rad (TM ) and φ S (TM ) = S (TM ). Hence, φ (ltr (TM )) = ltr (TM ) and φ ( S (TM ⊥ )) = S (TM ⊥ ). Then using (2.4), we obtain h(φ X , Y ) = φ h( X , Y ) for X , Y ∈ Γ(TM ), and consequently h(φ X , φY ) = − h( X , Y ).

(5.15)

Thus trace h

S (TM )

=

m−2r

m−2r

i =1

i =1

∑ ε i {h(ei , ei ) + h(φ ei ,φ ei )} =

∑ ε {h(e , e ) − h(e , e )} = 0 i

i

i

i

i

(5.16)

From (1.10), we get g ( h s ( X , Y ), W ) = − g ( D l ( X , W ), Y ) for X , Y ∈ Γ( RadTM ) W ∈ Γ ( S (TM ⊥ ) ).

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

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

(1) Aξ∗φξ and AN φ N have no components in D′, (2) D s (φ N , N ) has no components in L1⊥ , (3) trace AWa D0 ⊥φ L1 = 0, for N ∈ Γ(ltr (TM )) and ξ ∈ Γ( RadTM ), where D′ = φ (ltr (TM )) ⊥ φ ( L1 ). Proof. Suppose M is irrotational. From (1.2) and (1.7), we have g (hl (φξ , φξ ), ξ1 ) = − g (∇φξ ξ , φξ1 ). Then using (1.7) and (1.13), we obtain g (hl (φξ , φξ ), ξ1 ) = g ( Aξ∗φξ , φξ1 ),

∀ξ , ξ1 ∈ Γ( RadTM ).

(5.17)

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

g (h s (φξ , φξ ),W ) = g ( Aξ∗φξ , BW ), ∀ξ ∈ Γ( RadTM ), W ∈ Γ( S (TM ⊥ )). Now using (1.2), (1.7), (1.9), (3.8) and (3.9), we derive

(5.18)

h(φ N , φ N ) = −ω AN φ N + CD s (φ N , N ), ∀ N ∈ Γ (ltr (TM )). Then the proof follows from (5.17)~(5.19) and Theorem 5.2.

(5.19)


496

References [1] [2] [3] [4] [5] [6]

[7] [8] [9] [10] [11]

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

Received: June, 2009