Generalised Cauchy-Riemann Lightlike Submanifolds of
Indefinite Kenmotsu Manifolds
Ram Shankar Gupta
iUniversity School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Sector -16C, Dwarka, Delhi-110075, India.
ramshankar.gupta@gmail.com
A. Sharfuddin
Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia (Central University)
New Delhi-110025, India sharfuddin ahmad12@yahoo.com
Received: 16.11.2009; accepted: 8.2.2010.
Abstract. In this paper we introduce the notion of generalised Cauchy-Riemann (GCR) lightlike submanifolds of indefinite Kenmotsu manifold which includes invariant, contact CR, contact screen Cauchy-Riemann (contact SCR) lightlike subclasses [12]. A condition has been discussed for GCR-lightlike submanifold of an indefinite Kenmotsu manifold to be minimal.
We have also studied totally contact umbilical GCR-lightlike submanifolds. Examples of GCR- lightlike submanifold of an indefinite Kenmotsu manifold have also been given.
Keywords: Degenerate metric, Kenmotsu manifold, CR-submanifold.
MSC 2000 classification: primary 53C40; secondary 53C15, 53C50, 53D15
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]. They have also studied possible lightlike submanifolds of indefinite Kaehler manifolds.
On the other hand, a general notion of lightlike submanifolds of indefinite Sasakian man- ifolds was introduced by Duggal and Sahin [8]. Recently we defined the lightlike subman- ifolds of indefinite Kenmotsu manifolds [12] and have studied Cauchy-Riemann and screen Cauchy-Riemann lightlike submanifolds. Moreover, we obtained that there do not exist inclu-
i
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.
http://siba-ese.unisalento.it/ c 2010 Universit` a del Salento
sion relation between these two classes. The objective of this paper is to define a generalised Cauchy-Riemann lightlike submanifold of indefinite Kenmotsu manifolds, which includes in- variant, screen real, contact CR lightlike subcases and real hypersurfaces.
In section 1, we have collected the formulae and information which are useful in sub- sequent sections. In section 2, we have studied GCR-lightlike submanifolds of an indefinite Kenmotsu manifold. In section 3, we have obtained the existence and non-existence conditions for GCR-lightlike submanifolds of indefinite Kenmotsu manifolds and have given an example of GCR-lightlike submanifold of R
413. In section 4, we have studied minimal GCR-lightlike submanifolds of indefinite Kenmotsu manifolds and have given an example of minimal GCR lightlike submanifold in R
154.
1 Preliminaries
An odd-dimensional semi-Riemannian manifold M is said to be an indefinite almost con- tact 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
φ
2X = −X + η(X)V, η ◦ φ = 0, φV = 0, η(V ) = 1
g(φX, φY ) = g(X, Y ) − η(X)η(Y ), g(X, V ) = η(X) (1) for any X, Y ∈ T M, where T M denotes the Lie algebra of vector fields on M.
An indefinite almost contact metric manifold M is called an indefinite Kenmotsu manifold if [5],
(∇
Xφ)Y = −g(φX, Y )V + η(Y )φX, and ∇
XV = −X + η(X)V (2) for any X, Y ∈ T M, where ∇ denote the Levi-Civita connection on M.
A submanifold M
mimmersed in a semi-Riemannian manifold {M
m+k, g} is called a light- like submanifold if it admits a degenerate metric g induced from g whose radical distribution Rad(T M ) is of rank r, where 1 ≤ r ≤ m. Now, Rad(T M) = T M ∩ T M
⊥, where
T M
⊥= [
x∈M
{u ∈ T
xM : g(u, v) = 0, ∀v ∈ T
xM } (3)
Let S(T M ) be a screen distribution which is a semi-Riemannian comlementary distribution of Rad(T M ) in T M , that is, T M = Rad(T M )⊥S(T M).
We consider a screen transversal vector bundle S(T M
⊥), which is a semi-Riemannian com- plementary vector bundle of Rad(T M ) in T M
⊥. Since, for any local basis {ξ
i} of Rad(T M), there exists a local frame {N
i} of sections with values in the orthogonal complement of S(T M
⊥) in [S(T M )]
⊥such that g(ξ
i, N
j) = δ
ijand g(N
i, N
j) = 0, it follows that there exists a lightlike transversal vector bundle ltr(T M ) locally spanned by {N
i}(cf.[6], page144).
Let tr(T M ) be the complementary (but not orthogonal) vector bundle to T M in T M |
M.
Then
tr(T M ) = ltr(T M )⊥S(T M
⊥) T M |
M= S(T M )⊥[Rad(T M) L
ltr(T M )]⊥S(T M
⊥). (4) A submanifold (M, g, S(T M ), S(T M
⊥)) of M is said to be
(i) r-lightlike if r < min{m, k};
(ii) coisotropic if r = k < m, S(T M
⊥) = {0};
(iii) isotropic if r = m < k, S(T M ) = {0};
(iv) totally lightlike if r = m = k, S(T M ) = {0} = S(T M
⊥).
Let ∇ , ∇ and ∇
tdenote the linear connections on M , M and vector bundle tr(T M ), respec- tively. Then the Gauss and Weingarten formulae are given by
∇
XY = ∇
XY + h(X, Y ), ∀X, Y ∈ Γ(T M), (5)
∇
XU = −A
UX + ∇
tXU, ∀U ∈ Γ(tr(T M)), (6) where {∇
XY, A
UX} and {h(X, Y ), ∇
tXU } belong to Γ(T M) and Γ(tr(T M)), respectively and A
Uis the shape operator of M with respect to U . Moreover, according to the decomposition (4), h
l, h
sare Γ(ltr(T M ))-valued and Γ(S(T M
⊥))-valued lightlike second fundamental form and screen second fundamental form of M , respectively, then
∇
XY = ∇
XY + h
l(X, Y ) + h
s(X, Y ), ∀X, Y ∈ Γ(T M), (7)
∇
XN = −A
NX + ∇
lX(N ) + D
s(X, N ), N ∈ Γ(ltr(T M)), (8)
∇
XW = −A
WX + ∇
sX(W ) + D
l(X, W ), W ∈ Γ(S(T M
⊥)), (9) where D
l(X, W ) and D
s(X, N ) are the projections of ∇
ton Γ(ltr(T M )) and Γ(S(T M
⊥)), respectively and ∇
l, ∇
sare linear connections on Γ(ltr(T M )) and Γ(S(T M
⊥)), respectively.
We call ∇
l, ∇
sthe lightlike and screen transversal connections on M , and A
N, A
Ware shape operators on M with respect to N and W , respectively. Using (5) and (7)∼(9), we obtain
g(h
s(X, Y ), W ) + g(Y, D
l(X, W )) = g(A
WX, Y ), (10)
g(D
s(X, N ), W ) = g(N, A
WX). (11)
Let P denote the projection of T M on S(T M ) and let ∇
∗, ∇
∗tdenote the linear connec- tions on S(T M ) and Rad(T M ), respectively. Then from the decomposition of tangent bundle of lightlike submanifold, we have
∇
XP Y = ∇
∗XP Y + h
∗(X, P Y ), (12)
∇
Xξ = −A
∗ξX + ∇
∗tXξ, (13)
for X, Y ∈ Γ(T M) and ξ ∈ Γ(Rad T M), where h
∗, A
∗are the second fundamental form and shape operator of distributions S(T M ) and Rad(T M ), respectively. From (12) and (13), we get
g(h
l(X, P Y ), ξ) = g(A
∗ξX, P Y ), (14) g(h
∗(X, P Y ), N ) = g(A
NX, P Y ), (15) g(h
l(X, ξ), ξ) = 0, A
∗ξξ = 0. (16) In general, the induced connection ∇ on M is not a metric connection. Since ∇ is a metric connection, by using (7), we obtain
(∇
Xg)(Y, Z) = g(h
l(X, Y ), Z) + g(h
l(X, Z), Y ). (17) However, it is important to note that ∇
∗, ∇
∗tare metric connections on S(T M ) and Rad(T M ), respectively.
A plane section Π in T
xM of an indefinite Kenmotsu manifold M is called a φ-section if
it is spanned by a unit vector X orthogonal to V and φX, where X is non-null vector field on
M . The sectional curvature K(Π) with respect to Π 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]
R(X, Y )Z = c − 3
4 {g(Y, Z)X − g(X, Z)Y } + c + 1
4 {η(X)η(Z)Y − η(Y )η(Z)X + g(X, Z)η(Y )V − g(Y, Z)η(X)V + g(φY, Z)φX
+ g(φZ, X)φY − 2g(φX, Y )φZ} (18) for any X, Y and Z vector fields on M .
Definition 1. 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(T M)) on M, called the transversal curvature vector field of M , such that for all X, Y ∈ Γ(T M),
h(X, Y ) = Hg(X, Y ) (19)
Using (7) and (19), it is easy to see that M is totally umbilical if and only if on each coordinate neighbourhood ¨ U there exist smooth vector fields H
l∈ Γ(ltr(T M)) and H
s∈ Γ(ltr(T M
⊥)) such that
h
l(X, Y ) = H
lg(X, Y ) D
l(X, W ) = 0
h
s(X, Y ) = H
sg(X, Y ) ∀X, Y ∈ Γ(T M), W ∈ Γ(S(T M
⊥)). (20) Similar to the concept of contact totally umbilical submanifold of Sasakian manifold in- troduced in the book of Yano and Kon (cf.[10], page 374), we define:
Definition 2. If the second fundamental form h of a submanifold M , tangent to the structure vector field V , of an indefinite Kenmotsu manifold M , be of the form
h(X, Y ) = [g(X, Y ) − η(X)η(Y )]α (21) for any X, Y ∈ Γ(T M), where α is a vector field transversal to M. Then M is called contact totally umbilical submanifold of M . Further if α = 0, then it is called totally geodesic.
The above Definition also holds for a lightlike submanifold M . For a contact totally um- bilical submanifold M , we have
h
l(X, Y ) = [g(X, Y ) − η(X)η(Y )]α
lh
s(X, Y ) = [g(X, Y ) − η(X)η(Y )]α
s(22) where α
s∈ Γ(S(T M
⊥)) and α
l∈ Γ(ltr(T M)).
We have the following definition by Bejancu and Duggal [3].
Definition 3. A lightlike submanifold (M, g, S(T M )) isometrically immersed in a semi- Riemannian manifold (M , g) is minimal if
(i) h
s= 0 on Rad(T M );
(ii) trace h = 0, where trace is written with respect to g restricted to S(T M ).
Definition 4. A lightlike submanifold M of an indefinite Kenmotsu manifold M is a screen real submanifold if Rad(T M ) and S(T M ) are, respectively, invariant and anti-invariant with respect to φ.
The above definition is the lightlike version (cf. [7]) of totally real submanifold of an almost Hermitian (or contact) manifold [10].
The following result is important for our work.
Proposition 1. [6] The lightlike second fundamental forms of a lightlike submanifold M
do not depend on S(T M ), S(T M
⊥) and ltr(T M ).
2 Generalised Cauchy-Riemann (GCR) Lightlike Submanifolds
We have the following definition of a GCR-lightlike submanifold :
Definition 5. Let (M, g, S(T M ), S(T M
⊥)) be a real lightlike submanifold, tangent to structure vector field V , immersed in an indefinite Kenmotsu manifold (M , g). Then M is called a generalised Cauchy-Riemann lightlike submanifold of M if the following conditions are satisfied:
(A) There exist two subbundles D
1and D
2of Rad(T M ) on M such that
Rad T M = D
1⊕ D
2, φ(D
1) = D
1, φ(D
2) ⊂ S(T M) (23) (B) There exist two vector subbundles D
0and D
′of S(T M ) such that over
M
S(T M ) = {φ(D
2) ⊕ D
′}⊥D
0⊥{V },
φD
0= D
0, φ(D
′) = L
1⊥L
2(24)
where D
0is nondegenerate and L
1and L
2are vector subbundles of S(T M
⊥) and ltr(T M ), respectively.
Thus we have the following decomposition:
T M = D ⊕ D
′⊥{V }, D = Rad T M⊥φ(D
2)⊥D
0(25) A contact GCR-lightlike submanifold is said to be proper if D
06= {0}, D
16= {0}, D
26= {0}
and L
16= {0}. Thus, from Definition 5, we have (a) Condition (A) implies that dim(RadT M )≥ 3.
(b) Condition (B) implies that dim(D)≥ 2s≥ 6 and dim(D
2) = dim(L
2). Thus dim(M )≥ 9 and dim(M )≥ 13.
(c) Any proper 9-dimensional contact GCR-lightlike submanifold is 3-lightlike.
(d) (a) and contact distribution (η = 0) imply that index(M ) ≥ 4.
Proposition 2. A GCR-lightlike submanifold M of an indefinite Kenmotsu manifold M is a contact CR (respectively, contact SCR lightlike submanifold) if and only if D
1= {0}
(respectively, D
2= {0}).
Proof. Let M be a contact CR-lightlike submanifold. Then φRadT M is a distribution on M such that RadT M ∩ φRadT M = {0} imply that ltr(T M) ∩ φ(ltr(T M)) = {0}. Thus it follows that φ(ltr(T M )) ⊂ S(T M).
Conversely, suppose M is a GCR-lightlike submanifold of an indefinite Kenmotsu manifold such that D
1= {0}. Then from (23), we have D
2= RadT M . Therefore, RadT M ∩ φ(RadT M)
= {0}, implying that M is a contact CR-lightlike submanifold of an indefinite Kenmotsu manifold.
Similarly, it can be proved that GCR-lightlike submanifold of an indefinite Kenmotsu manifold is a contact SCR lightlike submanifold if and only if D
2= {0}. The following follows:
Proposition 3. There exists no coisotropic, isotropic or totally lightlike proper GCR- lightlike submanifold M of an indefinite Kenmotsu manifold M .
Proof. If M is isotropic or totally lightlike, then S(T M ) = {0} and if M is coisotropic then S(T M
⊥). Hence, conditions (A) and (B) of definition 5 are not satisfied.
It is easy to see that any contact CR-lightlike three-dimensional submanifold is 1-lightlike
real hypersurface [12]. Moreover, it is proved in the same paper that contact SCR-lightlike
submanifolds have invariant and screen real lightlike subcases. Thus, from Proposition 2 it
follows that GCR-lightlike submanifold is an umbrella of real hypersurfaces, invariant, screen real and contact CR-lightlike submanifolds.
Hereafter, (R
2m+1q, φ
0, V, η, g) will denote the manifold R
2m+1qwith its usual Kenmotsu structure given by
η = dz, V = ∂z,
g = η ⊗ η + e
2z(− P
q/2i=1
(dx
i⊗ dx
i+ dy
i⊗ dy
i) + P
mi=q+1
(dx
i⊗ dx
i+ dy
i⊗ dy
i), φ
0( P
mi=1
(X
i∂x
i+ Y
i∂y
i) + Z∂z) = P
mi=1
(Y
i∂x
i− X
i∂y
i)
(26)
where (x
i, y
i, z) are the Cartesian coordinates.
Example 1. Let M = (R
134, g) be a semi-Euclidean space, where g is of signature (-, -, +, +, +, +, -, -, +, +, +, +, +) with respect to the canonical basis
{∂x
1, ∂x
2, ∂x
3, ∂x
4, ∂x
5, ∂x
6, ∂y
1, ∂y
2, ∂y
3, ∂y
4, ∂y
5, ∂y
6, ∂z}. (27) Consider a submanifold M of R
134, defined by
x
4= x
1cos θ − y
1sin θ, y
4= x
1sin θ + y
1cos θ, x
2= y
3, x
5= p
1 + (y
5)
2, y
56= ±1 (28) Then a local frame of T M is given by
Z
1= e
−z(∂x
1+ cos θ∂x
4+ sin θ∂y
4), Z
2= e
−z(− sin θ∂x
4+ ∂y
1+ cos θ∂y
4), Z
3= e
−z(∂x
2+ ∂y
3), Z
4= e
−z(∂x
3− ∂y
2),
Z
5= e
−z∂x
6, Z
6= e
−z∂y
6, Z
7= e
−z(y
5∂x
5+ x
5∂y
5), Z
8= e
−z(∂x
3+ ∂y
2),
Z
9= V = ∂z,
(29)
Hence, RadT M = span{Z
1, Z
2, Z
3}. Moreover φ
0Z
1= −Z
2and φ
0Z
3= Z
4∈ ΓS(T M).
Thus D
1= span {Z
1, Z
2}, D
2= span {Z
3}. Hence, (A) holds. Next, φ
0Z
5= −Z
6, which implies that D
0= {Z
5, Z
6} is invariant with respect to φ
0. By direct calculations, we get
S(T M
⊥) = span {W = e
−z(x
5∂x
5− y
5∂y
5)}
such that φ
0(W ) = −Z
7. Hence L
1= S(T M
⊥) and
ltr(T M ) = span
N
1= e
−z(−∂x
1+ cos θ∂x
4+ sin θ∂y
4), N
2= e
−z(− sin θ∂x
4− ∂y
1+ cos θ∂y
4),
N
3= e
−z(−∂x
2+ ∂y
3),
such that φ
0(N
1) = −N
2and φ
0(N
3) = Z
8. Hence, L
2= span{N
3} and D
′= span{φ
0(N
3), φ
0W }.
Thus M is a contact GCR-lightlike submanifold of R
134.
3 Existence and Non-existence Theorems
We prove an existence Theorem for GCR-lightlike submanifolds in an indefinite Kenmotsu space form:
Theorem 1. Let M be a lightlike submanifold with structure vector field tangent to M of
an indefinite Kenmotsu space form M (c) with c 6= −1. Then, M is a GCR-lightlike submanifold
of M (c) if and only if
(i) The maximal invariant subspaces of T
pM , for p ∈ M, define a distribution D = D
1⊥D
2⊥φ(D
2)⊥D
0where RadT M = D
1⊕ D
2, and D
0is non-degenerate invariant distribution.
(ii) There exists a lightlike transversal vector bundle ltr(T M ) such that
g(R(X, Y )ξ, N ) = 0, ∀X, Y ∈ Γ(D
0), N ∈ Γ(ltr(T M)), ξ ∈ Γ(Rad T M).
(iii) There exists a vector subbundle D on M such that
g(R(X, Y )W
1, W
2) = 0, ∀W
1, W
2∈ Γ(D), where D is orthogonal to D and R is the curvature tensor of M (c).
Proof. Suppose that M is a GCR-lightlike submanifold of M (c) with c 6= - 1. Then D = D
1⊥D
2⊥φ(D
2)⊥D
0is a maximal invariant subspace. From (18), we have
g(R(X, Y )ξ, N ) = − c + 1
2 {g(φX, Y )g(φξ, N)}
∀X, Y ∈ Γ(D
0), N ∈ Γ(ltr(T M)), ξ ∈ Γ(D
2). Since c 6= −1, g(φX, Y ) 6= 0 and g(φξ, N) = 0, and theerefore, we get
g(R(X, Y )ξ, N ) = 0.
Similarly we have
g(R(X, Y )W
1, W
2) = − c + 1
2 {g(φX, Y )g(φW
1, W
2)} = 0,
∀X, Y ∈ Γ(D
0), and W
1, W
2) ∈ Γφ(L
1).
Conversely, assume that (i), (ii) and (iii) are satisfied. Then, (i) implies that D is a invari- ant. From (ii) and (18), we have
g(φξ, N ) = 0 (30)
which implies that φξ ∈ Γ(S(T M)).
From (30), we get that g(ξ, φN ) = 0. Hence, a part of φltr(T M ) also belongs to S(T M ).
Similarly from (iii) and (18), we get
g(φW
1, W
2) = 0 (31)
which implies that φ(D) is orthogonal to D. Since D is non-degenerate, g(φW
1, φW
2) = g(W
1, W
2) 6= 0
Also, we have g(φξ, W ) = −g(ξ, φW ) = 0 implies that φ(D) is orthogonal to Rad(T M).
This also implies that φ(D) does not belong to ltr(T M ). On the other hand, invariant and non- invariant D
0imply g(φW, X) = 0 for X ∈ Γ(D
0). Thus, D ⊥D
0and φ(D)⊥D
0. Moreover, from a result in [3], we know that the structure vector field V belongs to S(T M ). Then summing up the above arguments, we conclude that
S(T M ) = {φD
2⊕ M
1}⊥D⊥D
0⊥{V } where φ(M
1) ⊂ ltr(T M), which completes the proof.
For any X ∈ Γ(T M), we write
φX = P X + F X (32)
where P X and F X are the tangential and transversal parts of φX. Similarly,
φW = BW + CW, W ∈ Γ(ltr(T M)) (33)
where BW and CW are sections of T M and tr(T M ), respectively.
Following are the two non-existence Theorems for GCR-lightlike submanifolds.
Theorem 2. There exists an induced metric connection on a proper GCR-lightlike sub- manifold M of an indefinite Kenmotsu manifold with structure vector field tangent to M if and only if for X ∈ Γ(T M), the following hold
P (A
∗φξX − ∇
∗tXφξ) ∈ Γ(Rad T M), ξ ∈ Γ(D
1) P (h
∗(X, φξ) − ∇
∗Xφξ) ∈ Γ(Rad T M), ξ ∈ Γ(D
2), and Bh(X, φξ) = 0, ξ ∈ Γ(RadT M).
Proof. Assume that M admits a metric connection ∇. Then we show that the radical distribution is parallel with respect to ∇ (cf. [6], Theorem 2.4, p.161). From (2), we get
∇
Xφξ = φ∇
Xξ − g(φX, ξ)V or,
φ ∇
Xφξ = −∇
Xξ (34)
for X ∈ Γ(T M), and ξ ∈ Γ(RadT M).
Using (7) in (34) we obtain
φ(∇
Xφξ + h(X, φξ)) = −∇
Xξ − h(X, ξ) (35) Considering the tangential part of the above equation for ξ ∈ Γ(D
1) and using (13), (32) and (33), we get
∇
Xξ = P A
∗φξX − P ∇
∗tXφξ − Bh(X, φξ) (36) Similarly, for ξ ∈ Γ(D
2) and using (12), (32), (33) and (35), we get
∇
Xξ = P h
∗(X, φξ) − P ∇
∗Xφξ − Bh(X, φξ) (37) Thus, from (36), ∇
Xξ ∈ Γ(RadT M) if and only if
P (A
∗φξX − ∇
∗tXφξ) ∈ Γ(Rad T M) and Bh(X, φξ) = 0 (38) for X ∈ Γ(T M), and ξ ∈ Γ(D
1).
Similarly, from (37), ∇
Xξ ∈ Γ(RadT M) if and only if
P (h
∗(X, φξ) − ∇
∗Xφξ) ∈ Γ(Rad T M) and Bh(X, φξ) = 0 (39) for X ∈ Γ(T M) and ξ ∈ Γ(D
2).
Then, the proof follows from (38) and (39).
Theorem 3. There exists no contact totally umbilical proper GCR-lightlike submanifold M with structure vector field tangent to M of an indefinite Kenmotsu space form M (c) with c 6=- 1.
Proof: The proof is similar to the proof of Theorem 4.11 [12].
4 Minimal GCR-lightlike submanifolds
In this section, we study minimal GCR-lightlike submanifold of an indefinite Kenmotsu manifold.
Example 2. Let M = (R
154, g) be a semi-Euclidean space, where g is of signature (-, -, +, +, +, +, +, -, -, +, +, +, +, +, +) with respect to the canonical basis
{∂x
1, ∂x
2, ∂x
3, ∂x
4, ∂x
5, ∂x
6, ∂x
7, ∂y
1, ∂y
2, ∂y
3, ∂y
4, ∂y
5, ∂y
6, ∂y
7, ∂z} (40)
Suppose M is a submanifold of R
154, given by
x
1= u
1cosh β, y
1= −u
2cosh β x
2= u
3, y
2= u
8x
3= u
1sinh β + u
2, y
3= −u
2sinh β + u
1x
4= u
3, y
4= u
9x
5= cos u
4cosh u
5, y
5= sin u
4sinh u
5x
6= cos u
6cosh u
7, y
6= cos u
6sinh u
7x
7= sin u
6cosh u
7, y
7= sin u
6sinh u
7z = u
10(41)
Then it is easy to see that a local frame of T M is given by
Z
1= e
−z(cosh β∂x
1+ sinh β∂x
3+ ∂y
3) Z
2= e
−z(∂x
3− cosh β∂y
1− sinh β∂y
3)
Z
3= e
−z(∂x
2+ ∂x
4)
Z
4= e
−z(− sin u
4cosh u
5∂x
5+ cos u
4sinh u
5∂y
5) Z
5= e
−z(cos u
4sinh u
5∂x
5+ sin u
4cosh u
5∂y
5) Z
6= e
−z(− sin u
6cosh u
7∂x
6+ cos u
6cosh u
7∂x
7− sin u
6sinh u
7∂y
6+ cos u
6sinh u
7∂y
7) Z
7= e
−z(cos u
6sinh u
7∂x
6+ sin u
6sinh u
7∂x
7+ cos u
6cosh u
7∂y
6+ sin u
6cosh u
7∂y
7) Z
8= e
−z∂y
2, Z
9= e
−z∂y
4, Z
10= ∂z = V
(42)
We see that M is a 3-lightlike submanifold with RadT M = span{Z
1, Z
2, Z
3} and φ
0Z
1= Z
2and φ
0Z
3= −Z
8− Z
9∈ Γ(S(T M)). Thus D
1= span{Z
1, Z
2} and D
2= span{Z
3}. On the other hand, φ
0Z
4= Z
5and D
0= span{Z
4, Z
5} is invariant. Moreover, since φ
0Z
6and φ
0Z
7are perpendicular to T M and they are nonnull, we can choose S(T M
⊥) = span{φ
0Z
6, φ
0Z
7}
Furthermore, the lightlike transversal vector bundle ltr(T M ) spanned by
N
1= e
−z(− cosh β∂x
1− sinh β∂x
3+ ∂y
3) N
2= e
−z(∂x
3+ cosh β∂y
1+ sinh β∂y
3)
N
3= e
−z(−∂x
2+ ∂x
4)
(43)
and φ
0N
1= N
2, φ
0N
3= Z
8−Z
9∈ Γ(S(T M)). Thus, we have φ
0D
′= span{φ
0Z
6, φ
0Z
7, φ
0N
3}.
Hence, we conclude that M is a contact GCR-lightlike submanifold of R
154. Then a quasi orthonormal basis of M along M is given by
ξ
1= Z
1, ξ
2= Z
2, ξ
3= Z
3, φ
0ξ
3= −e
−z(∂y
2+ ∂y
4), φ
0N
3= e
−z(∂y
2− ∂y
4) e
1= √
1cosh2u5−cos2u4
Z
4, e
2= √
1cosh2u5−cos2u4
Z
5e
3= √
1cosh2u7+sinh2u7
Z
6, e
4= √
1cosh2u7+sinh2u7
Z
7V = Z
10W
1= √
1cosh2u7+sinh2u7
φ
0Z
6, W
2= √
1cosh2u7+sinh2u7
φ
0Z
7normal N
1, N
2, N
3(44)
where ε
1= g(e
1, e
1) = 1, ε
2= g(e
2, e
2) = 1, ε
3= g(e
3, e
3) = 1 and ε
4= g(e
4, e
4) = 1.
By direct calculation and using Gauss formula (7), we get
h(ξ
1, ξ
1) = h(ξ
2, ξ
2) = h(ξ
3, ξ
3) = h(e
1, e
1) = h(e
2, e
2) = 0, h(φ
0ξ
3, φ
0ξ
3) = h(φ
0N
3, φ
0N
3) = h
l(e
3, e
3) = h
l(e
4, e
4) = 0, h
s(e
3, e
3) = −
e−z(cosh2u7+sinh2u7)32
W
2, h
s(e
4, e
4) =
e−z(cosh2u7+sinh2u7)32
W
2(45)
Therefore,
traceh
g|S(T M)= ε
1h
s(e
3, e
3) + ε
2h
s(e
4, e
4) = h
s(e
3, e
3) + h
s(e
4, e
4) = 0 (46) Thus M is a minimal proper contact GCR-lightlike submanifold of R
154.
Now, we prove characterisation results for minimal proper contact GCR-lightlike subman- ifold. We use the quasi orthonormal frame given by
{ξ
1, ..., ξ
q, e
1, ..., e
m, V, W
1, ..., W
n, N
1, ..., N
q}
where {ξ
1, ..., ξ
q, e
1, ..., e
m, V } ∈ Γ(T M) such that {ξ
1, ..., ξ
2p}, {ξ
2p+1, ..., ξ
q} and {e
1, ..., e
2l} form a basis of D
1, D
2and D
0respectively. Moreover, take {W
1, ..., W
k} a basis of L
1and {N
2p+1, ..., N
q} a basis of L
2. Thus, we have quasi orthonormal basis of M as follows
{ξ
1, .., ξ
2p, ξ
2p+1, .., ξ
q, φξ
2p+1, .., φξ
q, e
1, .., e
l, φe
1, .., φe
l, φW
1, .., φW
k, φN
2p+1, .., φN
q}.
Theorem 4. Let M be a proper contact GCR-lightlike submanifold of an indefinite Ken- motsu manifold M . Then M is minimal if and only if
traceA
Wj|S(T M)= 0, traceA
∗ξk|S(T M)= 0, g(Y, D
l(X, W )) = 0 (47) for X, Y ∈ Γ(RadT M) and W ∈ Γ(S(T M
⊥).
Proof. We know that h
l= 0 on Rad(T M ) [3]. Hence, from Definition 3, a GCR-lightlike submanifold is minimal if and only if
X
2l i=1ε
ih(e
i, e
i) + X
q j=2p+1h(φξ
j, φξ
j) + X
q j=2p+1h(φN
j, φN
j) + X
k l=1ε
lh(φW
l, φW
l) = 0,
and h
s= 0 on RadT M .
Now from (10), we have h
s= 0 on RadT M if and only if g(Y, D
l(X, W )) = 0, for X, Y ∈ Γ(RadT M ), and W ∈ Γ(S(T M
⊥)).
On the other hand
traceh|
S(T M )=
1qP
q a=1P
qj=2p+1
g(h
l(φξ
j, φξ
j), ξ
a)N
a+ g(h
l(φN
j, φN
j), ξ
a)N
a+
n1P
q j=2p+1P
nb=1
ε
b{g(h
s(φξ
j, φξ
j), W
b)W
b+ g(h
s(φN
j, φN
j), W
b)W
b} + P
nb=1
ε
b1 n{ P
2li=1
g(h
s(e
i, e
i), W
b)W
b+ P
kl=1
g(h
s(φW
l, φW
l), W
b)W
b} + P
qc=11 q
{ P
2li=1
g(h
l(e
i, e
i), ξ
c)N
c+ P
kl=1
g(h
l(φW
l, φW
l), ξ
c)N
c}
(48)
Using (10) and (14), we get
traceh|
S(T M )=
1qP
q a=1P
qj=2p+1
g(A
∗ξaφξ
j, φξ
j)N
a+ g(A
∗ξaφN
j, φN
j)N
a+
n1P
q j=2p+1P
nb=1
ε
b{g(A
Wbφξ
j, φξ
j)W
b+ g(A
WbφN
j, φN
j)W
b} + P
nb=1
ε
b1 n{ P
2li=1
g(A
Wbe
i, e
i)W
b+ P
kl=1
g(A
WbφW
l, φW
l)W
b} + P
qc=1 1 q
{ P
2li=1
g(A
ξce
i, e
i)N
c+ P
kl=1
