New York Journal of Mathematics
New York J. Math. 21 (2015) 191–203.
Totally umbilical hemi-slant lightlike
submanifolds
Rashmi Sachdeva, Rakesh Kumar
and Satvinder Singh Bhatia
Abstract. We study hemi-slant lightlike submanifolds of indefinite
Kaehler manifolds and prove that there do not exist totally umbilical hemi-slant lightlike submanifolds in indefinite Kaehler manifolds
other than totally geodesic hemi-slant lightlike submanifolds. We consequently prove that the induced connection of a totally umbilical hemislant lightlike submanifold is a metric connection. We obtain a characterization theorem on the nonexistence of totally umbilical hemi-slant
lightlike submanifold of an indefinite complex space form and some characterization theorems on minimal hemi-slant lightlike submanifolds.
Contents
1. Introduction
2. Lightlike submanifolds
3. Hemi-slant lightlike submanifolds
References
191
192
194
203
1. Introduction
As a generalization of holomorphic and totally real submanifolds in complex geometry, Chen [5, 6] introduced the notion of slant submanifolds.
They are further studied by many authors, e.g., [4, 3, 11]. They all studied
the geometry of slant submanifolds with positive definite metric. Therefore
this geometry may not be applicable to the other branches of mathematics
and physics, where the metric is not necessarily definite. The geometry of
submanifolds is one of the most important topics of differential geometry.
The geometry of semi-Riemannian submanifolds has many similarities with
the Riemannian case but the geometry of lightlike submanifolds is different
since their normal vector bundle intersects with the tangent bundle making it more difficult. Recently Sahin [13] proved that there do not exist
totally umbilical slant submanifolds in Kaehler manifolds other than totally
Received May 7, 2013.
2010 Mathematics Subject Classification. 53C15, 53C40, 53C50.
Key words and phrases. Indefinite Kaehler manifolds, hemi-slant lightlike submanifolds, totally umbilical lightlike submanifolds, minimal lightlike submanifolds.
ISSN 1076-9803/2015
191
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RASHMI SACHDEVA, RAKESH KUMAR AND SATVINDER SINGH BHATIA
geodesic slant submanifolds. Haider et al. [10] introduced hemi-slant lightlike submanifolds of an indefinite Kaehler manifold. Moreover, the growing
importance of lightlike submanifolds in mathematical physics and relativity makes this a topic of current interest. Therefore, we extended a result
proved by Sahin in [13] to hemi-slant lightlike submanifolds of indefinite
Kaehler manifolds. In particular, we prove that there do not exist totally
umbilical hemi-slant lightlike submanifolds in indefinite Kaehler manifolds
other than totally geodesic hemi-slant lightlike submanifolds. We also obtain
a characterization theorem on the nonexistence of totally umbilical hemislant lightlike submanifold of an indefinite complex space form and some
characterization theorems on minimal hemi-slant lightlike submanifolds.
2. Lightlike submanifolds
Let (M̄ , ḡ) be a real (m + n)-dimensional semi-Riemannian manifold of
constant index q such that m, n ≥ 1, 1 ≤ q ≤ m + n − 1 and (M, g) be
an m-dimensional submanifold of M̄ and g the induced metric of ḡ on M .
If ḡ is degenerate on the tangent bundle T M of M then M is called a
lightlike submanifold of M̄ . For a degenerate metric g on M , T M ⊥ is a
degenerate n-dimensional subspace of Tx M̄ . Thus, both Tx M and Tx M ⊥
are degenerate orthogonal subspaces but no longer complementary. In this
case, there exists a subspace Rad Tx M = Tx M ∩ Tx M ⊥ which is known as
radical (null) subspace. If the mapping Rad T M : x ∈ M −→ Rad Tx M ,
defines a smooth distribution on M of rank r > 0 then the submanifold M
of M̄ is called an r-lightlike submanifold and Rad T M is called the radical
distribution on M (for details see [7]).
Let S(T M ) be a screen distribution which is a semi-Riemannian complementary distribution of Rad(T M ) in T M , that is,
(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 ) and ltr(T M ) be complementary (but not orthogonal) vector
bundles to T M in T M̄ |M and to Rad T M in S(T M ⊥ )⊥ respectively. Then
we have
(2)
tr(T M ) = ltr(T M )⊥S(T M ⊥ ),
(3) T M̄ |M = T M ⊕ tr(T M ) = (Rad T M ⊕ ltr(T M ))⊥S(T M )⊥S(T M ⊥ ).
Let u be a local coordinate neighborhood of M and consider the local
quasi-orthonormal fields of frames of M̄ along M , on u as
{ξ1 , . . . , ξr , Wr+1 , . . . , Wn , N1 , . . . , Nr , Xr+1 , . . . , Xm },
where {ξ1 , . . . , ξr }, {N1 , . . . , Nr } are local lightlike bases of Γ(Rad T M |u ),
Γ(ltr(T M ) |u ) and {Wr+1 , . . . , Wn }, {Xr+1 , . . . , Xm } are local orthonormal
bases of Γ(S(T M ⊥ ) |u ) and Γ(S(T M ) |u ) respectively. For this quasiorthonormal fields of frames, we have:
TOTALLY UMBILICAL HEMI-SLANT LIGHTLIKE SUBMANIFOLDS
193
Theorem 2.1 ([7]). Let (M, g, S(T M ), S(T M ⊥ )) be an r-lightlike submanifold of a semi-Riemannian manifold (M̄ , ḡ). Then there exists a complementary vector bundle ltr(T M ) of Rad T M in S(T M ⊥ )⊥ and a basis of
Γ(ltr(T M ) |u ) consisting of smooth section {Ni } of S(T M ⊥ )⊥ |u , where u
is a coordinate neighborhood of M , such that
(4)
ḡ(Ni , ξj ) = δij ,
ḡ(Ni , Nj ) = 0,
for any
i, j ∈ {1, 2, . . . , r},
where {ξ1 , . . . , ξr } is a lightlike basis of Γ(Rad(T M )).
We have four cases for a lightlike submanifold (M, g, S(T M ), S(T M ⊥ )):
• M is an r-lightlike submanifold if r < min{m, n}.
• M is a coisotropic lightlike submanifold if
r = n < m,
S(T M ⊥ ) = {0}.
• M is an isotropic lightlike submanifold if r = m < n, S(T M ) = {0}.
• M is a totally lightlike submanifold if
r = n = m,
S(T M ⊥ ) = {0} = S(T M ).
¯ be the Levi-Civita connection on M̄ . Then, according to the deLet ∇
composition (3), the Gauss and Weingarten formulas are given by
(5)
¯ X Y = ∇X Y + h(X, Y ),
∇
¯ X U = −AU X + ∇⊥ U,
∇
X
for any X, Y ∈ Γ(T M ) and U ∈ Γ(tr(T M )), where {∇X Y, AU X} and
{h(X, Y ), ∇⊥
X U } belongs 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, AU is linear a operator on
M , known as shape operator. According to (2), considering the projection
morphisms L and S of tr(T M ) on ltr(T M ) and S(T M ⊥ ), respectively then
(5) becomes
(7)
¯ X Y = ∇X Y + hl (X, Y ) + hs (X, Y ),
∇
¯ X N = −AN X + ∇l N + Ds (X, N ),
∇
(8)
¯ X W = −AW X + ∇s W + Dl (X, W ),
∇
X
(6)
X
where we put
hl (X, Y ) = L(h(X, Y )),
hs (X, Y ) = S(h(X, Y )),
∇lX N = L(∇⊥
X N ),
∇sX W = S(∇⊥
X W ),
Ds (X, N ) = S(∇⊥
X N ),
Dl (X, W ) = L(∇⊥
X W ),
for any X ∈ Γ(T M ), N ∈ Γ(ltr(T M )) and W ∈ Γ(S(T M ⊥ )).
As hl and hs are Γ(ltr(T M ))-valued and Γ(S(T M ⊥ ))-valued respectively,
therefore they are called as the lightlike second fundamental form and the
screen second fundamental form on M . Using (2)–(3) and (6)–(8), we obtain
(9)
ḡ(hs (X, Y ), W ) + ḡ(Y, Dl (X, W )) = g(AW X, Y ),
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RASHMI SACHDEVA, RAKESH KUMAR AND SATVINDER SINGH BHATIA
for any X, Y ∈ Γ(T M ), W ∈ Γ(S(T M ⊥ )) and N ∈ Γ(ltr(T M )).
Let P̄ is a projection of T M on S(T M ). Now, we consider the decomposition (4), we can write
(10)
∇X P̄ Y = ∇∗X P̄ Y + h∗ (X, P̄ Y ),
∇X ξ = −A∗ξ X + ∇∗t
X ξ,
for any X, Y ∈ Γ(T M ) and ξ ∈ Γ(Rad T M ), where {∇∗X P̄ Y , A∗ξ X} and
{h∗ (X, P̄ Y ), ∇∗t
X ξ} belong to Γ(S(T M )) and Γ(Rad T M ) respectively. Here
∗
∗t
∇ and ∇X are linear connections on S(T M ) and Rad T M respectively. By
using (7), (8) and (10), we obtain
(11) ḡ(hl (X, P̄ Y ), ξ) = g(A∗ξ X, P̄ Y ),
ḡ(h∗ (X, P̄ Y ), N ) = ḡ(AN X, P̄ Y ).
¯ ḡ) be an indefinite almost Hermitian manifold
Definition 2.2. Let (M̄ , J,
¯
and ∇ be the Levi-Civita connection on M̄ with respect to ḡ. Then M̄ is
¯ that
called an indefinite Kaehler manifold if J¯ is parallel with respect to ∇,
¯ X J)Y
¯ = 0, for any X, Y ∈ Γ(T M̄ ).
is, (∇
3. Hemi-slant lightlike submanifolds
Definition 3.1. Let M be a 2q-lightlike submanifold of an indefinite Kaehler
manifold M̄ of index 2q. Then the submanifold M is said to be hemi-slant
lightlike submanifold of M̄ if the following conditions are satisfied:
¯
(i) Rad T M is a distribution on M such that J(Rad
T M ) = ltr(T M ).
(ii) For all x ∈ U ⊂ M and for each nonzero vector field X tangent
¯ and the
to S(T M ), the angle θ(X), called slant angle, between JX
vector space S(T M ) is constant.
A hemi-slant lightlike submanifold is said to be proper if S(T M ) 6= 0 and
θ 6= 0, π/2.
Example 1. Let M̄ = (<82 , ḡ) be a semi-Riemannian manifold, where <82
is a semi-Euclidean space of signature (−, −, +, +, +, +, +, +) with respect
to the canonical basis {∂x1 , ∂x2 , ∂x3 , ∂x4 , ∂x5 , ∂x6 , ∂x7 , ∂x8 }. Let M be a
submanifold of <82 given by
X(s, t, u, v) = (s, t, sin u, cos u, −u sin v, −u cos v, t, s),
then the tangent bundle T M is spanned by
Z1
Z2
Z3
Z4
= ∂x1 + ∂x8 ,
= ∂x2 + ∂x7 ,
= cos u∂x3 − sin u∂x4 − sin v∂x5 − cos v∂x6 ,
= −u cos v∂x5 + u sin v∂x6 .
Then it is easy to see that M is a 2-lightlike submanifold with
Rad T M = span{Z1 , Z2 },
S(T M ) = span{Z3 , Z4 }
TOTALLY UMBILICAL HEMI-SLANT LIGHTLIKE SUBMANIFOLDS
195
and that M is Riemannian. It is easy to prove that S(T M ) is a slant
distribution with slant angle θ = π/4. Moreover, the screen transversal
bundle S(T M ⊥ ) is spanned by
W1 = sin u∂x3 + cos u∂x4 ,
W2 = sec u∂x3 + sin v∂x5 + cos v∂x6 .
The transversal lightlike bundle ltr(T M ) is spanned by
1
N1 = (−∂x1 + ∂x8 ),
2
1
N2 = (−∂x2 + ∂x7 ).
2
Hence M is a hemi-slant lightlike submanifold of M̄ .
Using the decomposition in (1), we denote P and Q the projection morphisms on the distributions S(T M ) and Rad T M , respectively. Then for
any X ∈ Γ(T M ), we can write
(12)
X = P X + QX.
Applying J¯ to (12) we obtain
(13)
¯ = T P X + F P X + F QX.
JX
Then using the definition of hemi-slant lightlike submanifolds, we get
(14) T P X ∈ Γ(S(T M )),
F P X ∈ Γ(S(T M ⊥ )),
F QX ∈ Γ(ltr(T M )).
Similarly, for any U ∈ Γ(tr(T M )), we have
(15)
¯ = BU + CU,
JU
where BU and CU are the tangential and the transversal components of
¯ , respectively. Differentiating (13) and using (6) to (8) and (15), for any
JU
X, Y ∈ Γ(T M ), we have
(16)
¯ l (X, Y ),
(∇X T )P Y = AF P Y X + AF QY X + Bhs (X, Y ) + Jh
(17)
(∇X F )P Y = Chs (X, Y ) − hs (X, T P Y ) − Ds (X, F QY ),
(18)
(∇X F )QY = −hl (X, T P Y ) − Dl (X, F P Y ).
where
(∇X T )P Y = ∇X T P Y − T P ∇X Y,
(∇X F )P Y = ∇sX F P Y − F P ∇X Y,
(∇X F )QY = ∇lX F QY − F Q∇X Y.
Theorem 3.2 ([10]). The necessary and sufficient conditions for a 2qlightlike submanifold M of an indefinite Kaehler manifold M̄ of index 2q
to be hemi-slant lightlike submanifold are:
¯
(a) J(ltr(T
M )) is a distribution on M .
(b) For any vector field X tangent to M, there exists λ ∈ [−1, 0] such
that (P T )2 X = λX, where X ∈ Γ(T M ) and λ = − cos2 θ.
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RASHMI SACHDEVA, RAKESH KUMAR AND SATVINDER SINGH BHATIA
Corollary 3.3 ([10]). Let M be a hemi-slant lightlike submanifold of an
indefinite Kaehler manifold M̄ of index 2q. Then we have
(19)
g(T P X, T P X) = cos2 θg(P X, P X),
(20)
ḡ(F P X, F P X) = sin2 θg(P X, P X),
for any X, Y ∈ Γ(T M ).
Definition 3.4 ([12]). Let M be a lightlike submanifold of an indefinite
Kaehler manifold M̄ . Then M is called a semi-transversal lightlike submanifold of M̄ if the following conditions are satisfied:
¯
(i) Rad T M is a distribution on M such that J(Rad
T M ) = ltr(T M ).
(ii) There exists a real nonnull distribution D ⊂ S(T M ) such that
¯ = D, J(D
¯ ⊥ ) ⊂ S(T M ⊥ ),
S(T M ) = D ⊕ D⊥ , JD
where D⊥ is orthogonal complementary to D in S(T M ).
Lemma 3.5 ([12]). Let M be a 2q-lightlike submanifold of an indefinite
Kaehler manifold M̄ with constant index 2q such that 2q < dim(M̄ ). Then
the screen distribution S(T M ) of lightlike submanifold M is Riemannian.
Theorem 3.6. Let M be a 2q-dimensional lightlike submanifold of an indefinite Kaehler manifold M̄ of index 2q. Then any coisotropic semi-transversal
lightlike submanifold is a hemi-slant lightlike submanifold with θ = 0. Moreover, any semi-transversal lightlike submanifold of M̄ with D = {0} is a
hemi-slant submanifold with θ = π/2.
Proof. Let M be a 2q-dimensional semi-transversal lightlike submanifold
of an indefinite Kaehler manifold M̄ . Then by the definition of semitransversal lightlike submanifold, Rad T M is a distribution on M such that
¯
J(Rad
T M ) = ltr(T M ). If M is coisotropic, then S(T M ⊥ ) = {0} thus
D⊥ = {0}. Using the Lemma (3.5), the screen distribution S(T M ) is Riemannian. Since D is invariant with respect to J¯ therefore it follows that
θ = 0. The second assertion is clear.
Theorem 3.7. There exist no proper hemi-slant totally lightlike or isotropic
submanifold in indefinite Kaehler manifolds.
Proof. Let M be a totally lightlike submanifold of an indefinite Kaehler
manifold then T M = Rad(T M ) and hence S(T M ) = {0}. The other assertion follows similarly.
Lemma 3.8. Let M be a hemi-slant lightlike submanifold of an indefinite
Kaehler manifold M̄ then F P X ∈ Γ(S(T M ⊥ )), for any X ∈ Γ(T M ).
Proof. Using (2) and (4) it is clear that F P X ∈ Γ(S(T M ⊥ )) if and only if
ḡ(F P X, ξ) = 0, for any ξ ∈ Γ(Rad T M ). Therefore
¯ X − T P X, ξ) = ḡ(JP
¯ X, ξ) = −ḡ(P X, Jξ)
¯ =0
ḡ(F P X, ξ) = ḡ(JP
gives the result.
TOTALLY UMBILICAL HEMI-SLANT LIGHTLIKE SUBMANIFOLDS
197
Thus from the (14) it follows that F (S(T M )) is a subspace of S(T M ⊥ ).
Therefore there exists an invariant subspace µp of Tp M̄ such that
(21)
S(Tp M ⊥ ) = F (S(Tp M ))⊥µp .
Thus
Tp M̄ = S(Tp M )⊥{Rad(Tp M ) ⊕ ltr(Tp M )}⊥{F (S(Tp M ))⊥µp }.
Definition 3.9 ([8]). A lightlike submanifold M of a semi-Riemannian manifold (M̄ , ḡ) is totally umbilical in M̄ if there is a smooth transversal vector
field H ∈ Γ(tr(T M )) on M , called transversal curvature vector field of M ,
such that, for all X, Y ∈ Γ(T M )
(22)
h(X, Y ) = Hg(X, Y ).
It is clear that M is totally umbilical if and only if on each coordinate
neighborhood u there exists smooth vector fields H l ∈ Γ(ltr(T M )) and
H s ∈ Γ(S(T M ⊥ )), such that
(23)
hl (X, Y ) = H l g(X, Y ),
Dl (X, W ) = 0,
hs (X, Y ) = H s g(X, Y ).
A lightlike submanifold is said to be totally geodesic if h(X, Y ) = 0, for any
X, Y ∈ Γ(T M ), that is, a lightlike submanifold is totally geodesic if H l = 0
and H s = 0.
Lemma 3.10. Let M be a totally umbilical proper hemi-slant lightlike submanifold of an indefinite Kaehler manifold M̄ then ∇X X ∈ Γ(S(T M )) for
any X ∈ Γ(S(T M )).
¯ = ξ then
Proof. Let X ∈ Γ(S(T M )) and N ∈ Γ(ltr(T M )) such that JN
using (6), (8), (13) and (23), we obtain
¯ JN
¯ ) = ḡ(∇
¯ X T X, ξ) + ḡ(∇
¯ X F X, ξ)
ḡ(∇X X, N ) = ḡ(∇X JX,
= ḡ(hl (X, T X), ξ) + ḡ(Dl (X, F X), ξ)
= g(X, T X)g(H l , ξ).
Since g(X, T X) = g(X, JX) = 0, we have ḡ(∇X X, N ) = 0. Hence using (1)
and (4), the assertion follows.
Theorem 3.11. Every totally umbilical proper hemi-slant lightlike submanifold M of an indefinite Kaehler manifold M̄ is totally geodesic.
Proof. Let X, Y ∈ Γ(S(T M )) then by adding (17) and (18) we obtain
∇sX F P Y − F P ∇X Y − F Q∇X Y
= Chs (X, Y ) − hs (X, T P Y ) − hl (X, T P Y ) − Dl (X, F P Y ).
Replace X by T X and Y by X, we get
F P ∇T X X + F Q∇T X X = ∇sT X F P X − Chs (T X, X) + hs (T X, T P X)
+ hl (T X, T P X) + Dl (T X, F P X).
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RASHMI SACHDEVA, RAKESH KUMAR AND SATVINDER SINGH BHATIA
Since M is totally umbilical hemi-slant lightlike submanifold therefore using
¯ X)H s = 0, we
(19), (23) and the fact hs (T X, X) = g(T X, X)H s = ḡ(JX,
2
s
get cos θg(X, X)H = F P ∇T X X +F Q∇T X X −∇T X F P X. Now taking the
scalar product both sides with respect to F P X ∈ (S(T M ⊥ )), we obtain
cos2 θg(X, X)ḡ(H s , F P X) = ḡ(F P ∇T X X, F P X) − ḡ(∇sT X F P X, F P X).
Further using (20), we get
(24)
cos2 θg(X, X)ḡ(H s , F P X)
= sin2 θḡ(P ∇T X X, P X) − ḡ(∇sT X F P X, F P X).
¯ T X , we get
Next, taking covariant derivative of (20) with respect to ∇
ḡ(∇sT X F P X, F P X) = sin2 θg(∇T X P X, P X)
and then using this in (24), we obtain cos2 θg(X, X)ḡ(H s , F P X) = 0. Since
M is a proper hemi-slant lightlike submanifold and g is a Riemannian metric
on S(T M ) therefore we have ḡ(H s , F P X) = 0. Thus using (14) and (21),
we obtain
H s ∈ Γ(µ).
(25)
Now, using the Kaehlerian property of M̄ for any X, Y ∈ Γ(S(T M )) with
(6), (8) and (22), we have
∇X T P Y + g(X, T P Y )H − AF P Y X + ∇sX F P Y + Dl (X, F P Y )
¯
= T ∇X Y + F ∇X Y + g(X, Y )JH.
¯ s and then using
Taking the scalar product of both sides with respect to JH
invariant property of µ with (25), we obtain
¯ s ) = g(X, Y )ḡ(H s , H s ).
(26)
ḡ(∇sX F P Y, JH
Since µ is an invariant subspace therefore using the Kaehlerian character of
M̄ for any H s ∈ Γ(µ), we get
s ¯ s
¯ s)
− AJH
+ Dl (X, JH
¯ s X + ∇X JH
¯ l (X, H s ).
= −T AH s X − F AH s X + B∇sX H s + C∇sX H s + JD
Taking the scalar product of both sides with respect to F P Y , we get
¯ s , F P Y ) = −ḡ(F AH s X, F P Y ) + ḡ(C∇s H s , F P Y ).
ḡ(∇s JH
X
X
From (15), we know that for any U ∈ Γ(tr(T M )), BU and CU are tangential
¯ , respectively. Thus if U ∈ Γ(ltr(T M ))
and transversal components of JU
¯
then JU = BU ∈ Γ(Rad(T M )) and CU = 0. Moreover, since
S(T M ⊥ ) = F (S(T M ))⊥µ,
therefore for any U ∈ Γ(S(T M ⊥ )), BU ∈ Γ(S(T M )) and CU ∈ Γ(µ). Since
∇sX αS ∈ Γ(S(T M ⊥ )) therefore C∇sX αS ∈ Γ(µ). Hence, we have
¯ s , F P Y ) = −ḡ(F AH s X, F P Y ) = − sin2 θg(AH s X, P Y ).
(27) ḡ(∇s JH
X
TOTALLY UMBILICAL HEMI-SLANT LIGHTLIKE SUBMANIFOLDS
199
¯ is a metric connection therefore (∇
¯ X g)(F P Y, JH
¯ s ) = 0 this further
Since ∇
s
s
s
s
¯
¯
implies that ḡ(∇X F P Y, JH ) = ḡ(∇X JH , F P Y ), therefore using (27), we
obtain
¯ s ) = − sin2 θg(AH s X, P Y ).
(28)
ḡ(∇sX F P Y, JH
From (26) and (28), we have g(X, Y )g(H s , H s ) = − sin2 θg(AH s X, P Y ) and
using (9) here, we obtain
g(X, Y )g(H s , H s ) = − sin2 θḡ(hs (X, P Y ), H s ) = − sin2 θg(X, Y )g(H s , H s ).
This implies that (1 + sin)2 θg(X, Y )g(H s , H s ) = 0. Since M is a proper
hemi-slant lightlike submanifold and g is a Riemannian metric on S(T M )
therefore we obtain
H s = 0.
(29)
Furthermore, using the Kaehler character of M̄ for any X ∈ Γ(S(T M )), we
have
∇X T X + h(X, T X) − AF X X + ∇sX F X + Dl (X, F X)
= T ∇X X + F ∇X X + Bh(X, X) + Chs (X, X).
Since M is totally umbilical hemi-slant lightlike submanifold therefore using
h(X, T X) = 0 and then comparing the tangential components, we obtain
∇X T X − AF X X = T ∇X X + Bh(X, X). Taking the scalar product both
¯ = ξ ∈ Γ(Rad(T M ))
sides with respect to N ∈ Γ(ltr(T M )) such that JN
and using the Lemma (3.10), we get
(30)
¯ l (X, X), N ).
ḡ(AF X X, N ) = ḡ(Jh
Now using (6), (8), (23) and the Lemma (3.10), we have
¯ ∇
¯ X N ) − ḡ(T X, ∇
¯ X N ) = ḡ(∇
¯ X X, ξ) + ḡ(∇
¯ X T X, N )
ḡ(AF X X, N ) = ḡ(JX,
= ḡ(hl (X, X), ξ).
Hence using this in (30), we have 2ḡ(hl (X, X), ξ) = 0, since M is a totally umbilical proper hemi-slant lightlike submanifold therefore we have
g(X, X)ḡ(H l , ξ) = 0. Since g is a Riemannian metric on S(T M ) therefore
ḡ(H l , ξ) = 0 then using (4), we obtain that
(31)
H l = 0.
Thus from (29) and (31), the proof is complete.
Theorem 3.12. Let M be a totally umbilical proper hemi-slant lightlike submanifold of an indefinite Kaehler manifold M̄ then the induced connection
∇ is a metric connection on M .
Proof. Using (23) and (31), we have hl = 0 then using the Theorem 2.2 in
[7], at page 159, the induced connection ∇ becomes a metric connection on
M.
200
RASHMI SACHDEVA, RAKESH KUMAR AND SATVINDER SINGH BHATIA
¯ and ∇ respectively then
Denote by R̄ and R the curvature tensors of ∇
by straightforward calculations ([7]), we have
(32) R̄(X, Y )Z = R(X, Y )Z + Ahl (X,Z) Y − Ahl (Y,Z) X + Ahs (X,Z) Y
− Ahs (Y,Z) X + (∇X hl )(Y, Z) − (∇Y hl )(X, Z)
+ Dl (X, hs (Y, Z)) − Dl (Y, hs (X, Z)) + (∇X hs )(Y, Z)
− (∇Y hs )(X, Z) + Ds (X, hl (Y, Z)) − Ds (Y, hl (X, Z)),
where
(33)
(∇X hs )(Y, Z) = ∇sX hs (Y, Z) − hs (∇X Y, Z) − hs (Y, ∇X Z).
(34)
(∇X hl )(Y, Z) = ∇lX hl (Y, Z) − hl (∇X Y, Z) − hl (Y, ∇X Z).
An indefinite complex space form is a connected indefinite Kaehler manifold of constant holomorphic sectional curvature c, denoted by M̄ (c), whose
curvature tensor R̄ is given by [1]
c
(35)
R̄(X, Y )Z = {ḡ(Y, Z)X − ḡ(X, Z)Y + ḡ(JY, Z)JX
4
− ḡ(JX, Z)JY + 2ḡ(X, JY )JZ},
for X, Y, Z vector fields on M̄ .
Theorem 3.13. There exists no totally umbilical proper hemi-slant lightlike
submanifold of an indefinite complex space form M̄ (c) such that c 6= 0.
Proof. Suppose M be a totally umbilical proper hemi-slant lightlike submanifold of M̄ (c) such that c 6= 0. Then using (35), for any X ∈ Γ(S(T M ))
and ξ, ξ 0 ∈ Γ(Rad(T M )), we obtain
c
0
¯
¯ 0 , ξ).
(36)
ḡ(R̄(X, JX)ξ
, ξ) = − g(X, X)g(Jξ
2
On the other hand using (23) and (32), we get
(37)
0
¯
¯ ξ 0 ), ξ) − ḡ((∇ ¯ hl )(X, ξ 0 ), ξ).
ḡ(R̄(X, JX)ξ
, ξ) = ḡ((∇X hl )(JX,
JX
Now, using (23) and (34), we have
(38)
¯ ξ 0 ) = −g(∇X JX,
¯ ξ 0 )H l − g(JX,
¯ ∇X ξ 0 )H l
(∇X hl )(JX,
¯ X T X, ξ 0 ) = ḡ(hl (X, T X), ξ 0 )
= ḡ(∇
l 0
¯
= g(X, JX)ḡ(H
, ξ ) = 0.
Similarly
l
0
0
l
0
l
¯ ¯ X, ξ 0 )
(39) (∇JX
¯ ξ )H = ḡ(∇
¯ h )(X, ξ ) = −g(∇JX
¯ X, ξ )H − g(X, ∇JX
JX
l ¯
0
l
0
¯ X)ḡ(H , ξ ) = 0.
= ḡ(h (JX, X), ξ ) = g(JX,
Thus from (36) to (40), we obtain
c
¯ 0 , ξ) = 0.
g(X, X)g(Jξ
2
TOTALLY UMBILICAL HEMI-SLANT LIGHTLIKE SUBMANIFOLDS
201
¯ 0 , ξ) 6= 0,
Since g is a Riemannian metric on S(T M ) and (4) implies that g(Jξ
therefore c = 0. This contradiction completes the proof.
In [7], a minimal lightlike submanifold M is defined when M is a hypersurface of a 4-dimensional Minkowski space. Then in [2], a general notion
of minimal lightlike submanifold of a semi-Riemannian manifold M̄ is introduced as follows:
Definition 3.14. A lightlike submanifold (M, g, S(T M )) isometrically immersed in a semi-Riemannian manifold (M̄ , ḡ) is minimal if:
(i) hs = 0 on Rad(T M ).
(ii) trace h = 0, where trace is written with respect to g restricted to
S(T M ).
We use the quasi orthonormal basis of M given by
{ξ1 , . . . , ξr , e1 , . . . , ek },
such that {ξ1 , . . . , ξr } and {e1 , . . . , ek } form a basis of Rad(T M ) and S(T M )
respectively.
Theorem 3.15. Let M be a totally umbilical proper hemi-slant lightlike
submanifold of an indefinite Kaehler manifold M̄ then M is minimal.
Proof. The assertion follows directly using the Theorem (3.11).
Lemma 3.16. Let M be a proper hemi-slant lightlike submanifold of an
indefinite Kaehler manifold M̄ such that dim(S(T M )) = dim(S(T M ⊥ )). If
{e1 , . . . , ek } is a local orthonormal basis of Γ(S(T M )) then
{csc θF e1 , . . . , csc θF ek }
is an orthonormal basis of S(T M ⊥ )).
Proof. Since e1 , . . . , ek is a local orthonormal basis of S(T M ) and S(T M )
is Riemannian therefore using (20), we obtain
ḡ(csc θF ei , csc θF ej ) = csc2 θ sin2 θg(ei , ej ) = δij ,
this proves the assertion.
Definition 3.17 ([9]). A lightlike submanifold is called irrotational if and
¯ X ξ ∈ Γ(T M ) for all X ∈ Γ(T M ) and ξ ∈ Γ(Rad(T M )).
only if ∇
Theorem 3.18. Let M be an irrotational hemi-slant lightlike submanifold
of an indefinite Kaehler manifold M̄ . Then M is minimal if and only if
trace AWq |S(T M ) = 0, trace A∗ξj |S(T M ) = 0,
where {Wq }lq=1 is a basis of S(T M ⊥ ) and {ξj }rj=1 is a basis of Rad(T M ).
202
RASHMI SACHDEVA, RAKESH KUMAR AND SATVINDER SINGH BHATIA
Proof. Since M is irrotational so this implies that hs (X, ξ) = 0 for X ∈
Γ(T M ) and ξ ∈ Γ(Rad(T M )). Thus hs vanishes on Rad(T M ). Hence M is
minimal if and only if trace h = 0 on S(T M ), that is, M is minimal if and
P
only ki=1 h(ei , ei ) = 0. Using (9) and (11) we obtain


k
r
l
k  X

X
X
X
1
1
∗
(40)
g(AWq ei , ei )Wq .
h(ei , ei ) =
g(Aξj ei , ei )Nj +

r
l
i=1
i=1
q=1
j=1
Thus the assertion follows from (40).
Theorem 3.19. Let M be a proper hemi-slant lightlike submanifold of an
indefinite Kaehler manifold M̄ . Then M is minimal if and only if
trace AWq |S(T M ) = 0, trace A∗ξj |S(T M ) = 0,
and
where {Wq }lq=1
ḡ(Dl (X, W ), Y ) = 0, ∀X, Y ∈ Γ(Rad(T M )),
is a basis of S(T M ⊥ ) and {ξj }rj=1 is a basis of Rad(T M ).
Proof. Let X, Y ∈ Γ(Rad(T M )) then using (9), it is clear that hs = 0 on
Rad(T M ) if and only if ḡ(Dl (X, W ), Y ) = 0. Moreover using the Proposition 3.1 of [2], hl = 0 on Rad(T M ). Therefore M is minimal if and
P
P
only if ki=1 h(ei , ei ) = 0. Using (9) and (11) we obtain ki=1 h(ei , ei ) =
Pk
P
P
r
l
1
1
∗
i=1 { r
j=1 g(Aξj ei , ei )Nj + l
q=1 g(AWq ei , ei )Wq }. Thus the assertion
follows.
Theorem 3.20. Let M be a proper hemi-slant lightlike submanifold of an
indefinite Kaehler manifold M̄ such that dim(S(T M )) = dim(S(T M ⊥ )).
Then M is minimal if and only if
trace Acsc θF ei |S(T M ) = 0, trace A∗ξj |S(T M ) = 0,
and
where {ei }ki=1
ḡ(Dl (X, F ei ), Y ) = 0, ∀X, Y ∈ Γ(Rad(T M )),
is a basis of S(T M ).
Proof. Let {ei }ki=1 is a basis of S(T M ). By Lemma (3.16) {csc θF ei }ki=1 is
a basis of S(T M ⊥ ). Therefore we can write
hs (X, X) =
(41)
k
X
λi csc θF ei ,
i=1
for any X ∈ Γ(T M ) and for some functions λi , i ∈ {1, . . . , k}. Using (9) we
have ḡ(hs (X, X), W ) = ḡ(AW X, X) for any X ∈ Γ(S(T M )). Then using
(20) and (41), we obtain λi = ḡ(Acsc θF ei X, X) and hence we get
s
h (X, X) =
k
X
csc θF ei ḡ(Acsc θF ei X, X),
i=1
for any X ∈ Γ(S(T M )). Then the assertion comes from Theorem (3.20). TOTALLY UMBILICAL HEMI-SLANT LIGHTLIKE SUBMANIFOLDS
203
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(Rashmi Sachdeva) School of Mathematics and Computer Applications, Thapar
University, Patiala, India.
rashmi.sachdeva86@gmail.com
(Rakesh Kumar) Department of Basic and Applied Sciences, Punjabi University,
Patiala, India
dr rk37c@yahoo.co.in
(Satvinder Singh Bhatia) School of Mathematics and Computer Applications,
Thapar University, Patiala, India.
ssbhatia@thapar.edu
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