On weakly Ricci symmetric lightlike hypersurfaces
of indefinite Sasakian manifolds
Fortun´e Massamba
(Received May 22, 2008; Revised June 27)
Abstract. This paper deals with weakly Ricci-symmetric lightlike hypersur-faces of indefinite Sasakian manifolds, tangent to the structure vector field. We prove that, in a weakly Ricci symmetric lightlike η-Einstein (or Einstein) hypersurface of an indefinite Sasakian manifold, the associated 1-forms α and β satisfy α + β = 0 (Theorem 4). Also, we show that there exist no weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurfaces of indefinite Sasakian manifolds with cyclic parallel Ricci tensor if α + β + γ is not everywhere zero (Theorem 5). A particular case of weakly Ricci symmetric condition is studied and we prove that a special weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurface cannot be η-Einstein (or Einstein) and under certain condition, it cannot be (D ⊥ hξi, D0)-mixed-totally
geodesic (Theorem 7).
AMS 2000 Mathematics Subject Classification. 53C07, 53C25, 53C50.
Key words and phrases.Weak Ricci symmetries, Lightlike hypersurfaces, Indef-inite Sasakian, Screen distribution, Ricci tensor.
§1. Introduction
The notion of weakly Ricci symmetric manifolds was considered in [10], [9] and others references therein. A non-flat semi-Riemannian manifold M is called weakly Ricci-symmetric if the Ricci tensor Ric is non-zero and satisfies the following condition, for any vector fields X, Y and Z in M ,
(∇XRic)(Y , Z) = α(X)Ric(Y , Z) + β(Y )Ric(X, Z) + γ(Z)Ric(Y , X),
(1.1)
where α, β and γ defined respectively by , g(X, ρ) = α(X), g(X, δ) = β(X), g(X, κ) = γ(X), are 1-forms called the associated 1-forms which are not zero
simultaneously and ∇ is the Levi-Civita connection for a semi-Riemannian metric g. In such case, ρ, δ and κ are called associated vector fields corre-sponding to the 1-forms α, β and γ respectively. If in (1.1) the 1-form α is replaced by 2α, then the semi-Riemannian manifold is called a generalized pseudo Ricci symmetric introduced by Chaky and Koley in [3]. So the defining condition of weakly Ricci symmetric manifold is weaker than the generalized pseudo Ricci symmetric manifold. If in (1.1) the 1-form α is replaced by 2α and β and γ are equal to α, then the semi-Riemannian manifold is called a special weakly Ricci symmetric and investigated by Singh and Kahan [9].
The purpose of this paper is to investigate the effect of weakly Ricci sym-metric condition on the lightlike geometry of hypersurfaces of an indefinite Sasakian manifold, tangent to the structure vector field ξ. Especially, we pay attention to lightlike hypersurfaces with symmetric Ricci tensor. This, because of the geometric point of view and also, physically, Ricci tensor symmetric is essential (see [5] for details and references therein). In the next paragraph, we summarize basic formulae concerning geometric objects on lightlike hyper-surfaces, using notations of [4]. In the last part of the paper, we consider a weakly Ricci symmetric lightlike hypersurface of an indefinite Sasakian man-ifold. We prove that, in a weakly Ricci symmetric lightlike η-Einstein (or Einstein) hypersurface of an indefinite Sasakian manifold, the associated 1-forms α and β satisfy α + β = 0. We also prove that there exist no weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurfaces of indefinite Sasakian manifolds with cyclic parallel Ricci tensor if α + β + γ is not everywhere zero. Finally, we prove that a special weakly Ricci sym-metric screen locally (or globally) conformal lightlike hypersurface cannot be η-Einstein (or Einstein) and if the trace of AN satisfies the partial differential
equation ξ · trAN − τ (ξ)trAN = 0, it cannot be (D ⊥ hξi, D0)-mixed-totally
geodesic.
§2. Preliminaries
A (2n + 1)-dimensional semi-Riemannian manifold (M , g) is said to be an indefinite Sasakian manifold if it admits an almost contact structure (φ, ξ, η), i.e. φ is a tensor field of type (1, 1) of rank 2n, ξ is a vector field, and η is a 1-form, satisfying
φ2 = −I + η ⊗ ξ, η(ξ) = 1, η ◦ φ = 0, φξ = 0,
η(X) = g(ξ, X), g(φ X, φ Y ) = g(X, Y ) − η(X) η(Y ), (∇Xη)Y = −g(φ X, Y ), (∇Xφ)Y = g(X, Y )ξ − η(Y )X, ∇Xξ = −φ(X), ∀ X, Y ∈ Γ(T M ))
where ∇ is the Levi-Civita connection for a semi-Riemannian metric g. A plane section σ in TpM is called a φ-section if it is spanned by X and φ X,
where X is a unit tangent vector field orthogonal to ξ. The sectional curvature of a φ-section σ is called a φ-sectional curvature. A Sasakian manifold M with constant φ-sectional curvature c, M is said to be a Sasakian space form and is denoted by M (c). The curvature tensor R of a Sasakian space form M (c) is given by [11]
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 )ξ − g(Y , Z)η(X)ξ + g(φ Y , Z)φ X − g(φ X, Z)φ Y − 2g(φ X, Y )φ Z , X, Y , Z ∈ Γ(T M ). (2.2)
Let (M , g) be a (2n + 1)-dimensional semi-Riemannian manifold with index s, 0 < s < 2n + 1 and let (M, g) be a hypersurface of M , with g = g|M. M is a lightlike hypersurface of M if g is of constant rank 2n − 1 and the normal bundle T M⊥ is a distribution of rank 1 on M [4]. A complementary
bundle of T M⊥ in T M is a rank 2n − 1 non-degenerate distribution over M . It is called a screen distribution and is often denoted by S(T M ). A lightlike hypersurface endowed with a specific screen distribution is denoted by the triple (M, g, S(T M )). As T M⊥ lies in the tangent bundle, the following result has an important role in studying the geometry of a lightlike hypersurface. Theorem 1. [4] Let (M, g, S(T M )) be a lightlike hypersurface of a semi-Riemannian manifold (M , g). Then, there exists a unique vector bundle N (T M ) of rank 1 over M such that for any non-zero section E of T M⊥ on a coor-dinate neighborhood U ⊂ M , there exist a unique section N of N (T M ) on U satisfying
g(N, E) = 1 and g(N, N ) = g(N, W ) = 0, ∀ W ∈ Γ(S(T M )|U).
(2.3)
Throughout the paper, all manifolds are supposed to be paracompact and smooth. We denote Γ(E) the smooth sections of the vector bundle E. Also by ⊥ and ⊕ we denote the orthogonal and nonorthogonal direct sum of two vector bundles. By Theorem 1 we may write down the following decomposition
T M = S(T M ) ⊥ T M⊥, (2.4)
T M = T M ⊕ N (T M ) = S(T M ) ⊥ (T M⊥⊕ N (T M ))
Let ∇ be the Levi-Civita connection on (M , g), then by using the second decomposition of (2.4), we have Gauss and Weingarten formulae in the form
∇XY = ∇XY + h(X, Y ), ∀X, Y ∈ Γ(T M ),
(2.5)
and ∇XV = −AVX + ∇⊥XV, ∀X ∈ Γ(T M ), V ∈ Γ(N (T M )),
where ∇XY, AVX ∈ Γ(T M ) and h(X, Y ), ∇⊥XV ∈ Γ(N (T M )). ∇ is a
sym-metric linear connection on M called an induced linear connection, ∇⊥ is a linear connection on the vector bundle N (T M ). h is a Γ(N (T M ))-valued symmetric bilinear form and AV is the shape operator of M concerning V .
Equivalently, consider a normalizing pair {E, N } as in Theorem 1. Then (2.5) and (2.6) take the form, for any X, Y ∈ Γ(T M |U),
∇XY = ∇XY + B(X, Y ) N and ∇XN = −ANX + τ (X)N.
(2.7)
It is important to mention that the second fundamental form B is independent of the choice of screen distribution, in fact, from (2.7), we obtain
B(X, Y ) = g(∇XY, E), ∀ X, Y ∈ Γ(T M |U),
(2.8)
τ(X) = g(∇⊥XN, E), ∀ X ∈ Γ(T M |U).
(2.9)
Let P be the projection morphism of T M on S(T M ) with respect to the orthogonal decomposition of T M . We have
∇XP Y = ∇∗XP Y + C(X, P Y )E and ∇XE= −A∗EX − τ (X)E,
(2.10)
where ∇∗XP Y and A∗EX belong to Γ(S(T M )). C, A∗E and ∇∗ are called the local second fundamental form, the local shape operator and the induced connection on S(T M ). In general, the induced linear connection ∇ is not a metric connection and we have
(∇Xg)(Y, Z) = B(X, Y )θ(Z) + B(X, Z)θ(Y ),
where θ is a differential 1-form locally defined on M by θ(·) := g(N, ·). Also, we have the following identities,
g(A∗EX, P Y) = B(X, P Y ), g(A∗EX, N) = 0, B(X, E) = 0.
(2.11)
Finally, using (2.7), R and R are the curvature tensor fields of M and M are related as
R(X, Y )Z = R(X, Y )Z + B(X, Z)ANY − B(Y, Z)ANX
+ {(∇XB)(Y, Z) − (∇YB)(X, Z) + τ (X)B(Y, Z) − τ (Y )B(X, Z)} N,
(2.12)
where (∇XB)(Y, Z) = X.B(Y, Z) − B(∇XY, Z) − B(Y, ∇XZ).
(2.13)
§3. Main Results
Let (M , φ, ξ, η, g) be an indefinite Sasakian manifold and (M, g) be its lightlike hypersurface, tangent to the structure vector field ξ (ξ ∈ T M ). If E is a
local section of T M⊥, then g(φE, E) = 0, and φE is tangent to M . Thus φ(T M⊥) is a distribution on M of rank 1 such that φ(T M⊥) ∩ T M⊥= {0} . This enables us to choose a screen distribution S(T M ) such that it contains φ(T M⊥) as vector subbundle. We consider a local section N of N (T M ). Since g(φ N, E) = −g(N, φ E) = 0, we deduce that φ E is also tangent to M and belongs to S(T M ). On the other hand, since g(φ N, N ) = 0, we see that the component of φ N with respect to E vanishes. Thus φ N ∈ Γ(S(T M )). From (2.1), we have g(φ N, φE) = 1. Therefore, φ(T M⊥) ⊕ φ(N (T M )) (direct sum but not orthogonal) is a nondegenerate vector subbundle of S(T M ) of rank 2. It is known [2] that if M is tangent to the structure vector field ξ, then, ξ belongs to S(T M ). Using this, and since g(φE, ξ) = g(φN, ξ) = 0, there exists a nondegenerate distribution D0 of rank 2n − 4 on M such that
S(T M ) =nφ(T M⊥) ⊕ φ(N (T M ))o⊥ D0 ⊥ hξi,
(3.1)
where hξi is the distribution spanned by ξ, that is, hξi = Span{ξ}. It is easy to check that the distribution D0 is invariant under φ, i.e. φ(D0) = D0.
Example 1. Let R7 be the 7-dimensional real number space. We consider
{xi}1≤i≤7as cartesian coordinates on R7and define with respect to the natural
field of framesn∂x∂ioa tensor field φ of type (1, 1) by its matrix:
φ( ∂ ∂x1 ) = − ∂ ∂x2 , φ( ∂ ∂x2 ) = ∂ ∂x1 + x4 ∂ ∂x7 , φ( ∂ ∂x3 ) = − ∂ ∂x4 , φ( ∂ ∂x4 ) = ∂ ∂x3 + x6 ∂ ∂x7 , φ( ∂ ∂x5 ) = − ∂ ∂x6 , φ( ∂ ∂x6 ) = ∂ ∂x5 , φ( ∂ ∂x7 ) = 0. (3.2)
The differential 1-form η is defined by
η= dx7− x4dx1− x6dx3.
(3.3)
The vector field ξ is defined by ξ = ∂
∂x7. It is easy to check (2.1) and thus
(φ, ξ, η) is an almost contact structure on R7. Finally we define metric g on R7 by
g = (x24− 1)dx21− dx22+ (x26+ 1)dx23+ dx24− dx25− dx26+ dx27 − x4dx1⊗ dx7− x4dx7⊗ dx1+ x4x6dx1⊗ dx3+ x4x6dx3⊗ dx1
− x6dx3⊗ dx7− x6dx7⊗ dx3,
(3.4)
with respect to the natural field of frames. It is easy to check that g is a semi-Riemannian metric and (φ, ξ, η, g) given by (3.2)-(3.4) is a Sasakian structure on R7.
We now define a hypersurface M of (R7, φ, ξ, η, g) as M =(x1, ..., x7) ∈ R7 : x5 = x4 .
(3.5)
Thus the tangent space T M is spanned by {Ui}1≤i≤6, where U1 = ∂x∂1, U2 =
∂ ∂x2, U3 = ∂ ∂x3, U4 = ∂ ∂x4 + ∂ ∂x5, U5 = ∂
∂x6, U6 = ξ and the 1-dimensional
distribution T M⊥ of rank 1 is spanned by E, where E = ∂ ∂x4 +
∂
∂x5. It
fol-lows that T M⊥ ⊂ T M . Then M is a 6-dimensional lightlike hypersurface of
R7. Also, the transversal bundle N (T M ) is spanned by N = 12∂x∂4 −∂x∂5. On the other hand, by using the almost contact structure of R7 and also by taking into account of the decomposition (3.1), the distribution D0 is
spanned by F, φF , where F = U2, φF = U1 + x4ξ and the
distribu-tions hξi, φ(T M⊥) and φ(N (T M )) are spanned, respectively, by ξ, φE =
U3−U5+x6ξ and φN = 12(U3+U5+x6ξ). Hence M is a lightlike hypersurface
of R7.
Moreover, from (2.4) and (3.1) we obtain the decomposition T M = nφ(T M⊥) ⊕ φ(N (T M ))o⊥ D0 ⊥< ξ >⊥ T M⊥,
(3.6)
T M = nφ(T M⊥) ⊕ φ(N (T M ))o⊥ D0 ⊥< ξ >⊥ (T M⊥⊕ N (T M )).
Now, we consider the distributions on M ,
D := T M⊥⊥ φ(T M⊥) ⊥ D0, D0:= φ(N (T M )).
(3.7)
Then D is invariant under φ and
T M = D ⊕ D0⊥ hξi. (3.8)
Let us consider the local lightlike vector fields U := − φ N, V := − φ E. Then, from (3.8), any X ∈ Γ(T M ) is written as X = RX + QX + η(X)ξ, QX = u(X) U, where R and Q are the projection morphisms of T M into D and D0,
respectively, and u is a differential 1-form locally defined on M by u(·) := g(V, ·). Applying φ to X and (2.1) (note that φ2N = −N ), we obtain φ X = φ X+ u(X) N, where φ is a tensor field of type (1, 1) defined on M by φ X := φ RX and we also have φ2X = − X + η(X)ξ + u(X) U, ∀ X ∈ Γ(T M ). Now applying φ to φ2X and since φU = 0, we obtain φ3 + φ = 0, which shows that φ is an f -structure [11] of constant rank. We have the following useful identities ∇Xξ = −φX, (3.9) B(X, ξ) = −u(X), (3.10) B(X, U ) = C(X, V ) (3.11) (∇Xu)Y = −B(X, φ Y ) − u(Y )τ (X). (3.12)
Lemma 1. Let M be a lightlike hypersurface of an indefinite Sasakian mani-fold M with ξ ∈ T M . Then, M is (D ⊥ hξi, D0)-mixed totally geodesic if and only if, for any X ∈ Γ(D ⊥ hξi),
ANX ∈ Γ(φ(T M⊥) ⊥ D0⊥ hξi).
(3.13)
Proof. By the definition, M is (D ⊥ hξi, D0)-mixed totally geodesic if and only
if, for any X ∈ Γ(D ⊥ hξi), B(X, U ) = 0, From (3.11) we obtain u(ANX) =
g(ANX, V) = 0. i.e. ANX ∈ Γ(D ⊥ hξi). Given that ANX has no component
in Γ(T M⊥), then A
NX ∈ Γ(φ(T M⊥) ⊥ D0 ⊥ hξi). The converse is obvious
by using (3.11).
From (3.10), we have B(ξ, U ) = −1. This means that the lightlike hyper-surface M of an indefinite Sasakian manifold M , with ξ ∈ T M , cannot be (hξi, D0)-mixed totally geodesic.
Lemma 2. Let M be a lightlike hypersurface of an indefinite Sasakian space M2n+1 with ξ ∈ T M . Then, for any X ∈ Γ(T M ),
∇XU = − 2n−4 X i=1 C(X, φFi) g(Fi, Fi) Fi− θ(X)ξ + C(X, U )E + τ (X)U (3.14)
Proof. From the definition of lightlike hypersurface of an indefinite Sasakian manifold through the local field of framesφE, φN, ξ, E, Fi
1≤i≤2n−4 on U ⊂
M, where {Fi}1≤i≤2n−4 is an orthonormal field of frames of D0, we have, for
any X ∈ Γ(T M ), ∇XU =P2n−4i=1 λiFi+ϕ1ξ+ϕ2E+ϕ3V+ϕ4U.From (2.7) and
(2.10), we obtain λig(Fi, Fi) = g(∇XU, Fi) = −g(ANX, φFi) = −C(X, φFi),
ϕ1 = g(∇XU, ξ) = −g(∇XφN, ξ) = −θ(X), ϕ2 = g(∇XU, N) = C(X, U ),
ϕ3 = g(∇XU, U) = 0 and ϕ4= g(∇XU, V) = τ (X) which prove (3.14).
We are now concerned with the structure equations of the immersions of a lightlike hypersurface in a semi-Riemannian manifold. Let us consider the pair {E, N } on U ⊂ M (see Theorem 1) and by using (2.2) and (2.12), we have
(∇XB)(Y, Z) − (∇YB)(X, Z) = τ (Y )B(X, Z) − τ (X)B(Y, Z)
+c− 1
4 g(φY, Z)u(X) − g(φX, Z)u(Y ) − 2g(φX, Y )u(Z) . (3.15)
In the sequel, we need the following proposition.
Proposition 3. Let M be a lightlike hypersurface of an indefinite Sasakian space form M (c) of constant curvature c with ξ ∈ T M . Then, the Lie deriva-tive of the second fundamental form B with respect to ξ is given by
(LξB)(X, Y ) = −τ (ξ)B(X, Y ), ∀ X, Y ∈ Γ(T M ).
(3.16)
Moreover, if τ (ξ) 6= 0, then ξ is a Killing vector field with respect to the second fundamental form B if and only if M is totally geodesic
Proof. By replacing Z with ξ into (2.13) and using (3.9), we obtain (∇ξB)(X, Y ) = (LξB)(X, Y ) + B(φX, Y ) + B(X, φY ).
(3.17)
Likewise, by replacing Z with X and X with ξ into (2.13) and also using (3.9) and (3.10), we have
(∇XB)(ξ, Y ) = −X.u(Y ) + B(φX, Y ) + u(∇XY).
(3.18)
Substracting (3.17) and (3.18), and using (3.12) we obtain
(∇ξB)(X, Y ) − (∇XB)(ξ, Y ) = (LξB)(X, Y ) − u(Y )τ (X).
(3.19)
From (3.15), the left hand side of (3.19) becomes
(∇ξB)(X, Y ) − (∇XB)(ξ, Y ) = −u(Y )τ (X) − τ (ξ)B(X, Y ).
(3.20)
The expressions (3.19) and (3.20) implies (LξB)(X, Y ) = −τ (ξ)B(X, Y ). If
τ(ξ) 6= 0, the equivalence follows.
Note that the 1-form τ in (2.9) depends on the vector field E and it is easy to see that if E = λE with λ a positive smooth function on M , the associated 1-form τ is related to τ by
τ(X) = τ (X) + X(ln λ), ∀ X ∈ Γ(T M |U).
(3.21)
The induced connection ∇ on the lightlike hypersurface M is not metric in general and the Ricci tensor associated is not symmetric, contrary to the case of semi-Riemannian manifolds. However, for η-Einstein (or Einstein) light-like hypersurfaces, that is, the Ricci tensor Ric tensor satisfies Ric(X, Y ) = k1g(X, Y ) + k2η(X)η(Y ) (or Ric(X, Y ) = kg(X, Y )), due to the symmetric of
the induced degenerate metric g and η ⊗ η, the Ricci tensor is symmetric, and the notion of η-Einstein (respectively Einstein) manifold does not depend on the choice of the screen distribution S(T M ). Consequently
Proposition 4. On a lightlike η-Einstein (respectively, Einstein) hypersur-face, the 1-form τ in (2.9) is closed.
Proof. Define the Ricci tensor Ric as
Ric(X, Y ) = trace(Z −→ R(X, Y )Z), ∀ X, Y ∈ Γ(T M ) where R is the curvature tensor of the induced connection ∇.
Consider a local quasi-orthogonal frame field {X0, N, Xi}i=1,...,2n−1 on M
where {X0, Xi} is a local frame field on M with respect to the
(2.3), and E = X0. Put Rls := Ric(Xs, Xl) and R0k := Ric(Xk, X0).
Us-ing the frame field {X0, N, Xi}, a direct calculation gives locally Rls− Rsl =
2dτ (Xl, Xs) and R0k−Rk0 = 2dτ (X0, Xk). Since the Ricci tensor is symmetric
on M which is η-Einstein (respectively, Einstein), we have dτ = 0.
By definition Ric(X, Y ) = trace(Z −→ R(X, Y )Z), we have, for any X, Y ∈ Γ(T M ),
Ric(X, Y ) =
2n−4
X
i=1
εig(R(Fi, X)Y, Fi) + g(R(ξ, X)Y, ξ) + g(R(E, X)Y, N )
+ g(R(φE, X)Y, φN ) + g(R(φN, X)Y, φE), (3.22)
where {Fi}1≤i≤2n−4 is an orthogonal basis of D0 and εi = g(Fi, Fi) 6= 0, since
the distribution D0 is non-degenerate. From Gauss and Codazzi equations,
we obtain g(R(Fi, X)Y, Fi) = c+ 3 4 {εig(X, Y ) − g(X, Fi)g(Y, Fi)} + c− 1 4 −εiη(X)η(Y ) + g(Fi, φY)g(φX, Fi) + g(φX, Y )g(φFi, Fi) + 2g(φX, Fi)g(φY, Fi) + B(X, Y )C(Fi, Fi) − B(Fi, Y)C(X, Fi), (3.23) g(R(ξ, X)Y, ξ) = c+ 3 4 {−η(Y )η(X) + g(X, Y )} + c− 1 4 {−g(X, Y ) + η(X)η(Y )} + B(X, Y ))C(ξ, ξ) − B(ξ, Y )C(X, ξ), (3.24) g(R(E, X)Y, N ) = c+ 3 4 g(X, Y ) + c− 1 4 {−η(X)η(Y ) + u(Y )θ(φX) − 2u(X)θ(φY )} , (3.25) g(R(φE, X)Y, φN ) = c+ 3 4 {−u(Y )v(X) + g(X, Y )} + c− 1 4 {−η(X)η(Y ) + 2u(φX)v(φY ) + B(X, Y )C(φE, φN) − B(φE, Y )C(X, φN ), (3.26) g(R(φN, X)Y, φE) = c+ 3 4 {g(X, Y ) − u(X)v(Y )} + c− 1
4 −η(X)η(Y ) − θ(Y )u(φX) + 2v(φX)u(φY ) + B(X, Y )C(φN, φE) − B(φN, Y )C(X, φE).
(3.27)
So substituting (3.23), (3.24), (3.25), (3.26) and (3.27) in (3.22) and by re-grouping like terms, we have the following result.
Proposition 5. Let M be a lightlike hypersurface of an indefinite Sasakian manifold M with ξ ∈ T M . Then the Ricci tensor Ric is given by, for any X, Y ∈ Γ(T M ),
Ric(X, Y ) = ag(X, Y ) − bη(X)η(Y ) + B(X, Y )trAN
− B(ANX, Y),
(3.28)
where a = (2n+1)(c+3)−84 , and b = (2n+1)(c−1)4 and trace tr is written with respect to g restricted to S(T M ).
By Proposition 5 and using (3.10), we have the following useful identities Ric(X, ξ) = 2(n − 1)η(X) − u(X)trAN + u(ANX),
(3.29)
Ric(ξ, Y ) = 2(n − 1)η(Y ) − u(Y )trAN− B(ANξ, Y).
(3.30)
From (3.28), we have
Ric(X, Y ) − Ric(Y, X) = B(ANX, Y) − B(ANY, X).
(3.31)
This means that the Ricci tensor of a lightlike hypersurface M of an indefinite Sasakian space form M (c) is not symmetric in general. So, only some privi-leged conditions on the local second fundamental form of M may enable the Ricci tensor to be symmetric. It is easy to check, from (3.28), that the Ricci tensor of M is symmetric if and only if the shape operator of M is symmetric with respect to the second fundamental form of M . Also, the Ricci tensor of the induced connection ∇ of any totally geodesic lightlike hypersurface is symmetric.
Are there any others, with symmetric induced Ricci tensors, but not neces-sarily totally geodesic or shape operator symmetric with respect to the second fundamental form ? Here is one such class. First, we recall the definition of screen conformal lightlike hypersurfaces of a semi-Riemannian manifold M .
A lightlike hypersurface (M, g, S(T M )) of a semi-Riemannian manifold is screen locally conformal if the shape operators AN and A∗E of M and its screen
distribution S(T M ), respectively, are related by [5] AN = ϕ A∗E
(3.32)
where ϕ is a non-vanshing smooth function on U in M . In case U = M the screen conformality is said to be global. Such a submanifold has some important and desirable properties, for instance, the integrability of its screen distribution (see [5] for details).
Theorem 2. Let (M, g, S(T M )) be a locally (or globally) screen conformal lightlike hypersurface of an indefinite Sasakian manifold (M (c), g) of constant curvature c with ξ ∈ T M . Then the Ricci tensor of the induced connection ∇ is symmetric.
Proof. From (3.31) and (3.32), we obtain
Ric(X, Y ) − Ric(Y, X) = B(ANX, Y) − B(ANY, X)
= ϕ (B(A∗EX, Y) − B(A∗EY, X)) = ϕg([A∗E, A∗E]Y, X) = 0. This complete the proof.
Example 2. Let M be a hypersurface of R7, of Example 1, given by x5 = x4,
where (x1, ..., x7) is a local coordinate system for R7. As explained in Example
1, M is a lightlike hypersurface of R7 having a local quasi-orthogonal field of
frames {U1, U2, U3, U4= E, U5, U6 = ξ, N } along M . Denote by ∇ the
Levi-Civita connection on R7. Then, by straightforward calculations, we obtain ∇U1N = − 1 4x4U1− 1 4(x 2 4+ 1)ξ, ∇U2N = ∇U4N = ∇U5N = 0, ∇U3N = − 1 4x6U1− 1 4x4x6ξ, ∇ξN = 1 4U1+ 1 4x4ξ, (3.33) ∇U1E = − 1 2x4U1− 1 2(x 2 4+ 1)ξ, ∇U2E = ∇U4E = ∇U5E = 0, ∇U3E = − 1 2x6U1− 1 2x4x6ξ, ∇ξE= 1 2U1+ 1 2x4ξ. (3.34)
Using these equations above, the differential 1-form τ vanishes i.e. τ (X) = 0, for any X ∈ Γ(T M ). So, from the Gauss and Weingarten formulae we infer
ANU1 = 1 4x4U1+ 1 4(x 2 4+ 1)ξ, ANU2 = ANU4= ANU5 = 0, ANU3 = 1 4x6U1+ 1 4x4x6ξ, ANξ= − 1 4U1− 1 4x4ξ, (3.35) A∗EU1= 1 2x4U1+ 1 2(x 2 4+ 1)ξ, A∗EU2= AE∗U4= A∗EU5 = 0, A∗EU3= 1 2x6U1+ 1 2x4x6ξ, A ∗ Eξ= − 1 2U1− 1 2x4ξ. (3.36)
From (3.35) and (3.36), ANX = 12A∗EX, for any X ∈ Γ(T M ) and trAN = 0,
i.e. the shape operator AN is trace-free. Therefore, the hypersurface M of R7
is screen conformal and minimal. So, its screen distribution is integrable. The non-vanishing components of the second fundamental form B are given by
B(U1, U1) = −x4, B(U1, U3) = − 1 2x6, B(U1, U6) = 1 2. (3.37)
From the above equations, it is easy to check that, B(ANUi, Uj) = B(ANUj, Ui),
for any i 6= j and i, j = 1, ..., 6. Consequently, Ricci tensor of the induced con-nection ∇ on the hypersurface M of R7 is symmetric.
Also, we have
Theorem 3. Let (M, g, S(T M )) be a totally contact geodesic lightlike hyper-surface of an indefinite Sasakian manifold (M , g) with ξ ∈ T M . If the second fundamental form B of M is parallel, then the Ricci tensor of the induced connection ∇ is symmetric.
Proof. M is said to be totally contact geodesic lightlike hypersurface of an indefinite Sasakian manifold (M , g) if the local second fundamental form B of M satisfies
B(X, Y ) = η(X)B(Y, ξ) + η(Y )B(X, ξ) = −η(X)u(Y ) − η(Y )u(X), for any X, Y ∈ Γ(T M ) and its covariant derivative is given by
(∇XB)(Y, Z) = (u(X)θ(Y ) + g(φX, Y ))u(Z)
+ (u(X)θ(Z) + g(φX, Z))u(Y ) + (τ (X)u(Y ) + B(X, φY ))η(Z) + (τ (X)u(Z) + B(X, φZ))η(Y ). (3.38)
B is parallel if (∇ZB)(X, Y ) = 0, for any X, Y , Z ∈ Γ(T M ). Using (3.38),
we have, for any X ∈ Γ(T M ), 0 = (∇XB)(ξ, U ) = τ (X).
Proceed as in the proof of Proposition 4. Consider a local quasi-orthogonal frame field {X0, N, Xi}i=1,...,2n−1 on M where {X0, Xi} is a local frame field
on M with respect to the decomposition (3.7) with N , the unique section of transversal bundle N (T M ) satisfying (2.3), and E = X0. Put Rls :=
Ric(Xs, Xl) and R0k := Ric(Xk, X0). Using the frame field {X0, N, Xi}, we
have locally Rls− Rsl = 2dτ (Xl, Xs) = 0 and R0k− Rk0 = 2dτ (X0, Xk) = 0
which complete the proof.
Example 3. Let (R5, g) be the 5-dimensional semi-Riemmannian manifold, where the metric g is given, with respect to the cartesian coordinates {xi}1≤i≤5
on R5 and the natural field of frames {∂x∂i}, by
g = (x23− 1)dx21− dx22+ dx32+ dx24+ dx25− x3dx1⊗ dx5
− x3dx5⊗ dx1,
(3.39)
We define with respect to the natural field of framesn∂x∂
i
o
a tensor field φ of type (1, 1) by its matrix:
φ( ∂ ∂x1 ) = − ∂ ∂x2 , φ( ∂ ∂x2 ) = ∂ ∂x1 + x3 ∂ ∂x5 , φ( ∂ ∂x3 ) = − ∂ ∂x4 , φ( ∂ ∂x4 ) = ∂ ∂x3 and φ( ∂ ∂x5 ) = 0. (3.40)
The differential 1-form η and the vector field ξ are defined, respectively, by η= dx5− x3dx1 and ξ = ∂ ∂x5 . (3.41)
It is easy to check (φ, ξ, η, g) given by (3.39)-(3.41) is a Sasakian structure on R5.
Consider a hypersurface (M, g) in R5 given by the equation x4 = x2, where
(x1, ..., x5) is a local coordinate system for R5. The tangent space T M is
spanned by {U1, U2, U3, U4}, where U1 = ∂x∂1, U2 = ∂ ∂x2 + ∂ ∂x4, U3 = ∂
∂x3, U4 = ξ, and the 1-dimensional distribution T M
⊥ of rank 1 is spanned
by E with E = ∂ ∂x2 +
∂
∂x4. Also, the transversal bundle N (T M ) is spanned
by N = 12−∂x∂2 +∂x∂4. It follows that T M⊥ ⊂ T M . Then M is a 4-dimensional lightlike hypersurface of R5 having a local quasi-orthogonal field of frames {U1, U2 = E, U3, U4 = ξ, N } along M . Denote by ∇ the Levi-Civita
connection on R5. Then, by straightforward calculations, we obtain
∇XN = 0, ∀ X ∈ Γ(T M ).
Using these equations above, the differential 1-form τ vanishes i.e. τ (X) = 0, for any X ∈ Γ(T M ). So, from the Gauss and Weingarten formulae we have
ANX = 0, A∗EX = 0 and ∇XE = 0, ∀ X ∈ Γ(T M ).
Therefore, by Duggal-Bejancu theorems (Theorem 2.2 and Theorem 2.7) in [4] the lightlike hypersurface M of R5is totally geodesic and its distribution is parallel. Also, from the above equations, it is easy to check that η(X)B(Y, ξ)+ η(Y )B(X, ξ) = 0 = B(X, Y ), for any X, Y ∈ Γ(T M ). So M is totally contact geodesic, parallel and admits a symmetric induced Ricci tensor.
On the other hand, by considering again the Example 1, since B(U1, U6) = 1
2 6= −η(U1)u(U6) − η(U6)u(U1) = 0, the hypersurface M of R7 defined in the
example 1 is not totally contact geodesic.
Based on discussion so far it would be appropriate to say that from the geometric point of view alone, the induced tensor Ric on M must be symmet-ric, as without this property one only obtains tensorial relations. Physically, Ric symmetric is essential. Consequently, as per above note, it is desirable to assume that only dτ vanishes locally (or globally) on M . Luckily, we have so far seen that there are large classes of hypersurfaces with symmetric Ricci tensor.
In particular, symmetric induced Ricci tensor has been useful in finding several good properties of lightlike hypersurfaces [5]. For these reasons, only
symmetric induced Ricci tensor will be considered in the sequel of this pa-per. In this case, the weakly Ricci-symmetric notion in lightlike hypersurfaces becomes valid.
Next, we investigate the effect of weakly Ricci symmetric condition on the geometry of lightlike hypersurfaces in an indefinite Sasakian manifold.
Suppose that Ricci tensor of a lightlike hypersurface M of an indefinite Sasakian manifold (M , g) with ξ ∈ T M is symmetric. A submanifold M is called a weakly Ricci symmetric if
(∇XRic)(Y, Z) = α(X)Ric(Y, Z) + β(Y )Ric(X, Z)
+ γ(Z)Ric(Y, X), (3.42)
where α, β and γ are defined respectively by , for any X ∈ Γ(T M ), g(X, ρ) = α(X), g(X, δ) = β(X), g(X, κ) = γ(X), are forms called the associated 1-forms which are not zero simultaneously. We denote this kind of 2n-dimensional submanifold by (W RS)2n.
Note that the covariant derivative of the induced Ricci tensor on M (3.28) is given by, for any X, Y , Z ∈ Γ(T M ),
(∇XRic)(Y, Z) = a (B(X, Y )θ(Z) + B(X, Z)θ(Y ))
+ b η(Y )g(φX, Z) + η(Z)g(φX, Y ) + (∇XB)(Y, Z)trAN
+ B(Y, Z)(X.trAN) − (∇XB)(ANY, Z).
(3.43)
Also, for a lightlike η-Einstein hypersurface M , that is, the Ricci tensor Ric tensor satisfies Ric(X, Y ) = k1g(X, Y ) + k2η(X)η(Y ), the functions k1 and k2
are not necessarily constant on M . Since M is tangent the structure vector field ξ in an indefinite Sasakian manifold, k1 and k2 satisfy
k1+ k2 = 2(n − 1).
(3.44)
Theorem 4. Let M be weakly Ricci symmetric lightlike η-Einstein (or Ein-stein) hypersurface of an indefinite Sasakian manifold M2n+1(n > 1) with ξ∈ T M . Then the 1-forms α and β satisfy α + β = 0.
Proof. Suppose that M is a (W RS)2n lightlike η-Einstein hypersurfaces of
an indefinite Sasakian manifold M2n+1(n > 1) with ξ ∈ T M . Since M is η-Einstein, Ric(Y, Z) = k1g(X, Y ) + k2η(X)η(Y ). So, from (3.42) and using
(2.11), we obtain
k1(B(X, Y )θ(Z) + B(X, Z)θ(Y )) + k2(η(Z)(∇Xη)Y + η(Y )(∇Xη)Z)
+(∇Xk1)g(Y, Z) + (∇Xk2)η(Y )η(Z) = α(X) (k1g(Y, Z) + k2η(Y )η(Z))
+β(Y ) (k1g(X, Z) + k2η(X)η(Z)) + γ(Z) (k1g(Y, X) + k2η(Y )η(X)) .
Putting Z = ξ in (3.45) and using (3.44), we have
−k1u(X)θ(Y ) + k2(∇Xη)Y = (k1+ k2)α(X)η(Y ) + (k1+ k2)β(Y )η(X)
+ γ(ξ) (k1g(Y, X) + k2η(Y )η(X)) .
(3.46)
Again, taking X = ξ in (3.46) and using the fact that k1+ k2 6= 0 (n > 1), we
get α(ξ)η(Y ) + β(Y ) + γ(ξ)η(Y ) = 0, that is, β(Y ) = −(α(ξ) + γ(ξ))η(Y ). (3.47)
On the other hand, by taking X = V in (3.46), we have k2(∇Vη)Y =
(k1 + k2)α(V )η(Y ) + k1γ(ξ)u(Y ) which implies, by taking Y = U , γ(ξ) = 0.
Use this and (3.47) in (3.46), we get −k1u(X)θ(Y ) + k2(∇Xη)Y = (k1 +
k2) (α(X) − α(ξ)η(X)) η(Y ), that is
−k1u(X)θ(Y ) + k2(∇Xη)Y = (k1+ k2) (α(X) + β(X)) η(Y )
which implies, for Y = ξ, α(X) = −β(X) and the proof is complete.
Example 4. Let M be a hypersurface of R5 (indefinite Sasakian manifold
defined in the Example 3) given by
x4 = x2, x3>0,
where (x1, ..., x5) is a local coordinate system for R5. By proceeding as in
Example 3, M is a totally geodesic lightlike hypersurface of R5 having a local quasi-orthogonal field of frames {U1, U2 = E, U3, ξ, N}, where U1 =
∂ ∂x1, E = ∂ ∂x2 + ∂ ∂x4, U3 = ∂ ∂x3, ξ = ∂ ∂x5, N = 1 2 −∂x∂2 + ∂x∂4 along M . Using (3.28), M is η-Einstein. This means that the induced Ricci tensor on M is symmetric and it is given by Ric(X, Y ) = ag(X, Y ) − bη(X)η(Y ), where nonzero constants a and b satisfy a − b = 2. The non-vanishing components of the induced Ricci tensor on M are given by
Ric(U1, U1) = ax23− a, Ric(U3, U3) = a,
Ric(ξ, ξ) = 2, Ric(U1, ξ) = −ax3.
(3.48)
Using (3.43) and a direct calculation, it is easy to check that, for any X, Y , Z ∈ Γ(T M ),
(∇XRic)(Y, Z) = α(X)Ric(Y, Z) + β(Y )Ric(X, Z)
+ γ(Z)Ric(Y, X), (3.49)
where the associated 1-forms α, β and γ are defined explicitly by α(ξ) = β(ξ) = γ(ξ) = 0, α(U1) = β(U1) = γ(U1) = 0,
α(U3) = β(U3) = γ(U3) = 0, α(E) = −β(E) =
b ax3
, γ(E) = 0. (3.50)
Note that a lightlike Einstein hypersurface of a 3-dimensional indefinite Sasakian manifold, tangent to the structure vector field ξ, is Ricci flat. So, that hypersurface cannot be (W RS)2.
A non-zero Ricci tensor Ric of lightlike hypersurface M is said to be cyclic parallel if C∇Ric = 0, namely, for any X, Y , Z ∈ Γ(T M ),
(∇XRic)(Y, Z) + (∇YRic)(Z, X) + (∇ZRic)(X, Y ) = 0.
(3.51)
Let M admits a cyclic parallel Ricci tensor. Then, we have
0 = (∇XRic)(Y, Z) + (∇YRic)(Z, X) + (∇ZRic)(X, Y )
= α(X)Ric(Y, Z) + (β(X) + γ(X))Ric(Z, Y ) + α(Y )Ric(Z, X) + (γ(Y ) + β(Y ))Ric(X, Z) + α(Z)Ric(X, Y ) + (γ(Z) + β(Z))Ric(Y, X). (3.52)
Taking Z = ξ in (3.52) and using (3.29) and (3.31) α(X) {2(n − 1)η(Y ) − u(Y )trAN + u(ANY)}
+(β(X) + γ(X)) {2(n − 1)η(Y ) − u(Y )trAN − B(ANξ, Y)}
+α(Y ) {2(n − 1)η(X) − u(X)trAN − B(ANξ, X)}
+(γ(Y ) + β(Y )) {2(n − 1)η(X) − u(X)trAN + u(ANX)}
+α(ξ)Ric(X, Y ) + (γ(ξ) + β(ξ))Ric(Y, X) = 0. (3.53)
Again, taking Y = ξ in (3.53), using (3.11), (3.29) and (3.31), we have (2n − 3)(α(X) + β(X) + γ(X)) + (α(ξ) + β(ξ) + γ(ξ)) {4(n − 1)η(X)
−2u(X)trAN + u(ANX) − B(ANξ, X)} = 0.
(3.54)
Putting X = ξ in (3.54) and using (3.10), we get 3(2n−3)(α(ξ)+β(ξ)+γ(ξ)) = 0, that is α(ξ) + β(ξ) + γ(ξ) = 0. (3.55) Using (3.55) in (3.54), we have, α(X) + β(X) + γ(X) = 0, ∀ X ∈ Γ(T M ). (3.56) Therefore, we have
Theorem 5. There exist no weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurfaces M of indefinite Sasakian manifolds M2n+1 with ξ ∈ T M and cyclic parallel Ricci tensor if α + β + γ is not everywhere zero.
By Theorem 4 and definition (W RS)2n, it is easy to see that there are
no weakly Ricci symmetric lightlike Einstein hypersurfaces, tangent to the structure vector field ξ, with cyclic parallel Ricci tensor.
If in (3.42) the 1-form α is replaced by 2α and β and γ are equal to α, then we have
(∇XRic)(Y, Z) = 2α(X)Ric(Y, Z) + α(Y )Ric(X, Z)
+ α(Z)Ric(Y, X), (3.57)
where α is a non-zero 1-form defined by α(X) = g(X, ρ). A submanifold which satisfies (3.57) is called a special weakly Ricci symmetric submanifold and denoted by (SW RS)2n.
Theorem 6. There exist no special weakly Ricci symmetric screen locally (or globally) conformal (or Einstein) lightlike hypersurfaces M of an indefinite Sasakian manifold M2n+1 with ξ ∈ T M and cyclic parallel Ricci tensor.
Proof. Suppose that M is a special weakly Ricci symmetric screen locally (or globally) conformal (or Einstein) lightlike hypersurface M of an indefinite Sasakian manifold M2n+1 with ξ ∈ T M . If M admits a cyclic parallel Ricci tensor, then, from (3.56), we have 2α(X) + α(X) + α(X) = 0, ∀ X ∈ Γ(T M ), that is α(X) = 0 which contradicts the definition of (SW RS)2n.
From the differential geometry of lightlike hypersurfaces, we recall the fol-lowing result. The submanifold M is (D ⊥ hξi, D0)-mixed totally geodesic if for any X ∈ Γ(D ⊥ hξi), Y ∈ Γ(D0), B(X, Y ) = 0. The Latter reduces to B(X, U ) = 0, since the distribution D is of rank 1 and spanned by U .
Theorem 7. Let M be a special weakly Ricci symmetric screen locally (or glob-ally) conformal lightlike hypersurface of an indefinite Sasakian space (M2n+1(c), n >1) of constant curvature c, with ξ ∈ T M . Then, M cannot be η-Einstein (or Einstein). Moreover, if the trace of AN satisfies the partial differential
equation ξ · trAN − τ (ξ)trAN = 0, M cannot be (D ⊥ hξi, D0)-mixed totally
geodesic.
Proof. Suppose that special weakly Ricci symmetric screen locally (or globally) conformal lightlike hypersurface M is η-Einstein (or Einstein). Then, from Theorem 4, for any M ∈ Γ(T M ), 2α(X) = −α(X), that is α(X) = 0 which is inadmissible by the definition of (SW RS)2n. So M cannot be Einstein.
Suppose now that M is (D ⊥ hξi, D0)-mixed totally geodesic.Since M is
obtain
(∇XRic)(Y, ξ) = 2α(X)Ric(Y, ξ) + α(Y )Ric(X, ξ) + α(ξ)Ric(Y, X)
= 2(2n − 1)α(X)η(Y ) + (2n − 1)α(Y )η(X)
− (2α(X)u(Y ) + α(Y )u(X)) trAN + 2α(X)u(ANY)
+ α(Y )u(ANX) + α(ξ)Ric(Y, X).
(3.58)
Replacing X with ξ and using (3.11), (3.58) becomes
(∇ξRic)(Y, ξ) = 2(2n − 1)α(ξ)η(Y ) + 2(n − 1)α(Y )
− 2α(ξ)u(Y )trAN+ 2α(ξ)u(ANY)
+ α(ξ) ((2n − 1)η(Y ) − u(Y )trAN + u(ANY))
= 3(2n − 1)α(ξ)η(Y ) + 2(n − 1)α(Y ) − 3α(ξ)u(Y )trAN+ 3α(ξ)u(ANY).
(3.59)
On the other hand, using φξ = φξ = 0,
(∇ξRic)(Y, ξ) = ξ · Ric(Y, ξ) − Ric(∇ξY, ξ) − Ric(Y, ∇ξξ)
= 2(n − 1)ξ · η(Y ) − ξ · u(Y )trAN− u(Y )ξ · trAN
+ ξ · u(ANY) − (2n − 1)η(∇ξY) + u(∇ξY)trAN − u(AN∇ξY)
= ξ · u(ANY) − g(∇ξV, Y)trAN − u(Y )ξ · trAN − u(AN∇ξY)
= u(Y )(τ (ξ)trAN − ξ · trAN) + ξ · u(ANY) − u(AN∇ξY).
(3.60)
From (3.59) and (3.60), we obtain
3(2n − 1)α(ξ)η(Y ) + 2(n − 1)α(Y ) − 3α(ξ)u(Y )trAN + 3α(ξ)u(ANY)
= u(Y )(τ (ξ)trAN− ξ · trAN) + ξ · u(ANY) − u(AN∇ξY).
(3.61)
Substituting Y with ξ in (3.61), we obtain 8(n − 1)α(ξ) = 0. Since n > 1, we have
α(ξ) = 0. (3.62)
Taking (3.62) in (3.61),
2(n − 1)α(Y ) = u(Y )(τ (ξ)trAN − ξ · trAN) + ξ · u(ANY)
− u(AN∇ξY).
(3.63)
Since M is a (D ⊥ hξi, D0)-mixed-totally geodesic, then, by Theorem 1, for any Y ∈ Γ(D), ANY ∈ Γ(φ(T M⊥) ⊥ D0). Moreover, for any Y ∈ Γ(D),
u(Y ) = 0 and since the distribution D is invariant under φ, using (3.10), we have, g(∇ξY, V) = g(A∗Eξ, φY) = −u(φY ) = 0, that is, ∇ξY ∈ Γ(D ⊥ hξi).
As g(∇ξY, ξ) = 0, then ∇ξY ∈ Γ(D) and AN∇ξY ∈ Γ(φ(T M⊥) ⊥ D0). So
(3.63) becomes 2(n − 1)α(Y ) = 0, ∀ Y ∈ Γ(D) and since n > 1, α(Y ) = 0, ∀ Y ∈ Γ(D).
(3.64)
Next, we compute α(U ). Using (3.11) and (3.16), the right hand side of (3.63) is reduced to
u(Y )(τ (ξ)trAN − ξ · trAN) + ξ · u(ANY) − u(AN∇ξY)
= u(Y )(τ (ξ)trAN− ξ · trAN) + ξ · C(Y, V ) − C(∇ξY, V)
= u(Y )(τ (ξ)trAN− ξ · trAN) + ξ · B(Y, U ) − B(∇ξY, U)
= u(Y )(τ (ξ)trAN− ξ · trAN) + (LξB)(Y, U ) + B(φY, U ) + B(∇ξU, Y)
= u(Y )(τ (ξ)trAN− ξ · trAN) − τ (ξ)B(Y, U ) + B(φY, U ) + B(∇ξU, Y).
(3.65)
From Lemma 2, we obtain
∇ξU = − 2n−4 X i=1 C(ξ, φFi) g(Fi, Fi) Fi− θ(ξ)ξ + C(ξ, U )E + τ (ξ)U = − 2n−4 X i=1 g(ANξ, φFi) g(Fi, Fi) Fi+ C(ξ, U )E + τ (ξ)U. (3.66)
As M is a (D ⊥ hξi, D0)-mixed totally geodesic, again by Theorem 1, ANξ ∈
Γ(φ(T M⊥⊥ D 0 ⊥ hξi). Writing ANξ = v(ANξ)V + 2n−4 X i=1 µiFi+ η(ANξ)ξ,
we have g(ANξ, φFi) = µig(Fi, φFi) = 0, since g(Fi, φFi) = −g(φFi, Fi) =
−g(Fi, φFi), i.e. 2g(Fi, φFi) = 0. So (3.66) becomes
∇ξU = C(ξ, U )E + τ (ξ)U
(3.67)
and if the trace trAN satisfies the partial differential equation ξ · trAN −
τ(ξ)trAN = 0, with the aid of (3.67) and B(E, ·) = 0, (3.65) becomes
2(n − 1)α(Y ) = B(φY, U ), ∀ Y ∈ Γ(T M ). (3.68)
As n > 1 and since φU = 0, so by taking Y = U in (3.68), we obtain α(U ) = 0,
(3.69)
From (3.62), (3.64) and (3.69), α(Y ) = 0, ∀ Y ∈ Γ(T M ) which is inadmissi-ble by the definition of (SW RS)2n. Thus a special weakly Ricci symmetric
Some particular cases of lightlike submanifolds of indefinite Sasakian man-ifolds have been studied by Duggal and Sahin in [6]. They showed that in a contact screen Cauchy-Riemann (SCR)-lightlike submanifolds or irrotational screen real lightlike submanifold of an indefinite Sasakian manifold, the min-imality notion is equivalent to the trace-free of the shape operator AN with
respect to g restricted to S(T M ). Therefore, there exist lightlike hypersurfaces of indefinite Sasakian manifolds whose the trace of AN satisfies the partial
dif-ferential equation above.
Finally, we note that Theorems 5, 6 and 7 are also valid for any lightlike hypersurface M of an indefinite Sasakian manifold, tangent to the structure vector field ξ and whose dτ (or τ ) vanishes locally (or globally) on M or shape operator AN symmetric with respect to its second fundamental form.
Aknowledgments
Main results were done when the author was visiting The Abdus Salam Inter-national Centre for Theoretical Physics (ICTP) in the summer of 2007. He thanks that centre for the support during this work. He is also grateful to the referee for helping him to improve the presentation.
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Fortun´e Massamba
Department of Mathematics, University of Botswana Private Bag 0022 Gaborone, Rep. of Botswana.
