We study totally contact umbilical lightlike hypersurfaces of indefinite 𝒮-manifolds and prove the nonexistence of totally contact umbilical lightlike
hypersurface in indefinite 𝒮-space form.

1. Introduction

The general theory of nondegenerate submanifolds of Riemannian or semi-Riemannian manifolds is one of the most important topics of differential geometry [1, 2]. But the theory of degenerate or lightlike submanifolds of semi-Riemannian manifolds is relatively new [3] and different due to the fact that their normal vector bundle intersects with the tangent bundle. Thus, the study of lightlike submanifolds becomes more difficult than the study of nondegenerate submanifolds and one cannot use, in the usual way, the classical theory of submanifold to define induced objects on a lightlike submanifold. The degenerate geometry rises within the semi-Riemannian context, due to the existence of causal character of geometrical objects: their spacelike, timelike, or lightlike nature implies the existence of three types of hypersurfaces and submanifolds.

A lightlike framed hypersurface of a Lorentz C-manifold, with an induced metric connection, is a Killing horizon and a globally hyperbolic spacetime; a de Sitter spacetime can carry a framed structure [3, 4]. Moreover, the contact geometry has a significant use in differential equations, optics, and phase spaces of dynamical systems [5–7]. All these motivated us to study lightlike hypersurfaces of indefinite globally framed f-manifold, in particular indefinite 𝒮-manifolds.

In this paper, we study totally contact umbilical lightlike hypersurfaces of indefinite 𝒮-manifolds and prove the nonexistence of totally contact umbilical lightlike hypersurface in indefinite 𝒮-space form.

2. Preliminaries

A manifold M- of dimension 2n+r is called a globally framed f-manifold (g.f.f-manifold) if it is endowed with a nowhere vanishing (1,1)-tensor field ϕ- of constant rank, such that kerϕ- is parallelizable; that is, there exist global vector fields ζ-α, α∈{1,…,r}, with their dual 1-forms η-α, satisfying
(1)ϕ-2=-I+η-α⊗ζ-α,η-α(ζ-β)=δβα.
If the metric g- is a semi-Riemannian metric with index ν, 0<ν<2n+r such that
(2)g-(ϕ-X,ϕ-Y)=g-(X,Y)-∑α=1rϵαη-α(X)η-α(Y),
for any X,Y∈Γ(TM-), ϵα=±1 accordingly ζ-α is either spacelike or timelike, then the g.f.f-manifold (M-2n+r,ϕ-,ζ-α,η-α) is called an indefinite metric g.f.f-manifold. Then clearly η-α(X)=ϵαg-(X,ζ-α), for any α∈{1,…,r}. An indefinite metric g.f.f-manifold is called an indefinite 𝒮-manifold if it is normal that is, the tensor field N=Nϕ-+2dη-α⊗ζ-α vanishes, where Nϕ- is Nijenhuis tensor of ϕ- and dη-α=Φ, for any α∈{1,…,r}, where Φ(X,Y)=g-(X,ϕ-Y), for any X,Y∈Γ(TM-). The Levi-Civita connection of an indefinite 𝒮-manifold satisfies
(3)(∇-Xϕ-)Y=g-(ϕ-X,ϕ-Y)ζ-+η-(Y)ϕ-2(X),
where ζ-=∑α=1rζ-α and η-=∑α=1rϵαη-α. Also
(4)∇-Xζ-α=-ϵαϕ-X,
and kerϕ- is integrable flat distribution, since ∇-ζ-αζ-β=0, for detail see [8].

We recall notations and fundamental equations for lightlike hypersurfaces, which are due to Duggal and Bejancu [3].

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 TM⊥ is a distribution of rank 1 on M. A complementary bundle of TM⊥ in TM is a rank 2n-1 nondegenerate distribution over M. It is denoted by S(TM) and known as a screen distribution. Therefore we have
(5)TM=S(TM)⊥TM⊥,TM-|M=S(TM)⊥S(TM)⊥,
where S(TM)⊥ is a orthogonal complementary vector bundle of S(TM) in TM-|M. The following theorem has important roles in studying the geometry of lightlike hypersurface.

Theorem 1.

Let (M,g,S(TM)) be a lightlike hypersurface of a semi-Riemannian manifold (M-,g-). Then there exists a unique vector bundle tr(TM) of rank 1 over M, such that for any nonzero section ξ of TM⊥ on a coordinate neighborhood 𝒰⊂M, there exists a unique section N of tr(TM) on 𝒰 satisfying
(6)g-(ξ,N)=1,g-(N,N)=g-(N,W)=0,∀W∈Γ(S(TM)|𝒰).

Hence, we have the following decompositions of TM-|M:
(7)TM-|M=S(TM)⊥(TM⊥⊕tr(TM))=TM⊕tr(TM).

Let ∇- be the Levi-Civita connection on M- with respect to g-. Then using the decompositions in (7), Gauss and Weingarten formulae are given as
(8)∇-XY=∇XY+h(X,Y),∇-XN=-ANX+∇XtN,
for any X,Y∈Γ(TM) and N∈Γ(tr(TM)), where ∇XY and ANX belong to Γ(TM) while h(X,Y) and ∇XtN belong to Γ(tr(TM)). Here ∇ is a torsion free linear connection on M, h is a Γ(tr(TM))-valued symmetric bilinear form on Γ(TM) and known as the second fundamental form. AN is a linear operator on Γ(TM) and known as the shape operator of lightlike immersion and ∇t is a linear connection on Γ(tr(TM)).

Locally, for the pair {ξ,N}, following the Duggal and Bejancu's notation [3], we recall local second fundamental form B and 1-form τ as
(9)B(X,Y)=g-(h(X,Y),ξ),thereforeh(X,Y)=B(X,Y)N.τ(X)=g-(∇XtN,ξ)therefore∇XtN=τ(X)N.
Hence locally, (8) becomes
(10)∇-XY=∇XY+B(X,Y)N,∇-XN=-ANX+τ(X)N.

If P denotes the projection morphism of TM on S(TM) then from (5), we have
(11)∇XPY=∇X*PY+h*(X,PY),∇Xξ=-Aξ*X+∇X*tξ,
for any X,Y∈Γ(TM), ξ∈Γ(TM⊥), where ∇X*PY and Aξ*X belong to Γ(S(TM)) while h*(X,PY) and ∇X*tξ belong to Γ(TM⊥). Here h* and Aξ* are called the second fundamental form and the shape operator of the screen distribution, respectively. It should be noted that the induced linear connection ∇ is not a metric connection as it satisfies
(12)(∇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,·). By direct calculation, we have
(13)g(ANY,PX)=g-(N,h*(Y,PX)),g-(ANY,N)=0,g(Aξ*X,PY)=g-(ξ,h(X,PY)),g-(Aξ*X,N)=0,
for any X,Y∈Γ(TM), ξ∈Γ(TM⊥), and N∈Γ(tr(TM)). According to Duggal and Bejancu's notation [3], locally we have know
(14)C(X,PY)=g-(h*(X,PY),N)thereforeh*(X,PY)=C(X,PY)ξ,ϵ(X)=g-(∇X*tξ,N)therefore∇X*tξ=ϵ(X)ξ.
Then (11) become
(15)∇XPY=∇X*PY+C(X,PY)ξ,(16)∇Xξ=-Aξ*X+ϵ(X)ξ=-Aξ*X-τ(X)ξ
and also give
(17)g(ANY,PW)=C(Y,PW),g-(ANY,N)=0,g(Aξ*X,PY)=B(X,PY),g-(Aξ*X,N)=0.

From (9), it is clear that
(18)B(X,ξ)=0,
that is, the second fundamental form of a lightlike hypersurface is degenerate. Following the notations of Duggal and Bejancu [3], for the curvature tensor R- of M-, we have R-(X,Y,Z,W)=g-(R-(X,Y)Z,W), for any X,Y,Z,W∈Γ(TM-).

3. Lightlike Hypersurface of Indefinite <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M144"><mml:mrow><mml:mi>𝒮</mml:mi></mml:mrow></mml:math></inline-formula>-Manifolds

Let (M,g,S(TM)) be a lightlike hypersurface of an indefinite g.f.f-manifold (M-2n+r,ϕ-,ζ-α,η-α,g-), and α∈{1,…,r} such that the characteristic vector fields ζ-α, α∈{1,…,r} are tangent to M. As the ambient manifold M- has an additional geometric structure ϕ-, we must look for a particular screen distribution on M. Since g-(ϕ-ξ,ξ)=0, for any ξ∈Γ(TM⊥) therefore ϕ-(TM⊥) is a distribution on M of rank 1 such that TM⊥∩ϕ-(TM⊥)={0}. Moreover g-(ϕ-N,ξ)=-g-(N,ϕ-ξ)=0, for any N∈Γ(tr(TM)), therefore ϕ-N is tangent to M. Since g-(ϕ-N,N)=0; that is, the component of ϕ-N with respect to ξ vanishes; this implies ϕ-N∈Γ(S(TM)). As ξ and N are null vector fields satisfying g-(ξ,N)=1, therefore from (2), we deduce that ϕ-ξ and ϕ-N are null vector fields satisfying g-(ϕ-ξ,ϕ-N)=1. Hence ϕ-(TM⊥)⊕ϕ-(tr(TM)) is a vector subbundle of S(TM) of rank 2. It is known [9] that if structure vector fields ζ-α, α∈{1…,r} are tangent to M then ζ-α belong to S(TM). Therefore g-(ϕ-ξ,ζ-α)=g-(ϕ-N,ζ-α)=0 implies that there exists a nondegenerate distribution D0 of rank 2n-4 on M such that
(19)S(TM)={ϕ-(tr(TM))⊕ϕ-TM⊥}⊥D0⊥{ζ-α},
therefore
(20)TM={ϕ-(tr(TM))⊕ϕ-TM⊥}⊥D0⊥{ζ-α}⊥TM⊥.
Let D=TM⊥⊥ϕ-TM⊥⊥D0, and let D′=ϕ-tr(TM) then D clearly is invariant and D′ is anti-invariant under ϕ-, and we have
(21)TM=D′⊕D⊥{ζ-α}.

Consider the local lightlike vector fields as
(22)U=-ϕ-N∈ϕ-(tr(TM))=D′,V=-ϕ-ξ∈D.
Using the decompositions in (21), any X∈Γ(TM) can be written as
(23)X=SX+QX+∑α=1rη-α(X)ζ-α,
where S and Q are the projection morphisms of TM into D and D′, respectively, and where QX=u(X)U, and u is a local 1-form on M defined by u(X)=g-(V,X). Therefore
(24)u(U)=1,u(X)=0,∀X∈Γ(D),ϕ-2N=-N.Applying ϕ- to (23) and then using (24), we obtain ϕ-X=ϕ-(SX)+u(X)N. By assuming ϕ-(SX)=ϕX for any X∈Γ(TM), we obtain a tensor field ϕ of type (1,1) on M and given by
(25)ϕ-X=ϕX+u(X)N.Applying ϕ- to (25) and then using the definition of ϕX, we obtain
(26)ϕ2X=-X+u(X)U+∑α=1rη-α(X)ζ-α.
Moreover, since ϕX=ϕ-(SX) therefore
(27)ϕU=0,η-α∘ϕ=0,u(ϕX)=0,
for any X∈Γ(TM). Thus we have the following theorem analogous to a theorem proved in [10].

Theorem 2.

Let (M-,ϕ-,ζ-α,η-α,g-) be an indefinite g.f.f-manifold and let (M,g,S(TM)) be a lightlike hypersurface of M-; then (M,ϕ,ζ-α,U,η-α,u) is also a g.f.f-manifold.

Since ϕU=0 therefore applying ϕ to (26), we get ϕ3+ϕ=0, this implies that ϕ is an f structure of constant rank on M. Now using (2) and (25), we obtain
(28)g-(ϕX,ϕY)=g-(X,Y)-u(X)ν(Y)-u(Y)ν(X)-∑α=1rϵαη-α(X)η-α(Y),
for any X,Y∈Γ(TM-), where ν is a 1 form locally defined on M by ν(·)=g-(U,·). Replace X by ϕX in (28) and using (26) and (27), we derive
(29)g-(ϕX,Y)+g-(X,ϕY)=-u(X)θ(Y)-u(Y)θ(X).

Let M be a lightlike hypersurface of an indefinite 𝒮-manifold M-; then by using (4), (10), and (25), we obtain
(30)∇Xζ-α+B(X,ζ-α)N=-ϵαϕX-ϵαu(X)N,
for any X∈Γ(TM-) then comparing the tangential and transversal components, we get
(31)∇Xζ-α=-ϵαϕX,(32)B(X,ζ-α)=-ϵαu(X),thisimpliesh(X,ζ-α)=-ϵαu(X)N.Using (15) and (31), we get ∇X*ζ-α+C(X,ζ-α)ξ=-ϵαϕX, taking inner product with N∈Γ(tr(TM)), we get C(X,ζ-α)=-ϵαg-(ϕX,N), and then using (29), we obtain
(33)C(X,ζ-α)=ϵαg(X,ϕN)+u(X)θ(N)-u(N)θ(X)=-ϵαg-(X,U)=-ϵαν(X).

Using (3), for any N∈Γ(tr(TM)), we get ∇-Xϕ-N=ϕ-∇-XN+g-(X,N)ζ-, therefore
(34)B(X,U)=g-(h(X,U),ξ)=-g-(∇-Xϕ-N,ξ)=g-(∇-X,ϕξ)=g(ANX,V)=C(X,V).

Here we will follow Bejancu's definition [11] of totally contact umbilical submanifolds of Sasakian manifolds to state totally contact umbilical lightlike hypersurfaces of indefinite 𝒮-manifolds.

Definition 3.

A lightlike hypersurface M of an indefinite 𝒮-manifold is said to be totally contact umbilical lightlike hypersurface if the second fundamental form h of M satisfies
(35)h(X,Y)={g(X,Y)-∑α=1rϵαη-α(X)η-α(Y)}H+∑α=1rη-α(X)h(Y,ζ-α)+∑α=1rη-α(Y)h(X,ζ-α),
for any X,Y∈Γ(TM), where H is a transversal vector field on M; that is, H=λN, where λ is a smooth function on 𝒰⊂M.

Remark 4.

We can also write (35) as
(36)h(X,Y)=B(X,Y)N={B1(X,Y)+B2(X,Y)}N,
where
(37)B1(X,Y)=λ{g(X,Y)-∑α=1rϵαη-α(X)η-α(Y)},
and using (32)
(38)B2(X,Y)=-η-(X)u(Y)-η-(Y)u(X),
for any X,Y∈Γ(TM). If λ=0, that is, B1=0; then lightlike hypersurface M is said to be totally contact geodesic.

Let R- be the curvature tensor fields of M;- then for an indefinite 𝒮-space form M-(c), we have (see [8])
(39)R-(X,Y,W,Z)=c+3ϵ4{g-(ϕ-Y,ϕ-Z)g-(ϕ-X,ϕ-W)-g-(ϕ-X,ϕ-Z)g-(ϕ-Y,ϕ-W)}+c-ϵ4{Φ(W,X)Φ(Z,Y)-Φ(Z,X)Φ(W,Y)+2Φ(X,Y)Φ(W,Z)}+{η-(W)η-(X)g-(ϕ-Z,ϕ-Y)-η-(W)η-(Y)g-(ϕ-Z,ϕ-X)+η-(Y)η-(Z)g-(ϕ-W,ϕ-X)-η-(Z)η-(X)g-(ϕ-W,ϕ-Y)},
for any X,Y,Z,W∈Γ(TM-), where Φ(X,Y)=g-(X,ϕ-Y) and ϵ=∑α=1rϵα. Using (39), we obtain
(40)R-(X,Y,W,ξ)=c-ϵ4{g-(V,X)g-(W,ϕ-Y)-g-(V,Y)g-(W,ϕ-X)+2g-(V,W)g-(X,ϕ-Y)}.

Also, for the pair {ξ,N} on 𝒰⊂M, from (3.8) of page no. 94 of [3], we have
(41)R-(X,Y,W,ξ)=(∇XB)(Y,W)-(∇YB)(X,W)+τ(X)B(Y,W)-τ(Y)B(X,W),X,Y,W∈Γ(TM) and ξ∈Γ(TM⊥), where
(42)(∇XB)(Y,W)=X·B(Y,W)-B(∇XY,W)-B(Y,∇XW).

Hence using (40) and (41), we obtain
(43)(∇XB)(Y,W)-(∇YB)(X,W)=c-ϵ4{g-(V,X)g-(W,ϕ-Y)-g-(V,Y)g-(W,ϕ-X)+2g-(V,W)g-(X,ϕ-Y)}-τ(X)B(Y,W)+τ(Y)B(X,W),
for any X,Y,W∈Γ(TM-).

Lemma 5.

Let (M,g) be a totally contact umbilical lightlike hypersurface of an indefinite 𝒮-manifold (M-,g-); then for any X,Y,W∈Γ(TM), one has
(44)(∇XB1)(Y,W)=(X·λ){g(Y,W)-∑α=1rϵαη-α(Y)η-α(W)}+λ{B(X,Y)θ(W)+B(X,W)θ(Y)}+λη-(W){g-(ϕX,Y)+u(X)θ(Y)}+λη-(Y){g-(ϕX,W)+u(X)θ(W)}.(∇XB2)(Y,W)=ϵu(W){g-(Y,ϕX)+u(X)θ(Y)}+ϵu(Y){g-(W,ϕX)+u(X)θ(W)}+η-(W){B(X,ϕY)+τ(X)u(Y)}+η-(Y){B(X,ϕW)+τ(X)u(W)}.

Proof.

By virtue of (37) and (42), we obtain
(45)(∇XB1)(Y,W)=(X·λ){g(Y,W)-∑α=1rϵαη-α(Y)η-α(W)}-λ{g(∇XY,W)+g(Y,∇XW)}+λ∑α=1rϵα{η-α(∇XY)η-α(W)+η-α(∇XW)η-α(Y)}.
Now using (12) in (45), we have
(46)(∇XB1)(Y,W)=(X·λ){g(Y,W)-∑α=1rϵαη-α(Y)η-α(W)}-λ{X(g(Y,W))-B(X,Y)θ(W)-B(X,W)θ(Y)}+λ∑α=1r{g(∇XY,ζ-α)η-α(W)+g(∇XW,ζ-α)η-α(Y)}.

Again using (12), we have
(47)g(∇XY,ζ-α)=X(g(Y,ζ-α))+ϵαg(Y,ϕX)+ϵαu(X)θ(Y).
Thus from (46) and (47), the first expression of the theorem follows. Next, using (38) and (42), we obtain
(48)(∇XB2)(Y,W)=∑α=1rϵα{η-α(∇XY)u(W)+η-α(∇XW)u(Y)+η-α(W)u(∇XY)+η-α(Y)u(∇XW)}.

Using (31) and (32), we get
(49)η-α(∇XY)=g(Y,ϕX)+u(X)θ(Y),
and from (2), (3), (10), (16), (17), (18), and (25), we obtain
(50)u(∇XY)=B(X,ϕY)+τ(X)u(Y).
Thus by substituting (49) and (50) in (48), we obtain (44).

Theorem 6.

Let M-(c) be an indefinite 𝒮-space form and M be a totally contact umbilical lightlike hypersurface of M-(c). Then c=-3ϵ, where ϵ=∑α=1rϵα and λ satisfies the following differential equations:
(51)ξ·λ+λτ(ξ)-λ2=0,PX·λ+λτ(PX)=0,
for any X∈Γ(TM).

Proof.

Let M be a totally contact umbilical lightlike hypersurface of an indefinite 𝒮-space form M-(c) of constant ϕ--sectional curvature c. Then using (35), we have B(X,ϕY)=λg-(X,ϕY). Substituting (29), (44) in (43), we obtain
(52)(X·λ){g-(Y,W)-∑α=1rϵαη-α(Y)η-α(W)}-(Y·λ){g-(X,W)-∑α=1rϵαη-α(X)η-α(W)}+λ{B(X,W)θ(Y)-B(Y,W)θ(X)}+2λ{g-(ϕX,Y)+u(X)θ(Y)}η-(W)+λ{(g-(W,ϕX)η-(Y)-g-(W,ϕY)η-(X)}+λ{u(X)η-(Y)-u(Y)η-(X)}θ(W)+2ϵ{g-(ϕX,Y)+u(X)θ(Y)}u(W)+{λg-(X,ϕY)+τ(X)u(Y)-λg-(Y,ϕX)-τ(Y)u(X)}η-(W)+λ{g-(X,ϕW)η-(Y)-g-(Y,ϕW)η-(X)}+{τ(X)η-(Y)-τ(Y)η-(X)}u(W)+ϵ{g-(W,ϕX)u(Y)-g-(W,ϕY)u(X)}=c-ϵ4{g-(V,X)g-(W,ϕ-Y)-g-(V,Y)g-(W,ϕ-X)+2g-(V,W)g-(X,ϕ-Y)}+τ(Y)B(X,W)-τ(X)B(Y,W),
for any X,Y,W∈Γ(TM-). Put X=ξ in (52); we get
(53)(ξ·λ){g-(Y,W)-∑α=1rϵαη-α(Y)η-α(W)}-λB(Y,W)-2λu(Y)η-(W)-λu(W)η-(Y)-2ϵu(Y)u(W)+τ(ξ)u(Y)η-(W)+2λu(Y)η-(W)+τ(ξ)u(W)η-(Y)-ϵu(W)u(Y)+λu(W)η-(Y)=c-ϵ4{u(Y)u(W)+2u(W)u(Y)}-τ(ξ)B(Y,W).
Put Y=W=U in (53) and then using g(U,U)=0, u(U)=1, η-α(U)=0; we obtain
(54)-λB(U,U)-3ϵ=3(c-ϵ)4-τ(ξ)B(U,U),
since B(U,U)=0, we get c=-3ϵ. Moreover, by putting Y=V and W=U in (52), we obtain
(55)ξ·λ+λτ(ξ)-λ2=0.
Finally, putting X=PX, Y=PY, and W=PW in (52) with c=-3ϵ and using that S(TM) is nondegenerate, we obtain
(56){PX·λ+λτ(PX)}(PY-∑α=1rη-α(PY)ζ-α)={PY·λ+λτ(PY)}(PX-∑α=1rη-α(PX)ζ-α).
Putting PX=ζ-α in (56), we get
(57){ζ-α·λ+λτ(ζ-α)}(PY-∑α=1rη-α(PY)ζ-α)=0.
By taking Y=V, we obtain
(58)ζ-α·λ+λτ(ζ-α)=0.
Since PX∈Γ(S(TM)) therefore using (19), we can write
(59)PX=u(PX)U+ν(PX)V+∑i=12n-4βiFi+∑α=1rϵαη-α(PX)ζ-α=PX′+∑α=1rϵαη-α(PX)ζ-α,
where {Fi}1≤i≤2n-4 is an orthogonal basis of D0, then using (58), we have
(60)PX·λ+λτ(PX)=(PX′+∑α=1rϵαη-α(PX)ζ-α)·λ+λτ(PX′+∑α=1rϵαη-α(PX)ζ-α)=PX′·λ+λτ(PX′),
which leads to get from (56)
(61){PX′·λ+λτ(PX′)}PY′={PY′·λ+λτ(PY′)}PX′.

Now, assume that there exists a vector field X0 on some neighborhood of M such that PX0′·λ+λτ(PX0′)≠0 at some point p in the neighborhood. Then from (61), it is clear that all the vectors of the fiber (S(TM)-{ζ-α})p are collinear with (PX0′)p. This contradicts dim(S(TM)-{ζ-α})>1. This implies the result.

From the previous mentioned theorem, we have the following corollary.

Corollary 7.

There exist no totally contact umbilical lightlike real hypersurfaces of indefinite 𝒮-space form M-(c)(c≠-3ϵ) with ζ-α∈TM.

Acknowledgment

The authors would like to thank the anonymous referee for his/her valuable suggestions that helped them to improve this paper.

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