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DEMONSTRATIO MATHEMATICA

Vol. XLIV No 2 2011

Pablo Alegre

SEMI-INVARIANT SUBMANIFOLDS

OF LORENTZIAN SASAKIAN MANIFOLDS

Abstract. In this paper we introduce the notion of semi-invariant submanifolds of

a Lorentzian almost contact manifold. We study their principal characteristics and the

particular cases in which the manifold is a Lorentzian Sasakian manifold or a Lorentzian

Sasakian space form.

1. Introduction

The study of CR-submanifolds of an Hermitian manifold was initiated

by A. Bejancu in [1]. They generalized both totally real and holomorphic

immersions. Given an almost Hermitian manifold, (f

M , J, g), a submanifold

Mis called CR-submanifold if there exists a diﬀerentiable distribution Don

Msuch it is holomorphic, and its complementary orthogonal distribution

D⊥is totally real (J(Dx)⊆ Dxand J(D⊥

x)⊆TxM⊥for all x∈M).

Since then, many authors have treated CR-submanifolds on diﬀerent en-

vironments and have ampliﬁed the deﬁnition to other decompositions of the

tangent bundle (semi-slant and almost semi invariant submanifolds). On

complex geometry, M. M. Tripathi deals with generalized complex space

forms in [13].

Later, the subject was considered for Riemannian manifolds with an al-

most contact structure. In this sense A. Benjacu and N. Papaghiuc study

semi-invariant submanifolds of a Sasakian manifold or a Sasakian space form

([2], [3], [11], [12]) and C.-L. Bejan, A. Cabras and P. Matzeu study them

on cosymplectic manifolds in [4] and [5].

2000 Mathematics Subject Classiﬁcation: 53C40, 53C50.

Key words and phrases: Lorentzian metric, contact and almost contact Lorentzian

manifolds, Lorentzian Sasakian, Lorentzian Sasakian space forms, invariant, anti-invariant

and semi-invariant submanifolds.

The author is partially supported by the PAI project (Junta de Andalucía, Spain,

2009).

10.1515/dema-2013-0307

392 P. Alegre

Recently, some authors have deﬁned similar concepts on a semi-deﬁned

metric environment. So have done Kalpana and G. Guha deﬁning semi-

invariant submanifolds for a Lorentzian para Sasakian manifold, [9], and H.

Gill and K. K. Dube for a trans Lorentzian para Sasakian manifold, [8].

Also these authors have studied generalized CR-submanifolds of an trans

hyperbolic Sasakian manifold.

Our purpose is deﬁning semi-invariant submanifolds of a Lorentzian man-

ifold endowed with an almost contact structure. This complete the study of

semi-invariant submanifolds of a Lorentzian manifold. We only specify for a

Sasakian manifold, but it could also be studied the case of a trans Sasakian

manifold.

In Section 2, we review basic formulas and deﬁnitions for almost con-

tact metric manifolds, which we shall use later. In Section 3, we deﬁne

semi-invariant submanifolds of a Lorentzian almost contact manifold. We

also present a way to build these submanifolds and present some examples.

In Section 4, we study the integrability of all the distributions involved in the

deﬁnition of a semi-invariant submanifold. We characterize semi-invariant

submanifolds of a Lorentzian Sasakian space form in Section 5. Finally,

Section 6 focus on totally umbilical and totally contact umbilical submani-

folds.

2. Preliminaries

An odd dimensional Riemannian manifold, (f

M2n+1, g), is called a Loren-

tzian almost contact manifold if it is endowed with a structure (φ, ξ, η, g),

where φis a (1,1) tensor, ξand ηa vector ﬁeld and a 1-form on f

M, respec-

tively, and gis a Lorentz metric, satisfying

(1) φ2X=−X+η(X)ξ, g(φX, φY ) = g(X, Y ) + η(X)η(Y),

η(ξ) = 1, η(X) = −g(X, ξ ),

for any vector ﬁelds X, Y in f

M. Let Φdenote the 2-form in f

Mgiven by

Φ(X, Y ) = g(X, φY ), if dη = Φ f

Mis called contact Lorentzian manifold. A

normal contact Lorentzian manifold is called Lorentzian Sasakian, [10], this

is a contact Lorentzian one verifying

(2) (e

∇Xφ)Y=−g(X, Y )ξ−η(Y)X.

In such a case, we have

(3) e

∇Xξ=φX.

Finally, a Lorentzian Sasakian space form is a Lorentzian Sasakian manifold

with constant φ-sectional curvature.

Semi-invariant submanifolds of Lorentzian Sasakian manifolds 393

From now on, let us consider a submanifold Mof a Lorentzian almost

contact metric manifold (f

M , φ, ξ, η, g), tangent to the structure vector ﬁeld ξ.

Put φX =T X +NX for any tangent vector ﬁeld X, where T X (resp.

NX) denotes the tangential (resp. normal) component of φX. Similarly,

φV =tV +nV for any normal vector ﬁeld Vwith tV tangent and nV

normal to M.

Given a submanifold Mof a Lorentzian almost contact manifold

(f

M , φ, ξ, η, g), we also use gfor the induced metric on M.

We denote by e

∇the Levi-Civita connection on f

Mand by ∇the induced

Levi-Civita connection on M. Thus, the Gauss and Weingarten formulas are

respectively given by

˜

∇XY=∇XY+h(X, Y ),(4)

˜

∇XV=−AVX+∇⊥

XV,(5)

for vector ﬁelds X, Y tangent to Mand a vector ﬁeld Vnormal to M, where

hdenotes the second fundamental form, ∇⊥the normal connection and AV

the shape operator in the direction of V. The second fundamental form and

the shape operator are related by

(6) g(h(X, Y ), V ) = g(AVX, Y ).

The submanifold Mis said to be totally geodesic if hvanishes identically.

Comparing tangential and normal components in (1), we obtain:

Lemma 2.1. For a submanifold of a Lorentzian almost contact manifold,

the following equations hold:

T2+tN =−I+η⊗ξ,(7)

NT +nN = 0,(8)

T t +tn = 0,(9)

n2+Nt =−I.(10)

We state the next two Lemmas whose proofs are straightforwardly de-

duced from (2), (4) and (5) and hence omitted.

Lemma 2.2. Let Mbe a submanifold of a Lorentzian Sasakian manifold

(f

M , φ, ξ, η, g). Then,

(∇XT)Y−th(X, Y )−AN Y X=−g(X, Y )ξ−η(Y)X,(11)

(∇XN)Y+h(X, T Y )−nh(X, Y ) = 0,(12)

for any tangent vector ﬁelds X, Y .

Lemma 2.3. Let Mbe a submanifold of a Lorentzian Sasakian manifold

(f

M , φ, ξ, η, g). Then,

394 P. Alegre

(∇Xt)V−AnV X+T AVX= 0,(13)

(∇⊥

Xn)V−h(X, tV ) + N AVX= 0,(14)

for any X∈T M and V∈T⊥M.

Similarly, comparing the tangential and the normal components of (3),

(4) and (5), it follows the next lemma.

Lemma 2.4. Let Mbe a submanifold of a Lorentzian Sasakian manifold.

Then,

∇Xξ=T X,(15)

h(X, ξ) = NX,(16)

∇ξξ= 0,(17)

h(ξ, ξ) = 0,(18)

AVξ=−tV,(19)

∇⊥

ξV=nV −[V, ξ ],(20)

for any X∈T M and V∈T⊥M.

We obtain the following result directly from Lemmas 2.2 and 2.3.

Lemma 2.5. Let Mbe a submanifold of a Lorentzian Sasakian manifold.

Then,

(∇XT)ξ=−T2X,(21)

(∇XN)ξ=−N T X,(22)

(∇ξT)X= 0,(23)

(∇ξN)X=−2N T X −2nNX,(24)

(∇ξt)V=−2tnV = 2T tV,(25)

(∇ξn)V= 0,(26)

for any X∈T M and V∈T⊥M.

3. Semi-invariant submanifolds, examples

In this section, we introduce the notion of semi-invariant submanifold of

a Lorentzian almost contact manifold. This deﬁnition generalizes the notions

of both invariant and anti-invariant submanifolds and it is the equivalent one

to semi-invariant submanifolds on the Riemannian setting. We study the

distributions involved and we characterize the anti-invariant case. Finally,

Theorem 3.6 provides us with a way to construct diﬀerent examples of semi-

invariant submanifolds in a Lorentzian almost contact manifold.

Semi-invariant submanifolds of Lorentzian Sasakian manifolds 395

A non degenerated submanifold, M, of a Lorentzian almost contact man-

ifold is called a semi-invariant submanifold if T M =D ⊕ D⊥⊕< ξ >, with

Dan invariant distribution and D⊥a anti-invariant distribution.

Note that ξis a timelike vector ﬁeld and all vector ﬁelds in D ⊕ D⊥are

spacelike.

A semi-invariant submanifold is also characterized by the decomposition

of the normal bundle.

Proposition 3.1. Let Mbe a submanifold of a Lorentzian almost contact

manifold (f

M , φ, ξ, η, g). Then, Mis a semi-invariant submanifold if and

only if T⊥M=D ⊕ D⊥; where Dis the maximal anti-invariant space in

T⊥Mand D⊥is its orthogonal complement on T⊥M.

Proof. If Mis a semi-invariant submanifold, the tangent space admits a

decomposition T M =D ⊕ D⊥⊕< ξ >. Let us consider D⊥=φD⊥⊆T⊥M

and Dthe orthogonal complement on T⊥M.

Firstly, given V∈D⊥, there exits X∈ D⊥such V=φX. Then,

φV =φ2X=−X+η(X)ξ=−X∈T M,

and therefore, D⊥is anti-invariant.

Secondly, let be W∈D, for all tangent vector ﬁeld X, we have NX ∈ D⊥

and then,

g(φW, X) = −g(W, φX ) = −g(W, NX) = 0,

so φW is normal. But also, given V∈D⊥

g(φW, V ) = −g(W, φV ) = 0.

Therefore φW ∈D, and Dis invariant.

Conversely, if T⊥M=D ⊕ D⊥with Dinvariant and D⊥anti-invariant,

let us call D⊥=φD⊥and Dthe orthogonal complement of D⊥⊕< ξ >

on T M . Proceeding as before, we can conclude that Dis invariant, D⊥is

anti-invariant, and therefore Mis a semi-invariant submanifold.

A semi-invariant submanifold is known to be invariant, anti-invariant and

proper if D⊥= 0,D= 0 and D 6= 0 6=D⊥respectively. And it is called

vertical proper if D 6= 0 6=φD⊥.

We can prove the next two propositions about the distributions involved

in the deﬁnition.

Proposition 3.2. For a semi-invariant submanifold of a Lorentzian al-

most contact manifold, the following equalities hold:

i) Ker(N) = D⊕ < ξ >,

ii) Ker(T) = D⊥⊕< ξ >,

396 P. Alegre

iii) TD=D,

iv) Ker(n) = ND⊥,

v) Ker(t) = D.

Proof. i) and ii) are directly deduced from the deﬁnition of semi-invariant

submanifold.

For iii), if X∈ D, by (8) N T X = 0, so TD ⊆ D. For the other inclusion,

by virtue of (7), T2X=−X, and then X=T(−T X)∈TD.

For iv), consider X∈ D⊥. From (8) nNX =−N T X = 0, so Ker(n)⊇

ND⊥. Conversely, let us put U∈Ker(n). From (9) and (10) it follows that

T tU =−tnU = 0 and U=−n2U−NtU =−NtU. From the ﬁrst equality,

tU ∈ D⊥and then the second one implies U∈ND⊥.

Lastly, from iv), as Ker(n) = D⊥, we deduce Ker(t)⊆ D. For the

other inclusion, notice that for every Vnormal to M,tV ∈ D⊥. Then

tnV =T tV = 0 and φtV =N tV =−V−n2V, so n2V=−Vand therefore

tV = 0.

Proposition 3.3. Let Mbe a semi-invariant submanifold of a Lorentzian

Sasakian manifold. Then, Dis even dimensional.

Proof. For any X∈ D,φX ∈ D so φX =T X , and therefore T2X=−X.

Then, given {X1,...,Xr}an orthonormal basis of D, let T Xi=Pr

j=1 aj

iXj

and A= (aj

i). Hence, det(A2) = det(−I) = (−1)rbut det(A2) = det(A)2

must be positive, so ris even.

Now we characterize the anti-invariant case:

Theorem 3.4. Let Mbe a submanifold of a Lorentzian Sasakian manifold.

Mis anti-invariant if and only if ∇T= 0.

Proof. If Mis anti-invariant, T= 0 and ∇T= 0. Conversely, given

Xx∈TxM, by (21) (∇XT)ξ=−T2X= 0, and since g(T X , T X) =

−g(T2X, X ) = 0, it follows that T X = 0.

The next theorem helps us to construct a number of examples of semi-

invariant submanifolds. They are a Lorentzian version of the ones oﬀered

in [6]. First we need the following lemma whose prove is straightforward

computation.

Lemma 3.5. Let (e

N2n, G, J)be an almost Hermitian manifold. Then

(e

N2n×R, φ, ξ, η, g)is a Lorentzian almost contact manifold with

φU, a ∂

∂t = (JU, 0), ξ =0,∂

∂t ,

ηU, a ∂

∂t =a, g U, a ∂

∂t ,V, b ∂

∂t =G(U, V )−ab.

Semi-invariant submanifolds of Lorentzian Sasakian manifolds 397

Theorem 3.6. Let Mbe a CR-submanifold of an almost Hermitian man-

ifold (f

M2n, G, J). Then, M×Ris a semi-invariant submanifold of (M2n×

R, φ, ξ, η, g).

Proof. Since Mis a CR-submanifold, the tangent space admits a decom-

position T M =D⊕D⊥with Dinvariant and D⊥anti-invariant. Then,

T(M×R) = D ⊕ D⊥⊕< ξ > with D={(U, 0)/U ∈D}and D⊥=

{(V, 0)/V ∈D⊥}. Let us now see how Dis invariant and D⊥is anti-invariant.

On the one hand, for any (U, V )∈ D

φ(U, 0) = (JU, 0) ∈ D

so Dis invariant. On the other hand, φ(V, 0) = (J V, 0) is normal to T(M×R)

so D⊥is anti-invariant. Hence, M×Ris a semi-invariant submanifold.

Moreover, we can provide some other direct examples, they are based

on semi-slant submanifolds of a Riemannian Sasakian manifold given on [6].

To do so, we consider on R2m+1 the following Lorentzian Sasakian structure

(φ0, ξ, η, g), given by

η=1

2dz −

m

X

i=1

yidxi, ξ = 2 ∂

∂z ,

g=−η⊗η+1

4

m

X

i=1

(dxi⊗dxi+dyi⊗dyi),

φ0m

X

i=1 Xi

∂

∂xi+Yi

∂

∂yi+Z∂

∂z =

m

X

i=1 Yi

∂

∂xi−Xi

∂

∂yi+

m

X

i=1

Yiyi∂

∂z ,

where {xi, yi, z},i= 1,...,m are the cartesian coordinates.

First, we present an example with odd dimD⊥:

Example 3.7. The equation x(u, v, w, s, t) = 2(u, 0, w, 0, v, 0,0, s, t), de-

ﬁnes a semi-invariant submanifold in R9with its Lorentzian Sasakian struc-

ture, (φ0, ξ, η, g). To prove this fact, we take the orthogonal basis

e1= 2 ∂

∂x1

+y1

∂

∂z , e2= 2 ∂

∂y1

, e3= 2 ∂

∂x3

+y3

∂

∂z ,

e4= 2 ∂

∂y4

, e5= 2 ∂

∂z =ξ,

and deﬁne the distributions D=he1, e2i,and D⊥=he3, e4i.It is clear that

T M =D ⊕ D⊥⊕ hξi.

We can also provide an example with even dimensional D⊥:

Example 3.8. The equation x(u, v, s, t) = 2(u, v, s, 0, t),deﬁnes a semi-in-

variant submanifold in R5with its Lorentzian Sasakian structure (φ0, ξ, η, g).

398 P. Alegre

In this case, T M =D ⊕ D⊥⊕ hξi, just taking

D=Span 2∂

∂x1

+y1

∂

∂z ,2∂

∂y1,

D⊥=Span 2∂

∂x2

+y2

∂

∂z .

4. Integrability conditions

In this section, we study the integrability of all the distributions involved

in the deﬁnition of semi-invariant submanifolds. We also deal with the local

structure of a semi-invariant submanifold of a Lorentzian Sasakian manifold.

Proposition 4.1. Let Mbe a semi-invariant submanifold of a Lorentzian

Sasakian manifold f

M. Then, D,D⊥and D ⊕ D⊥are ξ-parallel.

Proof. For any X∈ D and Y∈ D⊥

g(∇ξX, ξ) = ξg(X, ξ)−g(X, ∇ξξ) = 0,

g(∇ξX, Y ) = ξg(X, Y )−g(X, ∇ξY) = g(T2X, ∇ξY) = −g(T X, T ∇ξY)

=g(T X, ∇ξT Y ) = 0

so ∇ξX∈ D, that is Dis ξ-pararell.

Similarly, we can proceed for D⊥. Finally, if Dand D⊥are ξ-parallel,

D ⊕ D⊥also is.

Corollary 4.2. Let Mbe a semi-invariant submanifold of a Lorentzian

Sasakian manifold f

M, and K ∈ {D,D⊥,D ⊕ D⊥}. Then, for X∈ K,

[X, ξ]∈ K.

Proof. Given X∈ K, by virtue of the above proposition ∇ξX∈ K. More-

over, ∇Xξ= [X, ξ]− ∇ξXand by (3), ∇Xξ=φX so ∇Xξ=T X ∈ D

(0∈ D⊥, T X ∈ D ⊕ D⊥), if X∈ D (D⊥,D ⊕ D⊥respectively).

Then, if one of this distributions Kis integrable, K⊕ < ξ > will also be.

Now, for a Lorentzian contact metric manifold we have the following

results.

Lemma 4.3. Let Mbe a submanifold of a contact Lorentzian manifold.

Then, 2g(X, T Y ) = −η([X, Y ]) for all X, Y orthogonal to ξ.

Proof. For a Lorentzian contact manifold it holds that dη =−Φ. So

2g(X, T Y ) = 2g(X, φY ) = 2Φ(X, Y ) = −2dη(X, Y ) = −η[X, Y ].

Lemma 4.4. Let Mbe a semi-invariant submanifold of a contact Lorentzian

manifold. D⊥is integrable if and only if dΦ(X, Y , Z) = 0, for any Xtangent

to M,Y, Z ∈ D⊥.

Semi-invariant submanifolds of Lorentzian Sasakian manifolds 399

Proof. Consider Xtangent to Mand Y, Z ∈ D⊥. Then,

3dΦ(X, Y, Z) = g(φ[Y, Z ], X),

so dΦ(X, Y, Z) = 0 if and only if [Y, Z ]∈KerT=D⊥⊕hξi. This is equivalent

to

[Y, Z ] + η[Y, Z]ξ∈ D⊥,

but, using Lemma 4.3, η([Y, Z]) = −2g(X, T Y ) = 0.

Now we can state the following theorem:

Theorem 4.5. Let Mbe a semi-invariant submanifold of a contact Loren-

tzian manifold. Then, D⊥is always integrable.

Proof. If Mis a contact metric manifold, dΦ = d2η= 0 and so, the result

follows from Lemma 4.4.

Lemma 4.6. Let Mbe a semi-invariant submanifold of a contact Lorentzian

manifold. D⊥⊕< ξ > is integrable if and only if dΦ(X, Y, Z) = 0, for any

Xtangent to Mand Y, Z ∈ D⊥⊕ hξi.

Proof. Given X∈T M ,Y, Z ∈ D ⊕ D⊥, we have T Y =T Z = 0 and

3dΦ(X, Y, Z) = −g([X, Z], φX) = g(φ[Y, Z], X).

So [Y, Z ]is normal if and only dΦ(X, Y, Z ) = 0, for all Xtangent to M.

Again, from this lemma, for a contact metric manifold we deduce:

Theorem 4.7. Let Mbe a semi-invariant submanifold of a contact Loren-

tzian manifold. Then, D⊥⊕ hξiis always integrable.

On the other hand, we have

Theorem 4.8. Let Mbe a semi-invariant submanifold of a Lorentzian al-

most contact manifold with D 6= 0. Then Dand D⊕D⊥are never integrable.

Proof. Given K ∈ {D,D ⊕ D⊥}, and e

X∈ K such that Te

X6= 0,e

Xcan be

written as X+X⊥with X∈ D and X⊥∈ D⊥. Then,

Te

X=T X +T X ⊥=T X ∈TD=D ⊆ K.

But

η([X, T X ]) = −2g(T X, T X )6= 0,

because T X is an spacelike vector ﬁeld, so [X, T X ]is not normal to ξ, which

implies that [X, T X ]/∈ K, thus Kis not integrable.

Finally, we characterize the integrability of D ⊕ hξi.

Theorem 4.9. Let Mbe a semi-invariant submanifold of a Lorentzian

Sasakian manifold. Then, D⊕ < ξ > is integrable if and only if h(X , T Y )−

h(Y, T X ) = 0 for all X, Y ∈ D⊕ < ξ >.

400 P. Alegre

Proof. Given X, Y ∈ D ⊕ hξi,[X, Y ]belongs to D ⊕ hξiif and only if

N[X, Y ] = 0.

Using (12),

N[X, Y ] = N∇XY−N∇YX

=∇XNY +h(X, T Y )−nh(X, Y )

− ∇YNX −h(Y , T X) + nh(X, Y )

=h(X, T Y )−h(Y , T X ),

from which the proof follows.

Theorem 4.10. Let Mbe a semi-invariant submanifold of a Lorentzian

Sasakian manifold. Then, Mis locally the product M1×M2, where M1is a

leaf of D ⊕ hξiand M2is a leaf of D⊥if and only if h(X, T Y )∈ D, for all

Xtangent to M,Y∈ D⊥.

Proof. We shall prove that both D⊕ < ξ > and D⊥are involutive and their

leaves are totally geodesic immersed in M, so Mis locally the product of

these leaves.

For Y∈ D ⊕ hξi,Z∈ D⊥, by virtue of (2) and (4),

g(∇XY, Z ) = g(e

∇XY, Z ) = g(φe

∇XY, φZ )(27)

=g(e

∇XφY +g(X, Y )ξ−η(Y)X, N Z) = g(e

∇XT Y, N Z)

=g(h(X, T Y ), N Z).

So if X∈ D ⊕ hξi,∇XY∈ D ⊕ hξiif and only if h(X, T Y )∈ D. Then

D ⊕ hξiis involutive and its leaf is totally geodesic immersed in M.

Similarly, from (27), if X∈ D⊥, as g(∇XZ, Y ) = −g(Z, ∇XY) =

−g(h(X, T Y ), N Z), we have that ∇XY∈ D ⊕ hξiif and only if h(X, T Y )

∈ D. In this case, we obtain that D ⊕ hξiis also involutive and its leaf is

totally geodesic immersed in M.

5. Semi-invariant submanifolds of Lorentzian Sasakian space forms

The following theorems are the Lorentzian Sasakian space forms equiv-

alent to the ones in [13] for generalized complex space forms and [12] for

Sasakian space foms. For diﬀerent values of the φ-sectional curvature, we

characterize proper semi-invariant submanifolds of a contact Lorentzian

manifold with constant φ-sectional curvature via the Riemann curvature

over certain distributions.

In [10], T. Ikawa, gives the value of the Riemann curvature tensor of

a Lorentzian Sasakian manifold with constant φ-sectional curvature k:

Semi-invariant submanifolds of Lorentzian Sasakian manifolds 401

(28) e

R(X, Y )Z

=k−3

4{g(Y, Z )X−g(X, Z)Y}

+k+ 1

4{g(X, φZ)φY −g(Y, φZ)φX + 2g(X, φY )φZ}

+k+ 1

4{η(Y)η(Z)X−η(X)η(Z)Y+g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ}.

Theorem 5.1. Let Mbe a submanifold tangent to ξof a Lorentzian

Sasakian space form f

M(k)with k6=−1. Then, Mis a vertical proper

semi-invariant submanifold if and only if the maximal invariant subspaces

Dx=TxM∩φ(TxM),x∈M, deﬁne a non-trivial subbundle Dof T M such

that

e

R(D, D, D⊥, D⊥) = 0,

where D⊥denotes the non-trivial orthogonal complementary subbundle of D

in T M .

Proof. If Mis a semi-invariant submanifold, let X, Y ∈ D and Z, V ∈ D⊥.

From (28), R(X, Y, Z, W ) = 0.

Conversely, let Dxand D⊥

xbe on the hypothesis. For all X∈Dand

Z, W ∈D⊥,

e

R(X, φX, Z, W ) = −k+ 1

2g(φX, φX )g(φZ, W )

=−k+ 1

2g(X, X )g(φZ, W ) = 0.

So, as k6=−1and Xis space-like, g(φZ, W ) = 0 and φD⊥is orthogonal to

D⊥.

Since Dis an invariant distribution, g(X, φZ) = −g(φX, Z ) = 0, so

φD⊥is orthogonal to Dand also to ξ. Then, φD⊥⊆T M ⊥, that is D⊥is

anti-invariant and therefore Mis a semi-invariant submanifold.

Theorem 5.2. Let Mbe a submanifold tangent to ξof a Lorentzian

Sasakian space form f

M(k)with k6= 0. Then, Mis a proper semi-invariant

submanifold if and only if the maximal anti-invariant subspaces D⊥

xof TxM,

x∈M, deﬁne a non-trivial subbundle D⊥of T M such that

e

R(D, φD, D, D) = 0,

where Ddenotes the orthogonal complementary subbundle of D⊥⊕ hξiin

T M , and Ddenotes the orthogonal complementary of φD⊥in T⊥M.

Proof. Given V∈D,Z∈D⊥, and X, Y ∈D, it is just a simple computation

to check that R(X, φY, V, Z) = 0.

402 P. Alegre

Reciprocally, let X, Y , V be as above,

0 = e

R(X, φX, V , X) = −kg(X, X )g(φX, V ).

So, as k6= 0,g(φX, V ) = 0 for all X∈Dand N∈D, that is φD is

orthogonal to D. Since D⊥is anti-invariant, g(φX, Z ) = −g(X, φZ) = 0,

X∈D,Z∈D⊥, so φD is orthogonal to D⊥. And by hypothesis, Dis

orthogonal to ξ. Therefore, we deduce that φD ⊂T M , and then φD =D.

That is, Dis invariant and Mis a semi-invariant submanifold.

Also studying the relation between e

Rand the distributions on the normal

bundle we get some results.

Theorem 5.3. Let Mbe a submanifold tangent to ξof a Lorentzian

Sasakian space form f

M(k)with k6=−1. Then Mis a vertical proper

semi-invariant submanifold if and only if the maximal invariant subspaces

Dx=T⊥

xM∩φ(T⊥

xM),x∈M, of T⊥

xMdeﬁne a non-trivial subbundle D

of T⊥Msuch that

e

R(D, D, D⊥, D⊥) = 0,

where D⊥denotes the non-trivial orthogonal complementary subbundle of D

in T⊥M.

Proof. Given N, U ∈Dand V, W ∈D⊥, let us suppose that Mis a semi-

invariant submanifold. Then

R(N, U, V , W) = k+ 1

4(g(φU, V )g(φN , W ) + g(φN, V )g(φU, W )

−2g(φN, U )g(φV, W )) = 0,

because g(φU, V ) = g(φN , V ) = g(φV, W ) = 0.

Conversely, let Dx=T⊥

xM∩φ(T⊥

xM)deﬁne a non-trivial subbundle

satisfying the condition over R, and let N∈D, and V, W ∈D⊥be. Then,

0 = e

R(N, φN , V, W ) = −k+ 1

2g(N, N )g(φV, W ).

Since Dis not empty, there exits Na spacelike vector ﬁeld with g(N, N )6= 0,

and, as k6=−1, from the equation above we deduce that φD⊥is orthog-

onal to D⊥. We always have φD⊥orthogonal to D, because g(φV, N ) =

−g(V, φN ) = 0. So φD⊥⊂T M , that is D⊥is anti-invariant. Therefore,

T⊥M=D⊕D⊥where one is an invariant distribution and the other is an

anti-invariant one, and by virtue of Proposition 3.1, this implies that Mis

a semi-invariant submanifold.

Semi-invariant submanifolds of Lorentzian Sasakian manifolds 403

And ﬁnally,

Theorem 5.4. Let Mbe a submanifold tangent to ξof a Lorentzian

Sasakian space form f

M(k)with k6= 0. Then, Mis a proper semi-invariant

submanifold if and only if the maximal anti-invariant subspaces D⊥

xof T⊥

xM,

x∈M, deﬁne a non-trivial subbundle D⊥of T⊥Msuch that

e

R(D, φD, D, D) = 0,

where Ddenotes the non-trivial orthogonal complementary subbundle of

φD⊥⊕ hξiin T M , and Ddenotes the non-trivial orthogonal complemen-

tary of D⊥in T⊥M.

Proof. It is just a simple computation to prove the direct assertion. For the

converse, let us suppose that the maximal anti-invariant subspace D⊥and

its complementary Dsatisfy the equation above. Then,

0 = e

R(V, φV , X, V ) = −kg(V, V )g(φX, V )

for all X∈Dand V∈D, so φD ⊥D. Of course, φD is orthogonal to

Dand ξ. Finally, it is also orthogonal to φD⊥, because, given W∈D⊥,

we have g(φV , φW ) = g(V, W ) = 0.Thus, φD⊆D, and then invariant.

Therefore, T⊥M=D⊕D⊥, one is invariant and the other anti-invariant

and, by Proposition 3.1, Mis a semi-invariant submanifold.

6. Totally umbilical and totally geodesic submanifolds

In this last section, we ask ourselves about totally geodesic, totally con-

tact geodesic, totally umbilical and totally contact umbilical submanifolds

of a Lorentzian almost contact manifold.

Lemma 6.1. Let Mbe a submanifold of a Lorentzian Sasakian manifold,

and let Kbe a distribution on Mwith ξ∈ K. Then, if Mis K-umbilical, it

is also K-totally geodesic.

Proof. Let Mbe a K-umbilical submanifold, ξ∈ K. Then, h(X, Y ) =

g(X, Y )Kfor certain Knormal to M,X, Y ∈ K. In particular,

h(ξ, ξ) = g(ξ, ξ )K=−K,

but, from (16), h(ξ, ξ) = Nξ = 0, so K= 0. Then, h(X, Y ) = 0,X, Y ∈ K,

and it follows that Mis K-totally geodesic.

This implies the following:

Theorem 6.2. A totally umbilical submanifold of a Lorentzian Sasakian

manifold, tangent to ξ, is totally geodesic.

And for a totally geodesic submanifold we have:

404 P. Alegre

Theorem 6.3. A totally geodesic submanifold of a Lorentzian Sasakian

manifold, tangent to ξ, is invariant.

Proof. Let Mbe a totally geodesic submanifold, then h(X, ξ) = 0 for all X

tangent to T M . But by (16), N X =h(X, ξ) = 0 for all Xand therefore M

is invariant.

From these two theorems we deduce:

Corollary 6.4. Every totally umbilical submanifold of a Lorentzian Sasa-

kian manifold, tangent to ξ, is invariant.

So proper semi-invariant submanifolds which are totally geodesic or to-

tally umbilical do not exist.

For a contact Riemannian manifold it is usual to study totally contact

umbilical and totally contact geodesic submanifolds, [1], [14]. We now study

the equivalent cases for a contact Lorentzian manifold. A submanifold Mm

of an almost contact Lorentzian manifold is called totally contact umbilical if

there exits a normal vector ﬁeld Ksuch that the second fundamental form

of Mis given by

(29) h(X, Y ) = g(φX, φY )K+η(X)h(Y , ξ) + η(Y)h(X, ξ),

and it is called totally contact geodesic if K= 0, that is, the second funda-

mental form is given by

(30) h(X, Y ) = η(X)h(Y , ξ) + η(Y)h(X, ξ).

As h(ξ, ξ) = N ξ = 0 we see that K=1

mTr h=m+1

nH, where His the

mean curvature vector of M. So (29) and (30) imply the following Lemma.

Lemma 6.5. Let Mbe a totally contact umbilical submanifold, tangent to

the structure vector ﬁeld ξ, of a Lorentzian Sasakian manifold f

M. Then M

is totally contact geodesic if and only if Mis minimal.

Lemma 6.6. Let Mbe a proper semi-invariant submanifold of a Lorentzian

Sasakian manifold, tangent to ξ, then NT = 0 and T t = 0.

Proof. Firstly, let us point that both equalities are equivalent because from

(1) we get

g(N T X, V ) = g(X, T tV ),

for every X∈T M and V∈T⊥M. Secondly, as T M =D ⊕ D⊥⊕ hξi, it is

enough to prove this equalities for X∈ D ⊕ D⊥.

Let us consider πand π⊥the projections over Dand D⊥respectively.

Then the following holds

(31) π+π⊥=I, π2=π, π⊥2=π⊥and ππ⊥=π⊥π= 0.

Semi-invariant submanifolds of Lorentzian Sasakian manifolds 405

Moreover, as πX ∈ D and π⊥X∈ D⊥,NπX = 0 and T π ⊥X= 0. This

last equation, using the ﬁrst one of (31), implies

(32) T π =T.

Then applying πin (9), 0 = N T π +nNπ =N T π =N T , where we have

used (32).

Moreover, by (8) and (9), from this Lemma it is deduced that, for such

a submanifold, nN = 0 and tn = 0.

Theorem 6.7. Let Mbe a proper semi-invariant submanifold of a Loren-

tzian Sasakian manifold, tangent to ξand with dim(D⊥)≥2. If Mis totally

contact umbilical, then it is totally contact geodesic.

Proof. In order to prove that K= 0, ﬁrst we prove that tK = 0. From

Theorem 4.7, D⊥⊕ hξiis always integrable. Then, from (11) it can be

deduced that

(33) ANY X=ANX Y,

for every X, Y ∈ D⊥. From (29),

g(h(X, X ), φtK) = g(X, X )g(K, φtK).

But, from the above Lemma, T t = 0 so tK ∈ D⊥and therefore using (6)

and (29),

g(h(X, X ), φtK) = g(AN T K X, X ) = g(ANX tK, X )

=g(h(tK, X), N X) = g(tK, X)g(K, N X).

From both equations, it follows

(34) g(X, X)g(tK, tK ) = g(tK, X)2.

As dim(D⊥)≥2, we can choose X∈ D⊥with g(tK, X ) = 0, and then from

(34), as tK is spacelike, it is tK = 0. On the other hand using (13),

g((∇Xt)K, Y ) = g(AnK X, Y )−g(T AKX, Y ),

and as tK = 0,

−g(t∇⊥

XK, Y ) = g(h(X, Y ), nK ) + g(h(X, T Y ), K),

using (29) it follows

(35) g(∇⊥

XK, N Y ) = g(X, T Y )g(K, K).

Putting Y=T X in (35), and using NT = 0 from Lemma 6.6, the last

equation implies K= 0 which ﬁnishes the proof.

Theorem 6.8. A totally contact geodesic submanifold of a Lorentzian

Sasakian manifold, tangent to ξ, is invariant.

406 P. Alegre

Proof. Let Mbe a totally contact geodesic submanifold, by (29) and (16),

NX =h(X, ξ ) = (η(X)+ 1)h(ξ, ξ). But again, by (16), h(ξ, ξ) = Nξ = 0 for

all Xtangent to T M . Therefore N X = 0 for all X, and Mis invariant.

Acknowledgement. The author is grateful to the referee for his valu-

able suggestions.

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DEPARTAMENTO DE ECONOMÍA

MÉTODOS CUANTITATIVOS E HISTORIA ECONÓMICA

ÁREA DE ESTADÍSTICA E INVESTIGACIÓN OPERATIVA

UNIVERSIDAD PABLO DE OLAVIDE

CTRA. DE UTRERA, KM. 1

41013 – SEVILLA, SPAIN

E-mail: psalerue@upo.es

Received October 17, 2009.

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