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# Semi-invariant submanifolds of Lorentzian Sasakian manifolds

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## 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.
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
Dis totally real (J(Dx) Dxand J(D
x)TxMfor all xM).
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 = 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 )ξη(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+ηξ,(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)Yth(X, Y )AN Y X=g(X, Y )ξη(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)VAnV X+T AVX= 0,(13)
(
Xn)Vh(X, tV ) + N AVX= 0,(14)
for any XT M and VTM.
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 XT M and VTM.
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 XT M and VTM.
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 Da anti-invariant distribution.
Note that ξis a timelike vector ﬁeld and all vector ﬁelds in D Dare
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 TM=D D; where Dis the maximal anti-invariant space in
TMand Dis its orthogonal complement on TM.
Proof. If Mis a semi-invariant submanifold, the tangent space admits a
decomposition T M =D D< ξ >. Let us consider D=φDTM
and Dthe orthogonal complement on TM.
Firstly, given VD, there exits X Dsuch V=φX. Then,
φV =φ2X=X+η(X)ξ=XT M,
and therefore, Dis anti-invariant.
Secondly, let be WD, 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 VD
g(φW, V ) = g(W, φV ) = 0.
Therefore φW D, and Dis invariant.
Conversely, if TM=D Dwith Dinvariant and Danti-invariant,
let us call D=φDand Dthe orthogonal complement of D< ξ >
on T M . Proceeding as before, we can conclude that Dis invariant, Dis
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=Drespectively. 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 UKer(n). From (9) and (10) it follows that
T tU =tnU = 0 and U=n2UNtU =NtU. From the ﬁrst equality,
tU Dand then the second one implies UND.
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 =Vn2V, 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
XxTxM, 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 =DDwith Dinvariant and Danti-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 Dis 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 Dis 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=ηη+1
4
m
X
i=1
(dxidxi+dyidyi),
φ0m
X
i=1 Xi
∂xi+Yi
∂yi+Z
∂z =
m
X
i=1 Yi
∂xiXi
∂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,Dand D Dare 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 Dare ξ-parallel,
D Dalso 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 Drespectively).
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 =Φ. So
2g(X, T Y ) = 2g(X, φY ) = 2Φ(X, Y ) = 2(X, Y ) = η[X, Y ].
Lemma 4.4. Let Mbe a semi-invariant submanifold of a contact Lorentzian
manifold. Dis 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, Dis 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 XT 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 DDare never integrable.
Proof. Given K {D,D D}, and e
X K such that Te
X6= 0,e
Xcan be
written as X+Xwith 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 ] = NXYNYX
=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 Dif and only if h(X, T Y ) D, for all
Xtangent to M,Y D.
Proof. We shall prove that both D⊕ < ξ > and Dare 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 )ξη(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
=k3
4{g(Y, Z )Xg(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)Xη(X)η(Z)Y+g(X, Z)η(Y)ξg(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=TxMφ(TxM),xM, deﬁne a non-trivial subbundle Dof T M such
that
e
R(D, D, D, D) = 0,
where Ddenotes 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 XDand
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 φDis orthogonal to
D.
Since Dis an invariant distribution, g(X, φZ) = g(φX, Z ) = 0, so
φDis orthogonal to Dand also to ξ. Then, φDT M , that is Dis
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,
xM, deﬁne a non-trivial subbundle Dof 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 φDin TM.
Proof. Given VD,ZD, 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 XDand ND, that is φD is
orthogonal to D. Since Dis anti-invariant, g(φX, Z ) = g(X, φZ) = 0,
XD,ZD, 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
xMφ(T
xM),xM, of T
xMdeﬁne a non-trivial subbundle D
of TMsuch that
e
R(D, D, D, D) = 0,
where Ddenotes the non-trivial orthogonal complementary subbundle of D
in TM.
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
xMφ(T
xM)deﬁne a non-trivial subbundle
satisfying the condition over R, and let ND, and V, W Dbe. 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 φDis orthog-
onal to D. We always have φDorthogonal to D, because g(φV, N ) =
g(V, φN ) = 0. So φDT M , that is Dis anti-invariant. Therefore,
TM=DDwhere 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,
xM, deﬁne a non-trivial subbundle Dof TMsuch 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 Din TM.
Proof. It is just a simple computation to prove the direct assertion. For the
converse, let us suppose that the maximal anti-invariant subspace Dand
its complementary Dsatisfy the equation above. Then,
0 = e
R(V, φV , X, V ) = kg(V, V )g(φX, V )
for all XDand VD, so φD D. Of course, φD is orthogonal to
Dand ξ. Finally, it is also orthogonal to φD, because, given WD,
we have g(φV , φW ) = g(V, W ) = 0.Thus, φDD, and then invariant.
Therefore, TM=DD, 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 XT M and VTM. Secondly, as T M =D D hξi, it is
enough to prove this equalities for X D D.
Let us consider πand πthe projections over Dand Drespectively.
Then the following holds
(31) π+π=I, π2=π, π2=πand ππ=ππ= 0.
Semi-invariant submanifolds of Lorentzian Sasakian manifolds 405
Moreover, as πX D and πX 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 Dand 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 Dwith 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
CTRA. DE UTRERA, KM. 1
41013 SEVILLA, SPAIN
E-mail: psalerue@upo.es
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