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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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