Some lightlike submanifolds of almost complex manifolds with Norden metric

0.22 MB·English

by  Galia Nakova

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1
2
1
0
2

n
a SOME LIGHTLIKE SUBMANIFOLDS OF ALMOST
J
COMPLEX MANIFOLDS WITH NORDEN METRIC

7
1
GaliaNakova

] Universityof VelikoTarnovo ”St.Cyril and St. Metodius”
G Faculty of Mathematics and Informatics,
D T. Tarnovski 2 str.,
5000 VelikoTarnovo, Bulgaria
.
h E-mail:[email protected]
t
a
m InthispaperwestudysubmanifoldsofanalmostcomplexmanifoldwithNor-
den metric which are non-degenerate with respect to the one Norden metric
[
and lightlike with respect to the other Norden metric on the manifold. Rela-

1 tionsbetweentheinducedgeometricobjectsofsomeofthesesubmanifoldsare
v given.Examplesoftheconsideredsubmanifoldsareconstructed.
0 MSC:53B25,53C50, 53B50,53C42,53C15
0
Keywords:Lightlikesubmanifolds;Almostcomplexmanifolds;Nordenmetric;
6
3
. Introduction
1
0 The general theory of lightlike submanifolds has been developed in1 by K.
2
Duggal and A. Bejancu. The geometry of Cauchy-Riemann (CR) lightlike
1
: submanifoldsofindefiniteKaehlermanifoldswaspresentedin1 ,too.Many
v
new types of lightlike submanifolds of indefinite Kaehler manifolds, indef-
i
X
inite Sasakian and indefinite quaternion Kaehler manifolds are introduced
r in2 by K.DuggalandB.Sahin.In1 and2 ,differentapplicationsoflightlike
a
geometry in the mathematical physics are given. However, lightlike sub-
manifolds of almost complex manifolds with Norden metric (or B-metric)
havenotbeenconsideredyet.Thestudyofsuchsubmanifoldsisinteresting
becausethere existsa difference betweenthe geometryofa2n-dimensional
indefinitealmostHermitianmanifoldandthegeometryofa2n-dimensional
almost complex manifold with Norden metric. The difference arises due to
thefactthatinthefirstcase,thealmostcomplexstructureJ isanisometry
with respect to the semi-Riemannian metric g of index 2q(0 < 2q < 2n)
and in the second case, J is an anti-isometry with respect to the metric
g, which is necessarily of signature (n,n). This property of the pair (J,g)
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of an almost complex manifold with Norden metric M allows to define the
tensorfield g onM by g(X,Y)=g(JX,Y), whichis aNordenmetric,too.
LetM bearealm-dimensionalsubmanifoldofanalmostcomplexman-
e e
ifold witn Norden metric (M,J,g,g). The geometry of M depends on the
behaviour of the tangent bundle of M with respect to the action of the
e
almost complex structure J and the induced metric on M. Due to that
there exist two Norden metrics g and g of M we can consider two induced
metrics g and g onM by g and g, respectively.For a submanifold M of M
e
three cases with respect to the induced metrics g and g on M are possible:
e e
M is non-degenerate with respect to both g and g; M is degenerate with
e
respect to both g and g; M is non-degenerate with respect to g (resp. g)
e
and degenerate with respect to g (resp. g).
e e
In this paper we consider mainly submanifolds of the third type. In
e
Sections 1 and 2 we recall some preliminaries about lightlike submanifolds
of semi-Riemannian manifolds and almost complex manifolds with Norden
metric, respectively. The main results of the paper are given in Section
3. We prove that a necessary and sufficient condition for the submanifold
(M,g) of M to be a CR-submanifold is (M,g) to be a Radical transversal
lightlike submanifold of M. We obtain also that the submanifold (M,g) of
e
M isgeneric,totallyrealorLagrangianifandonlyif(M,g)isacoisotropic,
anisotropicoratotallylightlikesubmanifoldofM,respectively.InSection
e
4, in the case of totally real (M,g) and isotropic (M,g), relations between
the induced geometric objects are found. The structure equations of these
e
submanifolds of a Kaehler manifold with Norden metric are obtained. In
the last section we consider known matrix Lie subgroups of GL(2;C) as
examples of the submanifolds introduced in Section 3.
1. Lightlike submanifolds of semi-Riemannian manifolds
Follow1 we give some basic notions and formulas for lightlike submanifolds
of semi-Riemannian manifolds.
Let (M,g) be a real (m +n)-dimensional semi-Riemannian manifold
(m > 1, n 1), i.e. g is a semi-Riemannian metric of constant index
≥
q 1,...,m+n 1 and M be a submanifold of M of codimension n.
∈ { − }
Denote by g the induced tensor field on M of g and suppose that rankg =
const on M. If rankg = m, i.e. g is non-degenerate on the tangent bundle
TM of M, then M is called a non-degenerate submanifold of M. In the
case rankg < m, i.e. g is degenerate on TM, then M is called a lightlike
submanifold of M. We will note an orthogonaldirect sum by and a non-
⊥
orthogonaldirectsumby .ThetangentspaceT M andthe normalspace
x
⊕
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T M⊥ of a non-degenerate submanifold M of M are non-degenerate and
x
they are complementary orthogonal vector subspaces of T M. However, if
x
M is a lightlike submanifold of M, both T M and T M⊥ are degenerate
x x
orthogonal, but non-complementary subspaces of T M and there exists a
x
subspace RadT M =T M T M⊥ =RadT M⊥, where
x x x x
∩
RadT M = ξ T M :g(ξ ,X)=0, X T M .
x x x x x
{ ∈ ∀ ∈ }
The dimension of RadT M depends on x M. The submanifold M of M
x
∈
is said to be an r-lightlike (an r-degenerate, an r-null) submanifold if the
mapping
RadTM :x M RadT M,
x
∈ −→
defines a smooth distribution on M of rank r > 0 (it means
dim(RadT M) = r for all x M). RadTM is called the Radical (light-
x
∈
like, null) distribution on M.
Let S(TM) be a complementary distribution of RadTM in TM and
S(TM⊥) be a complementary vector bundle of RadTM in TM⊥. Both
S(TM)andS(TM⊥)arenon-degeneratewithrespecttogandthefollowing
decompositions are valid
TM =RadTM S(TM), TM⊥ =RadTM S(TM⊥). (1)
⊥ ⊥
ThedistributionS(TM)andthevectorbundleS(TM⊥)arecalledascreen
distribution and a screen transversal vector bundle of M, respectively. Al-
thoughS(TM)isnotunique,itiscanonicallyisomorphictothefactorvec-
tor bundle TM/RadTM.As S(TM)is a non-degeneratevector subbundle
of TM we have
TM =S(TM) S(TM)⊥,
⊥
where S(TM)⊥ is the complementary orthogonalvector bundle of S(TM)
in TM. As S(TM⊥) is a vector subbundle of S(TM)⊥ and since both are
non-degenerate we have the following orthogonaldirect decomposition
S(TM)⊥ =S(TM⊥) S(TM⊥)⊥.
⊥
Let tr(TM)andltr(TM)be complementary (but neverorthogonal)vector
bundles to TM in TM and to RadTM in S(TM⊥)⊥, respectively. Then
we have
tr(TM)=ltr(TM) S(TM⊥);
⊥ (2)
TM =TM tr(TM)=S(TM) S(TM⊥) (RadTM ltr(TM)).
⊕ ⊥ ⊥ ⊕
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The vector bundle ltr(TM)is called a lightlike transversal vector bundle of
M and tr(TM) is called a transversal vector bundle of M. There exists a
local quasi-orthonormalbasis1of M along M:
ξ ,N ,X ,W , i 1,...,r , a r+1,...,m , α r+1,...,n ,
i i a α
{ } ∈{ } ∈{ } ∈{ }
where ξ and N are lightlike basises of RadTM and ltr(TM) , re-
i i |U |U
{ } { }
spectively; X and W are basises of S(TM) and S(TM⊥) .
a α |U |U
{ } { }
The following possible four cases with respect to the dimension m and
codimension n of M and rank r of RadTM are studied:
(1) r-lightlike, if 0<r <min(m,n);
(2) coisotropic, if 1 r =n<m, S(TM⊥)= 0 ;
≤ { }
(3) isotropic, if 1<r =m<n, S(TM)= 0 ;
{ }
(4) totally lightlike, if 1<r =m=n, S(TM)= 0 =S(TM⊥).
{ }
2. Almost complex manifolds with Norden metric
Let (M,J,g) be a 2n-dimensional almost complex manifold with Norden
metric3 ,i.e. J is analmostcomplex structureandg is ametric onM such
that:
2
J X = X, g(JX,JY)= g(X,Y) (3)
− −
for arbitrary differentiable vector fields X,Y on M.
The tensor field g of type (0,2) on M defined by
e g(X,Y)=g(JX,Y) (4)
is a Norden metric on M,etoo. Both metrics g and g are necessarily of
signature (n,n). The metric g is saidto be anassociated metric ofM. The
e
Levi-Civitaconnectionofgisdenotedby .ThetensorfieldF oftype(0,3)
e ∇
on M is defined by F(X,Y,Z)= g(( J)Y,Z). Let be the Levi-Civita
X
∇ ∇
connection of g. Then
e
e Φ(X,Y)= Y Y (5)
X X
∇ −∇
isatensorfieldoftype(1,2)onM.Seince and aretorsionfreewehave
∇ ∇
Φ(X,Y)=Φ(Y,X). A classification of the almost complex manifolds with
e
Norden metric with respect to the tensor F is given in3 and eight classes
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areobtained.In4 theseclassesarecharacterizedbyconditionsforthetensor
Φ, too.
Analogously, as for an almost Hermitian manifold, for an almost com-
plex manifold with Norden metric (M,J,g,g) the following important
classes of non-degenerate submanifolds of M with respect to the induced
e
metric g (resp. g) on the submanifold can be considered:
(a) M is callede a holomorphic (or an invariant) submanifold of M if
J(T M) = T M, x M. The dimension m of an invariant submani-
x x
∀ ∈
fold M of M is necessarily even;
(b) M is called a totally real (or an anti-invariant) submanifold of M if
J(T M) T M⊥, x M. In this case dimM =m n;
x x
⊆ ∀ ∈ ≤
(c) A totally real submanifold M is called Lagrangian if dimM =m=n;
(d) M is called a CR-submanifold of M if M is endowed with two comple-
mentary orthogonal distributions D and D⊥ satisfying the conditions:
J(D )=D and J(D⊥) T M⊥, x M;
x x x ⊂ x ∀ ∈
(e) The CR-submanifold M of M is called a generic submanifold if
dimD⊥ =codimM =2n m.
−
The CR-submanifold M of M is called non-trivial (proper) if dimD > 0
and dimD⊥ > 0. The holomorphic and the totally real submanifolds are
particularcasesoftheclassofCR-submanifolds.Holomorphicsubmanifolds
of almost complex manifolds with Norden metric were studied by K. Grib-
achev. In5 ,6 and7 hypersurfaces of Kaehler manifolds with Norden metric
were considered.
3. Submanifolds of an almost complex manifold with
Norden metric which are non-degenerate with respect to
the one Norden metric and lightlike with respect to the
other Norden metric
Let(M,J,g,g)bea2n-dimensionalalmostcomplexmanifoldwithNorden
metric and M be an m-dimensional submanifold immersed by ϕ in M.
e
For simplicity we identify for each x M the tangent space T M with
x
∈
ϕ (T M) T M. Let g and g be two metrics of M. We assume that
x ϕ(x)
∗ ⊂
the immersionϕ is anisometrywith respectto both metrics g andg onM
e
and we identify the metrics g and g on M with the induced metrics on the
e
subspaceϕ (T M)ofg andg,respectively.Hence,foranyx M wehave
x
∗ e ∈
g(X ,Y )=g(X ,Y ), eg(X ,Y )=g(X ,Y ), X ,Y T M.
x x x x x x x x x x x
∀ ∈
e e
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We note that TM = T M is the tangent bundle of both submanifolds
x
x∈SM
(M,g) and (M,g) of M. We will denote: the normal bundle of (M,g) and
e
(M,g) by TM⊥ and TM⊥, respectively; an orthogonal direct sum with
e
respect to g (resp. g) by (resp. ) and a non-orthogonal direct sum by
e ⊥ ⊥
(resp. ).
⊕ ⊕ e e
In this section we give an answer to the question what type is the sub-
e
manifold (M,g) of M if (M,g) belongs to one of the basic classes of non-
degenerate submanifolds.
e
Lemma 3.1. Let (V,J,g,g) be a 2n-dimensional almost complex vector
spacewithNordenmetricandW bea2p-dimensionalholomorphic subspace
e
of V. Then for the induced metrics g and g on W of g and g respectively,
we have
e e
1) g is non-degenerate iff g is non-degenerate;
2) g is degenerate iff g is degenerate.
e
e
Proof. 1)Let g be non-degenerate.We assume that g is degenerate.Then
there exists ξ W, ξ = 0 such that g(ξ,X) = 0, X W. As W is a
∈ 6 ∀e ∈
holomorphic subspace of V and KerJ = 0 it follows that there exists
e { }
Jξ W,Jξ = 0 such that for X W we have g(Jξ,X) = g(Jξ,X) =
∈ 6 ∀ ∈
g(ξ,X) = g(ξ,X) = 0. So, we obtain that g is degenerate, which is a
contradiction. Analogously, we can check that if g is non-degenerate, then
e e
g is non-degenerate, too. The truth of 2) follows from 1).
e
Theorem 3.1. Let (M,J,g,g) be a 2n-dimensional almost complex man-
ifold with Norden metric and M be a 2p-dimensional submanifold of M.
e
The submanifold (M,g) is holomorphic iff the submanifold (M,g) is holo-
morphic.
e
Proof. The proof of the theorem follows from assertion 1) of Lemma 3.1
by replacing the space V and the subspace W by the tangent bundle TM
of M and the tangent bundle TM of the submanifolds (M,g) and (M,g),
respectively.
e
Radical transversal lightlike submanifolds of indefinite Kaehler manifolds
areintroducedin8 by Sahin. Further,we showthat suchsubmanifolds nat-
urallyariseonanalmostcomplexmanifoldwithNordenmetricM andthey
are related with CR-submanifolds of M. First, analogously as in8 we give
the following
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Definition 3.1. Let (M,g,S(TM),S(TM⊥)) be a lightlike submanifold
of an almost complex manifold with Norden metric (M,J,g,g). We say
that M is a Radical transversal lightlike submanifold of M if the following
e
conditions are satisfied:
J(RadTM)=ltr(TM), (6)
J(S(TM))=S(TM). (7)
Moreover, we say that a Radical transversal lightlike submanifold M is
proper if S(TM)=0.
6
It is important to note:
(1) Taking into account that for an isotropic and a totally lightlike sub-
manifold S(TM)=0 and Definition 3.1,it is clear that there exists no
proper Radical transversal isotropic and totally lightlike submanifold
of M.
(2) Contrary to the case when M is a Radical transversal lightlike sub-
manifold of an indefinite Kaehler manifold 8 , for an almost com-
plex manifold with Nordenmetric (M,J,g,g), M canbe an1-lightlike
Radical transversal lightlike submanifold of M. Indeed, let us suppose
e
that (M,g) is an 1-lightlike Radical transversal lightlike submanifold
of (M,J,g,g). Then there exist basises ξ and N of RadTM and
{ } { }
ltr(TM) respectively, such that g(ξ,N) = 1. On the other hand, (6)
e
implies that Jξ = αN Γ(ltr(TM)), α F(ltr(TM)). Thus, for the
∈ ∈
function α we obtain α=g(ξ,Jξ), which is not zero.
Proposition 3.1. Let (M,g)be aRadical transversal lightlike submanifold
of an almost complex manifold with Norden metric (M,J,g,g). Then the
distribution S(TM⊥) is holomorphic with respect to J.
e
Proof. As the tangent bundle TM and the transversal vector bundle
tr(TM) of (M,g) are complementary (but not orthogonal with respect to
g)vectorsubbundles ofTM,foranyX Γ(TM)andanyV Γ(tr(TM)),
∈ ∈
we have
JX =TX+FX; JV =tV +fV, (8)
where TX,tV belong to Γ(TM) and FX,fV belong to Γ(tr(TM)). Then
T andf areendomorphismsonTM andtr(TM),respectively;F andt are
transversalbundle-valued1-formonTM andtangentbundle-valued1-form
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on tr(TM), respectively. According to the decompositions (1), (2), we can
write any X Γ(TM) and any V Γ(tr(TM)) in the following manner
∈ ∈
X =PX +QX; V =LV +SV, (9)
where P and Q are the projection morphisms of TM on S(TM) and
RadTM, respectively; L and S are the projection morphisms of tr(TM)
onltr(TM)andS(TM⊥), repectively.So,for anyW ΓS(TM⊥) wehave
∈
JW =tW +fW =PtW +QtW +LfW +SfW. (10)
Now, if X ΓS(TM), ξ Γ(RadTM), N Γ(ltr(TM)), using (2), (6),
∈ ∈ ∈
(7), (10) and J(ltr(TM))=RadTM, we compute
0=g(W,JX)=g(JW,X)=g(PtW,X),
0=g(W,Jξ)=g(JW,ξ)=g(LfW,ξ), (11)
0=g(W,JN)=g(JW,N)=g(QtW,N).
Since g is non-degenerate on S(TM) and RadTM ltr(TM), from (11) it
⊕
follows that PtW = LfW = QtW = 0. Then (10) becomes JW = SfW,
which means that S(TM⊥) is holomorphic.
Theorem 3.2. Let (M,J,g,g) be a 2n-dimensional almost complex man-
ifold with Norden metric and M be an m-dimensional submanifold of M.
e
The submanifold (M,g) is a CR-submanifold with an r-dimensional totally
real distribution D⊥ iff (M,g) is an r-lightlike Radical transversal lightlike
submanifold.
e
Proof. As far as we know, CR-submanifolds of almost complex manifolds
with Norden metric are not studied. Therefore, first we give some prelimi-
naries about them.
Let(M,g)beaCR-submanifoldofM andassumethatitisnotgeneric.
Hence, (M,g) is endowed with two complementary orthogonal with re-
spect to g distributions D(dimD = 2p) and D⊥(dimD⊥ = r : 1 r <
≤
min(m,2n m)) such that JD = D, JD⊥ TM⊥. Following 9 , for any
− ⊂
X Γ(TM) we have
∈
JX =TX+FX, (12)
where TX is the tangentialpart of JX and FX is the normalpart of JX.
Then T is anendomorphismonthe tangentbundle TM andF is a normal
bundle-valued 1-form on TM. Similarly, for any V Γ(TM⊥) we have
∈
JV =tV +fV, (13)
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where tV is the tangential part of JV and fV is the normal part of JV.
Thenf isanendomorphismonthe normalbundle TM⊥ andt isatangent
bundle-valued 1-form on TM⊥. If we denote by P and Q the projection
morphisms of TM on D and D⊥ respectively, for any X Γ(TM) we can
∈
write
X =PX +QX, (14)
wherePX Γ(D)andQX Γ(D⊥).Using(12)and(13),for Y Γ(TM)
∈ ∈ ∈
and U Γ(TM⊥) we compute g(JX,Y) = g(TX,Y) and g(JV,U) =
∈
g(fV,U).SincethealmostcomplexstructureJ isananti-isometrywithre-
specttotheNordenmetricsgandg,weobtainthatT andf areself-adjoint
operators on TM and TM⊥ with respect to both metrics. We note that T
e
andf areskewself-adjoint 9 whenM isanalmostHermitianmanifold.We
also find g(FX,V) = g(X,tV). Moreover, applying J to the equality (14)
we obtain
JX =JPX +JQX, (15)
where JPX Γ(D) and JQX Γ(JD⊥). From (12) and (15) it follows
∈ ∈
thatT =JP andF =JQ.Due tothe factthatD⊥ isnon-degeneratewith
respect to g, there exists an orthonormalbasis ξ ,...,ξ of D⊥, i.e.
1 r
{ }
g(ξ ,ξ )=ǫ , ǫ = 1; g(ξ ,ξ )=0, i=j; i,j =1,2,...,r. (16)
i i i i i j
± 6
Since KerJ = 0 , Jξ ,...,Jξ is a basis of JD⊥. Moreover, it is an
1 r
{ } { }
orthonormalbasis such that
g(Jξ ,Jξ )= ǫ ; g(Jξ ,Jξ )=0, i=j; i,j =1,2,...,r. (17)
i i i i j
− 6
Consequently,JD⊥ isanr-dimensionalnon-degeneratesubbundleofTM⊥
with respect to g and we put
TM⊥ =JD⊥ (JD⊥)⊥, (18)
⊥
where (JD⊥)⊥ is the complementary orthogonalvectorsubbundle of JD⊥
in TM⊥. We denote by P and P the projection morphisms of TM⊥ on
1 2
JD⊥ and (JD⊥)⊥, respectively. Then for any V Γ(TM⊥) we put
∈
V =P V +P V, (19)
1 2
where P V Γ(JD⊥) and P V Γ((JD⊥)⊥). Now, we will show that
1 2
∈ ∈
the subbundle (JD⊥)⊥ is holomorphic with respect to J. Take W
∈
Γ((JD⊥)⊥), X Γ(TM) and according to (15) we compute g(JW,X) =
∈
g(W,JX) = 0, which means that JW Γ(TM⊥). On the other hand,
∈
for each N Γ(JD⊥) we have JN Γ(D⊥), which implies g(JW,N) =
∈ ∈
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10
g(W,JN) = 0, i.e. JW Γ((JD⊥)⊥). We continue by proving that the
e ∈
normalbundle TM⊥ ofthe submanifold(M,g)coincides with the subbun-
dle D⊥ (JD⊥)⊥ of TM. First, let us suppose that Y Γ(D⊥ (JD⊥)⊥).
⊥ e ∈ ⊥
Thenfor X Γ(TM)using(15)wehaveg(X,Y)=g(JX,Y)=0.Hence,
∀ e ∈
Y Γ(TM⊥), i.e. the following relation holds
∈ e
e
D⊥ (JD⊥)⊥ TM⊥. (20)
⊥ ⊆
e
Now, let Y Γ(TM⊥). Then Y Γ(TM) and g(X,Y) = 0 for
∈ ∈
X Γ(TM). The last is equivalent to g(X,JY) = 0 which shows that
∀ ∈ e
JY Γ(TM⊥). Since (JD⊥)⊥ is holomorphic with respect to J, apply-
∈
ing J to (18) we obtain J(TM⊥) = D⊥ (JD⊥)⊥. So, we conclude that
⊥
Y Γ(D⊥ (JD⊥)⊥) and we have
∈ ⊥
e
TM⊥ D⊥ (JD⊥)⊥. (21)
⊆ ⊥
e
From(20)and(21)itfollowsthatTM⊥ =D⊥ (JD⊥)⊥.Further,forX
⊥ ∈
Γ(D), ξ Γ(D⊥) and W Γ((JD⊥)⊥), we get g(ξ,X) = g(Jξ,X) = 0,
∈ ∈
g(ξ,W)=g(Jξ,W)=0.ThelasttwoequalitiesshowthatD⊥isorthogonal
e
toDand(JD⊥)⊥ withrespecttog.Thenthefollowingdecompositionsare
e
valid
e
TM =D D⊥, (22)
⊥
e e
TM⊥ =D⊥ (JD⊥)⊥. (23)
⊥
From (22) and (23) it is clear that theesmooth distribution D⊥ on (M,g)
of rankr > 0 is an intersection of the tangent bundle TM and the normal
e e
bundle TM⊥ of (M,g). Hence, (M,g) is an r-lightlike submanifold of M
with Radical distribution RadTM which coincides with D⊥. According
e e
to (22) and (23), the distribution D and the vector bundle (JD⊥)⊥ are a
e
screendistributionS(TM)andascreentransversalvectorbundleS(TM⊥)
of(M,g),respectively.Thebasis ξ ,...,ξ ofD⊥ satisfying(16)isabasis
1 r
{ }
ofRadTM andconsequentlywithrespecttog wehaveg(ξ ,ξ )=0forany
i j
e
i,j = 1,...,r. We put N = ǫ Jξ , i = 1,...,r. Then N ,...,N is a
i i i 1 r
− e e { }
basis of JD⊥ such that
g(N ,N )= ǫ ǫ g(Jξ ,ξ )=0, i,j =1,...,r, (24)
i j i j i j
−
e
g(N ,ξ )=ǫ g(ξ ,ξ )=1; g(N ,ξ )=ǫ g(ξ ,ξ )=0; i,j =1,...,r. (25)
i i i i i i j i i j
Freom (24) it follows that JDe⊥ is a lightlike vector bundle with respect to
g. The equalities (25) show that JD⊥ is not orthogonal to RadTM with
e