(Para-)Hermitian and (para-)Kähler Submanifolds of a para-quaternionic Kähler manifold

0.29 MB·English

by  Massimo Vaccaro

(PARA-)HERMITIAN AND (PARA-)KA¨HLER SUBMANIFOLDS
OF A PARA-QUATERNIONIC KA¨HLER MANIFOLD
2
1 MASSIMOVACCARO
0
2

n Abstract: On a para-quaternionic K¨ahler manifold (M4n,Q,g), which is first
a of all a pseudo-Riemannian manifold, a natural definition of (almost) K¨ahler and
J (almost) para-Ka¨hler submanifold (M2m,J,g) can be gifven wheere J = J1|M is
9
a (para-)complex structure on M which is the restriction of a section J of the
1 1
para-quaternionic bundle Q. In this paper, we extend to such a submanifold M

] mostoftheresultsprovedbyAlekseevskyandMarchiafava,2001,whereHermitian
G
and K¨ahler submanifolds of a quaternionic K¨ahler manifold have been studied.
D Conditions for the integrability of an almost (para-)Hermitian structure on M
. are given. Assuming that the scalar curvature of M is non zero, we show that
h
t any almost (para-)K¨ahler submanifold is (para-)K¨ahler and moreover that M is
a f
(para-)K¨ahler iff it is totally (para-)complex. Considering totally (para-)complex
m
submanifolds of maximal dimension 2n, we identify the second fundamental form
[ h of M with a tensor C = J ◦h ∈ TM ⊗S2T∗M where J ∈ Q is a compatible
2 2

1 para-complex structure anticommuting with J1. This tensor, at any point x∈M,
v belongs to the first prolongation S(1) of the space S ⊂ EndT M of symmetric
J J x
9
endomorphisms anticommuting with J. When M4n is a symmetric manifold the
9
9 condition for a (para-)K¨ahler submanifold M2n to be locally symmetric is given.
f
3 In the case when M is a para-quaternionicspace form, it is shown,by using Gauss
1. and Ricci equations,that a (para-)K¨ahlersubmanifold M2n is curvature invariant.
f
0 MoreoveritisalocallysymmetricHermitiansubmanifoldifftheu(n)-valued2-form
2 [C,C] is parallel. Finally a characterization of parallel K¨ahler and para-Ka¨hler
1
submanifold of maximal dimension is given.
:
v
i 1. Introduction
X
r A pseudo-Riemannian manifold (M4n,g) with the holonomy group contained
a in Sp (R) · Sp (R) is called a para-quaternionic K¨ahler manifold. This means
1 n
that there exists a 3-dimensionalparallel subbundle Q⊂EndTM of the bundle of
endomorphismswhichislocallygeneratedbythreeskew-symmetricanticommuting
endomorphisms I,J,K satisfying the following para-quaternionic relations
−I2 =J2 =K2 =Id, IJ =−JI =K.
The subbundle Q ⊂ End(TM) is called a para-quaternionic structure. Any para-
quaternionic K¨ahler manifold is an Einstein manifold [3].
Date:January20,2012.
2000 Mathematics Subject Classification. 53C40,53A35,53C15.
Key words and phrases. para-quaternionic K¨ahler manifold , (almost) Hermitian, (almost)
K¨ahlersubmanifold,(almost)para-Hermitian,(almost)para-Ka¨hlersubmanifold.
Work done under the programs of GNSAGA-INDAM of C.N.R. and PRIN07 ”Riemannian
metricsanddifferentiablestructures”ofMIUR(italy).
1
2 MASSIMOVACCARO
Let ǫ = ±1; a submanifold (M2m,Jǫ = Jǫ| ,g) of the para-quaternionic
TM
K¨ahler manifold (M4n,Q,g), where M ⊂M is a submanifold, the induced metric
g = g| is non-degenerate, and Jǫ is a section of the bundle Q →M such that
JǫTMM=TM, (Jǫf)2 =ǫIde, is called an almfost ǫ-Hermitian su|bMmanifold.
Aenalmostǫ-Hermitian submanifold(M2m,Jǫ,g)ofapara-quaternionicK¨ahler
manifold (M4n,Q,g) is called ǫ-Hermitian if the almost ǫ-complex structure Jǫ is
integrable, almost ǫ-K¨ahler if the K¨ahler form F =g◦Jǫ is closed and ǫ-K¨ahler if
F is parallefl. Noteethat ǫ-K¨ahler submanifolds are minimal ([2]).
We will always assume that M4n has non zero reduced scalar curvature ν =
scal/(4n(n+2)).
f
In section 3 we study an almost ǫ-Hermitian submanifold (M2m,Jǫ,g) of the
para-quaternionicK¨ahler manifold M4n and give the necessary and sufficient con-
dition to be ǫ-Hermitian. If furthermore M is analytic, we show that a sufficient
f
condition for integrability is that codimT M > 2 at some point x ∈ M where by
x
T M we denote the maximal Q -invariant subspace of T M. Then, as an applica-
x x x
tion, we prove that, if the set U of points x∈M where the Nijenhuis tensor of Jǫ
ofanalmostǫ-Hermitian submanifoldofdimension4kisnotzeroisopenanddense
in M and T M is non degenerate, then M is a para-quaternionic submanifold.
x
Infact,byextendingaclassicalresultofquaternionicgeometry(see[1],[12]),we
show that a non degenerate para-quaternionic submanifold of a para-quaternionic
K¨ahlermanifoldis totallygeodesic,henceapara-quaternionicK¨ahlersubmanifold.
In section 4, we give two equivalent necessary and sufficient conditions for an
almost ǫ-Hermitian manifold to be ǫ-K¨ahler. We prove that an almost ǫ-K¨ahler
submanifold M2m of a para-quaternionic K¨ahler manifold M4n is ǫ-K¨ahler and,
hence, a minimal submanifold (see [2]) and give some local characterizations of
f
such a submanifold (Theorem 4.2). In Theorem 4.3 we prove that the second
fundamentalformhofaǫ-K¨ahlersubmanifoldM satisfiesthefundamentalidentity
h(JǫX,Y)=Jǫh(X,Y) ∀X,Y ∈TM
andthat,conversely,iftheaboveidentityholdsonanalmostǫ-Hermitian submani-
foldM2m ofM4n thenM2m iseitheraǫ-K¨ahlersubmanifoldorapara-quaternionic
(Ka¨hler)submanifoldandthesecasescannothappensimultaneously. Inparticular,
f
we prove that an almost ǫ-Hermitian submanifold M is ǫ-K¨ahler if and only if it
is totally ǫ-complex, i.e. it satisfies the condition J T M⊥T M ∀x ∈ M, where
2 x x
J ∈Q is a compatible para-complex structure anticommuting with Jǫ.
2
In section 5, we study an ǫ-K¨ahler submanifold M of maximal dimension 2n
in a para-quaternionic K¨ahler manifold (M4n,Q,g) (still assuming ν 6= 0). Using
the field of isomorphisms J : TM → T⊥M between the tangent and the normal
bundle, we identify, as in [52], the second ffundameental form h of M with a tensor
C = J ◦h ∈ TM ⊗S2T∗M. This tensor, at any point x ∈ M, belongs to the
2
first prolongation SJ(1ǫ) of the space SJǫ ⊂ EndTxM of symmetric endomorphisms
anticommuting with Jǫ. Using the tensor C, we present the Gauss-Codazzi-Ricci
equationsinasimpleformandderivefromitthenecessaryandsufficientconditions
for the ǫ-K¨ahler submanifold M to be parallel and to be curvature invariant (i.e.
R Z ∈ TM, ∀X,Y,Z ∈ TM). In subsection 5.4 we study a maximal ǫ-K¨ahler
XY
submanifold M of a (locally) symmetric para-quaternionic K¨ahler space M4n and
e
getthenecessaryandsufficientconditionsforM tobealocallysymmetricmanifold
f
(PARA-)HERMITIANAND(PARA-)KA¨HLERSUBMANIFOLDOFAPQKMANIFOLD 3
in terms of the tensor C. In particular, if M4n is a quaternionic space form, then
the ǫ-K¨ahlersubmanifoldM is curvature invariant. Inthis case, M is symmetric if
f
and only if the 2-form
[C,C]:X ∧Y 7→[C ,C ] X,Y ∈TM,
X Y
withvaluesintheunitaryalgebraoftheǫ-Hermitianstructureandthatsatisfiesthe
firstandthesecondBianchiidentity,isparallel. MoreoverM isatotallyǫ-complex
totally geodesic submanifold of the quaternionic space form M4n if and only if
ν
Ric = (n+1)g f
M
2
(see Proposition 5.14).
In Section 6 we characterize a maximal ǫ-K¨ahler submanifold M of the para-
quaternionicK¨ahler manifoldM4n withparallelnonzerosecondfundamentalform
h, or shortly, parallel ǫ-K¨ahler submanifold. In terms of the tensor C, this means
f
that
∇ C =−ǫω(X)Jǫ◦C, X ∈TM
X
where ω = ω | and ∇ is the Levi-Civita connection of M. When (M2n,J,g),
1 TM
where J = Jǫ,ǫ = −1, is a parallel not totally geodesic K¨ahler submanifold, the
covariant tensor g ◦ C has the form gC = q + q where q ∈ S3(T∗1,0M) (resp.
x
q¯∈ S3(T∗0,1M)) is a holomorphic (resp. antiholomorphic) cubic form. We prove
x
that any parallel, not totally geodesic, K¨ahler submanifold (M2n,J,g) of a para-
quaternionicK¨ahler manifold(M4n,Q,g)withν 6=0admitsapairofparallelholo-
morphic line subbundle L = span (q) of the bundle S3T∗1,0M and L = span (q)
C C
f
of the bundle S3T∗0,1M such that the connection induced on L (resp. L) has
the curvature RL = −iνg ◦ J = −iνF (resp. RL = iνg ◦J = iνF). In case
(M2n,J,g) where J = Jǫ,ǫ = +1, is a parallel not totally geodesic para-Ka¨hler
submanifoldof(M4n,Q,g)wehavegC =q++q− ∈S3(T∗+M)+S3(T∗−M) where
TM = T+ +T− is the bi-Lagrangean decomposition of the tangent bundle. We
prove that, in thfis case,ethe pair of real line subbundle L+ := Rq+ ⊂ S3(T∗+M)
and L− := Rq− ⊂ S3(T∗−M)) are globally defined on M and parallel w.r.t the
Levi-Civita connection which defines a connection ∇L+ on L+ (resp. ∇L− on L−)
whose curvature is
RL+ =νF, (resp. RL− =−νF).
2. Para-quaternionic Ka¨hler manifolds
Foramoredetailedstudyofpara-quaternionicK¨ahlermanifoldssee[15],[2],[9],
[8], [14]. Moreover for a survey on para-complex geometry see [4], [7].
Definition 2.1. ([2]) Let (ǫ ,ǫ ,ǫ ) = (−1,1,1) or a permutation thereof. An
1 2 3
almost para-quaternionic structure on a differentiable manifold M (of dimen-
sion 2m) is a rank 3 subbundle Q ⊂ EndTM, which is locally generated by three
f
anticommuting fields of endomorphism J ,J ,J = J J , such that J2 = ǫ Id.
1 f2 3 1 2 α α
Such a triple will be called a standard basis of Q. A linear connection ∇ which
preservesQiscalledanalmost para-quaternionicconnection. Analmostpara-
e
quaternionic structure Q is called a para-quaternionic structure if M admits a
para-quaternionic connection i. e. a torsion-free connection which preserves Q. An
f
4 MASSIMOVACCARO
(almost) para-quaternionic manifold is a manifold endowed with an (almost)
para-quaternionic structure.
Observe that J J =ǫ ǫγJ where (α,β,γ) is a cyclic permutation of (1,2,3).
α β 3 γ
Definition 2.2. ([2]) An (almost) para-quaternionic Hermitian manifold
(M,Q,g) is a pseudo-Riemannian manifold (M,g) endowed with an (almost) para-
quaternionic structure Q consisting of skew-symmetric endomorphisms. The non
f e f e
degeneracy of the metric implies that dimM =4n and the signature of g is neutral.
(M4n,Q,g), n>1, is called a para-quaternionic K¨ahler manifold if the Levi-
f e
Civita connection preserves Q.
f e
Proposition 2.3. ([3]) The curvature tensor R of a para-quaternionic K¨ahler ma-
nifold (M,Q,g), of dimension 4n>4, at any point admits a decomposition
e
(1) f e R=νR0+W,
where ν = scal is the reduced scealar curvature,
4n(n+2)
(2)
1 1
R (X,Y):= ǫ g(J X,Y)J + (X∧Y − ǫ J X∧J Y), X,Y ∈T M,
0 α α α α α α p
2 4
Xα Xα
e
is the curvaturetensor of the para-quaternionic projective space of the same dimen-
sion as M and W is a trace-free Q-invariant algebraic curvature tensor, where Q
acts by derivations. In particular, R is Q-invariant.
f
We define a para-quaternioniec K¨ahler manifold of dimension 4 as a
pseudo-Riemannian manifold endowed with a parallel skew-symmetric para-qua-
ternionic K¨ahler structure whose curvature tensor admits the decomposition (1).
SincetheLevi-Civitaconnections∇ofapara-quaternionicK¨ahlermanifoldpre-
serves the para-quaternionic K¨ahler structure Q, one can write
e
(3) ∇J =−ǫ ω ⊗J +ǫ ω ⊗J ,
α β γ β γ β γ
wheretheωα, α=1,2,3earelocallydefined1-formsand(α,β,γ)isacyclicpermu-
tation of (1,2,3). We shall denote by F := g(J ·,·) the K¨ahler form associated
α α
with J and put F′ :=−ǫ F .
α α α α
e
WerecalltheexpressionfortheactionofthecurvatureoperatorR(X,Y), X,Y ∈
TM of M, on J :
α e
(4f) f [R(X,Y),Jα]=ǫ3ν(−ǫβFγ′(X,Y)Jβ +ǫγFβ′(X,Y)Jγ)
where (α,β,γ) ies a cyclic permutation of (1,2,3).
Proposition2.4. ([2])Thelocally definedK¨ahler formssatisfythefollowing struc-
ture equations
(5) νF′ :=−ǫ νF =ǫ (dω −ǫ ω ∧ω ),
α α α 3 α α β γ
where (α,β,γ) is a cyclic permutation of (1,2,3).
By taking the exterior derivative of (5) we get
νdF′ =ǫ d(dω −ǫ ω ∧ω )=−ǫ (ǫ dω ∧ω −ǫ ω ∧dω ) .
α 3 α α β γ 3 α β γ α β γ
Since dω =ǫ νF′ +ǫ ω ∧ω and dω =ǫ νF′ +ǫ ω ∧ω , we get
β 3 β β γ α γ 3 γ γ α β
νdF′ =−ǫ [(ǫ ǫ νF′ ∧ω )−(ǫ ω ∧ǫ νF′,)]
α 3 α 3 β γ α β 3 γ
(PARA-)HERMITIANAND(PARA-)KA¨HLERSUBMANIFOLDOFAPQKMANIFOLD 5
that is ν[dF′ −ǫ (−F′ ∧ω +ω ∧F′)]=0. Hence we have the following result.
α α β γ β γ
Proposition 2.5. On a para-quaternionic K¨ahler manifold the following integra-
bility conditions hold
(6) ν[dF′ −ǫ (−F′ ∧ω +ω ∧F′)]=0, (α,β,γ)=cycl(1,2,3).
α α β γ β γ
3. Almost ǫ-Hermitian submanifolds of M4n
The definitionofan(almost)complexstructureonadifferenftiablemanifoldand
theconditionforits integrabilityarewellknown. We justrecallthefollowingother
definitions (see [2]).
Definition 3.1. An (almost)para-complex structure on a differentiable mani-
fold M is a field of endomorphisms J ∈ EndTM such that J2 = Id and the ±1-
eigenspace distributions T±M of J have the same rank. An almost para-complex
structure is called integrable, or para-complex structure, if the distributions
T±M are integrable or, equivalently, the Nijenhuis tensor N , defined by
J
N (X,Y)=[JX,JY]−J[JX,Y]−J[X,JY]+[X,Y], X,Y ∈TM
J
vanishes. An(almost)para-complex manifold (M,J) is amanifold M endowed
with an (almost) para-complex structure.
Definition 3.2. An (almost) ǫ-complex structure ǫ ∈ {−1,1} on a differen-
tiable manifold M of dimension 2n is a field of endomorphisms J ∈EndTM such
that J2 =ǫId and moreover, for ǫ=+1 the eigendistributions T±M are of rank n.
An ǫ-complex manifold is a differentiable manifold endowed with an integrable
(i.e. N =0) ǫ-complex structure.
J
Consequently, the notation (almost) ǫ-Hermitian structure, (almost) ǫ-K¨ahler
structure, etc.. will be used with the same convention.
Letrecallthatasubmanifoldofapseudo-Riemannianmanifoldisnondegenerate
if it has non degenerate tangent spaces.
Definition 3.3. Let (M4n,Q,g) be a para-quaternionic K¨ahler manifold. A g-non
degeneratesubmanifoldM2m ofM iscalledanalmost ǫ-Hermitian submanifold
f e e
of M if there exists a section Jǫ :M →Q such that
f |M
f JǫTM =TM (Jǫ)2 =ǫId.
We will denote such submanifold (M2m,Jǫ,g) where (g =g| , Jǫ =Jǫ| ).
M M
For a classification of almost (resp. para-)Hermitian meanifolds see [13], (resp.
[6],[11]).
Notice (see[20],[21],[22])thatinanypointx∈M theinducedmetric g =<,>
x x
ofan (almost) Hermitiansubmanifold has signature2p,2q with p+q =m whereas
the signature of the metric of an (almost) para-Hermitian submanifold is always
neutral(m,m). Inbothcasesthentheinducedmetricispseudo-Riemannian(and
Hermitian). Keeping in mind this fact, we will not use the suffix ”pseudo” in the
following.
For any point x ∈ M2m, we can always include Jǫ into a local frame (J =
1
Jǫ,J ,J =J J =−J J )ofQdefinedinaneighbourhoodU ofxinM suchthat
2 3 1 2 2 1
J2 = Id. Such frame will be called adapted to the submanifold M and in fact,
2 e f
6 MASSIMOVACCARO
sinceourconsiderationsarelocal,wewillassumeforsimplicity thatU ⊃M2m and
put
e
F =F =g◦Jǫ, ω =ω .
1|M 1|M
Moreover,we have
(7) ∇Jǫ =−ω ⊗J −ǫω ⊗J
3 2 2 3
where∇indicatestheLevi-eCivitaconnectiononM,andincomplexcase(ǫ=−1),
from (ǫ ,ǫ ,ǫ ) = (−1,1,1), we have J J = −J , J J = J whereas in para-
e1 2 3 2 3 f1 3 1 2
complex case, where (ǫ ,ǫ ,ǫ )=(1,1,−1), we have J J =−J , J J =−J .
1 2 3 2 3 1 3 1 2
For any x ∈ M we denote T M the maximal para-quaternionic (Q-invariant)
x
subspace of the tangentspace T M. Note that if (J ,J ,J ) is anadapted basis in
x 1 2 3
a point x∈M then T M =T M ∩J T M.
x x 2 x
We allow T M to be degenerate (even totally isotropic), hence its dimension
x
is even (not necessarily a multiple of 4) and the signature of g| is (2k,2s,2k)
TxM
where 2s = dimkerg (see [20]). We recall that a subspace of a para-quaternionic
vector space (V,Q) is pure if it contains no non zero Q-invariant subspace. We
write then
T M =T M ⊕D
x x x
where D is any Jǫ-invariant pure supplement (the existence of such supplement
x
is proved in [20]).
RecallthatifM isanondegeneratesubmanifoldofapseudo-Riemannianmani-
fold(M,g)andT M =T M⊕T⊥M istheorthogonaldecompositionofthetangent
x x x
space T M at point x ∈ M then the Levi-Civita covariant derivative ∇ of the
fxe f X
metric g in the direction of a vector X ∈T M can be written as:
x
f e
∇ −A
e ∇ ≡ X X .
X (cid:18) At ∇⊥ (cid:19)
X X
e
that is
(8) ∇ Y =∇ Y +h(X,Y), ∇ ξ =−AξX +∇⊥ξ
X X X X
for any tangeent (resp. normal) vector field Y e(resp. ξ) on M. Here ∇X is the
covariant derivative of the induced metric g on M, ∇⊥ is the normal covariant
X
derivative in the normal bundle T⊥M which preserves the normal metric
g⊥ =g|T⊥M,AtXY =h(X,Y)∈T⊥M wherehisthesecond fundamental form
and A ξ = AξX, where Aξ ∈ End TM is the shape operator associated with a
X
e
normal vector ξ.
Theorem 3.4. Let (M2m,Jǫ,g), m > 1, be an almost ǫ-Hermitian submanifold
of the para-quaternionic K¨ahler manifold (M4n,Q,g). Then
(1) the almost ǫ-complex structure Jǫ is integrable if and only if the local 1-
f e
form ψ =ω ◦Jǫ−ω on M2m associated with an adapted basis H =(J )
3 2 α
vanishes.
(2) Jǫ is integrable if one of the following conditions holds:
a) dim(D )>2 on an open dense set U ⊂M;
x
b) (M,Jǫ) is analytic and dim(D )>2 at some point x∈M;
x
Proof. (1)Letproceedasin[5],Theorem1.1. Remarkthatif(M,Jǫ)isanalmostǫ-
complex submanifoldof analmostǫ-complex manifold (M,Jǫ) then the restriction
of the Nijenhuis tensor NJǫ to the submanifold M coincides with the Nijenhuis
f
(PARA-)HERMITIANAND(PARA-)KA¨HLERSUBMANIFOLDOFAPQKMANIFOLD 7
tensorNJǫ ofthealmostcomplexstructureJǫ =Jǫ|TM. ThenforanyX,Y ∈TM,
we can write
12NJǫ(X,Y) = [JǫX,JǫY]−Jǫ[JǫX,Y]−Jǫ[X,JǫY]+ǫ[X,Y]=
21NJǫ(X,Y) = [∇JǫX(JǫY)−∇JǫY(JǫX)]−Jǫ[∇JǫXY −∇Y(JǫX)]
−eJǫ[∇X(JǫY)−e∇JǫYX]+ǫ[∇XYe −∇YX]e
= (∇JǫXeJǫ)Y −(∇JeǫYJǫ)X +Jǫ(∇YJǫ)X −Jǫ(∇XJǫ)Y
and hence, from (3) e e e e
12NJǫ(X,Y)= −[ω3(JǫX)−ω2(X)]J2Y +[−ǫω2(JǫX)+ω3(X)]J3Y
+[ω (JǫY)−ω (Y)]J X −[−ǫω (JǫY)+ω (Y)]J X
3 2 2 2 3 3
where (J ,J ,J ) is an adapted local basis. This implies (1) in one direction.
1 2 3
Viceversa, let NJǫ(X,Y)=0, ∀X,Y ∈TxM. By applying J2 to both members
of the above equality, this is equivalent to the identity
(9) ψ(X)Y +ǫψ(JǫX)JǫY =ψ(Y)X +ǫψ(JǫY)JǫX, ∀X,Y ∈T M.
x
Let assume that there exists a non zero vector X ∈T M such that ψ(X)6=0. We
x
show that this leads to a contradiction. Let consider a vector 06=Y ∈T M which
x
is not en eigenvectorof Jǫ and such that span(X,JǫX)∩span(Y,JǫY)=0. It is
easyto check that such a vector Y alwaysexists. Then the vectors in both sides of
(9) must be zero which implies in particular that ψ(X)=0. Contradiction.
(2)WeassumethatJǫ isnotintegrable. Thenthe1-formψ =(ω ◦Jǫ−ω )|
3 2 TM
is not identically zero, by (1). Denote by a = g−1ψ the local vector field on M
associatedwith the 1-formψ and let a=a+a′ with a∈TM and a′ ∈D. Now we
need the following
Lemma 3.5. Let (M2m,Jǫ,g),m > 1, be an almost ǫ-Hermitian submanifold of
a para-quaternionic K¨ahler manifold (M4n,Q,g). Then in any point x ∈ M2m
where the Nijenhuis tensor N(Jǫ) 6= 0, or equivalently the vector a 6= 0, any
x x
Jǫ-invariant supplementary subspace Dfis spanened by a′ and Jǫa′:
x x x
D =span{a′,Jǫa′}.
x x x
Moreover if T M is not para-quaternionic (i.e. dimD 6=0) then ψ(T M)≡0.
x x x
Proof. Remark that
(10)
21NJǫ(X,Y) =−ψ(X)J2Y +ǫψ(JǫX)J3Y +ψ(Y)J2X−ǫψ(JǫY)J3X
=−J {ψ(X)Y +ǫψ(JǫX)JǫY −ψ(Y)X −ǫψ(JǫY)JǫX},
2
that is NJǫ(X,Y)∈J2TM ∩TM =TM for any X,Y ∈TM. Hence
(11) ψ(X)Y +ǫψ(JǫX)JǫY −ψ(Y)X −ǫψ(JǫY)JǫX ∈TM ∀X,Y ∈TM.
(cid:20) (cid:21)
TakingX ∈T M and06=Y ∈D thefirsttwotermsof(11)areinD andthelast
x x x
two in T M. We conclude that ψ(T M) ≡ 0 if dimD 6= 0. For X = a = g−1ψ,
x x x
since g(a,Jǫa)=0, the last condition says that
b :=|a|2Y −ψ(Y)a−ǫψ(JǫY)Jǫa∈TM ∀ Y ∈TM
Y
Considering the D-component of the vector b for Y =Y ∈TM and Y =Y′ ∈D
Y
respectively, we get the equations:
(12) −ψ(Y)a′−ǫψ(JǫY)Jǫa′ =0, ∀ Y ∈TM
8 MASSIMOVACCARO
(13) |a|2Y′−ψ(Y′)a′−ǫψ(JǫY′)Jǫa′ =0 ∀ Y′ ∈D.
The last equation shows that D = {a′,Jǫa′} when a 6= 0 (whereas (12) confirms
x
that ψ(T M)≡0 when dimD6=0). Observe that a′ is never an eigenvector of the
x
para-complex structure J.
(cid:3)
Continuing the proof of Theorem (3.4): The Lemma implies statements (2a)
and (2b) since in the analytic case the set U of points where the analytic vector
field a 6= 0 is open (complementary of the close set where a = 0) and dense (since
otherwise it would exist an open set U with a(U)=0 which, by the analiticy of a
it would imply a=0 everywhere) and dimD ≤2 on U. (cid:3)
x
e e
From (10) it follows the
Corollary 3.6. In case T M is pure ǫ-complex i.e. T M =0 in an open dense set
x x
in M than the almost Hermitian submanifold is Hermitian.
This is a generalization of the 2-dimensional case where clearly, by the non
degeneracy hypotheses, T M is pure for any x∈M.
x
Definition3.7. AsubmanifoldM ofanalmostpara-quaternionicmanifold (M,Q)
isanalmost para-quaternionicsubmanifoldifitstangentbundleisQ-invariant.
f
Then (M,Q| ) is an almost para-quaternionic manifold.
TM
The following proposition is the extension to the para-quaternionic case of a
basic result in quaternionic case.
Proposition3.8. Anondegeneratealmostpara-quaternionicsubmanifold M4m of
apara-quaternionicK¨ahlermanifold(M4n,Q,g)isatotallygeodesicpara-quaternionic
K¨ahler submanifold.
f e
Proof. Let A be the shape operator of the para-quaternionic submanifold. Then,
for any X,Y ∈Γ(TM), ξ ∈Γ(T⊥M),
g(Aξ(J X),Y)=−g(∇ ξ,Y)=−g(∇ ξ,J X)
α JαX Y α
=g(ξ,∇ (J X))=g(ξ,(∇ J )X +J ∇ X).
e Y α e e Y α eαe Y
Moreover e e e e e e e
g(ξ,J ∇ X) =−g(J ξ,∇ X)=−g(J ξ,∇ Y −[X,Y])
α Y α Y α X
=−g(J ξ,∇ Y)=g(ξ,J ∇ Y)=g(ξ,∇ (J Y)−(∇ J )Y)
e e e e eα eX e eα Xe X α X α
=g(ξ,∇ (J Y))=−g(∇ ξ,J Y)=−g(J AξX,Y)
e eX eα e eXe α e e α e e e
and e e e e e
g(ξ,(∇ J )X)=g(ξ,−ǫ ω (Y)J X +ǫ ω (Y)J )=0
Y α β γ β γ β γ
since J X,J X ∈Γ(TM). It follows that AJ =−J A, α=1,2,3. Computing
β γe e e e α α
AJ = −J A = −ǫ ǫ J J A = −ǫ ǫ AJ J = −(ǫ ǫ )2AJ = −AJ we get
α α 3 α β γ 3 α β γ 3 α α α
A = 0 i.e. h = 0. Now it is immediate to deduce that (M4m,Q| ,g) is also
TM
para-quaternionic K¨ahler. (cid:3)
Corollary 3.9. Let (M4k,Jǫ,g) be an almost ǫ-Hermitian submanifold of dimen-
sion 4k of a para-quaternionic K¨ahler manifold M4n. Assume that the set U of
points x ∈ M where the Nijenhuis tensor of Jǫ is not zero is open and dense
f
in M and that, ∀x ∈ U, T M is non degenerate. Then M is a totally geodesic
x
para-quaternionic K¨ahler submanifold.
(PARA-)HERMITIANAND(PARA-)KA¨HLERSUBMANIFOLDOFAPQKMANIFOLD 9
Proof. As in [5] by taking into account that, by the non degeneracy hypotheses of
T M, it is necessarily dimD =0. (cid:3)
x x
4. Almost ǫ-Ka¨hler, ǫ-Ka¨hler and totally ǫ-complex submanifolds
Definition 4.1. The almost ǫ-Hermitian submanifold (M2m,Jǫ,g) of a para-
quaternionic K¨ahler manifold (M4n,Q,g) is called almost ǫ-K¨ahler (resp., ǫ-
K¨ahler)iftheK¨ahlerformF =F | =g◦Jǫ isclosed(resp. parallel). Moreover
M is called totally ǫ-complex fif1 TM e
J T M ⊥T M ∀x∈M
2 x x
where(J ,J ,J )isanadaptedbasis(notethatJ T M ⊥T M ⇔J T M ⊥T M).
1 2 3 2 x x 3 x x
For a study of (almost)-K¨ahler and totally complex submanifolds of a quater-
nionic manifold see [5],[10],[17],[18].
In case M is the n-dimensional para-quaternionic numerical space Hn, the pro-
totype of flat para-quaternionic K¨ahler spaces (see [21]), typical examples of such
f e
submanifolds are the flat K¨ahler (resp. para-Ka¨hler) submanifolds M2k = Ck
(resp. Ck) obtained by choosing the first k para-quaternionic coordinates as com-
plex(resp. para-complex)numbersandthe remainingn−k equalsto zero. Incase
e
M4n = HPn is the para-quaternionic projective space endowed with the standard
para-quaternionicK¨ahler metric (see [8]), examples of non flat K¨ahler (resp. para-
f e
K¨ahler)submanifolds are given by the immersions of the projective complex (resp.
para-complex)spacesCPk−1 (resp. CPk−1)induced bythe immersionsconsidered
above in the flat case.
e
From (3) one has
(14) (∇ Jǫ)Y = −ω (X)Id−ǫω (X)Jǫ J Y T X,Y ∈TM.
X 3 2 2
(cid:2) (cid:3)(cid:2) (cid:3)
and then, by arguing as in [5], the following theorem is deduced.
Theorem 4.2. Let (M4n,Q,g) be a para-quaternionic K¨ahler manifold.
1) A totally ǫ-complex submanifolds of M is ǫ-K¨ahler.
f e
2) If ν 6=0, for an almost ǫ-Hermitian submanifold (M2m,Jǫ,g), m>1, of M the
f
following conditions are equivalent:
f
k ) M is ǫ-K¨ahler,
1
k ) ω | =ω | =0 ∀x∈M,
2 2 TxM 3 TxM
k ) M is totally ǫ-complex.
3
Proof. Thefirststatementfollowsfrom(14). Thesecondstatementisprovedin[2]
Proposition 20. (cid:3)
Theorem 4.3. Let (M4n,Q,g) be a para-quaternionic K¨ahler manifold with non
vanishing reduced scalar curvature ν and (M2m,Jǫ,g) an almost ǫ-Hermitian sub-
manifold of M4n. f e
a) If (M2m,Jǫ,g) is ǫ-K¨ahler then the second fundamental form h of M sa-
f
tisfies the identity
(15) h(X,JǫY)=h(JǫX,Y)=Jǫh(X,Y) ∀X,Y ∈TM.
In particular h(JǫX,JǫY)=ǫh(X,Y).
10 MASSIMOVACCARO
b) Conversely, iftheidentity(15)holdsonanalmostǫ-Hermitian submanifold
M2m ofM4n thenitiseitheraǫ-K¨ahlersubmanifoldorapara-quaternionic
(K¨ahler) submanifold and these cases cannot happen simultaneously.
f
Proof. (a) Let (M2m,Jǫ,g) be an almost ǫ-Hermitian submanifold of M. By (3),
(∇ Jǫ)Y =(∇ Jǫ)Y +h(X,JǫY)−Jǫh(X,Y) f
(16) X X
=−ω (X)J Y −ǫω (X)J Y, X,Y ∈TM.
e 3 2 2 3
From Theorem (4.2), we get
0=(∇ Jǫ)Y +h(X,JǫY)−Jǫh(X,Y), ∀X,Y ∈TM
X
and, from (∇ Jǫ)Y =0 it is clear that if (M,Jǫ) is ǫ-K¨ahler then (15) holds.
X
(b)Conversely,letassumethat(15)holdsonthealmostǫ-Hermitian submanifold
(M,Jǫ,g). Then for any X,Y ∈T M, from (16) we have
x
(∇ Jǫ)Y =(∇ Jǫ)Y.
X X
Hence, ∀X,Y ∈T M, e
x
(∇ Jǫ)Y =−ω (X)J Y −ǫω (X)J Y =(−ω (X)Id−ǫω (X)Jǫ)J Y ∈T M.
X 3 2 2 3 3 2 2 x
Then,eitherJ T M =T M i.e. T M isapara-quaternionicvectorspaceorω | =
2 x x x 2 x
ω | = 0 and by Theorem (4.2) the two conditions cannot happen simultaneously.
3 x
The setM ={x∈M |J T M =T M}is a closedsubsetandthe complementary
1 2 x x
open subset M = {x ∈ M | ω | = ω | = 0} is a closed subset as well since,
2 2 x 3 x
from Theorem (4.2), M = {x ∈ M | J T M ⊥ T M}. Then, either M = 0 and
2 2 x x 2
M = M is a para-quaternionic K¨ahler submanifold or M = 0 and M = M is
1 1 2
ǫ-K¨ahler. (cid:3)
Corollary 4.4. A totally geodesic almost ǫ-Hermitian submanifold (M,Jǫ,g) of
a para-quaternionic K¨ahler manifold (M4n,Q,g) with ν 6= 0 is either a ǫ-K¨ahler
submanifoldorapara-quaternionicsubmanifoldandtheseconditionscannothappen
simultaneously. f e
Proof. ThestatementfollowsdirectlyfromTheorem(4.3)since(15)certainlyholds
for a totally geodesic submanifold (h=0). (cid:3)
The following results have been proved in [2].
Proposition4.5. ([2])TheshapeoperatorAofanǫ-K¨ahlersubmanifold(M2m,Jǫ,g)
of a para-quaternionic K¨ahler manifold (M4n,Q,g) anticommutes with Jǫ, that is
AJǫ =−JǫA.
f e
Corollary 4.6. ([2]) Any ǫ-K¨ahler submanifold of a para-quaternionic K¨ahler ma-
nifold is minimal.
We conclude this section with the following result concerning almost ǫ-K¨ahler
submanifolds.
Theorem 4.7. Let (M4n,Q,g) be a para-quaternionic K¨ahler manifold with non
vanishing reduced scalar curvature ν. Then any almost ǫ-K¨ahler submanifold
(M2m,Jǫ,g) of M isfǫ-K¨ahleer.
f