An Almost Contact Structure on $G_2$-Manifolds

0.22 MB·English

by  Albert J. Todd

ON AN ALMOST CONTACT STRUCTURE ON G -MANIFOLDS
2
A.J.TODD
Abstract. In this article, we study an almost contact metric structure on a G2-manifoldconstructed by
Arikan,ChoandSalurin[2]viatheclassificationofalmostcontact metricstructures givenbyChineaand
5
Gonzalez[14]. Inparticular,wecharacterize whenthisalmostcontact metricstructure iscosymplecticand
1
narrow downthe possibleclasses inwhich this almostcontact metric structure couldlie. Finally,we show
0
that any closed G2-manifoldadmits an almost contact metric 3-structure by constructing itexplicitlyand
2
characterize whenthisalmostcontactmetric3-structure is3-cosymplectic.

n
a
J

8
2 1. Introduction

Let M be a smooth manifold of dimension 2n+1. Let Ξ denote a 2n-dimensional distribution on M
]
G which is maximally non-integrable. Locally, we have that Ξ = kerα for some 1-form α; the “maximally
D non-integrable” condition then means that a∧(dα)n 6= 0. If such a Ξ exists, then the structure group of
the tangent bundle to M admits a reductionto the groupU(n)×1. Conversely,if M is an odd-dimensional
.
h manifold suchthat the structure groupof the tangentbundle to M admits a reduction to U(n)×1, then M
t
a is said to have an almost contact structure. Recently Borman, Eliashberg and Murphy [5] have shown that
m in fact any closed, odd-dimensional manifold with an almost contact structure admits a contact structure.
[ Our interest here is not in the contact structure itself, but rather the underlying almost contact structure.

The reason is that such a reduction of the structure group of the tangent bundle to M is equivalent to the
1
existence of a (1,1)- tensor φ, a vector field ξ and a 1-form η satisfying
v
6 φ2 =−I+η⊗ξ (1)
6
9 η(ξ)=1 (2)
6
0 Often, we will generally refer to the triple (φ,ξ,η) as the almost contact structure. It is straight-forwardto
. show that equations (1) and (2) imply that φ(ξ) ≡ 0, η ◦φ ≡ 0 and that φ has rank 2n. A Riemannian
1
metric g is said to be compatible with the almost contact structure (φ,ξ,η) if
0
5
g(φX,φY)=g(X,Y)−η(X)η(Y) (3)
1
: foranyvectorfieldsX andY onM. Inthiscase,the quadruple(φ,ξ,η,g)iscalledanalmost contact metric
v
i structure. Using equations (2) and (3) and that φξ =0 yields
X
g(ξ,X)=η(X) (4)
r
a
for any vector field X on M. On any manifold with an almost contact structure, there always exists a
compatible metric; however, such a compatible metric is not in general unique. Using φ and g, we can now
define a smooth differential 2-form ω by
ω(X,Y)=g(X,φY) (5)
This 2-form is called the fundamental 2-form of the almost contact metric structure. Note that ω satisfies
η∧ωn 6=0 (6)
showing that almost contact metric manifolds are orientable. A reference for this material and more infor-
mationoncontactandalmostcontactgeometrycanbefoundin[4]. Variousconditionscanbeplacedonthe
almost contact metric structure (φ,ξ,η,g) to give various classes of almost contact metric structures such
as cosymplectic, Sasakian and Kenmotsu geometries. These types of almost contact metric structures, and
several others, have been studied extensively by many authors, see, e. g., [3], [4], [6], [11], [12], [13], [18],
[20], [21], [24], [26], [27], [28], [29], [30] and [31]. The classification of the various types of almost contact
metricstructureswasundertakenbyChineaandGonzalez[14]whereintheyclassifiedalmostcontactmetric
1
2 A.J.TODD
structures based on the covariantderivative of a certain differential 2-form associated to the almost contact
metric structure. We will discuss their classification in Section 2.
Let M be a 7-dimensional manifold admitting a smooth differential 3-form ϕ such that, for all p ∈ M,
the pair (T M,ϕ) is isomorphic as an oriented vector space to the pair (R7,ϕ ) where
p 0
ϕ =dx123+dx145+dx167+dx246−dx257−dx347−dx356 (7)
0
with dxijk = dxi ∧dxj ∧dxk. In [8], it is shown that the Lie group G can be defined as the set of all
2
elements ofGL(7,R) thatpreserveϕ ,so fora manifoldadmitting sucha3-form,thereis a reductionin the
0
structure group of the tangent bundle to the exceptional Lie group G ; hence, the pair (M,ϕ) is called a
2
manifold with G -structure whereby an abuse of terminology we often refer to ϕ as the G -structure. Using
2 2
the theory of G-structures and the inclusion of G in SO(7), all manifolds with G -structure are necessarily
2 2
orientable and spin, any orientable 7-manifold with spin structure admits a G -structure, and associated to
2
a given G -structure ϕ are a metric g called the G -metric, satisfying
2 ϕ 2
(Xyϕ)∧(Yyϕ)∧ϕ=6g (X,Y)Vol (8)
ϕ ϕ
for any vector fields X and Y on M and a 2-fold vector cross product ×; these three structures are related
via
ϕ(X,Y,Z)=g (X ×Y,Z) (9)
ϕ
forallvectorfieldsX,Y,Z. Itisimportanttonotethat,onecanalsoshowthatstartingwitheitherthetwo-
fold vector crossproduct or the G -metric g , one can alwaysobtain the other two structures (see e. g. [22]
2 ϕ
or [17]; thus in particular,if the flow of a vectorfield preservesone of these structures, it must also preserve
the other two. A natural geometric requirement is that the 3-form ϕ be covariant constant with respect to
the Levi-Civita connection of the G -metric g ; if this is so, we say that the G -structure is integrable and
2 ϕ 2
call the pair (M,ϕ) a G -manifold. Note that for a G -manifold (M,ϕ), the 2-fold vector cross product ×
2 2
is also covariant-constantwith respect to the Levi-Civita connection of the G -metric g . It is a nontrivial
2 ϕ
fact thatthe integrabilityof the G -structure is equivalentto the holonomyof g being a subgroupof G as
2 ϕ 2
well as ϕ being simultaneously closed and coclosed, that is, dϕ= 0 and δϕ= 0 respectively, where δ is the
adjoint to the exterior derivative d defined in terms of the Hodge star ⋆ of g . References for G -geometry
ϕ 2
andconstructionsofmanifoldswith G -holonomyinclude [17],[7],[8],[9],[10],[17],[19],[22],[23],[25],[32],
2
[34] and [35].
Note that, by Berger’s classification of Riemannian holonomy groups, the only nontrivial subgroups of
G which can occur as the holonomy of a Riemannian manifold are SU(2) and SU(3). From [22] (or [23]),
2
if M is a G -manifold with holonomy equal to SU(2), then M is given by one of the Cartesian products
2
N ×R3 or N ×T3 where N is a Calabi-Yau 2-fold and T3 is the 3-torus; on the other hand, if M is a
G -manifold with holonomy equal to SU(3), then M is given by one of the Cartesian products N ×R or
2
N ×S1 where N is a Calabi-Yau 3-fold and S1 is the unit circle. In this way, we see that there is a direct
relation between G -geometry and symplectic geometry; however,in the case where M has holonomy equal
2
to G ,we seethatG -geometryreallyis its owngeometry. AnimportantpropertyofG -manifoldswithfull
2 2 2
G -holonomy is that they are simply connected, so in particular, the first Betti number of such a manifold
2
is 0. Moreover,all G -manifolds, regardlessof holonomy, are Ricci-flat. It is well known that on a Ricci-flat
2
manifold, the sets of parallel (with respect to the Levi-Civita connection) vector fields, Killing vector fields
andharmonicvectorfields coincide (avectorfieldis called harmonic ifits metric-dual1-formis harmonicin
the sense of Hodge theory); hence, on a G -manifold with full G -holonomy there cannot exist any nonzero
2 2
parallelvector fields and hence no nonzero Killing vectorfields nor nonzero harmonic vector fields. Further,
the existence of such a vector field (nonzero) immediately implies that the holonomy must either be trivial,
SU(2) or SU(3).
With this juxtaposition, a natural question occurs. Do there exists 7-dimensional manifolds which admit
both a G -structure and a contact structure, and if so, how do these geometries interact? This particular
2
line of researchhas only just begun. In the literature, there is work of Matzeu and Munteanu [29] on vector
cross products and almost contact structures focusing in particular on hypersurfaces of R8, and a similar
construction appears in [4]; however, work of Arikan, Cho and Salur [2], [1] has really brought this issue
to light. Using techniques of spin geometry, they show [2, Cor. 3.2] that every manifold with G -structure
2
admits an almost contact structure. Note that this result does not depend on the manifold being closed,
compactornoncompactnordoesitdependontheintegrabilityornon-integrabilityoftheG -structureitself.
2
ON AN ALMOST CONTACT STRUCTURE ON G2-MANIFOLDS 3
Unfortunately, this result is nonconstructive; however, with this, we now have that there always exists at
least one non-vanishing vector field, call it ξ, on a manifold with G -structure. Assume that ξ has been
2
normalized to have length 1 everywhere using the G -metric g , then using ξ and the two-fold vector cross
2 ϕ
product, define a (1,1)-tensor φ by
φ(X)=ξ×X (10)
and a 1-form η by
η(X)=g (ξ,X) (11)
ϕ
Since the cross product on a 7-manifold satisfies
X ×(X ×X )=−g (X ,X )X +g (X ,X )X (12)
1 1 2 ϕ 1 1 2 ϕ 1 2 1
we immediately have that φ2 = −I +η⊗ξ; that η(ξ) = 1 follows immediately from the definitions. Thus,
(φ,ξ,η)asdefinedheregiveanalmostcontactstructure;moreover,thisalmostcontactstructureiscompatible
with g since
ϕ
g (φX,φY)=g (ξ×X,ξ×Y)=ϕ(ξ,X,ξ×Y)=−ϕ(ξ,ξ×Y,X)=−g (ξ×(ξ×Y),X)
ϕ ϕ ϕ
(13)
=−g (−Y +η(Y)ξ,X)=g (Y,X)−η(Y)g (ξ,X)=g (X,Y)−η(X)η(Y)
ϕ ϕ ϕ ϕ
Hence,we seethat(φ,ξ,η,g )is analmostcontactmetric structure. Throughoutwewillreferto this asthe
ϕ
standard G almost contact metric structure
2
Let (M,Ξ) be a contact manifold where Ξ is the 2n-dimensionalcontact distribution. An almost contact
metricstructure(φ,ξ,η,g)issaidtobetheassociated almostcontactmetricstructureofthecontactstructure
ifΞ=kerη andω =dη whereω isgivenby(5). Itisinterestingtonote,especiallyinlightoftheresultsfrom
[5], that the above constructedalmostcontactmetric structure ona closed7-manifoldM with G -structure
2
cannotbe the associatedalmostcontactmetric structure of a contactmetric structure if the G -structure is
2
closed, i. e., dϕ=0 (see [2, Corollary 5.8]). For if so
dη(X,Y)=g (X,φY)=g(X,ξ× Y)=ϕ(ξ,Y,X)=−(ξyϕ)(X,Y) (14)
ϕ ϕ
and hence, by equation (8) and an application of Stokes’ Theorem,
1
Vol(M)= Vol = ((ξyϕ)∧(ξyϕ)∧ϕ)
ϕ
6
ZM ZM (15)
1 1 1
= (dη∧dη∧ϕ)= d(η∧dη∧ϕ)= (η∧dη∧ϕ)=0
6 6 6
ZM ZM Z∂M
In particular, this is the case on a closed G -manifold.
2
Hencethegoalofthecurrentarticleistofurtherthislineofresearchbysheddinglightontheclassification
of the standard G almost contactstructure focusing specifically onclosed G -manifolds; also,this article is
2 2
part of a larger research goal to use the well-established areas of symplectic and contact geometries to gain
a better understanding of G -geometry, see [16], [15], [33]. In Section 2, we will discuss the classification
2
scheme of Chinea and Gonzalez [14]; in Section 3, we will use their classification to better understand the
standard G almost contact structure. In Section 4, we give a brief introduction to almost contact metric
2
3-structuresfromtheworkKuo[26]. Finally,inSection5wewillconstructanalmostcontact3-structureon
a closed 7-manifold with (possibly non-integrable) G -structure and use our results from Section 3 to give
2
some fundamental results on these structures. Our main results are as follows.
Theorem. dω =0 if and only if ∇ξ =0. Further, in this case, dη =0, and the almost contact structure is
normal and hence is cosymplectic.
Corollary. If (M,ϕ) is a G -manifold with full G -holonomy, then dω 6= 0 (or equivalently, ∇ξ 6= 0). In
2 2
particular, the standard G almost contact metric structure cannot be cosymplectic, almost cosymplectic nor
2
quasi-Sasakian.
Corollary. If (M,ϕ) is a G -manifold, then the standard G almost contact metric structure cannot be a
2 2
contact metric structure. In particular, it cannot be Sasakian (or more generally, a-Sasakian where a is
constant).
Theorem. Let M be a G -manifold with the standard G almost contact metric structure. If ∇ξ 6=0, then
2 2
the almost contact metric structure is of class G where
4 A.J.TODD
(1) In general, we have
G ⊆C ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C
5 6 7 8 9 10 12
(2) In the case that δη =0, then
G ⊆C ⊕C ⊕C ⊕C ⊕C ⊕C
6 7 8 9 10 12
(3) In the case that ∇ ξ =0, then
ξ
G ⊆C ⊕C ⊕C ⊕C ⊕C ⊕C
5 6 7 8 9 10
(4) In the case that both δη =0 and ∇ ξ =0,
ξ
G ⊆C ⊕C ⊕C ⊕C ⊕C
6 7 8 9 10
(5) In the case that the almost contact structure is normal, then
G ⊆C ⊕C ⊕C ⊕C
5 6 7 8
(6) In the case that the almost contact structure is normal and δη =0, then
G ⊆C ⊕C ⊕C
6 7 8
Theorem. Let (M,ϕ) be a closed 7-manifold with (not necessarily integrable) G -structure ϕ. Then M
2
admits an almost contact metric 3-structure which is compatible with the G -metric.
2
Corollary. On a closed G -manifold (M,ϕ) with full G -holonomy, the almost contact metric 3-structure
2 2
constructed above cannot be an almost cosymplectic 3-structure and hence cannot be a cosymplectic 3-
structure.
Corollary. On a closed G -manifold (M,ϕ) the almost contact metric 3-structureconstructed above cannot
2
be a contact metric 3-structure (equivalently, it cannot be 3-Sasakian).
2. Classification of Almost Contact Metric Structures
In this section, we review the classification of almost contact metric structures given by Chinea and
Gonzalez [14].
As mentioned above, one can require many different types of conditions of a given almost contact metric
structure. Of particular importance are normal almost contact structures. Let (M,φ,ξ,η) be an almost
contact manifold. Then on M ×R we can define the almost complex structure J given by
d d
J X,f = φX −fξ,η(X) (16)
dt dt
(cid:18) (cid:19) (cid:18) (cid:19)
Using the Nijenhuis torsion tensor defined for any (1,1)-tensor by
[T,T](X,Y)=T2[X,Y]−T[TX,Y]−T[X,TY]+[TX,TY] (17)
where the bracket on the right-hand side of the formula is the standard Lie bracket of vector fields, it is
straight-forward to show that the integrability of J, that is, the vanishing of [J,J], is equivalent to the
simultaneous vanishing of the following four tensors on M:
N(1)(X,Y)=[φ,φ](X,Y)+2dη(X,Y)ξ (18)
where [φ,φ] is the Nijenhuis torsion tensor of φ;
N(2)(X,Y)=(L η)Y −(L η)X (19)
φX φY
where L is the Lie derivative operator;
N(3)(X)=(L φ)X (20)
ξ
and
N(4) =(L η)X (21)
ξ
Moreover,it can be shown that if N(1) is identically zero then N(2), N(3) and N(4) must be identically zero
as well [4]. Thus, the integrability of J is equivalent to the vanishing of the (1,2)-tensor N(1), and in this
case, that is, if N(1) ≡0, we call the almost contact structure (φ,ξ,η) normal.
Other types of almost contact metric manifolds that have appeared in the literature are given as follows
[14]:
ON AN ALMOST CONTACT STRUCTURE ON G2-MANIFOLDS 5
• Almost cosymplectic: dω =0=dη
• Cosymplectic: almost cosymplectic and normal
• Quasi-Sasakian: dω =0 and normal
• Almost a-Kenmotsu: dη = 0 and dω = 2a[η(X)ω(Y,Z)+η(Y)ω(Z,X)+η(Z)ω(X,Y)] where a is
3
a differentiable function on M
• a-Kenmotsu: almost a-Kenmotsu and normal
• Almost a-Sasakian: aω =dη where a is a differentiable function on M
• a-Sasakian: Almost a-Sasakian and normal
• Nearly K-cosymplectic: (∇ φ)Y +(∇ φ)X =0 and ∇ ξ =0 for all vector fields X,Y.
X Y X
• Quasi-K-cosymplectic: (∇ φ)Y +(∇ φ)(φY)=η(Y)∇ ξ
X φX φX
• Semi-cosymplectic: δω =0=δη
• Trans-Sasakian:
1
(∇ ω)(Y,Z)=− [(g(X,Y)η(Z)−g(X,Z)η(Y))δω(ξ)+(g(X,φY)η(Z)−g(X,φZ)η(Y))δη]
X
2n
• Nearlytrans-Sasakian: (∇ ω)(X,Y)=− 1 [g(X,X)δω(Y)−g(X,Y)δω(X)+g(φX,Y)η(Y)δη]and
X 2n
(∇ η)Y =− 1 [g(φX,φY)δη+g(φX,Y)δω(ξ)]
X 2n
• Almost K-contact: ∇ φ=0
ξ
The classificationofChineaandGonzalezdepends onadecompositionofthe spaceC ofcovarianttensors
of degree 3 which possess the same symmetry properties as ∇ω, the covariantderivative of the fundamental
2-form (5). As we will need various pieces of their theory, we give here a sketch of the details of their
decomposition and refer the reader to [14] for the full proof.
Let V denote a real vector space of dimension 2n+1 with an almost contact structure (φ,ξ,η) and a
compatible metric h·,·i. Then C(V) is a subspace of the space of covariant tensors of degree 3, denoted
0V, defined by
3
N
0
C(V)= α∈ V :α(x,y,z)=−α(x,z,y)=−α(x,φy,φz)+η(y)α(x,ξ,z)+η(z)α(x,y,ξ) (22)
( )
3
O
Then the space of quadratic invariants of C(V) is generated by the following 18 invariants:
2
i1(α)= Xα(ei,ej,ek) ; i10(α)=Xα(ei,ei,ξ)α(ej,ej,ξ);
i,j,k i,j
i2(α)= Xα(ei,ej,ek)α(ej,ei,ek); i11(α)=Xα(ei,ej,ξ)α(ej,φei,ξ);
i,j,k i,j
i3(α)= Xα(ei,ej,ek)α(φei,φej,ek); i12(α)=Xα(ei,ej,ξ)α(φej,φei,ξ);
i,j,k i,j
i4(α)= Xα(ei,ei,ek)α(ej,ej,ek); i13(α)=Xα(ξ,ej,ek)α(φej,ξ,ek);
i,j,k j,k
2
i5(α)=Xα(ξ,ej,ek) ; i14(α)=Xα(ei,φei,ξ)α(ej,φej,ξ);
j,k i,j
2
i6(α)=Xα(ei,ξ,ek) ; i15(α)=Xα(ei,φei,ξ)α(ej,ei,ξ);
i,k i,j
i7(α)=Xα(ξ,ej,ek)α(ej,ξ,ek); i16(α)=Xα(ξ,ξ,ek)2;
j,k k
i8(α)=Xα(ei,ej,ξ)α(ej,ei,ξ); i17(α)=Xα(ei,ei,ek)α(ξ,ξ,ek);
i,j i,k
i9(α)=Xα(ei,ej,ξ)α(φei,φej,ξ); i18(α)=Xα(ei,ei,φek)α(ξ,ξ,ek);
i,j i,k
6 A.J.TODD
where {e ,...,e ,ξ} is an orthonormal basis of V and α ∈ C(V). They then give a decomposition of
1 2n
C(V) into orthogonalirreducible factors where orthogonality is with respect to the inner product given by
2n+1
hα,α˜i= α(e ,e ,e )α˜(e ,e ,e ) (23)
i j k i j k
i,j,k=1
X
where α,α˜ ∈C(V)and{e } is anorthonormalbasisofV. First, they decomposeC(V) intothree orthogonal
i
subspaces D , D and C defined by
1 2 12
D ={α∈C(V):α(ξ,x,y)=α(x,ξ,y)=0} (24)
1
D ={α∈C(V):α(x,y,z)=η(x)α(ξ,y,z)+η(y)α(x,ξ,z)+η(z)α(x,y,ξ)} (25)
2
C ={α∈C(V):α(x,y,z)=η(x)η(y)α(ξ,ξ,z)+η(x)η(z)α(ξ,y,ξ)} (26)
12
Next, they further decompose D into four orthogonal irreducible subspaces given by
1
C ={α∈C(V):α(x,x,y)=α(x,y,ξ)=0} (27)
1
C ={α∈C(V):α(x,y,z)+α(y,z,x)+α(z,x,y)=0, α(x,y,ξ)=0} (28)
2
C ={α∈C(V):α(x,y,z)−α(φx,φy,z)=0, c α=0} (29)
3 12
1
C ={α∈C(V):α(x,y,z)= [(hx,yi−η(x)η(y))c α(z)−(hx,zi−η(x)η(z))c α(y)
4 12 12
2(n−1) (30)
−hx,φyic α(φz)+hx,φzic α(φy)], c α(ξ)=0}
12 12 12
for any x,y,z ∈ V and c α(x) = α(e ,e ,x) where {e } is an orthonormal basis of V. Finally, they
12 i i i
decompose D into seven orthogonalirreducible subspaces given by
2
P
1
C ={α∈C(V):α(x,y,z)= [hx,φziη(y)c α(ξ)−hx,φyiη(z)c α(ξ)]} (31)
5 12 12
2n
1
C ={α∈C(V):α(x,y,z)= [hx,yiη(z)c α(ξ)−hx,ziη(y)c α(ξ)]} (32)
6 12 12
2n
C ={α∈C(V):α(x,y,z)=η(z)α(y,x,ξ)−η(y)α(φx,φz,ξ), c α(ξ)=0} (33)
7 12
C ={α∈C(V):al(x,y,z)=−η(z)α(y,x,ξ)−η(y)α(φx,φz,ξ), c α(ξ)=0} (34)
8 12
C ={α∈C(V):α(x,y,z)=η(z)α(y,x,ξ)+η(y)α(φx,φz,ξ)} (35)
9
C ={α∈C(V):α(x,y,z)=−η(z)α(y,x,ξ)+η(y)α(φx,φz,ξ)} (36)
10
C ={α∈C(V):α(x,y,z)=−η(x)α(ξ,φy,φz)} (37)
11
for any x,y,z ∈ V where c (ξ) = α(e ,φe ,ξ) and {e } is an orthonormal basis of V. Thus, we have 12
12 i i i
invariant subspaces C of C(V) which are orthogonaland irreducible with respect to the U(n)×1 action.
i
P
Using the definition of the inner product on C(V) together the quadratic invariants, we see that
||α||2 =i (α)+i (α)+2i (α)+i (α) (38)
1 5 6 16
||c (α)||2 =i (α)+i (α)+i (α)+2i (α) (39)
12 4 10 16 17
||c (α)||2 =i (α)+i (α) (40)
12 4 14
Thenthere existlinear relationsamongthe quadraticinvariantsfor eachofthe subspacesC whichareused
i
to prove that each of the subspaces is irreducible. These relations will prove useful to us, so we give them
ON AN ALMOST CONTACT STRUCTURE ON G2-MANIFOLDS 7
below. Let A={1,2,3,4,5,7,11,13,15,16,17,18}.
C : i (α)=−i (α)=−i (α)=||α||2; i (α)=0 for m≥4;
1 1 2 3 m
C : i (α)=2i (α)=−i (α)=||α||2; i (α)=0 for m≥4;
2 1 2 3 m
C : i (α)=i (α)=||α||2; i (α)=i (α)=0 for m≥4;
3 1 3 2 m
2n
n n
C : i (α)=i (α)= i (α)= c2 (α)(e ); i (α)=i (α)=0 for m>4;
4 1 3 (n−1)2 4 (n−1)2 12 k 2 m
k
X
1
C : i (α)=−i (α)=i (α)=−i (α)= i (α); i (α)=i (α)=0 for m∈A;
5 6 8 9 12 14 10 m
2n
1
C : i (α)=i (α)=i (α)=i (α)= i (α); i (α)=i (α)=0 for m∈A;
6 6 8 9 12 10 14 m
2n
||α||2
C : i (α)=i (α)=i (α)=−i (α)= ; i (α)=i (α)=i (α) for m∈A;
7 6 8 9 12 10 14 m
2
||α||2
C : i (α)=−i (α)=i (α)=−i (α)= ; i (α)=i (α)=i (α) for m∈A;
8 6 8 9 12 10 14 m
2
||α||2
C : i (α)=i (α)=−i (α)=−i (α)= ; i (α)=i (α)=i (α) for m∈A;
9 6 8 9 12 10 14 m
2
||α||2
C : i (α)=−i (α)=−i (α)=i (α)= ; i (α)=i (α)=i (α) for m∈A;
10 6 8 9 12 10 14 m
2
C : i (α)=||α||2; i (α)=0 for m6=5;
11 5 m
C : i (α)=||α||2; i (α)=0 for m6=16;
12 16 m
Now,letM beanodd-dimensionalmanifoldwithanalmostcontactmetricstructure;then,forallp∈M,
T M is an odd-dimensional vector space with the almost contact metric structure (φ ,ξ ,η ,g ). Hence,
p p p p p
we can decompose the vector space C(T M) according to the decomposition above, and we say that the
p
almost contact structure is of class C , where C is one of the invariant subspaces of C(T M), if (∇ω) ∈C
j j p p j
for all p ∈ M. Note that α(x,y,z) from above, becomes (∇ ω)(Y,Z) for vector fields X,Y,Z on M, that
X
c (∇ω)(X)=−δω(X) and that c (∇ω)(ξ)=δη where we have used the formula
12 12
n
δ·=− e y(∇ ·) (41)
i ei
i=1
X
for the codifferential where {e } is an orthonormalframe.
i
ChineaandGonzalezthengivetherelationshipbetweentheclassesC andthevarioustypesofalmostcon-
i
tactmetricmanifoldsdefinedabove. Again,wewillrefertovariousoftheserelationshipsinourclassification,
so we will give them here.
• Almost cosymplectic: C ⊕C
2 9
• Quasi-Sasakian: C ⊕C
6 7
• a-Kenmotsu: C
5
• a-Sasakian: C
6
• Nearly K-cosymplectic: C
1
• Quasi-K-cosymplectic: C ⊕C ⊕C ⊕C
1 2 9 10
• Semi-cosymplectic: C ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C
1 2 3 7 8 9 10 11 12
• Trans-Sasakian: C ⊕C
5 6
• Nearly trans-Sasakian: C ⊕C ⊕C
1 5 6
• Almost K-contact: 10 C
j=1 j
• Normal: C ⊕C ⊕C ⊕C ⊕C ⊕C
3 4 5 6 7 8
L
3. Classification Theorems for the Standard G Almost Contact Metric Structure on
2
G -Manifolds
2
Throughoutthissection(M,ϕ)willdenoteaclosed,i. e.,compactwithemptyboundary,G -manifoldwith
2
thestandardG almostcontactstructure(φ,ξ,η)and{e ,e ,e ,e ,e ,e ,ξ}willdenotealocalorthonormal
2 1 2 3 4 5 6
8 A.J.TODD
frame aboutanarbitrarypoint. Also,we denote the G -metric g simply by g asit is the only metric under
2 ϕ
consideration in this section.
We beginby notingsome helpfulformulae. RecallthatonaG -manifold,the crossproductisparallel,so
2
(∇ φ)(Y)=∇ (φY)−φ(∇ Y)=∇ (ξ×Y)−ξ×(∇ Y)=∇ ξ×Y (42)
X X X X X X
In particular, ∇φ=0 if and only if ∇ξ =0. Further, since g(ξ,ξ)=1, we find
0=∇ g(ξ,ξ)=2g(∇ ξ,ξ) (43)
X X
which implies that ∇ ξ is perpendicular to ξ for all vector fields X on M, and finally, for any vector fields
X
X,Y,Z on M, we have
(∇ ω)(Y,Z)=∇ ω(Y,Z)−ω(∇ Y,Z)−ω(Y,∇ Z)=∇ g(Y,φZ)−g(∇ Y,φZ)−g(Y,φ(∇ Z))
X X X X X X X
=g(∇ Y,φZ)+g(Y,(∇ φ)Z)+g(Y,φ(∇ Z))−g(∇ Y,φZ)−g(Y,φ(∇ Z))
X X X X X
=g(Y,∇ ξ×Z)
X
(44)
Theorem 3.1. dω =0 if and only if ∇ξ =0. Further, in this case, dη =0, and thealmost contact structure
is normal and hence is cosymplectic.
Proof. We note that
ω(X,Y)=g(X,φY)=g(X,ξ×Y)=g(ξ×Y,X)=ϕ(ξ,Y,X)=−ϕ(ξ,X,Y)=(−ξyϕ)(X,Y) (45)
Thus, using the fact that dϕ=0, we see that
−dω =d(ξyϕ)=ξydϕ+d(ξyϕ)=L ϕ (46)
ξ
From this, we conclude that dω =0 if and only if L ϕ=0; recall from the Introduction we have L ϕ=0 if
ξ ξ
andonly if L g =0,that is, if and only if ξ is Killing with respectto the G -metric. Since G -manifolds are
ξ 2 2
Ricci-flat, ξ is Killing if and only if ∇ξ =0.
We assume for the remainder of this proof that ∇ξ =0. Then
2dη(X,Y)=Xη(Y)−Yη(X)−η([X,Y])=Xg(ξ,Y)−Yg(ξ,X)−g(ξ,[X,Y])
=g(∇Xξ,Y)+g(ξ,∇XY)−g(∇Yξ,X)−g(ξ,∇YX)−g(ξ,∇XY −∇YX) (47)
=g(ξ,∇ Y −∇ X −∇ Y +∇ X)=0
X Y X Y
Finally, to see that in this case the almost contact metric structure is normal, we first recall that ∇ξ =0
implies ∇φ=0. Then calculating
[φ,φ](X,Y)+2dη(X,Y)ξ
=φ2[X,Y]−φ[φX,Y]−φ[X,φY]+[φX,φY]
=−[X,Y]+η([X,Y])ξ−φ[φX,Y]−φ[X,φY]+[φX,φY]
=−[X,Y]+g(ξ,[X,Y])ξ−φ[φX,Y]−φ[X,φY]+[φX,φY]
=−[X,Y]+g(ξ,[X,Y])ξ−φ{∇ Y −(∇ φ)X −φ∇ X}−φ{(∇ φ)Y +φ∇ Y −∇ X}
φX Y Y X X φY
+(∇ φ)Y +φ∇ Y −(∇ φ)X −φ∇ X
φX φX φY φY
=−[X,Y]+g(ξ,[X,Y])ξ−φ∇ Y +φ2∇ X −φ2∇ Y +φ∇ X +φ∇ Y −φ∇ X
φX Y X φY φX φY
=−[X,Y]+g(ξ,[X,Y])ξ−∇ X +η(∇ X)ξ+∇ Y −η(∇ Y)ξ
Y Y X X
=−∇ Y +∇ X +∇ Y −∇ X +g(ξ,∇ Y −∇ X +∇ X −∇ Y)ξ =0
X Y X Y X Y Y X
(48)
(cid:3)
Corollary 3.2. If (M,ϕ) is a G -manifold with full G -holonomy, then dω 6= 0 (or equivalently, ∇ξ 6= 0).
2 2
In particular, the standard G almost contact metric structure cannot be cosymplectic, almost cosymplectic
2
nor quasi-Sasakian.
Proof. The first statement follows since the existence of a parallel vector field on a G -manifold implies a
2
reduction in the holonomy group to a proper subgroup of G . The second statement follows since all three
2
geometries require dω =0. (cid:3)
ON AN ALMOST CONTACT STRUCTURE ON G2-MANIFOLDS 9
Corollary 3.3. If (M,ϕ) is a G -manifold, then the standard G almost contact metric structure cannot be
2 2
a contact metric structure. In particular, it cannot be Sasakian (or more generally, a-Sasakian where a is
constant).
Proof. Recallthatacontactmetricstructurerequiresω =dη whichimmediatelyyields dω =0;however,by
the abovetheorem, this implies that dη =0 which gives ω =0, contradicting the fact that η∧(ω)3 6=0. (cid:3)
Note that Chinea and Gonzalez’ result means that we can decompose ∇ω as follows:
(∇ ω)(Y,Z)=α (X,Y,Z)+α (X,Y,Z)+β (X,Y,Z)
X 1 2 12
=(β1(X,Y,Z)+β2(X,Y,Z)+β3(X,Y,Z)+β4(X,Y,Z))+(β5(X,Y,Z)+β6(X,Y,Z) (49)
+β (X,Y,Z)+β (X,Y,Z)+β (X,Y,Z)+β (X,Y,Z)+β (X,Y,Z))+β (X,Y,Z)
7 8 9 10 11 12
where α ∈ D , α ∈ D , β ∈ C for all i = 1,2,...,12 with α = β +β +β +β and α = 11 β .
1 1 2 2 i i 1 1 2 3 4 2 i=5 i
Since we have classified the almost contact metric structures where ξ is parallel, we now consider the case
P
that ∇ξ 6=0. We begin with the following observation.
Lemma 3.4. (∇ ω)(ξ,Y)=0 if and only if ∇ξ =0.
X
Proof. We note that
(∇ ω)(ξ,Y)=g(ξ,∇ ξ×Y)=g(ξ×∇ ξ,Y) (50)
X X X
Thus,(∇ ω)(ξ,Y)=0forallvectorfieldsX,Y ifandonlyifξ×∇ ξ =0forallvectorfieldsX. Ofcourse,
X X
if ∇ ξ =0 for all vector fields X, then ξ×∇ ξ =0 for all vector fields X. Conversely, if ξ×∇ ξ =0 for
X X X
all vector fields X, then
0=ξ×(ξ×∇ ξ)=−∇ ξ+g(ξ,∇ ξ)ξ =−∇ ξ (51)
X X X X
where the first equality follows from (12) and the secondfollows fromthe fact that ∇ ξ is perpendicular to
X
ξ for all X, and hence the result follows. (cid:3)
Since α requires in particular that α (X,ξ,Y)=0 for all X,Y, this immediately yields the following.
1 1
Corollary 3.5. If ∇ξ 6=0, then α =0.
1
Now, that ∇ξ 6= 0 implies that ξ is not a harmonic vector field, so in particular, we must have either
dη 6=0 or δη 6=0 or both. Note that since ||δη||2 at each point is given by i (∇ω)+i (∇ω), δη =0 for all
4 14
p ∈ M, for all classes C except when j = 4,5; since we already have that β = 0 because α = 0, we see
j 4 1
that, if ∇ξ 6=0, then δη 6=0 if and only if β 6=0.
5
To continue with the classification, we note that under the condition that ∇ξ 6=0, there are two natural
cases: either ∇ ξ = 0 (in which case there exists some vector field X 6= ξ with ∇ ξ 6= 0) or ∇ ξ 6= 0. The
ξ X ξ
following lemmas will prove useful.
Lemma 3.6. i (∇ω)=0 if and only if ∇ ξ =0 for all j =1,...,6.
6 ej
Proof. Note that, from the definitions of the quadratic invariants given in Section 2
6
2
0=i (∇ω)= (∇ ω)(ξ,e ) (52)
6 ej k
j,k=1
X (cid:0) (cid:1)
if and only if for all j,k =1,...,6
0=(∇ ω)(ξ,e )=g(ξ,∇ ξ×e )=g(∇ ξ×e ,ξ)=ϕ(∇ ξ,e ,ξ)=ϕ(ξ,∇ ξ,e )=g(ξ×∇ ξ,e )
ej k ej k ej k ej k ej k ej k
(53)
Recalling that ξ × ∇ ξ is also perpendicular to ξ, we see, by nondegeneracy of the metric, that g(ξ ×
ej
∇ ξ,e )=0 for all j,k =1,...,6 if and only if for all j =1,...,6
ej k
ξ×∇ ξ =0 (54)
ej
Since ξ×(ξ×∇ ξ)=−∇ ξ+g(ξ,∇ ξ)ξ =−∇ ξ, we thus find that ξ×∇ ξ =0 for all j =1,...,6 if
ej ej ej ej ej
and only if ∇ ξ =0 for all j =1,...,6 as we wanted to show. (cid:3)
ej
Using similar techniques, we can show
Lemma 3.7. i (∇ω)= ((∇ ω)(ξ,e ))2 =0 if and only if ∇ ξ =0.
16 k ξ k ξ
P
10 A.J.TODD
From this, we see that ∇ ξ 6= 0 if and only if i 6= 0; however, the only class that allows i 6= 0 is C
ξ 16 16 12
and hence we conclude that β 6= 0 if and only if ∇ ξ 6= 0. Further, we note that since class C requires
12 ξ 11
both i (β )6=0 and i (β )=0; however,by the above, i (β )=0 implies ∇ ξ =0, so
5 11 16 11 16 11 ξ
i (β )= g(e ,∇ ξ×e )=0 (55)
5 11 j ξ k
j,k
X
Hence, we must have β =0 always. Summarizing these results, we have the following theorem.
11
Theorem 3.8. Let M be a G -manifold with the standard G almost contact metric structure. If ∇ξ 6= 0,
2 2
then the almost contact metric structure is of class G where
(1) In general, we have
G ⊆C ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C (56)
5 6 7 8 9 10 12
(2) In the case that δη =0, then
G ⊆C ⊕C ⊕C ⊕C ⊕C ⊕C (57)
6 7 8 9 10 12
(3) In the case that ∇ ξ =0, then
ξ
G ⊆C ⊕C ⊕C ⊕C ⊕C ⊕C (58)
5 6 7 8 9 10
(4) In the case that both δη =0 and ∇ ξ =0,
ξ
G ⊆C ⊕C ⊕C ⊕C ⊕C (59)
6 7 8 9 10
(5) In the case that the almost contact structure is normal, then
G ⊆C ⊕C ⊕C ⊕C (60)
5 6 7 8
(6) In the case that the almost contact structure is normal and δη =0, then
G ⊆C ⊕C ⊕C (61)
6 7 8
4. Almost Contact 3-Structures
Wenowdefinealmostcontact3-structuresasdefinedoriginallybyKuo[26]. LetM beasmoothmanifold
of dimension 4m+3 [26, Thm. 4]. Then an almost contact 3-structure on M is a reduction of the structure
group of the tangent bundle of M to Sp(m)×I where I is the 3×3 identity matrix [26, Thm. 5]. Such
3 3
a reduction of the structure group of the tangent bundle of M is equivalent to the existence of 3 almost
contact structures (φ ,ξ ,η ), i=1,2,3 satisfying
i i i
η (ξ )=η (ξ )=0 (62)
i j j i
φ ξ =−φ ξ =ξ (63)
i j j i k
η ◦φ =−η ◦φ =η (64)
i j j i k
φ φ −η ⊗ξ =−φ φ +η ⊗ξ =φ (65)
i j j i j i i j k
for any cyclic permutation(i,j,k) of(1,2,3). Of fundamental importanceis that if M4m+3 admits 2 almost
contact structures (φ ,ξ ,η ), i=1,2 satisfying
i i i
η (ξ )=η (ξ )=0 (66)
1 2 2 1
φ ξ =−φ ξ (67)
1 2 2 1
η ◦φ =−η ◦φ (68)
1 2 2 1
φ φ −η ⊗ξ =−φ φ +η ⊗ξ (69)
1 2 2 1 2 1 1 2
theninfactM admitsanalmostcontact3-structurebysettingφ =φ φ −η ⊗ξ ,ξ =φ ξ andη =η ◦φ ;
3 1 2 2 1 3 1 2 3 1 2
see [26, Thm. 1]. If there exists a single Riemannian metric g that is compatible with all 3 of the almost
contact structures, then (φ ,ξ ,η ,g), i = 1,2,3 is called an almost contact metric 3-structure. Again, of
i i i
fundamental importance is that if M4m+3 admits 2 almost contact metric structures (φ ,ξ ,η ,g), i = 1,2,
i i i
then (66), (67) and (68) follow from (69); see [26, Thm. 2]. Almost contact 3-structures were introduced
in order to give a structure of contact-type that is similar to an almost quaternionic structure in the same
waythatanalmostcontactstructureissimilartoanalmostcomplexstructure. Almostcontact3-structures
have been the subject of many articles and have been shown to have importance in physics as well.