Almost Contact Metric Structures Induced by $G_2$ Structures

0.15 MB·

by  Nülifer Özdemir

Almost Contact Metric Structures Induced by
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1 G Structures
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n Nu¨lifer O¨zdemir∗
a
J
Department of Mathematics, Anadolu University, 26470 Eski¸sehir, Turkey
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]
G Mehmet Solgun†
D
Department of Mathematics, Bilecik Seyh Edebali University, 11210 Bilecik, Turkey
.
h
t
a
S¸irin Aktay‡
m
[ Department of Mathematics, Anadolu University, 26470 Eski¸sehir, Turkey

1
v
7
3
5
1 Abstract
0
.
1 We study almost contact metric structures induced by 2-fold vector
0 cross products on manifolds with G structures. We get some results
6 2
on possible classes of almost contact metric structures. Finally we
1
: give examples.
v
i
X
r
a
∗E.mail: [email protected]
†E.mail: [email protected]
‡E.mail: [email protected]
This study was supported by Anadolu University Scientific Research Projects Com-
mission under the grant no: 1501F017.
1
1 Introduction
A recent research area in geometry is the relation between manifolds with
structure group G and almost contact metric manifolds. A manifold with
2
G structure has a 3-form globally defined on its tangent bundle with some
2
properties. Such manifolds are classified into sixteen classes by Ferna´ndez
and Gray in [10] according to the properties of the covariant derivative of
the 3-form.
On an almost contact metric manifold, there exists a global 2-form and
the properties of the covariant derivative of this 2-form yields 212 classes of
almost contact metric manifolds, see [3, 9].
Recently Matzeu and Munteanu constructed almost contact metric struc-
ture induced by the 2-fold vector cross product on some classes of manifolds
with G structures [12]. Arikan et.al. proved the existence of almost contact
2
metric structures on manifolds with G structures [4]. Todd studied almost
2
contact metric structures on manifolds with parallel G structures [14].
2
Ouraimistostudyalmostcontactmetricstructuresonmanifoldswithar-
bitrary G structures. We eliminate some classes that almost contact metric
2
structure induced from a G structure may belong to according to proper-
2
ties of characteristic vector field of the almost contact metric structure. In
particular, we also investigate the possible classes of almost contact met-
ric structures on manifolds with nearly parallel G structures. In addition,
2
we give examples of almost contact metric structures on manifolds with G
2
structures induced by the 2-fold vector cross product.
2 Preliminaries
Consider R7 with the standard basis {e ,...,e }. The fundamental 3-form on
1 7
R7 is defined as
ϕ = e123 +e145 +e167 +e246 −e257 −e347 −e356
0
where {e1,...,e7} is the dual basis of the standard basis and eijk = ei∧ej∧ek.
Then compact, simple and simply connected 14-dimensional Lie group G is
2
G := {f ∈ GL(7,R) | f∗ϕ = ϕ }.
2 0 0
A manifold with G structure is a 7-dimensional oriented manifold whose
2
structure group reduces to the group G . In this case, there exists a global
2
1
3-form ϕ on M such that for all p ∈ M, (T M,ϕ ) ∼= (R7,ϕ ). This 3-
p p 0
form is called the fundamental 3-form or the G structure on M and gives a
2
Riemannian metric g, a volume form and a 2-fold vector cross product P on
M defined by ϕ(x,y,z) = g(P(x,y),z) for all vector fields x,y on M [8].
Manifolds (M,g) with G structure ϕ were classified according to prop-
2
erties of the covariant derivative of the fundamental 3-form. The space
W = {α ∈ (R7)∗ ⊗Λ3(R7)∗|α(x,y ∧z ∧P(y,z)) = 0 ∀x,y,z ∈ R7}
of of tensors having the same symmetry properties asthe covariant derivative
ofϕwaswritten, andthenthisspacewasdecomposedintofourG -irreducible
2
subspaces using the representation of the group G on W. Since
2
(∇ϕ)∈W ={α∈T∗M ⊗Λ3(T∗M)|α(x,y∧z∧P(y,z))=0 ∀x,y,z∈T M}
p p p p p
and there are 16 invariant subspaces of W , each subspace corresponds to
p
a different class of manifolds with G structure. For example, the class P,
2
in which the covariant derivative of ϕ is zero, is the class of manifolds with
parallel G structure. A manifold which is in this class is sometimes called a
2
G manifold. W corresponds to the class of nearly parallel manifolds, which
2 1
are manifolds with G structure ϕ satisfying dϕ = k∗ϕ for some constant k
2
[10].
Let M2n+1 be a differentiable manifold of dimension 2n+1. If there is a
(1,1) tensor field φ, a vector field ξ and a 1-form η on M satisfying
φ2 = −I +η ⊗ξ, η(ξ) = 1,
then M is said to have an almost contact structure (φ,ξ,η). A manifold with
an almost contact structure is called an almost contact manifold.
If in addition to an almost contact structure (φ,ξ,η), M also admits a
Riemannian metric g such that
g(φ(x),φ(y)) = g(x,y)−η(x)η(y)
for all vector fields x,y, then M is an almost contact metric manifold with
the almost contact metric structure (φ,ξ,η,g). The Riemannian metric g is
called a compatible metric. The 2-form Φ defined by
Φ(x,y) = g(x,φ(y))
2
for all x,y ∈ Γ(TM) is called the fundamental 2-form of the almost contact
metric manifold (M,φ,ξ,η,g).
In [9], a classification of almost contact metric manifolds was obtained via
the study of the covariant derivative of the fundamental 2-form. Let (ξ,η,g)
be an almost contact metric structure on R2n+1. A space
C = {α ∈ ⊗0R2n+1|α(x,y,z) = −α(x,z,y) = −α(x,φy,φz)
3
+η(y)α(x,ξ,z)+η(z)α(x,y,ξ)}
having the same symmetries as the covariant derivative of the fundamental
2-form was given. First this space was written as a direct sum of three
subspaces
D = {α ∈ C|α(ξ,x,y) = α(x,ξ,y) = 0},
1
D = {α ∈ C|α(x,y,z) = η(x)α(ξ,y,z)+η(y)α(x,ξ,z)+η(z)α(x,y,ξ)}
2
and
C = {α ∈ C|α(x,y,z) = η(x)η(y)α(ξ,ξ,z)+η(x)η(z)α(ξ,y,ξ)}
12
and then, D , D were decomposed into U(n) × 1 irreducible components
1 2
C ,...,C and C ,...,C , respectively. Thus there are 212 invariant sub-
1 4 5 11
spaces, denoted by C ,...,C , each corresponding to a class of almost con-
1 12
tact metric manifolds. For example, the trivial class such that ∇Φ = 0 cor-
responds to the class of cosymplectic [5] (called co-Ka¨hler by some authors)
manifolds, C is the class of nearly-K-cosymplectic manifolds, etc.
1
In the classification of Chinea and Gonzales, it was shown that the space
of quadratic invariants of C is generated by the following 18 elements:
i (α) = α(e ,e ,e )2 i (α) = α(e ,e ,e )α(e ,e ,e )
1 i j k 2 i j k j i k
P P
i,j,k i,j,k
i (α) = α(e ,e ,e )α(φe ,φe ,e ) i (α) = α(e ,e ,e )α(e ,e ,e )
3 i j k i j k 4 i i k j j k
P P
i,j,k i,j,k
i (α) = α(ξ,e ,e )2 i (α) = α(e ,ξ,e )2
5 j k 6 i k
P P
j,k i,k
i (α) = α(ξ,e ,e )α(e ,ξ,e ) i (α) = α(e ,e ,ξ)α(e ,e ,ξ)
7 j k j k 8 i j j i
P P
j,k i,j
i (α) = α(e ,e ,ξ)α(φe ,φe ,ξ) i (α) = α(e ,e ,ξ)α(e ,e ,ξ)
9 i j i j 10 i i j j
P P
i,j i,j
i (α) = α(e ,e ,ξ)α(e ,φe ,ξ) i (α) = α(e ,e ,ξ)α(φe ,φe ,ξ)
11 i j j i 12 i j j i
P P
i,j i,j
3
i (α) = α(ξ,e ,e )α(φe ,ξ,e ) i (α) = α(e ,φe ,ξ)α(e ,φe ,ξ)
13 j k j k 14 i i j j
P P
j,k i,j
i (α) = α(e ,φe ,ξ)α(e ,e ,ξ) i (α) = α(ξ,ξ,e )2
15 i i j j 16 k
P P
i,j k
i (α) = α(e ,e ,e )α(ξ,ξ,e ) i (α) = α(e ,e ,φe )α(ξ,ξ,e )
17 i i k k 18 i i k k
P P
i,k i,k
where {e ,e ,...,e ,ξ} is a local orthonormal basis. Also following relations
1 2 6
among quadratic invariants were expressed for manifolds having dimensions
≥ 7, where α ∈ C and A = {1,2,3,4,5,7,11,13,15,16,17,18}:
C1 :i1(α) = −i2(α) = −i3(α) = ||α||2; im(α) = 0 (m ≥ 4)
C2 :i1(α) = 2i2(α) = −i3(α) = ||α||2; im(α) = 0 (m ≥ 4)
C3 :i1(α) = i3(α) = ||α||2; i2(α) = im(α) = 0 (m ≥ 4)
2n
C4 :i1(α) = i3(α) = (n−n1)2i4(α) = (n−n1)2 Pc212(α)(ek);
k
i (α) = i (α) = 0 (m > 4)
2 m
C5 :i6(α) = −i8(α) = i9(α) = −i12(α) = 21ni14(α);
i (α) = i (α) = 0 (m ∈ A)
10 m
C6 :i6(α) = i8(α) = i9(α) = i12(α) = 21ni10(α);
i (α) = i (α) = 0 (m ∈ A)
14 m
C7 :i6(α) = i8(α) = i9(α) = −i12(α) = ||α2||2;
i (α) = i (α) = i (α) = 0 (m ∈ A)
10 14 m
C8 :i6(α) = −i8(α) = i9(α) = −i12(α) = ||α2||2;
i (α) = i (α) = i (α) = 0 (m ∈ A)
10 14 m
C9 :i6(α) = i8(α) = −i9(α) = −i12(α) = ||α2||2;
i (α) = i (α) = i (α) = 0 (m ∈ A)
10 14 m
C10 :i6(α) = −i8(α) = −i9(α) = i12(α) = ||α2||2;
i (α) = i (α) = i (α) = 0 (m ∈ A)
10 14 m
C11 :i5(α) = ||α||2;im(α) = 0 (m 6= 5)
C12 :i16(α) = ||α||2;im(α) = 0 (m 6= 16)
For details, refer to [9].
We give below most studied classes of almost contact metric structures
as direct sum of spaces C :
i
| C |= the class of cosymplectic manifolds.
C = the class of nearly-K-cosymplectic manifolds.
1
C ⊕C = the class of almost cosymplectic manifolds.
2 9
4
C = the class of α-Kenmotsu manifolds.
5
C = the class of α-Sasakian manifolds.
6
C ⊕C = the class of trans-Sasakian manifolds.
5 6
C ⊕C = the class of quasi-Sasakian manifolds.
6 7
C ⊕C ⊕C = the class of semi-cosymplectic and normal manifolds.
3 7 8
C ⊕C ⊕C = the class of nearly trans-Sasakian manifolds.
1 5 6
C ⊕C ⊕C ⊕C = the class of quasi-K-cosymplectic manifolds.
1 2 9 10
C ⊕C ⊕C ⊕C ⊕C ⊕C = the class of normal manifolds.
3 4 5 6 7 8
D ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C = the class of almost-K-contact manifolds.
1 5 6 7 8 9 10
C ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C ⊕C = the class of semi-cosymplectic manifolds.
1 2 3 7 8 9 10 11
Let (M,g) be a 7-dimensional Riemannian manifold with G structure ϕ
2
and the associated 2-fold vector cross product × and let ξ be a nowhere zero
vector field of unit length on M. Then for
φ(x) := ξ ×x η(x) := g(ξ,x),
(φ,ξ,η,g) is an almost contact metric structure on M [12, 4]. Throughout
thisstudy, (φ,ξ,η,g)willdenotethealmostcontactmetricstructure(a.c.m.s)
induced by the G structure ϕ on M and Φ the fundamental 2-form of the
2
a.c.m.s.
3 Almost contact metric structures obtained
from G Structures
2
Let M be a manifold with G structure ϕ and ξ a nowhere zero unit vector
2
field on M and (φ,ξ,η,g) the a.c.m.s. with the fundamental form Φ induced
by the G structure ϕ.
2
If ∇ϕ = 0, then it can be seen that ∇Φ = 0 if and only if ∇ξ = 0 [2, 14].
If ξ is a Killing vector field on a manifold with any G structure, then
2
dη(x,y) = 1{(∇ η)(y)−(∇ η)(x)}
2 x y
= 1{g(∇ ξ,y)−g(∇ ξ,x)}
2 x y
= g(∇ ξ,y),
x
5
which implies
dη = 0 ⇔ ∇ξ = 0.
Therefore if the Killing vector field ξ is not parallel, then the a.c.m.s. can
not be nearly-K-cosymplectic (C ).
1
To deduce further results, we focus on the covariant derivative of the
fundamental 2-form Φ, where the a.c.m.s. (φ,ξ,η,g) is obtained from a G
2
structure of any class and ξ is any nonzero vector field. Direct calculation
gives
(∇ Φ)(y,z) = g(y,∇ (ξ ×z))+g(∇ z,ξ ×y). (3.1)
x x x
We also compute some of i (∇Φ),(k = 1,...,18) to understand which class
k
∇Φ may belong to.
Proposition 3.1 Let ϕ be a G structure on M of an arbitrary class and
2
(φ,ξ,η,g) an a.c.m.s. obtained from ϕ. Then
a. i (∇Φ) = 0 if and only if ∇ ξ = 0 for i = 1,··· ,6 (Note that ∇ ξ
6 ei ξ
need not be zero),
b. i (∇Φ) = 0 if and only if ∇ ξ = 0.
16 ξ
Proof By direct calculation, for any i,k ∈ {1,2,...,6}
(∇ Φ)(ξ,e ) = g(ξ,∇ (ξ ×e ))+g(∇ e ,ξ ×ξ)
ei k ei k ei k
= g(ξ,∇ (ξ ×e ))
ei k
= −g(∇ ξ,ξ ×e )
ei k
and thus, we obtain
i (∇Φ) = ((∇ Φ)(ξ,e ))2 = g(∇ ξ,ξ ×e )2.
6 X ei k X ei k
i,k i,k
Since ξ ×e is also a frame element, i (∇Φ) = 0 if and only if ∇ ξ is zero.
k 6 ei
Similarly,
(∇ Φ)(ξ,e ) = g(ξ,∇ (ξ ×e ))+g(∇ e ,ξ ×ξ)
ξ k ξ k ξ k
= −g(∇ ξ,ξ ×e )
ξ k
for any k ∈ {1,2,...,6}, and we get
i (∇Φ) = (∇ Φ)(ξ,e )2 = g(∇ ξ,ξ ×e )2.
16 X ξ k X ξ k
k k
6
Note that g(∇ ξ,ξ) = 0 since ξ is of unit length. As a result, i (∇Φ) = 0 if
ξ 16
and only if ∇ ξ = 0.
ξ
Proposition 3.2 Let (φ,η,ξ,g) be an almost contact metric structure in-
duced by a G structure ϕ. Then,
2
• i (∇Φ) = 0 if and only if div(ξ) = 0.
14
6
• i (∇Φ) = −div(ξ)g(ξ,v), where v = e ×(∇ ξ).
15 P j ej
j=1
Proof For any i,j ∈ {1,2,...,6} we have
(∇ Φ)(φe ,ξ) = g(ξ ×e ,∇ (ξ ×ξ))+g(∇ ξ,ξ ×(ξ ×e ))
ei i i ei ei i
= −g(∇ ξ,e )
ei i
= g(ξ,∇ e ).
ei i
On the other hand,
6 6
∇ e = − div(e )e −div(ξ)ξ −∇ ξ
X ei i X i i ξ
i=1 i=1
and thus
g(ξ, ∇ e ) = −g(ξ, div(e )e )−g(ξ,div(ξ)ξ)−g(ξ,∇ ξ)
X ei i X i i ξ
i i
= −div(ξ).
Then
i (∇Φ) = (∇ Φ)(φe ,ξ)(∇ Φ)(φe ,ξ)
14 P ei i ej j
i,j
= g(ξ, ∇ e ) g(ξ, ∇ e ) = (div(ξ))2.
(cid:16) P ei i (cid:17)(cid:16) P ej j (cid:17)
i j
Therefore, i (∇Φ) is zero if and only if div(ξ) is zero.
14
Similarly, from equations
(∇ Φ)(φe ,ξ) = −g(∇ ξ,e ) and (∇ Φ)(e ,ξ) = g(∇ ξ,ξ ×e )
ei i ei i ej j ej j
7
we have,
i (∇Φ) = (∇ Φ)(φe ,ξ) (∇ Φ)(e ,ξ)
15 X ei i ej j
i,j
= g(ξ,∇ e )g(∇ ξ,ξ ×e )
X ei i ej j
i,j
= g(ξ, ∇ e ) g(ξ,∇ (e ×ξ))
(cid:16) X ei i (cid:17)(cid:16)X ej j (cid:17)
i j
= g(ξ,−div(ξ)ξ)−g(ξ, div(e )e ) g(ξ,e ×∇ ξ)
(cid:16) X i i (cid:17)(cid:16)X j ej (cid:17)
i j
= −div(ξ).g(ξ,v).
Now consider in particular an a.c.m.s. induced by a nearly parallel G
2
structure.
Proposition 3.3 Let (φ,η,ξ,g) be an almost contact metric structure in-
duced by a nearly parallel G structure. Then,
2
• i (∇Φ) = 0 if and only if ∇ ξ = 0.
5 ξ
• If ∇ ξ = 0, then i (∇Φ) = i (∇Φ) = 0.
ξ 17 18
Proof Since ϕ is nearly parallel, for any j,k ∈ {1,2,...,6} we have
(∇ Φ)(e ,e ) = g(e ,∇ (ξ ×e ))+g(∇ e ,ξ ×e )
ξ j k j ξ k ξ k j
= g(e ,∇ ξ ×e )+g(e ,ξ ×∇ e )+g(∇ e ,ξ ×e )
j ξ k j ξ k ξ k j
= −g(∇ ξ,e ×e ).
ξ j k
So,
i (∇Φ) = ((∇ Φ)(e ,e ))2 = (g(∇ ξ,e ×e ))2
5 X ξ j k X ξ j k
j,k j,k
which is zero if and only if ∇ ξ is zero. Here, e ×e is also a frame element.
ξ j k
Similarly, For any i,k ∈ {1,2,...,6},
(∇ Φ)(e ,φe ) = g(e ,∇ (ξ ×(ξ ×e ))+g(∇ (ξ ×e ),ξ ×e )
ei i k i ei k ei k i
= g(e ,∇ (−e ))+g(∇ (ξ ×e ),ξ ×e )
i ei k ei k i
= g(∇ e ,e )+g(∇ (ξ ×e ),ξ ×e )
ei i k ei k i
8
(∇ Φ)(ξ,e ) = g(ξ,∇ (ξ ×e ))+g(∇ e ,ξ ×ξ)
ξ k ξ k ξ k
= g(ξ,∇ ξ ×e )+g(ξ,ξ ×∇ e )
ξ k ξ k
= −g(e ,(∇ ξ)×ξ).
k ξ
Then
i (∇Φ) = ((∇ Φ)(e ,φe ))((∇ Φ)(ξ,e ))
18 X ei i k ξ k
i,k
= − g(∇ e ,e )+g(∇ (ξ ×e ),ξ ×e ) g(e ,(∇ ξ)×ξ)
X(cid:16) ei i k ei k i (cid:17)(cid:16) k ξ (cid:17)
i,k
= − g(∇ e ,e )g(e ,(∇ ξ)×ξ)
X(cid:16) ei i k k ξ (cid:17)
i,k
− g(∇ (ξ ×e ),ξ ×e )g(e ,(∇ ξ)×ξ)
X(cid:16) ei k i k ξ (cid:17)
i,k
= − g(∇ e ,e )g(e ,(∇ ξ)×ξ)
X(cid:16) ei i k k ξ (cid:17)
i,k
+ g(∇ e ,e )g(e ,(∇ ξ)×ξ)
X(cid:16) ei i k k ξ (cid:17)
i,k
− g(ξ ×e ,e ×∇ ξ)g(e ,(∇ ξ)×ξ
X(cid:16) k i ei k ξ (cid:17)
i,k
= − g(ξ ×( g((∇ ξ)×ξ,e )e +g((∇ ξ)×ξ,ξ)ξ),e ×∇ ξ)
X X ξ k k ξ i ei
i k
= − g(ξ ×((∇ ξ)×ξ),e ×∇ ξ)
X ξ i ei
i
= −g(∇ ξ, (e ×∇ ξ)).
ξ X i ei
i
Thus, if ∇ ξ is zero, so is i (∇Φ).
ξ 18
For i we compute
17
(∇ Φ)(e ,e ) = g(e ,∇ (ξ ×e ))+g(∇ e ,ξ ×e )
ei i k i ei k ei k i
and
(∇ Φ)(ξ,e ) = g(ξ,∇ (ξ ×e ))+g(∇ e ,ξ ×ξ)
ξ k ξ k ξ k
= −g(∇ ξ,ξ ×e )
ξ k
= g(e ,ξ ×(∇ ξ))
k ξ
9