Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy
By using the fiber-base decompositions of manifolds, the definition of warped-like product is regarded as a generalization of multiply warped product manifolds, by allowing the fiber metric to be not block diagonal. We consider the (3 + 3 + 1) decomposition of 7-dimensional warped-like product manif...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508394715086848 |
|---|---|
| author | Uğuz, S. Угуз, С. |
| author_facet | Uğuz, S. Угуз, С. |
| author_sort | Uğuz, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:16:44Z |
| description | By using the fiber-base decompositions of manifolds, the definition of warped-like product is regarded as a generalization of multiply warped product manifolds, by allowing the fiber metric to be not block diagonal. We consider the (3 + 3 + 1) decomposition of 7-dimensional warped-like product manifolds, which is called a special warped-like product of the form $M = F × B$; where the base $B$ is a onedimensional Riemannian manifold and the fiber $F$ has the form $F = F_1 × F_2$ where $F_i ; i = 1, 2$, are Riemannian 3-manifolds. If all fibers are complete, connected, and simply connected, then they are isometric to $S_3$ with constant curvature $k > 0$ in the class of special warped-like product metrics admitting the (weak) $G_2$ holonomy determined by the fundamental 3-form. |
| first_indexed | 2026-03-24T02:24:31Z |
| format | Article |
| fulltext |
UDC 517.91
S. Uğuz (Harran Univ., Turkey)
SPECIAL WARPED-LIKE PRODUCT MANIFOLDS
WITH (WEAK) G2 HOLONOMY
СПЕЦIАЛЬНИЙ СПОТВОРЕНИЙ ДОБУТОК МНОГОВИДIВ
ЗI (СЛАБКОЮ) G2 ГОЛОНОМIЄЮ
By using fiber-base decomposition of the manifolds, the definition of warped-like product is considered as a generalization
of multiply-warped product manifolds, by allowing the fiber metric to be not block diagonal. We consider (3 + 3 + 1)
decomposition of 7-dimensional warped-like product manifolds, which is called a special warped-like product of the form
M = F × B, where the base B is a one-dimensional Riemannian manifold and the fibre F is of the form F = F1 × F2
where Fi, i = 1, 2, are Riemannian 3-manifolds. If all fibers are complete, connected, and simply connected, then the
fibers are isometric to S3 with constant curvature k > 0 in the class of special warped-like product metrics admitting the
(weak) G2 holonomy determined by the fundamental 3-form.
З використанням волоконних розкладiв многовидiв розглянуто визначення спотвореного добутку як узагальнення
багаторазово спотворених добуткiв многовидiв, при цьому волоконна метрика може не бути блочно-дiагональною.
Вивчено (3 + 3 + 1) розклади 7-вимiрних спотворених добуткiв многовидiв, що називаються спецiальними спотво-
реними виду M = F ×B, де база B — одновимiрний рiманiв многовид, а волокно F має фому F = F1 ×F2, де Fi,
i = 1, 2, — рiмановi 3-многовиди. Якщо всi волокна є повними i однозв’язними, то вони є iзометричними до S3 зi
сталою кривиною k > 0 у класi спецiальних спотворених метрик добутку, що допускають (слабку) G2 голономiю,
визначену фундаментальною 3-формою.
1. Introduction. The notion of holonomy group was introduced by Elie Cartan in 1923 [3, 4] and
proved to be an efficient tool for the classification of Riemannian manifolds. The list of possible
restricted holonomy groups of irreducible, simply-connected nonsymmetric spaces was given by
M. Berger in 1955 [5]. Berger’s list (refined later by the work of [6, 7]) includes the groups SO(n),
U(n), SU(n), Sp(n), Sp(n)Sp(1) that could occur in dimensions n, 2n and 4n respectively and
two special cases, G2 holonomy in 7 dimensions and Spin (7) holonomy in 8 dimensions. Manifolds
with holonomy SO(n) constitute the generic case, all others are denoted as manifolds with “special
holonomy”and the last two cases are described as manifolds with “exceptional holonomy”.
The existence of manifolds with exceptional holonomy was first demonstrated by R. Bryant [9],
then complete examples were given by R. Bryant and S. Salamon [10] and the first compact examples
were found by D. Joyce in 1996 [11]. The study of manifolds with exceptional holonomy and the
construction of explicit examples is still an active research area both for mathematics and physics
[12 – 20] (see also references of papers). The concept of weak holonomy was introduced by A. Gray in
1971 as an extension of holonomy [21]. Manifolds with weak holonomy groups are also investigated
as given in [22, 23].
The motivation for our work was firstly the explicit Spin (7) metric on S3 × S3 × R2 given
by Yasui and Ootsuka [24] and secondly the explicit G2 metric on SU(2) × SU(2) × R given by
Konishi and Naka [2]. We investigated whether one could obtain other solutions by relaxing some of
their assumptions, in particular without requiring the three dimensional submanifolds to be S3. We
noticed that their metric ansatzs were a generalization of warped products and we called “warped-
like product”as a general framework for multiply warped product manifolds. In this paper we study
special warped-like product manifolds with (weak) G2 holonomy.
c© S. UĞUZ, 2013
1126 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
SPECIAL WARPED-LIKE PRODUCT MANIFOLDS WITH (WEAK) G2 HOLONOMY 1127
The outline of paper is given as follows: we set up the basic definitions of our study in Section 2.
We defined “warped-like product metrics”as a general framework for our metrical ansatz and for a
special case we present (3 + 3 + 1) warped-like product manifolds in Section 3. As studied Spin (7)
case in [17, 18], we show that the requirement that the fundamental 3-form ϕ and the Hodge dual
of ϕ be closed forms, determines the connection and with suitable global assumptions, hence the
three manifolds (fibers) we started with are three spheres and we recover the Konishi and Naka
solution after gauge transformations in Section 4. In Section 5 weak holonomy in 7-dimensional case
is investigated for the special warped-like product metrics. Using the fundamental 3-form ϕ and its
relation with weak holonomy, we prove also that the fibers are isometric to S3 with constant curvature
k > 0 as obtained in Section 4. Conclusions of the study with further remarks are summarized in
Section 6.
2. Technical preliminaries. 2.1. G2 manifolds and the fundamental 3-form. As G2 is a
subgroup of SO(7), a manifold M with G2 holonomy is a real orientable 7-dimensional manifold,
called a G2 manifold which is classified by the existence of a certain 3-form ϕ which is called
fundamental form, denoted by ϕ [9].
2.1.1. G2-structure. A 7-dimensional manifold M admits a G2-structure if the structure group
of the frame bundle reduces to the exceptional Lie group G2 ⊂ SO(7) ⊂ GL(7) [27]. The existence
of a G2-structure on M is equivalent to the existence of a positive nondegenerate 3-form ϕ defined
on the whole manifold and using this 3-form it is possible to define a Riemannian metric gϕ on M [8]
gϕ(X,Y )vol =
1
6
iXϕ ∧ iY ϕ ∧ ϕ. (2.1)
If ϕ is parallel with respect to the Levi – Civita connection, i.e., ∇ϕ = 0, then the holonomy group
is contained in G2, the G2-structure is called parallel and the corresponding manifolds are called
G2-manifolds [27]. In this case the induced metric gϕ is Ricci-flat [25].
2.1.2. Manifolds with G2 holonomy. The definition of G2 manifold by using holonomy is
presented in the following definition.
Definition 2.1. Let (M, g) be a Riemannian manifold. If the holonomy group of g is contained
in G2, then M is called a G2 manifold.
In the present paper we are interested in 7-dimensional real oriented manifolds whose holonomy
group is a subgroup of G2. These manifolds are characterized by the existence of a closed, and G2
invariant 3-form called the “fundamental 3-form ϕ” [9]. Conversely, if the fundamental form and its
Hodge dual are closed, then the manifold has G2 holonomy, as given by the following theorem of
Fernandez and Gray.
Proposition 2.1 [27]. The holonomy group of a Riemannian metric (as given in (2.1)) defined
by the fundamental 3-form ϕ is contained in G2 if and only if dϕ = d ∗ ϕ = 0.
The proposition above implies that, assuming the existence of a globally defined fundamental 3-
form (i.e., existence of a positive nondegenerate 3-form), the problem of provingM hasG2 holonomy
is reduced to the local problem of checking that ϕ and ∗ϕ are closed forms. We shall do this under a
simplifying assumption, that we call “warped-like product”metric ansatz. As a special case in seven
dimensions, we shall consider product manifolds M = F1×F2×B, where F1 and F2 are 3-manifolds
and B is diffeomorphic to R. Since all 3-manifolds are paralellizable, the first assumption ensures
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1128 S. UĞUZ
the existence of independent sections of the fiber in the product decomposition M = F ×B and the
second assumption is made for convenience.
2.2. Weak holonomy group G2. The concept of weak holonomy group was introduced by Alfred
Gray in [7]. Much of the early work of Gray was concerned with the study of Riemannian manifolds
with special holonomy groups. We present Alfred Gray’s definition of the weak holonomy group of a
seven dimensional Riemannian manifold. Gray shows the following important result about the weak
holonomy group G2.
Theorem 2.1 [7]. Let (M, g) be a 7-dimensional Riemannian manifold with weak holonomy
group G2. Then M is an Einstein manifold.
It is well known that a manifold with holonomy G2 is Ricci-flat [25]. Thus, it follows from this
result that the weak holonomy G2 is indeed a more general notion.
Then the manifold with weak G2 holonomy can be obtained by the following definition.
Definition 2.2 [21]. A G2-structure ϕ is said to be weak holonomy G2 if dϕ = λ ∗ ϕ with
constant λ.
From the definition above, it is clear that d ∗ ϕ = 0 and thus this may indeed be considered as
a generalization of the holonomy equations dϕ = 0, d ∗ ϕ = 0. Our notation is given as follows: ei
and ei, i = 1, . . . , n, denote respectively local orthonormal frames for the tangent and the cotangent
bundles. This gives rise to local bases for k-forms denoted by
eij = ei ∧ ej , eijk = ei ∧ ej ∧ ek, eijkl = ei ∧ ej ∧ ek ∧ el . . . . (2.2)
In the following we shall omit the wedge symbol in exterior products. The explicit expression of the
fundamental 3-form ϕ is chosen as [2]
ϕ = e123 − e156 + e246 − e345 + e147 + e367 + e257. (2.3)
And the Hodge dual of the fundamental 3-form is written as follows:
∗ϕ = e4567 − e2347 + e1357 − e1267 + e2356 + e1245 + e1346. (2.4)
3. Warped-like product manifolds. Let (F, gF ), (B, gB) be Riemannian manifolds and f > 0
be smooth function on B. A warped product manifold is a product manifold M = F ×B equipped
with the metric
Let (F, gF ), (B, gB) be Riemannian manifolds and f > 0 be smooth function on B. A warped
product manifold is a product manifold M = F ×B equipped with the metric
g = π∗2gB + (f ◦ π2)2π∗1gF ,
where π1 : F ×B −→ F and π2 : F ×B −→ B are the natural projections [30]. A generalization of
the notion of warped product metrics is the “multiply-warped products defined as follows [31]. Let
(Fi, gi), i = 1, 2, . . . , k, and (B, gB) be Riemannian manifolds and fi > 0 be smooth functions on
B. A multiply-warped product manifold is the product manifold F1 × F2 × ...× Fk × B, equipped
with the metric
g = π∗BgB +
k∑
i=1
(fi ◦ πB)2π∗i gi,
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SPECIAL WARPED-LIKE PRODUCT MANIFOLDS WITH (WEAK) G2 HOLONOMY 1129
where πB : F1 × F2 × ...× Fk × B −→ B and πi : F1 × F2 × ...× Fk × B −→ Fi are the natural
projections on B and Fi respectively. In this scheme, the metric is block diagonal, with the metrics
of the Fi’s are multiplied by a conformal factor depending on the coordinates of the base. We further
generalize this concept by allowing nondiagonal blocks in the fiber space [32].
Remark 3.1. We can define a metric on M by choosing linearly independent local sections of
the cotangent bundle T ∗M and declaring these to be orthonormal.
By using fiber-base decomposition, we see that warped-like product is considered as a generaliza-
tion of multiply-warped product manifolds, by allowing the fiber metric to be non block diagonal [32].
Definition 3.1 [32]. Let M be the topologically product manifold M = F1×F2× ...×Fk ×B,
where dim Fa = na, a = 1, ..., k, dim B = n. Assume that these manifolds are equipped with
Riemannian metrics gFa and gB respectively. Let Ua ⊂ Fa and V ⊂ B be coordinate neighborhoods
on Fa and B respectively, and let U1×U2× ...×Uk × V. Denote the local sections of the cotangent
bundle of each Fa respectively by {θia}
na
i=1, the local coordinates of each Fa by {yia}
na
i=1, and the
local coordinates on B by x1, x2, ..., xn. If the metric on M is defined by the following orthonormal
frame:
eia =
k∑
b=1
nb∑
j=1
Abi
ajθ
j
b , i = 1, ..., na, a = 1, .., k,
eiB =
n∑
j=1
aiBj dx
j , i = 1, ..., n,
where
Abi
aj = Abi
aj(x
1, x2, ..., xn), aiBj = aiBj(x
1, x2, ..., xn),
then (M, ei) is called as a “warped-like product”manifold.
3.1. 7-Dimensional special warped-like product manifolds. For a special case, we will define
7-dimensional special warped-like product manifolds in the following section.
Definition 3.2. Let M = F1 × F2 × B be an 7-dimensional topologically product manifold
where F1, F2 are 3-manifolds and B is a one dimensional manifold, each equipped with Riemannian
metrics. Let θi, θî be orthonormal sections of the cotangent bundles of F1 and F2 respectively and x
be local coordinate on B. If the metric on M is defined by the following orthonormal frame:
ei = A(x)θi +B(x)θî, eî = Â(x)θi + B̂(x)θî, e7 = a(x)dx, i = 1, 2, 3, (3.1)
then we call (M, ei) i = 1, 2, ..., 7, a ”special warped-like product”on 7-dimensional manifold.
3.2. Fundamental 3-form and 7-dimensional special warped-like product structure. When re-
labeling the indices 1̂ = 4, 2̂ = 5 and 3̂ = 6, we get the following forms which are more suitable for
our purposes:
ϕ = (e11̂ + e22̂ + e33̂)e7 + e123 − e12̂3̂ − e1̂23̂ − e1̂2̂3, (3.2)
∗ϕ = e121̂2̂ + e131̂3̂ + e232̂3̂ + (e1̂2̂3̂ − e1̂23 − e12̂3 − e123̂)e7. (3.3)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1130 S. UĞUZ
When we introduce the exterior forms β, µ and ν
β = e11̂ + e22̂ + e33̂, µ = e123 − e12̂3̂ − e1̂23̂ − e1̂2̂3, ν = e1̂2̂3̂ − e1̂23 − e12̂3 − e123̂, (3.4)
we can write ϕ and ∗ϕ as
ϕ = βe7 + µ, ∗ϕ = νe7 − 1
2
β2.
Proposition 3.1. Let F be a 6-dimensional Riemannian manifold of the form F = F1 × F2
and Fi, i = 1, 2, be 3-manifolds. Let θi, θî, i = 1, 2, 3, be orthonormal sections of the cotangent
bundles of F1 and F2 respectively. Let (M = F × R, ei) be a 7-dimensional special warped-like
product manifold given in Definition 3.2. Then the fundamental form and its Hodge dual are written
as
ϕ = fωe7 + φ+1 m1 + φ+2 m2 + φ−1 n1 + φ−2 n2,
∗ϕ = −1
2
f2ω2 +
(
φ+1 m̃1 + φ+2 m̃2 + φ−1 ñ1 + φ−2 ñ2
)
e7,
where
ω = θ11̂ + θ22̂ + θ33̂, φ+1 = θ123, φ−1 = θ1̂2̂3̂,
φ+2 = θ12̂3̂ + θ1̂23̂ + θ1̂2̂3, φ−2 = θ1̂23 + θ12̂3 + θ123̂,
(3.5)
and f, mi, ni, m̃i, ñi, i = 1, 2, are given
f = AB̂ −BÂ, m1 = [A3 − 3AÂ2], m2 = [AB2 − 2BÂB̂ −AB̂2],
n1 = [B3 − 3BB̂2], n2 = [A2B − 2AÂB̂ −BÂ2],
(3.6)
m̃1 = [Â3 − 3A2Â], m̃2 = [ÂB̂2 − 2ABB̂ − ÂB2],
ñ1 = [B̂3 − 3B2B̂], ñ2 = [Â2B̂ − 2ABÂ−A2B̂].
(3.7)
Proof. When we substitute the special warped-like product structure, we get µ and ν as
µ = [A3 − 3AÂ2]θ123 + [AB2 − 2BÂB̂ −AB̂2](θ12̂3̂ + θ1̂23̂ + θ1̂2̂3)+
+[B3 − 3BB̂2]θ1̂2̂3̂ + [A2B − 2AÂB̂ −BÂ2](θ12̂3 + θ1̂23 + θ123̂),
ν = [Â3 − 3A2Â]θ123 + [ÂB̂2 − 2ABB̂ −B2Â](θ12̂3̂ + θ1̂23̂ + θ1̂2̂3)+
+[B̂3 − 3B2B̂]θ1̂2̂3̂ + [Â2B̂ − 2ABÂ−A2B̂](θ12̂3 + θ1̂23 + θ123̂).
We introduce new variables to simplify the notation φ±i , i = 1, 2, as
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SPECIAL WARPED-LIKE PRODUCT MANIFOLDS WITH (WEAK) G2 HOLONOMY 1131
φ+1 = θ123, φ+2 = θ12̂3̂ + θ1̂23̂ + θ1̂2̂3,
φ−1 = θ1̂2̂3̂, φ−2 = θ1̂23 + θ12̂3 + θ123̂.
Then we can write
µ = φ+1 m1 + φ+2 m2 + φ−1 n1 + φ−2 n2,
ν = φ+1 m̃1 + φ+2 m̃2 + φ−1 ñ1 + φ−2 ñ2,
where the coefficient functions mi and ni, i = 1, 2, are given by the equations (3.6). Hence we write
the fundamental 3-form ϕ and its dual form ∗ϕ on M as follows:
ϕ = fωe7 + φ+1 m1 + φ+2 m2 + φ−1 n1 + φ−2 n2,
∗ϕ = −1
2
f2ω2 +
(
φ+1 m̃1 + φ+2 m̃2 + φ−1 ñ1 + φ−2 ñ2
)
e7.
Proposition 3.1 is proved.
3.3. Fibre-base decomposition of 7-dimensional special warped-like product manifolds. We
consider the decomposition of the manifold M as “base”and “fiber” , then we decompose the exterior
algebra as
Λp(M) =
⊕
a+k=p
Λ(a,k)(M),
where a = 1, . . . , 6 and k = 1. Under the exterior derivative these summands are mapped as
d : Λ(a,k)(M) −→ Λ(a+1,k) ⊕ Λ(a,k+1).
We can refine this decomposition by splitting the components for each fiber as
Λp(M) =
⊕
a+b+k=p
Λ(a,b,k)(M),
where a and b range from 1 to 3 and k = 1 as before. The effect of the exterior derivative is given
by
d : Λ(a,b,k)(M) −→ Λ(a+1,b,k) ⊕ Λ(a,b+1,k) ⊕ Λ(a,b,k+1).
By using the structure of 7-dimensional special warped-like product manifolds, we investigate
G2 and the weak G2 holonomy metrics on these type of manifolds and prove a main theorem related
to the special warped-like product manifolds with these G2 structures in the following sections.
4. Special warped-like product manifolds with G2 holonomy. In this section we consider the
case where the seven dimensional manifold has a (3 + 3 + 1) decomposition, i.e., the base is one
dimensional and the fiber is a product of 3-manifolds. As all 3-manifolds are paralellizable [33] we
work with global sections of the cotangent bundles of the fibers and for simplicity we assume that
the base is R. Here we will prove that under suitable global assumptions the fibers are isometric to
3-spheres S3 with constant curvature k > 0.
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1132 S. UĞUZ
Theorem 4.1. Let M be diffeomorphic to F × B, where the base B is a one dimensional
Riemannian manifold diffeomorphic to R, the fibre F is a 6-manifold of the form F = F1 × F2, and
Fi, i = 1, 2, are complete, connected and simply connected 3-manifolds. Let the metric on M be a
special warped-like product with the following orthonormal frame:
ei = A(x)θi +B(x)θî, i = 1, 2, 3,
eî = Â(x)θi + B̂(x)θî, i = 1, 2, 3,
e7 = a(x)dx.
Let ϕ be the fundamental 3-form on M given by
ϕ = fωe7 + φ+1 m1 + φ+2 m2 + φ−1 n1 + φ−2 n2
and its dual
∗ϕ = −1
2
f2ω2 +
(
φ+1 m̃1 + φ+2 m̃2 + φ−1 ñ1 + φ−2 ñ2
)
e7.
If dϕ = d ∗ ϕ = 0, then F1 and F2 are isometric to S3 with constant curvature k > 0.
Before proving the above theorem, we present two propositions which give the closeness prop-
erties of ϕ and ∗ϕ respectively. The crucial step in the proof of this theorem is to find projections of
the 4-form dϕ into subspaces of Λ4(M) determined by the special warped-like product structure.
Proposition 4.1. Let (M, ei) be a 7-dimensional special warped-like product manifold as in
Theorem 4.1. If dϕ = 0, then the following two conditions must be satisfied:
fdωe7 = φ+1 dm1 + φ+2 dm2 + φ−1 dn1 + φ−2 dn2, (4.1)
dφ+2 m2 + dφ−2 n2 = 0, (4.2)
where f, ω, φ±i ,mi, ni, i = 1, 2, are given in equations (3.5), (3.6).
Proof. We substitute ei and eî given by the equations (3.1) into the expressions of β, µ and ν
given in equations (3.4), we obtain
ϕ =
[
fωe7
]
+
[
φ+1 m1 + φ+2 m2 + φ−1 n1 + φ−2 n2
]
,
as in Proposition 3.1. The terms in the brackets belong to subspaces Λ2,1, and Λ3,0 respectively. Note
that dfe7 = de7 = 0 since the base of the multi-warped product is one dimensional. Similarly, as
each Fi is three dimensional, their volume forms are closed, i.e.,
dφ+1 = dφ−1 = 0.
Then dϕ = 0 reduces to
dϕ =
[
fdωe7 − φ+1 dm1 − φ+2 dm2 − φ−1 dn1 − φ
−
2 dn2
]
+
+
[
dφ+2 m2 + dφ−2 n2
]
, (4.3)
where the terms in the brackets belong respectively to Λ3,1(M) and Λ4,0(M).
Proposition 4.1 is proved.
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SPECIAL WARPED-LIKE PRODUCT MANIFOLDS WITH (WEAK) G2 HOLONOMY 1133
Proposition 4.2. Let (M, ei) be a 7-dimensional special warped-like product manifold as in
Theorem 4.1. If d ∗ ϕ = 0, then the following two conditions must be satisfied:
ωdω = 0, (4.4)
fdfω2 =
(
dφ+2 m̃2 + dφ−2 ñ2
)
e7, (4.5)
where f, ω, φ±i , m̃i, ñi, i = 1, 2, are given in equations (3.5), (3.6).
Proof. We can write
∗ϕ =
[
−1
2
f2ω2
]
+
[
φ+1 m̃1 + φ+2 m̃2 + φ−1 ñ1 + φ−2 ñ2
]
e7,
as in Proposition 3.1. The terms in the brackets belong to subspaces Λ4,0 and Λ3,1 respectively. By
using the previous proposition arguments, d ∗ ϕ = 0 reduces to
d ∗ ϕ =
[
−f2ωdω
]
+
[
−fdfω2 +
(
dφ+2 m̃2 + dφ−2 ñ2
)
e7
]
,
where the terms in the brackets belong respectively to Λ5,0(M) and Λ4,1(M).
Proposition 4.2 is proved.
Here we prove that the equation (4.1) given in Proposition 4.1 fixes the exterior derivatives of
the θi’s and θî’s completely for the manifold M in Theorem 4.1.
Proposition 4.3. Let (M, ei) be a 7-dimensional special warped-like product manifold as in
Theorem 4.1. If
fdωe7 − φ+1 dm1 − φ+2 dm2 − φ−1 dn1 − φ
−
2 dn2 = 0,
then
dθ1 = λ1θ
23, dθ2 = −λ1θ13, dθ3 = λ1θ
12,
dθ1̂ = λ2θ
2̂3̂, dθ2̂ = −λ2θ1̂3̂, dθ3̂ = λ2θ
1̂2̂,
(4.6)
where λi, i = 1, 2, are arbitrary nonzero constants.
Proof. Let us write the exterior derivative mi, ni, i = 1, 2, are of the following form:
dm1 = u1e
7, dm2 = u2e
7,
dn1 = v1e
7, dn2 = v2e
7,
where u1, u2, v1, v2 are functions on B. Then we can factorize e7 in the condition and obtain
[fdω]−
[
φ+1 u1
]
−
[
φ+2 u2
]
−
[
φ−1 v1
]
−
[
φ−2 v2
]
= 0. (4.7)
In (4.7) the terms in the brackets belong to subspaces Λ(2,1,0)⊕Λ(1,2,0), Λ(3,0,0), Λ(1,2,0), Λ(0,3,0) and
Λ(2,1,0) respectively. This implies that u1 = v1 = 0, that is,
dm1 = dn1 = 0.
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1134 S. UĞUZ
Thus we obtain
fdω = φ+2 u2 + φ−2 v2.
If we write explicitly ω, φ+2 and φ−2 , then
fd(θ11̂ + θ22̂ + θ33̂) = (θ12̂3̂ + θ1̂23̂ + θ1̂2̂3)u2 + (θ1̂23 + θ12̂3 + θ123̂)v2.
When we rearrange the equality,
(fdθ1 − v2θ23)θ1̂ − (fdθ1̂ + u2θ
2̂3̂)θ1+
+(fdθ2 + v2θ
13)θ2̂ − (fdθ2̂ − u2θ1̂3̂)θ2+
+(fdθ3 − v2θ12)θ3̂ − (fdθ3̂ + u2θ
1̂2̂)θ3 = 0,
we obtain
dθ1 =
v2
f
θ23, dθ2 = −v2
f
θ13, dθ3 =
v2
f
θ12, (4.8)
dθ1̂ = −u2
f
θ2̂3̂, dθ2̂ =
u2
f
θ1̂3̂, dθ3̂ = −u2
f
θ1̂2̂. (4.9)
If we take the exterior derivative of dθ1 =
v2
f
θ23, we get
d
(
v2
f
)
θ23 +
v2
f
dθ2θ3 − v2
f
θ2dθ3 = 0.
Using the equations (4.8), it is seen that d
(
v2
f
)
= 0, in similar way d
(
u2
f
)
= 0, that is,
v2
f
,
u2
f
are constants. This proves the Proposition 4.3 if the nonzero constants are chosen as λ1 and λ2.
We complete the proof of Theorem 4.1 by using the following result.
Theorem 4.2 [34]. Any two connected, simply connected complete Riemannian manifolds of
constant curvature k are isometric to each other.
Proof of Theorem 4.1. One can see that the equations (4.6) describes the Lie algebra su(2), it
follows that if the fibers are connected and simply connected, then they are diffeomorphic to S3 [36,
p. 127] (Section 3.65). Using the equations (4.6), it is seen that the sectional curvatures of F1 and F2
are positive, i.e., K(Fi) =
λ2i
4
> 0. Then by the Theorem 4.2, it follows that F1 and F2 are isometric
to S3 with constant curvature k > 0.
Theorem 4.1 is proved.
Remark 4.1. For the existence of the solution, we have to find A,B, Â, B̂ and a(x) such that
the equations in Propositions 4.1, 4.2 are satisfied. From the exterior derivatives of the basis 1-forms
θi and θî, it is seen that the equation (4.4) of Proposition 4.2 holds identically. The other equations
are to be solved, but instead of this computation, we will use Konishi – Naka solution in the following
section.
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SPECIAL WARPED-LIKE PRODUCT MANIFOLDS WITH (WEAK) G2 HOLONOMY 1135
4.1. Konishi – Naka solution. The aim of this section is to prove that Konishi – Naka met-
ric ansatz is unique in the class of special warped-like product metrics admitting the G2 structure
determined by the fundamental 3-form given in the equation (2.3).
Now we recall that the Konishi – Naka solution [24] on
M = SU(2)× SU(2)×R
is given by the following (global) orthonormal frame:
ei = A(x)θi, i = 1, 2, 3,
eî = Â
(
θî − 1
2
θi
)
, i = 1, 2, 3,
e7 = dx,
(4.10)
where the local sections of the cotangent bundle of each SU(2) respectively by θi, θî and the
functions A(x), Â satisfy the differential equations
dA
dx
=
Â
2A
,
dÂ
dx
= 1− Â2
4A2
. (4.11)
Thus the metric is
g = A(x)2
3∑
i=1
(θi)2 + Â(x)2
(
3∑
i=1
[
θî − 1
2
θi
])2
+ dx2. (4.12)
We can take e7 = dx, as in [2]. We will show that we can also set B = 0 in the equation
(3.1) by a frame transformation and obtain exactly the Konishi – Naka metrical ansatz. An orthogonal
transformation of the cotangent frame {ei, eî} is given by
ẽi = P i
je
j +Qi
je
ĵ , i = 1, 2, 3,
ẽî = P̂ i
je
j + Q̂i
je
ĵ , i = 1, 2, 3,
where P, Q, P̂ , Q̂ satisfy
PP t +QQt = I, P P̂ t +QQ̂t = 0, P̂ P̂ t + Q̂Q̂t = I.
The new basis elements ẽi, ẽî can be written now as
ẽi = Ãθi + B̃θî, ẽî =
˜̂
Aθi +
˜̂
Bθî, (4.13)
where
à = AP + ÂQ, B̃ = BP + B̂Q,
˜̂
A = AP̂ + ÂQ̂,
˜̂
B = BP̂ + B̂Q̂.
(4.14)
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1136 S. UĞUZ
We will now show that we can set B̃ = 0 by an orthogonal transformation. Note that if B is
nonzero, but B̂ is zero, then, B̃ = 0 gives BP = 0, and since B is a scalar, the matrix P is identically
zero. From (4.13) it follows that Q is a unitary hence nonsingular matrix and Q̂ is identically zero.
Finally the last equation in (4.14) implies that P̂ is also a unitary matrix. But since in (4.14), the
quantities Ã, B̃, ˜̂
A, ˜̂
A are scalars, it follows that the orthogonal matrices P̂ and Q are proportional
to identity. It follows that the transformation interchanges the roles of the subspaces.
Assuming now that both B and B̂ are nonzero, the equation B̃ = BP + B̂Q = 0 implies that
the matrix P is proportional to the matrix Q, i.e., P = −B̂
B
Q. Substituting this in Ã, we see that
ÃI =
(
Â− AB̂
B
)
Q hence Q = Q0(x, y)I , that is, Q is the proportional to identity. Then from the
first equation in (4.13), we can determine Q0 and obtain P and Q as
Q = ± B√
B2 + B̂2
I, P = ∓ B̂√
B2 + B̂2
I.
As P̂ =
B
B̂
Q̂ and substituting in  we see that Q̂ is also proportional to identity and determine P̂
and Q̂ as
Q̂ = ε
B̂√
B2 + B̂2
I and P̂ = ε
B√
B2 + B̂2
I,
where ε2 = 1. The transformation matrix(
P Q
P̂ Q̂
)
=
1√
B2 + B̂2
(
∓B̂I ±BI
εBI εB̂I
)
is clearly orthogonal and the coefficients of the new frame are
à = ∓ f√
B2 + B̂2
, B̃ = 0,
˜̂
A = ε
AB + ÂB̂√
B2 + B̂2
,
˜̂
B = ε
√
B2 + B̂2.
If we choose the (global) orthonormal frame as in the equation (4.10), then we can see that
A = A(x), B = 0,
 = −1
2
Â(x), B̂ = Â(x),
a = a(x) = 1,
where A(x), Â(x) satisfy the condition given in (4.11). By a straight forward computation using the
equations (4.11), it can be seen that the conditions given in Propositions 4.1, 4.2 are satisfied, hence
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SPECIAL WARPED-LIKE PRODUCT MANIFOLDS WITH (WEAK) G2 HOLONOMY 1137
we obtain a direct proof that the solution given in [2] is a G2 metric. Thus we have the following
corollary which implies that the Konishi – Naka solution is unique up to gauge transformations.
Corollary 4.1. Let M be a special warped-like product manifold. Consider the G2 holonomy
structure determined by the fundamental 3-form given in the equations (2.3) on M . Then there
exists a unique metric in the class of special warped-like product metrics admitting this special G2
structure and the metric is obtained as given in the equations (4.12) up to gauge transformation.
Let us consider the extension of the holonomy concept in 7-dimensional manifolds, that is, if we
replace the condition from G2 holonomy to weak G2 holonomy on M , then we obtain that the fibers
are the same (S3) for this special warped-like product as in Section 4.
5. Special warped-like product manifolds with weak G2 holonomy. We now consider the
weak holonomy G2 for (3+3+1) decomposition. It is proved that under the same global assumptions
in Section 4, the fibers are also isometric to S3.
Theorem 5.1. Let (M, ei) be 7-dimensional special warped-like product manifold as in Theo-
rem 4.1. If dϕ = λ ∗ϕ with λ 6= 0, then F1 and F2 are also isometric to S3 with constant curvature
k > 0.
Let us find the projections of the 4-form dϕ into subspaces of Λ4(M) under the warped-like
product structure.
Proposition 5.1. Let (M, ei) be a 7-dimensional special warped-like product manifold as in
Theorem 4.1. If dϕ = λ ∗ ϕ with λ 6= 0, then the following two conditions must be satisfied:
fdωe7 −
2∑
i=1
(
φ+i dmi + φ−i dni
)
= λ
2∑
i=1
(
φ+i m̃i + φ−i ñi
)
e7, (5.1)
dφ+2 m2 + dφ−2 n2 = −1
2
λf2ω2, (5.2)
where f, ω, φ±i ,mi, ni, i = 1, 2, are given in equations (3.5), (3.6).
Proof. As similarly obtained before, the exterior derivative of ϕ can be written
dϕ =
[
fdωe7 − φ+1 dm1 − φ+2 dm2 − φ−1 dn1 − φ
−
2 dn2
]
+
+
[
dφ+2 m2 + dφ−2 n2
]
,
where the terms in the brackets belong respectively to Λ3,1(M) and Λ4,0(M). Also
∗ϕ =
[
−1
2
f2ω2
]
+
[(
φ+1 m̃1 + φ+2 m̃2 + φ−1 ñ1 + φ−2 ñ2
)
e7
]
,
has the terms in the brackets belong respectively to Λ4,0(M) and Λ3,1(M) . If we impose the
conditions, this gives us the two equations of Proposition 5.1.
In the following, it is proved that the equation (5.1) given in Proposition 5.1 fixes the exterior
derivatives of the θi’s and θî’s completely for the manifold M in Theorem 5.1.
Proposition 5.2. Let (M, ei) be a 7-dimensional special warped-like product manifold as in
Theorem 4.1. If
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1138 S. UĞUZ
fdωe7 −
2∑
i=1
(
φ+i dmi + φ−i dni
)
= λ
2∑
i=1
(
φ+i m̃i + φ−i ñi
)
e7,
then
dθ1 = λ1θ
23, dθ2 = −λ1θ13, dθ3 = λ1θ
12,
dθ1̂ = λ2θ
2̂3̂, dθ2̂ = −λ2θ1̂3̂, dθ3̂ = λ2θ
1̂2̂,
where λ1 and λ2 are arbitrary nonzero constants.
Proof. Consider the exterior derivative mi, ni, i = 1, 2, as
dm1 = u1e
7, dm2 = u2e
7,
dn1 = v1e
7, dn2 = v2e
7,
where u1, u2, v1, v2 are functions on B. Then we can factorize e7 in the condition and obtain
[fdω]−
[
φ+1 (u1 + λm̃1)
]
−
[
φ+2 (u2 + λm̃2)
]
−
−
[
φ−1 (v1 + λñ1)
]
−
[
φ−2 (v2 + λñ2)
]
= 0. (5.3)
In (5.3) the terms in the brackets belong to subspaces Λ(2,1,0)⊕Λ(1,2,0), Λ(3,0,0), Λ(1,2,0), Λ(0,3,0) and
Λ(2,1,0) respectively. This implies that
u1 + λm̃1 = v1 + λñ1 = 0.
Thus we obtain
fdω = φ+2 (u2 + λm̃2) + φ−2 (v2 + λñ2).
If we write explicitly ω, φ+2 and φ−2 , then
fd(θ11̂ + θ22̂ + θ33̂) = (θ12̂3̂ + θ1̂23̂ + θ1̂2̂3)(u2 + λm̃2)+
+(θ1̂23 + θ12̂3 + θ123̂)(v2 + λñ2).
When we rearrange the equality,(
fdθ1 − (v2 + λñ2)θ
23
)
θ1̂ −
(
fdθ1̂ + (u2 + λm̃2)θ
2̂3̂
)
θ1+
+
(
fdθ2 + (v2 + λñ2)θ
13
)
θ2̂ −
(
fdθ2̂ − (u2 + λm̃2)θ
1̂3̂
)
θ2+
+
(
fdθ3 − (v2 + λñ2)θ
12
)
θ3̂ −
(
fdθ3̂ + (u2 + λm̃2)θ
1̂2̂
)
θ3 = 0,
we obtain
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SPECIAL WARPED-LIKE PRODUCT MANIFOLDS WITH (WEAK) G2 HOLONOMY 1139
dθ1 =
v2 + λñ2
f
θ23, dθ2 = −v2 + λñ2
f
θ13, dθ3 =
v2 + λñ2
f
θ12, (5.4)
dθ1̂ = −u2 + λm̃2
f
θ2̂3̂, dθ2̂ =
u2 + λm̃2
f
θ1̂3̂, dθ3̂ = −u2 + λm̃2
f
θ1̂2̂. (5.5)
If we take the exterior derivative of dθ1 =
(
v2 + λñ2
f
)
θ23, we get
d
(
v2 + λñ2
f
)
θ23 +
(
v2 + λñ2
f
)
dθ2θ3 −
(
v2 + λñ2
f
)
θ2dθ3 = 0.
Using the equation (5.4), it is seen that d
(
v2 + λñ2
f
)
= 0, in similar way d
(
u2 + λm̃2
f
)
= 0,
that is,
v2 + λñ2
f
,
u2 + λm̃2
f
are constants. If the nonzero constants are chosen as λi, i = 1, 2, this
proves the Proposition 5.2.
Proof of Theorem 5.1. It can be proved in similar way in the proof of Theorem 4.1.
Finally we obtain the following main result for the 7-dimensional special warped-like product
manifolds with (weak) G2 holonomy.
Theorem 5.2. Let M be diffeomorphic to F × B, where the base B is a one dimensional
Riemannian manifold diffeomorphic to R, the fibre F is a 6-manifold of the form F = F1 × F2, and
Fi, i = 1, 2, are complete, connected and simply connected 3-manifolds. Let the metric on M be a
special warped-like product metric (3.1). If M is the manifold with the G2 holonomy or with the
weak G2 holonomy determined by the fundamental 3-form (3.2), then the fibers Fi’s are isometric to
S3 with constant curvature k > 0. Also there exists a unique metric in the class of special warped-like
product metrics admitting the G2 holonomy, and the metric is written as given (4.12) up to gauge
transformation.
6. Conclusions. In this paper we define warped-like product metrics as a generalization of mul-
tiply warped products and study special type of these metrics for G2 cases. Different types of fibers-
base decompositions will be investigated in the next studies. We believe that our approach of the
warped-like product metrics will be an important notion for the manifolds with special holonomies.
Some other interesting results and further connections for the other holonomies wait to be explored.
Acknowledgments. We would like to thank Prof. Dr. Ayse H. Bilge for many fruitful discussions.
This work was initiated in the Humboldt University (visiting research), Department of Mathematics
in Berlin. The author also thanks Prof. Dr. Thomas Friedrich and Prof. Dr. Ilka Agricola for providing
support and all facilities.
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Received 20.11.11,
after revision — 20.05.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
|
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| record_format | ojs |
| resource_txt_mv | umjimathkievua/be/b691d15641dac17735f00d519c7ba2be.pdf |
| spelling | umjimathkievua-article-24952020-03-18T19:16:44Z Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy Cпеціальний спотворений добуток многовидю зі (слабкою) $G_2$ голономiєю Uğuz, S. Угуз, С. By using the fiber-base decompositions of manifolds, the definition of warped-like product is regarded as a generalization of multiply warped product manifolds, by allowing the fiber metric to be not block diagonal. We consider the (3 + 3 + 1) decomposition of 7-dimensional warped-like product manifolds, which is called a special warped-like product of the form $M = F × B$; where the base $B$ is a onedimensional Riemannian manifold and the fiber $F$ has the form $F = F_1 × F_2$ where $F_i ; i = 1, 2$, are Riemannian 3-manifolds. If all fibers are complete, connected, and simply connected, then they are isometric to $S_3$ with constant curvature $k > 0$ in the class of special warped-like product metrics admitting the (weak) $G_2$ holonomy determined by the fundamental 3-form. З використанням волоконних розкладiв многовидів розглянуто визначення спотвореного добутку як узагальнення багаторазово спотворених добутків многовидів, при цьому волоконна метрика може не бути блочно-діагональною. Вивчено (3 + 3 + 1) розклади 7-вимірних спотворених добутків многовидів, що називаються спеціальними спотвореними виду $M = F × B$, де база $B$ — одновимірний ріманів многовид, а волокно F має форму $F = F_1 × F_2$, де $F_i ; i = 1, 2$, — ріманові 3-многовиди. Якщо всі волокна є повними i однозв'язними, то вони є ізометричними до $S_3$ зі сталою кривиною $k > 0$ у класі спеціальних спотворених метрик добутку, що допускають (слабку) $G_2$ голономію, визначену фундаментальною 3-формою. Institute of Mathematics, NAS of Ukraine 2013-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2495 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 8 (2013); 1126–1140 Український математичний журнал; Том 65 № 8 (2013); 1126–1140 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2495/1756 https://umj.imath.kiev.ua/index.php/umj/article/view/2495/1757 Copyright (c) 2013 Uğuz S. |
| spellingShingle | Uğuz, S. Угуз, С. Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy |
| title | Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy |
| title_alt | Cпеціальний спотворений добуток многовидю зі (слабкою) $G_2$ голономiєю |
| title_full | Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy |
| title_fullStr | Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy |
| title_full_unstemmed | Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy |
| title_short | Special Warped-Like Product Manifolds with (Weak) $G_2$ Holonomy |
| title_sort | special warped-like product manifolds with (weak) $g_2$ holonomy |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2495 |
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