Manifolds admitting a G̃2-structure.
Hông-Vân Lê
Abstract
We find a necessary and sufficient condition for a compact 7-manifold to admit a
G̃2-structure. As a result we find a sufficient condition for an open 7-manifold to admit
a closed 3-form of G̃2-type.
MSC: 55S35, 53C10
1 Introduction
Recently a new class of geometries related with stable forms has been discovered [Hitchin2000],
[Hitchin2001], [Witt2005], [Le2006], [LPV2007]. In some cases we can define easily a nec-
essary and sufficient condition for a manifold M to admit a stable form of type ω in terms
of topological invariants of M , for example if ω is a 3-form of G2-type [Gray1969]. But
in general there is no method to solve the question how to find a necessary and sufficient
condition for a manifold to admit a stable form. In a previous note [Le2006] we have
wrongly stated a sufficient condition for an open manifold to admit a closed stable 3-form
of G̃2-type. We recall that [Bryant1987] a 3-form on R
7 is called of G̃2-type, if it lies on
the Gl(R7)-orbit of a 3-form
3 = θ1 ∧ θ2 ∧ θ3 + α1 ∧ θ1 + α2 ∧ θ2 + α3 ∧ θ3
Here αi are 2-forms on V
7 which can be written as
α1 = y1 ∧ y2 + y3 ∧ y4, α2 = y1 ∧ y3 − y2 ∧ y4, α3 = y1 ∧ y4 + y2 ∧ y3
and (θ1, θ2, θ3, y1, y2, y3, y4) is an oriented basis of (V
The group G̃2 can be defined as the isotropy group of ω
3 under the action of Gl(R7).
Bryant proved that [Bryant1987] G̃2 coincides with the automorphism group of the split
octonians.
In this note we prove the following
http://arxiv.org/abs/0704.0503v2
Main Theorem. Suppose that M7 is a compact 7-manifold. Then M7 admits a 3-form
of G̃2-type, if and only if M
7 is orientable and spinnable. Equivalently the first and second
Stiefel-Whitney classes of M7 vanish. Suppose that M7 is an open manifold which admits
an embedding to a compact orientable and spinnable 7-manifold. Then M7 admits a closed
3-form of G̃2-type.
2 Proof of Main Theorem
Our proof is based on the following simple fact on G̃2.
2.1. Lemma. We have π1(G̃2) = Z2. Hence its maximal compact Lie group is SO(4).
This Lemma is well-known, (Bryant mentioned it but he omitted a proof in [Bryant1987]),
but I did not find an explicit proof of it in popular lectures on Lie groups, though it could
be given as an exercise. For a hint to a solution of this exercise we refer to [HL1982], p.115,
for an explicit embedding of SO(4) into G2. The reader can also check that the image of
this group is also a subgroup of G̃2 ⊂ Gl(R
7). We shall denote this image by SO(4)3,4.
The Cartan theory on symmetric spaces implies that SO(4)3,4 is a maximal compact Lie
subgroup of G̃2.
Now let us return to proof of our Main theorem. Clearly if M7 admits a G̃2-structure,
then it must be orientable and spinnable, since a maximal compact Lie subgroup SO(4)3,4
of G2 is also a compact subgroup of G2.
2.2. Lemma. Assume that M7 is compact, orientable and spinnable. Then M7 admits a
G̃2-structure.
Proof. SinceM7 is compact, orientable and spinable,M7 admits a SU(2)-structure [Friedrich1997].
Now it is easy to see that it admits a SO(4)3,4-structure, where SO(4)3,4 is a maximal com-
pact Lie subgroup of G2. Hence M
7 admits a G̃2-structure. ✷
To prove the last statement of the Main Theorem we shall use the following theorem due
to Eliashberg-Mishachev to deform the 3-form ω3 to a closed 3-form ω̄3 of G̃2-type on
For a subspaceR ⊂ ΛpM we denote by CloaR a subspace of the space SecR which consists
of closed p-forms ω : M → R in the cohomology class a ∈ Hp(M).
Eliashberg-Mishashev Theorem [E-M2002,10.2.1] Let M be an open manifold, a ∈
Hp(M) a fixed cohomology class and R an open Diff M-invariant subset. Then the inclu-
CloaR →֒ SecR
is a homotopy equivalence. In particular,
- any p-form ω : M → R is homotopic in R to a closed form ω̄.
- any homotopy of p-form ωt : M :→ R which connects two closed forms ω0, ω1 ∈ a can be
deformed in R into a homotopy of closed forms ω̄t connecting ω0 and ω1 ∈ a.
Let R be the space of all 3-forms of G̃2-type on M = M
7. Clearly this space is an open
DiffM7-invariant subset of Λ3M7. Now we apply the Eliashberg-Mishashev theorem to
our 3-form ω3 of G̃2-type whose existence has been proved above. Hence M
7 admits a
closed 3-form ω̄3 of G̃2-type. ✷
2.3. Remark. It seems that we can drop the closedness condition in our Main Theorem
and use the classical obstruction theory to prove the main Theorem.
Acknowledgement.
This note is partially supported by grant of ASCR Nr IAA100190701.
References
[Adams1996] J.F. Adams, Lectures on exceptional Lie groups, The Chicago University
Press, 1996.
[Bryant1987] R. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2), 126
(1987), 525-576.
[E-M2002] Y. Eliashberg and N. Mishachev, Introduction to the h-Principle, AMS
2002.
[Gray1969] A. Gray, Vector cross products on manifolds, TAMS 141, (1969), 465-504,
(Errata in TAMS 148 (1970), 625).
[Friedrich1997] Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, On nearly
parallel G2-manifolds, Journal Geom. Phys. 23 (1997), 259-286.
[HL1982] , R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. (182),
47-157.
[Hitchin2000] N. Hitchin, The geometry of three-forms in 6 and 7 dimensions, J.D.G. 55
(2000), 547-576.
[Hitchin2001] N. Hitchin, Stable forms and special metrics, Contemporean math., (2001),
288, 70-89.
[Le2006] H. V. Le, The existence of symplectic 3-forms on 7-manifolds,
arXiv:math.DG/0603182.
[LPV2007] H.V.Le, M. Panak and J. Vanzura, Manifolds admitting stable forms, in
preparation.
[Witt2005] F. Witt, Special metric structures and closed forms, Ph.D. Thesis ,
arxiv:math.DG/0502443.
Hong Van Le, Institute of Mathematics, Zitna 25, 11567 Praha 1, hvle@math.cas.cz,
http://arxiv.org/abs/math/0603182
http://arxiv.org/abs/math/0502443
	Introduction
	Proof of Main Theorem

  We find a necessary and sufficient condition for a compact 7-manifold to
admit a $\tilde G_2$-structure. As a result we find a sufficient condition for
an open 7-manifold to admit a closed 3-form of $\tilde G_2$-type.

Introduction
Recently a new class of geometries related with stable forms has been discovered [Hitchin2000],
[Hitchin2001], [Witt2005], [Le2006], [LPV2007]. In some cases we can define easily a nec-
essary and sufficient condition for a manifold M to admit a stable form of type ω in terms
of topological invariants of M , for example if ω is a 3-form of G2-type [Gray1969]. But
in general there is no method to solve the question how to find a necessary and sufficient
condition for a manifold to admit a stable form. In a previous note [Le2006] we have
wrongly stated a sufficient condition for an open manifold to admit a closed stable 3-form
of G̃2-type. We recall that [Bryant1987] a 3-form on R
7 is called of G̃2-type, if it lies on
the Gl(R7)-orbit of a 3-form
3 = θ1 ∧ θ2 ∧ θ3 + α1 ∧ θ1 + α2 ∧ θ2 + α3 ∧ θ3
Here αi are 2-forms on V
7 which can be written as
α1 = y1 ∧ y2 + y3 ∧ y4, α2 = y1 ∧ y3 − y2 ∧ y4, α3 = y1 ∧ y4 + y2 ∧ y3
and (θ1, θ2, θ3, y1, y2, y3, y4) is an oriented basis of (V
The group G̃2 can be defined as the isotropy group of ω
3 under the action of Gl(R7).
Bryant proved that [Bryant1987] G̃2 coincides with the automorphism group of the split
octonians.
In this note we prove the following
http://arxiv.org/abs/0704.0503v2
Main Theorem. Suppose that M7 is a compact 7-manifold. Then M7 admits a 3-form
of G̃2-type, if and only if M
7 is orientable and spinnable. Equivalently the first and second
Stiefel-Whitney classes of M7 vanish. Suppose that M7 is an open manifold which admits
an embedding to a compact orientable and spinnable 7-manifold. Then M7 admits a closed
3-form of G̃2-type.
2 Proof of Main Theorem
Our proof is based on the following simple fact on G̃2.
2.1. Lemma. We have π1(G̃2) = Z2. Hence its maximal compact Lie group is SO(4).
This Lemma is well-known, (Bryant mentioned it but he omitted a proof in [Bryant1987]),
but I did not find an explicit proof of it in popular lectures on Lie groups, though it could
be given as an exercise. For a hint to a solution of this exercise we refer to [HL1982], p.115,
for an explicit embedding of SO(4) into G2. The reader can also check that the image of
this group is also a subgroup of G̃2 ⊂ Gl(R
7). We shall denote this image by SO(4)3,4.
The Cartan theory on symmetric spaces implies that SO(4)3,4 is a maximal compact Lie
subgroup of G̃2.
Now let us return to proof of our Main theorem. Clearly if M7 admits a G̃2-structure,
then it must be orientable and spinnable, since a maximal compact Lie subgroup SO(4)3,4
of G2 is also a compact subgroup of G2.
2.2. Lemma. Assume that M7 is compact, orientable and spinnable. Then M7 admits a
G̃2-structure.
Proof. SinceM7 is compact, orientable and spinable,M7 admits a SU(2)-structure [Friedrich1997].
Now it is easy to see that it admits a SO(4)3,4-structure, where SO(4)3,4 is a maximal com-
pact Lie subgroup of G2. Hence M
7 admits a G̃2-structure. ✷
To prove the last statement of the Main Theorem we shall use the following theorem due
to Eliashberg-Mishachev to deform the 3-form ω3 to a closed 3-form ω̄3 of G̃2-type on
For a subspaceR ⊂ ΛpM we denote by CloaR a subspace of the space SecR which consists
of closed p-forms ω : M → R in the cohomology class a ∈ Hp(M).
Eliashberg-Mishashev Theorem [E-M2002,10.2.1] Let M be an open manifold, a ∈
Hp(M) a fixed cohomology class and R an open Diff M-invariant subset. Then the inclu-
CloaR →֒ SecR
is a homotopy equivalence. In particular,
- any p-form ω : M → R is homotopic in R to a closed form ω̄.
- any homotopy of p-form ωt : M :→ R which connects two closed forms ω0, ω1 ∈ a can be
deformed in R into a homotopy of closed forms ω̄t connecting ω0 and ω1 ∈ a.
Let R be the space of all 3-forms of G̃2-type on M = M
7. Clearly this space is an open
DiffM7-invariant subset of Λ3M7. Now we apply the Eliashberg-Mishashev theorem to
our 3-form ω3 of G̃2-type whose existence has been proved above. Hence M
7 admits a
closed 3-form ω̄3 of G̃2-type. ✷
2.3. Remark. It seems that we can drop the closedness condition in our Main Theorem
and use the classical obstruction theory to prove the main Theorem.
Acknowledgement.
This note is partially supported by grant of ASCR Nr IAA100190701.
References
[Adams1996] J.F. Adams, Lectures on exceptional Lie groups, The Chicago University
Press, 1996.
[Bryant1987] R. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2), 126
(1987), 525-576.
[E-M2002] Y. Eliashberg and N. Mishachev, Introduction to the h-Principle, AMS
2002.
[Gray1969] A. Gray, Vector cross products on manifolds, TAMS 141, (1969), 465-504,
(Errata in TAMS 148 (1970), 625).
[Friedrich1997] Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, On nearly
parallel G2-manifolds, Journal Geom. Phys. 23 (1997), 259-286.
[HL1982] , R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. (182),
47-157.
[Hitchin2000] N. Hitchin, The geometry of three-forms in 6 and 7 dimensions, J.D.G. 55
(2000), 547-576.
[Hitchin2001] N. Hitchin, Stable forms and special metrics, Contemporean math., (2001),
288, 70-89.
[Le2006] H. V. Le, The existence of symplectic 3-forms on 7-manifolds,
arXiv:math.DG/0603182.
[LPV2007] H.V.Le, M. Panak and J. Vanzura, Manifolds admitting stable forms, in
preparation.
[Witt2005] F. Witt, Special metric structures and closed forms, Ph.D. Thesis ,
arxiv:math.DG/0502443.
Hong Van Le, Institute of Mathematics, Zitna 25, 11567 Praha 1, hvle@math.cas.cz,
http://arxiv.org/abs/math/0603182
http://arxiv.org/abs/math/0502443
	Introduction
	Proof of Main Theorem
