Invariant torsion and G_2-metrics

We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G_2 that arises from SO(3)-structures with invariant intrinsic torsion.


Introduction
It was Bryant and Salamon [10] who constructed the first known examples of complete, irreducible metrics with Riemannian holonomy equal to the exceptional Lie group G 2 . All their metrics asymptotically look like that of a Riemannian cone over one of the homogeneous nearly-Kähler sixmanifolds CP(3), F 1,2 (C 3 ) or S 3 × S 3 . Whilst G 2 -manifolds have received much attention over the past 25 years, additional examples of complete, non-compact manifolds with G 2 -holonomy are still relatively few [8,16,4]. It therefore seems sensible to return to the original examples so as to better understand what makes them so special. The starting point of this paper is the Bryant-Salamon metric approaching the cone on S 3 × S 3 . This can be viewed as a metric on the spinor bundle of S 3 which we may write as Spin(4) × Sp(1) H; here the action of Spin(4) commutes with the right action of Sp(1) on H. As a consequence, the spinor bundle has the structure of a cohomogeneity one manifold with principal orbit Spin(4) × Sp(1)/ Sp (1). The isotropy representation is the sum of two copies of the adjoint representation sp (1); this factors through the quotient SO(3) = Sp(1)/Z 2 so as to give rise to SO(3)-structures on hypersurfaces of the spinor bundle. The structure determined by this cohomogeneity one action corresponds to the nearly-Kähler structure on S 3 × S 3 , but other invariant SO(3)-structures can be obtained by composing with an SO(3)-equivariant isomorphism of R 3 ⊕ R 3 . More generally, SO(3)-structures come in GL(2, R)-families, since GL(2, R) is the centralizer of SO (3) in GL (6, R). This is at the origin of the flexibility required to construct the Bryant-Salamon metric. Another way of viewing this fact is by observing that SO(3) preserves more than one metric on R 3 ⊕ R 3 .
We are grateful to Simon Salamon for many invaluable discussions and suggestions. 1 The SO(3)-structures associated with the Bryant-Salamon metric are homogeneous. In particular, they come with an Ambrose-Singer connection, characterized by having parallel torsion and curvature. Their intrinsic torsion, which is a map from the reduced frame bundle to (R 6 ) * ⊗ (so(6)/ so(3)), is therefore parallel and so determines an SO(3)-invariant element. In this sense, SO(3)-structures with invariant intrinsic torsion generalize the structures on S 3 × S 3 that give rise to the Bryant-Salamon metric.
A natural source of SO(3)-structures comes from Lie groups. Lie algebras of dimension six are characterized by the Chevalley-Eilenberg operator d : Λ(R 6 ) * → Λ(R 6 ) * , and we may think of d as representing a Lie algebra with a fixed frame and so a natural SO(3)-structure. If d is SO(3)equivariant, the associated SO(3)-structure has invariant intrinsic torsion. It turns out that any SO(3)-structure with invariant intrinsic torsion is locally equivalent to one obtained in this way. More precisely, non-flat SO(3)-structures with invariant intrinsic torsion are parameterized, up to scale, by an element of RP 5 lying in a subvariety M cut out by the condition d 2 = 0. Via the Segre embedding, this subvariety corresponds to RP 1 × RP 2 . If completeness is assumed, the classification result can be made global and turns out to be closely related to the classification of naturally reductive homogeneous spaces of dimension six, recently obtained by Agricola, Ferreira and Friedrich [1]. Indeed, to each element of M we can associate a unique naturally reductive homogeneous structure; our approach then results in a geometric way of interpreting the related part of their classification in terms of algebraic subvarieties.
In order to exploit the GL(2, R)-invariance, it is convenient to identify RP n with the space of homogeneous degree n polynomials in two variables. The intrinsic torsion is an element of R 4 . Projectively, it is given by a natural GL(2, R)-equivariant map: which is determined by polynomial multiplication (x, y) → xy. The actual connected, simply-connected Lie group that is obtained from the pair (x, y) depends on the power of x that divides y as well as the discriminant ∆ = ∆(y).
x 2 | y ∆ = 0 (0, 0, 0, 12, 13, 23) Each row in Table 1.1 represents a single GL(2, R)-orbit. In this way there are, up to this symmetry, precisely five non-Abelian models. However, metric properties are not invariant under GL(2, R). In particular, we show that the Einstein metrics correspond to three U(1)-orbits in RP 3 ; the geometry is S 3 × S 3 with either its nearly-Kähler metric or its bi-invariant metric, and the latter can appear in two ways.
The GL(2, R)-symmetry is also lost when computing the local G 2 -metric. In fact, the Bryant-Salamon metric is obtained by integrating the Hitchin flow, which is a flow on the space of half-flat SU(3)-structures. The halfflat condition reads as a linear condition on the SO(3)-intrinsic torsion and determines a subset in RP 1 × RP 2 which can be interpreted as the blowup of RP 2 at a point; in the non-projective setup, which also includes the Abelian case, this gives an R 3 ⊂ R 4 . The Hitchin flow determines a flow on this R 3 . Here the GL(2, R)-symmetry breaks down to a discrete group, namely the symmetric group Σ 3 formed of permutations in three letters. The maximally invariant solution gives rise to the nearly-Kähler S 3 × S 3 , whose evolution is simply the cone. The Bryant-Salamon metric appears as a Σ 2 -invariant solution. On the open subset ∆ < 0, one can recover the metrics of [6].
Strictly speaking, what we are evolving is not an SU(3)-structure, but rather the intrinsic torsion (of an SO(3)-structure). The actual solution, in terms of structures, is obtained by lifting the integral curves under the non-projective analogue of (1.1). This approach allows us to give a precise explanation of the "triality" symmetry appearing in [3]; this symmetry is the reason why Brandhuber et al. [8] were able to construct, seemingly, different solutions whose evolution equations look exactly the same. From our point of view, there is little reason to distinguish between these cases: we obtain the same G 2 -metric on the spinor bundle, but written in terms of three different cohomogeneity one actions of SU(2) × SU (2).
By choosing appropriate one-parameter subgroups, the GL(2, R)-symmetry can also be used to generate flows that interpolate between the different Lie algebras of M; such flows are sometimes referred to as Lie algebra "contractions". Motivated by their occurrence in the physics literature on G 2 -metrics [17,13], we classify the associated orbits of half-flat structures. Thinking of the half-flat structures as R 3 ⊂ R 4 , the union of these orbits comes in three families, each of which forms a two-plane. It turns out that only one of these planes is preserved by the Hitchin flow, meaning that the flow itself may be viewed as a contraction up to rescaling in this case. In the other two cases, contractions can be used as a way of generating initial data for Hitchin's flow equations that give rise to different solutions.
In the final result of the paper, we classify the complete holonomy G 2metrics which are determined by a half-flat SO(3)-structure with invariant intrinsic torsion. We show that the only full holonomy G 2 -metric is the Bryant-Salamon metric. This classification result is different from, but consistent with, the uniqueness result by Karigiannis and Lotay [20,Corollary 6.4]. It also supplements the classification of cohomogeneity one G 2 -structures obtained by Cleyton and Swann [14] who considered only actions of simple Lie groups.

The hypersurfaces: SO(3)-structures
Let V denote the three-dimensional irreducible representation of SO(3). In this paper, we are concerned with six-manifolds M whose tangent space at each point is modelled on (2.1) As we are free to rescale the metric on each summand V, this representation theoretical definition of an SO(3)-structure does not determine a canonical inclusion SO(3) ⊂ SO (6). We shall address this subtlety shortly. Before doing so, however, we will explain a geometric way of achieving (2.1). We start out by considering the non-degenerate 2-form together with a one-parameter family of simple 3-forms obtained by letting SO(2) act on the two-planes e 2i−1 , e 2i : η θ = (cos θ e 1 + sin θ e 2 ) ∧ (cos θ e 3 + sin θ e 4 ) ∧ (cos θ e 5 + sin θ e 6 ).
The centralizer of SO(3) in GL(6, R) is a copy of GL(2, R) which contains the above rotation group SO (2). A first clear indication that this maximal commuting subgroup should not be ignored is the fact that it can be used to remedy the ambiguity in our choice of inclusion SO(3) ⊂ SO (6).
Having fixed such an inclusion, the identification allows us to consider the module so(3) ⊥ which is the orthogonal complement of so(3) inside so (6). This latter module is reducible with irreducible summands given by R ⊕ 2V ⊕ S 2 0 (V). Here S 2 0 (V) denotes the fivedimensional irreducible representation of SO(3) that is defined as the kernel of the natural trace map S 2 (V) → R. Similarly, we shall, subsequently, denote by S 3 0 (V) the seven-dimensional irreducible representation, defined as the kernel of the trace map S 3 (V) → V.
In order to exploit the additional GL(2, R)-symmetry, it is useful to consider representations of the enlarged group SO(3) × SL(2, R); restricting the attention to SL(2, R) is harmless in our case, since the full symmetry group SO(3) × GL(2, R) would result in the same decompositions into irreducible modules. The irreducible components of this enlargement have the form A p ⊗ B q where A p , B p denote the p + 1-dimensional irreducible representations of SO(3) and SL(2, R), respectively.
The intrinsic torsion of an SO(3)-structure is, by definition, the projection of the torsion of any connection on the SO(3)-structure on the cokernel of the alternating map This alternating map extends to an isomorphism hence the space of intrinsic torsion can be identified with T * ⊗ so(3) ⊥ . It follows that any SO(3)-structure with intrinsic torsion τ has a unique connection with torsion ∂(τ). Subsequently, we shall refer to this as the "canonical connection"; its associated exterior covariant derivative operator will be denoted by D.

Lemma 2.2. The intrinsic torsion of an SO(3)-structure on a six-manifold belongs to the 72-dimensional SO(3)-module
As SO(3) × SL(2, R)-modules, this space can be expressed in the form Proof. The decomposition (2.4) follows from standard computations using the Clebsch-Gordan formula. Hence, only the expression (2.5) needs to be explained. The intrinsic torsion takes values in the cokernel of the alternating map (2.3) and this map is injective because of the inclusion so(3) ⊂ so (6). As SO(3) × SL(2, R)-modules, we therefore have that the intrinsic torsion belongs to the quotient The quadruplet (σ, η i ) which defines an SO(3)-structure can be viewed as a refinement of an associated SU(3)-structure. The latter can be realized by averaging over the forms η i : the 3-form has GL + (6, R)-stabilizer SL(3, C) which intersects with the stabilizer of σ in a copy of SU (3). As γ has stabilizer SL(3, C), it determines an almost complex structure J which can be used to define the dual 3-form: The space of SU(3)-intrinsic torsion was studied in great details in [12] and can be related to the space (2.4) via . This relation is schematically represented in Table 2.1.
We have already encountered several SO(3)-invariant forms. Amongst these, the 2-form σ and its square σ 2 exhaust the invariant forms of degree two and four:  [12], observe that We have also introduced five invariant 3-forms. These are not all independent as is evident from the decomposition: The exterior derivatives of the invariant forms detect part of the intrinsic torsion. In fact, they determine all of it apart from the 14-dimensional module A 6 B 1 of (2.5). (i) dσ determines the subspace 3R ⊕ 2V ⊕ 2S 2 0 (V); (ii) each dη i determines a subspace R ⊕ 2V ⊕ S 2 0 (V), and the three resulting subspaces are in direct sum; (iii) dγ determines the subspace R ⊕ 3V ⊕ S 2 0 (V). In terms of SO(3) × SL(2, R)-modules, this means that dσ determines the subspace 7) and the dη i , dγ altogether determine the modules (2.8) In particular, the only intrinsic torsion component not determined by the exterior derivatives of invariant forms is the 14-dimensional module A 6 B 1 .
Proof. We consider the alternating map (2.9) and assume its restriction to the direct sum W ⊕ T * ⊗ so(3) is an isomorphism; any such subspace W can be identified with the space of intrinsic torsion. We may assume W is invariant under SO(3) × GL(2, R). Given any SO(3)-invariant form α in Λ k T * , the infinitesimal action of gl(6, R) determines an equivariant linear map Denote by W α a maximal submodule on which l α is injective. As l α is zero on both T * ⊗ so(3) and the kernel of (2.9), we can assume W α ⊂ W. By construction, W α represents the component of W that is determined by dα.
Straightforward computations verify that l σ is surjective. Since it is also SO This proves (i) and (2.7).
The maps l γ and lγ are also surjective, proving (iii), whereas the image of l η i is isomorphic to R ⊕ 2V ⊕ S 2 0 (V). For example, we find that the image of l η 0 , inside Λ 4 T * , is the orthogonal complement of e 1246 , e 2346 , e 2456 ∼ = V.
The space of SO(3)-invariant 3-forms is the SO(3) × SL(2, R)-module γ, η 0 , η 1 , η 2 ∼ = B 3 . It follows that the part of intrinsic torsion ∑ i W η i + Wγ, determined by these forms, is a submodule of In fact, the module A 2 B 5 does not appear in the decomposition (2.5), and ∑ i W η i + Wγ contains at least one copy of R and S 2 0 V as an SO(3)-module. It must therefore contain B 3 ⊕ A 4 B 3 . In order to prove (2.8), it then suffices to prove that it also contains In order to prove this, we consider the symmetric group Σ 3 which can be viewed as a subgroup of GL(2, R) via Its irreducible representations are the trivial and alternating representations R and U (both one-dimensional) and the standard representation S (which has dimension two). The latter is induced by the inclusion (2.10). As SO(3) × Σ 3 -modules, we have: Since this is is an orthogonal group, we have Λ 2 T * ∼ = so (6). Moreover, Σ 3 acts trivially on so(3) because SO(3) commutes with GL(2, R). It follows that we may write The SO(3)-modules isomorphic to S 2 0 (V) are identified by the SO(3) × Σ 3 -equivariant maps, g 1 , g 2 , g 3 : T * → T * ⊗ so(3) ⊥ , , and the two SO(3)-equivariant maps g 4 , g 5 : V → T * ⊗ so(3) ⊥ , given by: These images are isomorphic to A 2 U and A 2 , respectively. Setting we compute This shows that each W η i contains an SO(3)-module isomorphic to 2V whose projections to 3A 2 S, A 2 U and A 2 have maximal rank, hence (2.12) which was what remained to be proved.
It is clear from the above that SO(3)-structures are admissible but not strongly admissible: SO(3) is the largest subgroup that acts trivially on the space of invariant forms, but the closedness of the invariant forms fails to ensure the vanishing of the intrinsic torsion.
Despite this gap between "closedness" and "parallelness", SO(3)-structures enjoy a rather remarkable feature which ties the curvature with the intrinsic torsion. The following result is, in some sense, an analogue of Bonan's results about Ricci-flatness of metrics with exceptional holonomy [7]. In the setting of SO(3)-structures, however, the curvature information encoded in the intrinsic torsion is not limited to the Ricci curvature. The following result implies that a torsion-free SO(3)-structure is, in fact, flat. Proposition 2.4. There are two SO(3)-equivariant linear maps R so(6) , R so (3) such that whenever P is an SO(3)-structure with intrinsic torsion τ, the composition is the curvature of the Levi-Civita connection, and is the curvature of the canonical connection.
Proof. Let ω so (6) denote the restriction of the Levi-Civita connection oneform to the (reduced frame bundle) P. The canonical connection form ω is the orthogonal projection of ω so(6) onto so(3), and the intrinsic torsion τ is the difference ω − ω so (6) . We can express the curvature of ω so (6) as (3) . The Bianchi identity Ω so(6) ∧ θ = 0 determines a subspace R in Λ 2 ⊗ so(6) given as the kernel of the natural map It is easy to check that R intersects trivially with S 2 (so(3)). It follows that the projection which gives the map R so (6) . Similarly, R so(3) is determined by . Remark 2.5. The statement of Proposition 2.4 holds more generally. If we replace SO(3) with G ⊂ SO(n), then the statement remains valid so long as the projection R → Λ 2 ⊗ g ⊥ is injective. (3) is not strongly admissible, we cannot express the curvature solely in terms of the exterior derivative of the defining forms. A way to circumvent this fact is achieved by using the inclusion of SO(3) in SU(3). Since the latter structure group is strongly admissible, the Ricci curvature can be expressed in terms of the exterior derivatives of the invariant forms σ and γ + iγ. As explained in [5] this, in particular, applies to half-flat SU(3)-structures, meaning those which have

As SO
equivalently, those structures whose intrinsic torsion belongs to the 21-

Invariant torsion
A natural generalization of torsion-free SO(3)-structures are those which have invariant intrinsic torsion τ, meaning Whilst the invariance refers to the action of SO(3), a significant role is played by the commuting subgroup SL(2, R) ⊂ GL(6, R). With respect to this group, the module (T * ⊗ so(3) ⊥ ) SO(3) appears as B 3 . Explicitly, we can choose coordinates u 1 , u 2 on R 2 such that the map is SL(2, R)-equivariant, and then we may think of B k as the space R k [u 1 , u 2 ] of homogeneous polynomials of degree k in u 1 and u 2 . This space is naturally mapped to a subspace of the tensorial algebra over B 1 via The intrinsic torsion, strictly speaking, takes values in a quotient of Λ 2 T * ⊗ T. The subspace of this quotient being fixed by SO (3) is identified via the following: The statement is then obtained by considering the map It follows that we can identify the intrinsic torsion τ with a quadruplet of functions λ 1 , λ 2 , λ 3 , λ 4 ∈ C ∞ (M). These then govern a differential complex, which is obtained by restriction of the exterior derivative to the span of the set of invariant forms; it is similar to the situation considered in [11]. This complex is completely determined by Expressing dσ this way allows us to easily determine the components of the SU(3)-intrinsic torsion: For SO(3)-structures, it turns out that invariant intrinsic torsion is the same as constant intrinsic torsion, as in [15]: Proof. Since there are no invariant 5-forms, σ 2 is necessarily closed. Applying d to the equations (3.1), we find that dλ i ∧ σ 2 vanishes, for i = 1, 2, 3, 4. As σ is non-degenerate, we conclude dλ i = 0, and the statement follows.
If we view the intrinsic torsion as taking values in T * ⊗ so(3) ⊥ , there is a cost, since this is not an SL(2, R)-module. Nevertheless, this way of viewing things will be useful subsequently.

Lemma 3.4. The intrinsic torsion τ λ can be written as
where τ λ ∈ T * ⊗ so(3) ⊥ is given by: here the second map is the projection, and the inclusion B 3 ⊕ B 1 ⊂ Λ 2 T * ⊗ T is the one given in Lemma 3.1. Explicit computation of the inverse then gives the stated formula.
Combining the above observations with Proposition 2.4, we see that invariant intrinsic torsion structures are locally uniquely determined by the torsion. Proof. Let P → M be an SO(3)-structure with invariant intrinsic torsion. By Lemma 3.3, the intrinsic torsion is constant which means the function τ : P → T * ⊗ so(3) ⊥ , defined in Lemma 3.4, is constant and so parallel. Thus, as a tensorial form τ ∈ Ω 1 (P, so(3) ⊥ ), we have where Θ ∈ Ω 2 (P, T) denotes the torsion, and represents the contraction showing that the curvature of the canonical connection is completely determined by the intrinsic torsion. The curvature is therefore constant as a map This implies that Ω so(3) is parallel, and the same applies to Θ.
Consider now two such structures P → M and P ′ → M ′ satisfying τ(u) = τ ′ (u ′ ). Then, by [21,Theorem VI.7.4], there is a local affine isomorphism mapping u to u ′ . Since it is affine with respect to an SO(3) connection, it maps P into P ′ , thereby giving a local equivalence.
If M and M ′ are connected, simply-connected and complete, then [21,Theorem VI.7.8] implies that the equivalence can be extended globally.
Remark 3.6. In the constant intrinsic torsion setting, the statement of Theorem 3.5 holds under the conditions mentioned in Remark 2.5.
By the proof of the above theorem, the canonical connection ∇ is an Ambrose-Singer connection, meaning its torsion and curvature tensors are parallel. In other terms [2], (M, ∇) is locally homogeneous. The following result, whose proof we shall defer, implies that M is, in fact, locally isometric to a homogeneous space. Following [25], there are eight classes of homogeneous structures which are defined according to the action of the orthogonal group on , we see that (M, ∇) can either have mixed type J 2 ⊕ J 3 or pure type J 3 , also referred to as naturally reductive; the latter case happens precisely when Before addressing the proof of Corollary 3.7, we should like to emphasize that the interesting invariant intrinsic torsion SO(3)-structures are precisely those which are not symplectic, in the following sense: Proof. The space T * R 3 can be equipped with a natural SO(3)-structure, compatible with the canonical symplectic form. This structure has vanishing intrinsic torsion and is, by Theorem 3.5, locally uniquely determined by this condition.
In general, we note that the exterior derivative of σ completely determines the component of the intrinsic torsion isomorphic to 4R. Therefore, a symplectic manifold with invariant SO(3)-intrinsic torsion must be torsion-free. The statement of the proposition now follows by local uniqueness.
3.1. The invariant torsion variety M. Let e 1 , . . . , e 6 be an SO(3) adapted basis of the dual of the Lie algebra g * of G, or, correspondingly, think of this as a coframe on G that determines a left-invariant SO(3)-structure. The torsion of the flat connection can then be expressed as and the intrinsic torsion of the structure is invariant precisely when (3.3) is an element of B 3 ⊕ ∂(T * ⊗ so (3)).
We shall now investigate the Lie algebras that are determined by elements κ λ,µ as in Lemma 3.1. By the above, these have invariant intrinsic torsion. In order to appropriately parameterize this family of Lie algebras, we introduce the following algebraic variety: follows from the computation: ; the coefficient of the second summand has been chosen so as to obtain M as the image of the induced projective map RP 1 × RP 2 → RP 5 . Up to a change of coordinates, this latter map is the Segre embedding and therefore an isomorphism.
Prompted by Lemma 3.9, we will represent elements of M by formal products of polynomials:  In addition, the associated flat connection has invariant torsion, and the intrinsic torsion is represented by the expanded product In particular the intrinsic torsion determines a surjective PSL(2, R)-equivariant Proof. After relabelling we find that an element κ λ,µ , given as in Lemma 3.1, corresponds to the structural equations (3.5) and so forth, where Note that rescaling of the metric amounts to the change e i → zẽ i , for a non-zero constant z. Then, lettingã = a/z, . . . ,r = r/z, we have that [ã : . . . :r] = [a : . . . : r] in RP 5 and dẽ 1 =ãẽ 35 +bẽ 46 +c(ẽ 36 +ẽ 45 ). Also observe that (3.5) becomes (3.4) upon the substitution: These equations define a Lie algebra provided d 2 = 0, corresponding to the Jacobi identity. Computing d 2 e i , using (3.5), we find that this condition can be rephrased in terms of the set of equations Having excluded the Abelian case, these constraints are equivalent to the condition that giving the asserted correspondence with points of M. By construction, the subset cut out by d 2 = 0 is preserved by the action of SL(2, R), since the natural map Hom(T * , Λ 2 T * ) → Hom(Λ 2 T * , Λ 3 T * ) is GL(6, R)-equivariant. Hence, the Jacobi identity defines a GL(6, R)invariant subvariety of Hom(T * , Λ 2 T * ). Its intersection with the SL(2, R)module B 1 ⊕ B 3 is necessarily preserved by SL(2, R). By Lemma 3.9, this action of SL(2, R) is the standard action on RP 1 × RP 2 .
By construction, the projection on B 3 that gives the intrinsic torsion is given by polynomial multiplication. This is a surjective map because every third degree polynomial has a linear factor. Motivated by Proposition 3.10, we shall refer to M as the invariant torsion variety. It is the properties of this variety that allows us to complete the proof of Corollary 3.7.
Proof of Corollary 3.7. Surjectivity of the polynomial multiplication map tells us that any value of invariant intrinsic torsion can be realized in terms of a left-invariant SO(3)-structure on a Lie group; this gives the last part of the statement. In addition, by Theorem 3.5, this implies that any invariant intrinsic torsion SO(3)-structure must be locally isometric to a left-invariant structure on a Lie group.
By the isomorphism M ∼ = RP 1 × RP 2 of Proposition 3.10, it follows that, up to the action of GL(2, R), there are five cases of invariant intrinsic torsion: Lemma 3.11. There are precisely five GL(2, R)-orbits in RP 1 × RP 2 , determined by the elements: . Proof. Third degree polynomials are classified according to the number and multiplicity of their real roots and the formal products are determined by choosing a linear factor. Using the action of GL(2, R), the statement follows.
In relation to Lemma 3.11, we have five naturally defined subvarieties of M. Indeed, we can think of the invariant torsion variety in terms of pairs of polynomials, and let denote the discriminant of y and the resultant of x, y, respectively. Then we can we define M + ⊂ M as the open subvariety corresponding to the case when y has two different real roots and gcd(x, y) = 1, meaning it is defined by ∆ > 0 and R = 0. Similarly, M − ⊂ M corresponds to the situation when y has no real roots and gcd(x, y) = 1, so that we have ∆ < 0, R = 0 as defining conditions. A third subvariety, S 1 ∼ = RP 1 × RP 1 , occurs when y has a double root, meaning ∆ = 0. Likewise, we can consider the subvariety S 2 ∼ = RP 1 × RP 1 corresponding to the case when x and y have a common root, i.e., R = 0. The intersection C = S 1 ∩ S 2 ∼ = RP 1 corresponds to y being a multiple of x 2 . A natural question is what are the isomorphism classes of Lie algebras constituting these five varieties. The list of Lie algebras that appear can be anticipated from [1,Section 8], where the authors classify naturally reductive homogeneous structures in dimension six. In order to understand this, note that 2R ⊂ T * ⊗ so(3) is transverse to Λ 3 T * . As a consequence, there is a unique equivariant map This means we can modify the canonical connection, using p(τ λ ), so as to obtain invariant skew-symmetric torsion and, in particular, a naturally reductive homogeneous structure. The adjusted Ambrose-Singer connection has torsion given by the 3-form We find that the correspondence between our five varieties and the five different Lie algebras of [1, Section 8] is as follows: (iv) points of S 2 \ C correspond to the direct sum so(3) ⊕ R 3 ; (v) points of C = S 1 ∩ S 2 correspond to the nilpotent Lie algebra (0, 0, 0, 12, 13, 23).
Proof. By Lemma 3.11, each of the listed subvarieties is a GL(2, R)-orbit and therefore connected. Following [1], one uses the Killing form to distinguish the various Lie algebras and with respect to the basis e 1 , e 3 , e 5 , e 2 , e 4 , e 6 , it can be expressed in terms of a block form matrix F; this 6 × 6 matrix consists of four blocks each of which is proportional to the identity matrix, diag(1, 1, 1) and, in addition, the off-diagonal blocks are identical. It follows that F has at most two distinct eigenvalues, and therefore its rank equals a multiple of three.
The rank of F is maximal precisely when its determinant does not vanish. Referring to (3.7), we find det F = (4∆R 2 ) 3 .
From this expression, it is clear that maximal rank corresponds to the open subvarieties M ± of M. In this case g is semi-simple and so, by the classification of semi-simple Lie algebras, must be a real form of sl(2, C) ⊕ sl(2, C). Hence g is either so(3, C) or the direct sum of two three-dimensional Lie algebras, each isomorphic to either so(3) or sl(2, R). Since the signature of the Killing form is either (6, 0) or (3,3), depending on whether ∆ is positive or negative, we see that the two situations correspond to so(3) ⊕ so(3) or so(3, C), respectively.
The intersection S 1 ∩ S 2 is the orbit of u 1 · u 2 1 . It follows from (3.4) that each element of this curve in M corresponds to the nilpotent Lie algebra g = (0, 0, 0, 12, 13, 23).
The complement of C in S 1 is the orbit of u 1 · u 2 2 . This consists of perfect Lie algebras whose radical is a three-dimensional ideal r = [g, g] ⊥ = g ⊥ spanned by e 2 , e 4 , e 6 . Computations show that r is Abelian. The semisimple quotient g /r is a three-dimensional semi-simple Lie algebra whose Lie bracket is SO(3)-invariant, hence necessarily so(3). Therefore any point of S 1 \ C corresponds to a Lie algebra which is isomorphic to the semi-direct product so(3) ⋉ R 3 . In this situation, so(3) acts non-trivially on R 3 , since [g, g] = g.
Finally, note that the complement of C in S 2 is the orbit of u 1 · u 1 u 2 . In this case, points of the orbit have first Lie algebra Betti number equal to three, and [g, g] is easily seen to be isomorphic to so (3). In addition, the radical of these Lie algebras, [g, g] ⊥ , is an ideal that intersects the derived algebra trivially and so coincides with the center. In conclusion, we have an orbit consisting of Lie algebras g = so(3) ⊕ R 3 .
The upshot of Lemma 3.11 and Theorem 3.12 is the list of model geometries mentioned in the introduction of the paper: 3.2. Curvature properties. By Theorem 2.3, one would expect curvature computations to be particularly simple for invariant intrinsic torsion structures. We now illustrate how this works by characterizing the locally conformally flat and Einstein metrics.
It is well known that S 3 × S 3 admits two left-invariant Einstein metrics, namely the bi-invariant metric and the nearly-Kähler metric. The basis of one-forms h 1 , . . . , h 6 satisfying dh 1 = −h 35 , dh 2 = −h 46 , and so forth, is orthonormal for the bi-invariant metric. In terms of this basis, an orthonormal coframe for the nearly-Kähler metric is given by There is a, seemingly, different Einstein metric which corresponds to the orthonormal coframe This metric can be distinguished from the first two by considering the ratio between the scalar curvature and the trace of the Killing form; this quantity is an invariant for Lie group isomorphisms that are also isometries and is different for the three metrics. In terms of the orthonormal frame for the third metric, the structure constants appear as (3.8) Remark 3.14. The arguments of [22] imply that the first and third metrics are isometric up to homothety. The isometry, however, is not an isomorphism of Lie groups.
Verifiably each of these Einstein metrics corresponds to an SO(3)-structure with invariant intrinsic torsion. The converse statement also holds by the second part of the following: Proof. Whilst the group SL(2, R) does not preserve metric properties, it contains a one-dimensional torus which certainly does and under which we have Here C k denotes the real irreducible representation of U(1) of weight k.
A Riemannian six-manifold has curvature that takes values in the module R = W ⊕ S 2 (R 6 ), which corresponds to the decomposition into the Weyl and Ricci tensors, respectively.
By Proposition 2.4 and the argument used in the proof of Theorem 3.5, the curvature of an SO(3)-structure with invariant intrinsic torsion τ has the form R so(6) (τ ⊗ τ, ∂(τ) τ). More precisely, we have Indeed, computations show that By its very construction, the Weyl tensor is a U(1)-equivariant map, which, in particular, means that its two scalar components come from a linear map 2R ⊂ S 2 (C 3 ) ⊕ S 2 (C 1 ) → 2R. The latter map is injective as can verified by choosing a basis. Consequently any P which is not (locally) flat will have Weyl tensor with non-trivial component in 2R ⊂ W SO(3) . In particular, P is locally conformally flat if and only if it is flat and therefore, by Proposition 3.8, locally equivalent to T * R 3 .
One verifies that the preimage in M of each component is also connected, and U(1) acts transitively on it since it acts transitively on both the base and fibre. The projection has degree two and one, according to the multiplicity of the defining equation as a factor in the resultant. Thus, up to U(1)-symmetry, we find the possibilities ). The first of these points yields the Killing metric, the second point gives (3.8) and, in terms of (3.5), the third solution reads  [23]. The fact that Pope's metric on SO(4) is the nearly-Kähler metric can be verified by setting p = 3, q = 1 in [23, Equation (16)].

Half-flat structures and contraction limits. According to (3.2) the SU(3)-structure associated with an invariant intrinsic torsion SO(3)-structu-
re is half-flat if and only if λ 2 = λ 4 . In this case, the SU(3)-intrinsic torsion takes values in 3R ⊂ W − 1 ⊕ W 3 which is not an SL(2, R)-module; the largest subgroup of GL(2, R) that fixes this subspace is Σ 3 × R * with Σ 3 acting as in (2.10).
In terms of the invariant torsion variety, we have the following geometric description of the half-flat structures: Proof. Considering the formal products of (non-zero) polynomials x, y as in (3.6), the half-flat structures constitute the set {x · y : In terms of the corresponding set of RP 1 × RP 2 , the projection map onto the second factor is an isomorphism away from [1 : 0 : 1] ∈ RP 2 . By choosing an affine chart centered at this point, the first part of the proposition is readily verified.
For the last assertion, we note that the product x · (y 1 u 2 1 + y 2 u 2 2 ) has ∆ = −4, R = x 2 1 + x 2 2 and λ 1 = x 1 = λ 3 . It follows that points in the preimage of [1 : 0 : 1] correspond to so(3, C) and that the associated SU(3)structure has W − 1 = 0, hence is Hermitian. We shall now describe a way of deforming half-flat structures on M. It is a type of Abelianization procedure which is sometimes referred to as Lie algebra contractions (see, for instance, [17]). To this end, we think of Lie algebras as points of the variety consisting of Chevalley-Eilenberg operators d : Λ(R) * → Λ(R 6 ) * . Regarding elements of D as linear isomorphisms R 6 → g to a Lie algebra, we note that GL(6, R) acts naturally on D in two ways; we can use either left or right composition on the isomorphism. From our view point, the right action is distinguished by being independent of the underlying Lie algebra and compatible with the natural action on Hom((R 6 ) * , Λ 2 (R 6 ) * ). In addition, the intrinsic torsion map is equivariant with respect to this action. In these terms, a contraction limit is a procedure by which one obtains an element of D as the limit point of a curve t → d · g t , g t ∈ GL(6, R).
A tractable subclass of contraction limits are those curves that arise as one-parameter subgroups. We restrict our attention further, namely to subgroups of GL(2, R); this group preserves the invariant intrinsic torsion condition. It is then harmless to consider only subgroups of SL(2, R), since the rescalings commute with SL(2, R) so as to give rise to homothetic structures on the same Lie algebra.
The action of a one-parameter subgroup is entirely determined by the fundamental vector field associated to a generator Explicitly, the fundamental vector field on B 3 associated to A a,b,c takes the form: Obviously, X a,b,c will generally fail to preserve the half-flat condition. In fact, up to the action of Σ 3 × R * , there are precisely three elements of sl(2, R) that generate a flow of half-flat structures: Up to the action of Σ 3 × R * there are three one-parameter subgroups in GL(2, R) with a non-trivial orbit of half-flat structures. These are generated by the vector fields X 1,0,0 , X 0,1,3 and X 0,1,−1 . The union of half-flat orbits is in each case a two-plane in B 3 which is given by

respectively.
The plane Π 1,0,0 contains a representative of each Lie algebra appearing in Theorem 3.12. The second plane, Π 0,1,3 , contains only so(3, C) and the nilpotent algebra. The third, Π 0,1,−1 , runs through SU(3) reductions of a fixed Hermitian structure on so(3, C) that only differ by a rotation of the complex volume form.
Proof. As B 3 has weights one and three the Cartan subalgebra generated by A a,b,c must act with multiplicities two and four. It follows that the largest subspace of B 3 which is both contained in {λ 2 = λ 4 } and is invariant under X a,b,c (if non-trivial) must be two-dimensional. This maximal subspace is contained in the two-plane where X a,b,c (λ 2 − λ 4 ) vanishes, leading to the following: The condition on the vector field X a,b,c to be tangent to this plane is found from the expression This gives us three situations to consider: (i) if c = 0, we must have 3b 2 − c 2 + 2bc − 12a 2 = 0 and a(b − c) = 0 which implies one of the following possibilities: (iii) alternatively, for c = 0, we can have b = 0. In that case, we find that a = ± √ 3 2 b. Considering the first of the three possibilities from (i), we have generating vector field X 0,1,−1 which means λ 1 = λ 3 . The corresponding plane, Π 0,1,−1 , then consists of half-flat structures on so(3, C) that have W ± 1 = 0, and the one-parameter group generated by X 0,1,−1 is the standard U(1) action which rotates the complex volume form γ + iγ.
Up to the action of Σ 3 × R * , the vector field X 1,± √ 3/2,± √ 3/2 appearing in the final subcase of (i) is equivalent to X 1,0,0 of case (ii). Let us therefore consider the generating vector field X 1,0,0 . This has λ 2 = 0 and therefore gives rise to the following possibilities . The first of these points has ∆ = −4y 1 y 3 and R = y 3 , so that we obtain all the types of Lie algebras appearing in M, except SO(3) × R 3 . Similarly, the second point gives ∆ = x 2 2 and R = 2x 2 2 x 1 which leads to all the possible types of Lie algebras apart from SO(3) ⋉ R 3 .
Acting with such an element maps u 1 · (u 2 1 ± u 2 2 ) to u 1 · (λ 2 u 2 1 ± λ −2 u 2 2 ). In this way we obtain the following two curves Both of these correspond to the semi-direct product so(3) ⋉ R 3 as λ → 0 and the nilpotent Lie algebra as λ → +∞. Points in λ ∈ (0, +∞) correspond to either SO(3, C) or S 3 × S 3 depending on whether they are in C + or C − , respectively.
On the other hand, if we let diag(λ, λ −1 ) act on . In this way, we obtain two curves that connect the nilpotent algebra and so(3) ⊕ R 3 through so(3) ⊕ so(3).

Remark 3.21.
In the next section we shall discuss another flow of half-flat structures which is also generated by the action of GL(2, R). This flow is unrelated to the above contractions in the sense that it preserves Π 1,0,0 , but not the other two-planes of Proposition 3.19.

Uniqueness of the Bryant-Salamon metric
A six-manifold M with an SU(3)-structure that is half-flat can, at least if real analytic, be embedded in a manifold N with holonomy contained in G 2 [19] (see [9] for counterexamples outside the real analytic setting). The metric on N is Ricci-flat and can be found by solving a system of evolution equations which is usually referred to as the Hitchin flow. Explicitly, we have As already discussed, the half-flat structures are characterized by the equations (2.14) whose symmetries consist of elements in GL(6, R) that preserve σ 2 and γ . This group contains SU(3), dilations R + and the group of order two corresponding to complex conjugation.
The algebraic variety of six-dimensional Lie algebras with a fixed SU(3)structure can be identified with the SU(3) quotient of the space D defined in (3.9). Focusing on the subvariety of half-flat structures leaves us with a quotient of N = d ∈ D | dγ = 0 = dσ 2 .

4.1.
Integrating the flow. Fixing a point d ∈ N SO(3) determines a Lie algebra g together with a fixed frame u : R 6 → g. If we denote also by u the induced maps Λ k (R 6 ) * → Λ k g * , then we have induced forms on g given by where the subscript refers to the "standard" forms on R 6 . For each g ∈ GL(2, R), gd obviously represents the same Lie algebra as g, but with a different frame ug. Consequently, we obtain a map GL(2, R) → Λ 3 g * , g → (ug)(γ R 6 ) = R g (γ), and, similarly, GL(2, R) → Λ 2 g * , g → (ug)(σ R 6 ) = R g (σ).
We can use the discriminant of a third degree polynomial , so as to describe the Hitchin flow as a linear flow on the space of polynomials; the discriminant then determines the velocity. Explicitly, we define the map Q : whose image is the space of polynomials with either three distinct roots or one triple root. Indeed, if xu 1 + yu 2 and zu 1 + wu 2 are linearly dependent, the image has a triple root. If they are independent, then the image of Q has three distinct roots. Conversely, any polynomial with three distinct roots can be written as the product of three linear factors f 1 f 2 f 3 such that f 1 + f 2 + f 3 = 0 and the equations a unique matrix, up to the Σ 3 action that permutes the linear factors. It follows that the restriction of Q to the set of invertible matrices defines a 3-fold covering map This map is equivariant with respect to the right action of GL(2, R), and the subgroup Σ 3 generated by (2.10) and acting on the left is its group of automorphisms.
Lemma 4.4. Let (ug −1 (t), g) be a one-parameter family of half-flat structures that satisfy Hitchin's evolution equations and let p be the intrinsic torsion of (u, g). If we set q(t) = Q(g(t)) then where (det g(t)) 6 = 3 4 ∆(q(t)). In particular, q(t) describes a line interval of the form (q + s − p, q + s + p). If s ± is finite, ∆(q + s ± p) = 0.
In addition, if the line interval does not contain any Hermitian structure, then ∆(q(t)) is strictly monotonic.
Proof. A solution, u(t) = ug −1 (t), to the evolution equations (4.1) satisfies The second equation can be rewritten in the form (det g(t)) 2 det(g(t)) ′ σ 2 = (λ 1 (t) − λ 3 (t))σ 2 /2, which leads to 2 3 ((det g) 3 by (4.4) the right-hand side vanishes if and only if the structure is Hermitian. Explicit computations show that the coefficients of q(s) represent the coefficients of L g γ, with respect to a suitable basis, and that the discriminant is related to det g as stated.
The fact that q evolves inside the affine line, follows from the equations. In fact, it ranges in a connected component of this space (hence a line interval), by surjectivity of (4.6). Clearly, the second evolution equation shows that the discriminant is either strictly increasing or strictly decreasing away from the Hermitian locus (4.4).

Remark 4.5 (Classifying according to symmetry).
Suppose that (ug −1 (t), g) is a solution to the Hitchin flow and (u, g) has intrinsic torsion p. For any k ∈ GL(2, R) the same solution can be written as (uk −1 (gk −1 ) −1 (t), g), which means the reference Lie algebra is changed from (u, g) to (uk −1 , g) and the integral line in B 3 from q + sp to kq + skp. Provided kp = p, this is also an integral line relative to (u, g). In particular, integral lines on (u, g) come in families determined by the stabilizer of p in GL(2, R).
Since p has three distinct linear factors, its stabilizer is the Σ 3 that permutes them; we just identified this with the group of automorphisms of the map Q, or, more explicitly, the Σ 3 of (2.10) acting on GL(2, R) on the left. The action on B 3 , however, comes from right multiplication on GL(2, R), and we therefore obtain a Σ 3 which is only conjugated to our "standard" one.
One can then categorize integral lines {q + sp} according to their stabilizer in Σ 3 , which depends only on the action on q. In this terminology, the nearly-Kähler metric is characterized by Σ 3 -invariance, and the Bryant-Salamon metric is Σ 2 -invariant.
When fixing −σ 3 0 /2 as a reference volume form, the canonical symplectic struc- The Hamiltonian for the Hitchin flow is then given by where V denotes the volume associated to a stable form as in [19]. Explicitly these volumes are V (σ 2 /2) = (1 + a 2 ) 3/2 and where we have imposed the half-flat condition, meaning λ 2 = λ 4 . The associated Hamiltonian equations now read and in the notation of Lemma 4.4 we have det g = √ 1 + a 2 . This means the "velocity" determined by the discriminant of q ∈ R 3 [u 1 , u 2 ] is, in fact, the time derivative of the position variable in the phase space where γ represents position and σ 2 /2 conjugate momentum.
On the other hand, when t < −6, we have this goes to infinity at the boundary point.

The classification.
Let G be a Lie group with a half-flat SO(3)-structure that has invariant intrinsic torsion. A maximal solution of the Hitchin flow determines a cohomogeneity one G 2 -metric on a product G ×(t − , t + ). We want to determine necessary conditions for G ×(t − , t + ) to embed in a complete cohomogeneity one manifold.
The possibility that (t − , t + ) is the whole real line can be ruled out by the following lemma; it allows us to deduce there are no non-flat complete solutions to the Hitchin flow which are defined on G ×R.
Then the affine line contains at least one polynomial with a root of multiplicity greater than one, equivalently with vanishing discriminant.
Proof. Suppose for contradiction that ∆ > 0 on the whole affine line (4.7).
Then we obtain a solution to the Hitchin flow for initial data which has intrinsic torsion given by p.
If the line (4.7) does not contain any Hermitian structures, then ∆ is strictly monotonic by Lemma 4.4. This implies that ∆( 1 3 u 3 1 − u 1 u 2 2 + sp) is a monotonic polynomial in s with no roots, which is absurd.
If the line contains a Hermitian structure, then this can be chosen as the initial point of the flow. In terms of polynomials this amounts to acting by an element of GL(2, R) which preserves the sign of ∆. Then p has the form is a fourth degree polynomial in s with negative leading coefficient, so it cannot be positive for all s.
The curve (t − , t + ) → G ×(t − , t + ) given by t → ϑ(t) = (e, t) is a geodesic which is orthogonal to the orbits. In order to obtain non-trivial examples of complete cohomogeneity one metrics, we therefore need to assume that at least one of the boundary points, say, t − is finite. In this case, ϑ(t − ) belongs to a special orbit which necessarily must be singular. Otherwise, it would be possible to extend the flow past t − in contradiction with the maximality assumption.
The special stabilizer H is a closed subgroup which is determined, at the Lie algebra level, by the null space of the limit metric. In the language of Lemma 4.8, we need g(t) to be defined on [t − , t + ), say, h is then the SO(3)-invariant subspace of g that contains the null space of (xe 1 + ye 2 ) 2 + (ze 1 + we 2 ) 2 . It follows that H is either three-dimensional or G = H. As G preserves the metric, the exponential map defines a local diffeomorphism is an open ball inside the normal space, V, to the orbit at ϑ(t − ). Since we are concerned with smoothness about the special orbit, there is no harm done by working with G × H V rather than G × H B.
Up to considering a connected component, we can assume that G is connected. Up to taking a covering space we may, in fact, assume G is simply-connected. In addition, we can assume that H is connected, since the action of H / H e on G × H e V is free and properly discontinuous, making G × H e V → G × H V into a covering map.
In summary, we are left to consider a manifold of the form G × H V, where ρ : H → GL(V) is a sphere transitive representation. In particular, this implies that dim H = dim V − 1.
If G = H the group must act sphere transitively on R 7 . For dimensional reasons this implies G is diffeomorphic to S 6 , which is absurd. We can therefore focus on the case when H is three-dimensional and V is a polar four-dimensional representation. Then we have V = H as a representation of H = SU (2).
H acts on G ×V on the right with fundamental vector fields of the form this action makes G ×V into an H structure on G × H V. In particular, the metric determines an H-equivariant mapping G ×V → S 2 (g / h ⊕V) * . By G-invariance it suffices to consider the restriction to V ∼ = {e} × V, and this gives rise to a map V → S 2 (g / h ⊕V) * . Similarly the G 2 -form determines a smooth map V → Λ 3 (g / h ⊕V) * . It is now a matter of identifying the G 2 -metric and 3-form as maps defined on V \ {0} and to determine whether these extend to all of V. To this end let us fix a vector v ∈ V so as to get a map which identifies the tangent space T [g,v] G × H V with g ⊕R. Explicitly, by choosing an H-invariant decomposition g = h ⊕ m, the map g ⊕R → m ⊕V is given by In other words, we have: We are now in a position to prove the main result of this section: the Bryant-Salamon metric is the only complete full holonomy G 2 -metric which arises from invariant intrinsic torsion SO(3)-structures. In particular, the Bryant-Salamon metric is the only one with holonomy equal to G 2 .
Proof. A six-manifold with an SO(3)-structure that has zero intrinsic torsion is flat and so evolves trivially under the Hitchin flow. This means we obtain the flat metric on a seven-dimensional manifold N. Assuming N is simply-connected and complete we obtain flat R 7 .
Let us next consider the more interesting case of non-zero intrinsic torsion. By Lemma 4.10 the maximal solution, s → p + qs, to the Hitchin flow is defined on an open interval I R.
If I = (s − , s + ) is bounded the fact that ∆(q + sp) is a polynomial of fourth degree in s implies that the integral s + s − ( 4 3 ∆) −1/6 ds is finite. In particular, the orthogonal geodesic has finite length. Completeness then requires that both boundary points s ± corresponds to special orbits.
By the same argument, if I = (s − , +∞) there is a special orbit at s − , and similarly for (−∞, s + ).
In conclusion completeness in the non-flat case is only possible if there is at least one special orbit.
Up to Σ 3 action we may assume det g is positive on I (corresponding to points on the principal orbits).
According to Lemma 4.8 there are now three cases we need to consider. (i) If p = u 1 (u 2 1 + u 2 2 ) the endpoint is necessarily of the form q = λu 3 1 . A computation gives ∆(q + sp) = −4s 3 (λ + s), so that the maximal interval is either s ∈ (−λ, 0) or (0, −λ), depending on the sign of λ. Both s = −λ and s = 0 should correspond to special orbits. However, q − λp = −λu 1 u 2 2 does not satisfy the conditions of Lemma 4.8. It follows that no complete metric arises in this case.
This observation means that we can assume q = λu 3 1 with λ = 0 (the case when q = 0, meaning h = g, was ruled out in section 4.2).
The cases (uℓ, g) and (uℓ 2 , g) also give rise to the Bryant-Salamon metric on (SU(2) × SU(2)) × SU(2) H. Indeed, this metric is preserved by the action of SU(2) (induced by right multiplication on H) that commutes with the action of SU(2) × SU(2) (induced by left multiplication in the group itself). This means we obtain three isometric cohomogeneity one actions of SU(2) 2 ; they correspond to its different inclusions in SU(2) 3 . A change in the choice of the inclusion determines an automorphism of order three which in concrete terms is given by (g, h) → (h −1 , gh −1 ). (4.11) For left-invariant tensors that are also right-invariant under the diagonal action of SU(2), the map (4.11) induces an identification which at the Lie algebra level reads ℓ : su(2) ⊕ su(2) → su(2) ⊕ su (2) Identifying su(2) ⊕ su(2) with R 6 through the frame u, we see that ℓ is represented by the matrix (4.9).
In summary, the three metrics arising from the evolution of (u, g), (uℓ, g) and (uℓ 2 , g) all correspond to the Bryant-Salamon metric, but they are realized with respect to different choices of cohomogeneity one action.
Remark 4.12. The group of order three generated by (4.9) is the "triality" symmetry mentioned in [3]. By taking an appropriate U(1) quotient, this symmetry relates a deformation and two small resolutions of the same conifold.
This triality stems from the fact that the principal orbit SU(2) × SU(2) is a three-symmetric space, but the SO(3)-structures we are considering are not invariant under this symmetry, unlike the nearly-Kähler metric.

Remark 4.13.
In [20,Corollary 6.4], Karigiannis and Lotay showed that the Bryant-Salamon metric is the only complete G 2 -metric that approaches the cone over S 3 × S 3 sufficiently fast. In our result, we do not make any assumptions on the asymptotical behaviour of the metric. A priori we could have found complete metrics that were not conical at infinity.
Regarding our implicit cohomogeneity one assumption, this can be relaxed if we ask for a complete holonomy G 2 -manifold with a real analytic hypersurface which carries a compatible invariant intrinsic torsion SO(3)-structure. Then uniqueness follows by combining Theorem 3.5 and the Cauchy-Kovalevskaya theorem. Remark 4.14. In case (i) of the above proof, meaning when p = u 1 (u 2 1 + u 2 2 ), we showed that the corresponding metrics are not complete, because they do not extend smoothly at one of the boundary points. In [6], however, it was shown that they are "half-complete", in the sense that they extend smoothly at the other boundary.