Multi-moment maps on nearly K\"ahler six-manifolds

We study multi-moment maps induced by a two-torus action on the four homogeneous nearly K\"ahler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly K\"ahler examples.

Haskins [7] proved the existence of one cohomogeneity-one nearly Kähler structure on S 6 and one on S 3 × S 3 . In this case the Lie group acting is SU(2) × SU (2).
All the spaces above have symmetry rank at least two, so it is a sensible question to ask whether there is a theory of nearly Kähler six-manifolds with a two-torus symmetry. A contribution in this direction was given in [14] making use of multi-moment maps. Assume a two-torus T 2 acts on a nearly Kähler six-manifold (M, σ, ψ ± ) preserving the SU(3)-structure. The action induces vector fields U and V on M , thus we have a smooth, T 2 -invariant, real-valued global function given by (1.2) and its differential can be computed by Cartan's formula obtaining (1. 3) Identity (1.3) and invariance imply ν M is a multi-moment map for the torus action [12]. The objective of the present article is to specialise this construction to the four homogeneous examples listed above and compute critical orbits of the resulting functions. This integrates the description of general properties of multi-moment maps initiated in [14] and provides concrete examples to work with. We explain the relation between points with non-trivial stabilisers and critical orbits, proving a general result on the configurations of the latter. It turns out that this information may be encoded in trivalent graphs, which represent the topological structure of the subspaces of M where the multi-moment map and its differential vanish. We construct these graphs in each homogeneous case, and use them to clarify or even replace the algebraic computations hiding the geometry.
The paper is organised as follows. Since the discussion of the special cases is fairly technical we give the abstract results first. One may then study the homogeneous cases bearing in mind the general picture described in Section 2. In all the remaining sections we quickly recall the homogeneous structure of each space, introduce two-torus actions, and then construct multi-moment maps. Critical points are found directly by imposing the condition ψ + (U, V, · ) = 0, according to (1.3).
Acknowledgements. This work is supported by the Danish Council for Independent Research -Natural Sciences Project DFF -6108-00358, and by the Danish National Research Foundation grant DNRF95 (Centre for Quantum Geometry of Moduli Spaces). The material contained in this paper is part of my PhD thesis [13]. I am grateful to Andrew Swann for his help and constant guidance. Further, I wish to thank Andrei Moroianu for useful discussions on the topic.
2. The vectors U p and V p are linearly dependent over C if and only if ν 2 M = h 2 at p. Proof. Since dν M = 3ψ + (U, V, · ), the expressions in (2.4) imply that a point p is critical if and only V p is a linear combination of U p and JU p . Therefore g UU V p = g UV U p + ν M JU p . If ν M vanishes at p then U p and V p are linearly dependent over R. The converse implication is obvious.
The second equivalence follows from the expression of V p in (2.4).
Remark 2.2. Note that the value 0 lies in the interior of ν M (M ) by [14,Proposition 3.2].
Observe that if U p and V p are linearly dependent over the reals the stabiliser H p of p cannot be trivial. Let us describe all possible stabilisers H p . In the following result M need not be nearly Kähler. Theorem 2.3. Let (M, σ, ψ ± ) be a six-dimensional manifold with an SU(3)-structure admitting a twotorus symmetry. Assume the T 2 -action is effective on M . Let p be a point in M and H p its stabiliser in T 2 .
1. If dim H p = 2 then H p = T 2 and there is a neighbourhood W of p in M with the following properties: the stabiliser of each point of W is either trivial or a circle S 1 < T 2 , and the set of points in W with one-dimensional stabilisers is a disjoint union of three totally geodesic twodimensional submanifolds which are complex with respect to J and whose closures only meet at p.

2.
If dim H p = 1 then H p = S 1 < T 2 and there is a neighbourhood W of p in M with the following properties: the stabiliser of each point of W is either trivial or H p and the set of points {q ∈ W : Stab T 2 (q) = H p } is a smooth totally geodesic submanifold of dimension two which is complex with respect to J.
3. If dim H p = 0 and H p is non-trivial, then H p ∼ = Z k for some k > 1. The T 2 -orbit E through p is a totally geodesic two-dimensional submanifold, complex with respect to J, and there is a neighbourhood W of this orbit where T 2 acts freely on W \ E.
Proof. Let g ∈ T 2 and denote by ϑ g the diffeomorphism of M mapping q to gq. Its differential T p ϑ g is an isomorphism between T p M and T gp M . In particular, when g ∈ H p , then T p ϑ g is an automorphism of T p M , and T p ϑ g ∈ SU(3) by assumption. Up to conjugation, T p ϑ g is an element of a maximal torus in SU(3), so for concreteness we take T p ϑ g = diag(e iϑ , e iϕ , e −i(ϑ+ϕ) ) with respect to the standard basis of C 3 . When dim H p = 2 then H p is exactly T 2 by the Closed Subgroup Theorem, and by the Equivariant Tubular Neighbourhood Theorem there is an open neighbourhood of p equivariantly diffeomorphic to the twisted product A point q = p in this neighbourhood coincides with a vector X in C 3 , and by equivariance the requirement gq = q in M translates to T q ϑ g X = X in C 3 . Denote X by (z 1 , z 2 , z 3 ) ∈ C 3 . Then we look for points fixed by a non-trivial element of the torus by imposing the condition diag(e iϑ , e iζ , e −i(ϑ+ζ) ) · (z 1 , z 2 , z 3 ) = (z 1 , z 2 , z 3 ), where e iϑ , e iζ = 1. One can solve the equation explicitly and find there are three S 1 -invariant directions F 1 , F 2 , F 3 corresponding to the standard basis of C 3 . Thus the lines zF 1 , zF 2 , zF 3 correspond to three two-dimensional invariant subspaces in C 3 whose points have onedimensional stabiliser. This proves that points p with stabiliser T 2 are isolated when they exist, and there are three two-dimensional, disjoint submanifolds in a neighbourhood of p in M , intersecting at p, and whose points are fixed by one-dimensional stabilisers. The fact that they are totally geodesic follows e.g. from [11, Chapter II, Theorem 5.1].
Assume now p has one-dimensional stabiliser H p . Choosing U in the Lie algebra of H p and V such that Span{U, V } = t 2 , we have U p = 0, V p = 0. So T p M ∼ = Span{V p , JV p } ⊕ R 4 ∼ = C ⊕ C 2 orthogonally, and C 2 gets an induced SU(2)-structure. Since V p and JV p are H p -invariant, T p ϑ g ∈ SU(2) for g ∈ H p . We claim H p ∼ = S 1 : the connected component of the identity in H p is conjugate to S 1 , so up to a change of basis its elements are diagonal matrices of the form diag(e iα , e −iα ). But T p ϑ g and diag(e iα , e −iα ) for all α commute because H p is Abelian. Thus T p ϑ g must be diagonal, hence in S 1 , and the claim is proved. Therefore, p has a neighbourhood diffeomorphic to can be chosen as follows: S 1 + acts on S 1 , and S 1 − acts on R ⊕ C 2 trivially on R and as the standard maximal torus in SU(2) on C 2 . But an element in S 1 − preserves JV p , so a point q in the neighbourhood S 1 × (R ⊕ C 2 ) is fixed by an element ℓ in the two-torus when the corresponding component in R ⊕ C 2 is fixed, namely T q ϑ ℓ X = X in R⊕ C 2 . Since the action of H p on R is trivial, this condition translates to a condition on Thus the set of points with non-trivial stabiliser is an invariant two-dimensional, totally geodesic submanifold containing p.
Finally, when p has zero-dimensional stabiliser H p there are two invariant independent directions U p , V p = 0. Two cases may occur: either V p ∈ Span{U p , JU p } or V p ∈ Span{U p , JU p }.
In the former case, T p M = Span{U p , JU p } ⊕ C 2 , so H p ≤ SU(2) is a discrete subgroup. But H p is compact and Abelian, so it is finite in SU(2) and is then conjugate to Z k for some integer k ≥ 1. Hence p has a neighbourhood diffeomorphic to Now, assume a point q in this neighbourhood is fixed by Z k . Since the action of Z k is trivial on T 2 /Z k and is free on C 2 \ {0}, q belongs to T 2 /Z k ∼ = T 2 , so it lies in the orbit of p.
In the case V p ∈ Span{U p , JU p } then H p fixes all of T p M , so it is a subgroup of SU(1) = {1}, and is then trivial.
Remark 2.4. Note that when H p has positive dimension the generators of the action are linearly dependent over the reals, whereas when H p is zero-dimensional and non-trivial they are linearly dependent over the complex numbers.
Remark 2.5. Consider the projection π : M → M/T 2 . Theorem 2.3 implies that by mapping fixedpoints and two-submanifolds of points with one-dimensional stabiliser to M/T 2 we obtain trivalent graphs, namely graphs where three edges depart from each vertex. In the first two cases W/T 2 is homeomorphic to R 4 . That C 3 /T 2 is homeomorphic to R 4 follows from the homeomorphism between S 5 /T 2 and S 3 [10] and by taking the cones on the respective spaces. For the second case the homeomorphism is obtained by looking at C 2 as a cone over S 3 and at the sphere S 3 as a principal S 1 -bundle over S 2 : In the third case the image of the exceptional orbit is an orbifold point in M/T 2 .
The shape of the graphs for the examples constructed by Foscolo and Haskins [7] are the same as for the homogeneous cases since the tori act in the same way, but the general critical sets may be different.

The six-sphere
Let us recall a few basic concepts from G 2 geometry to treat this case, detailed sources for what we need are e.g. [2] and [6]. Let V be a seven-dimensional vector space over the reals. Let The general linear group GL(7, R) acts by left-multiplication on V and therefore induces canonically an action on Λ 3 V * . One may define the Lie group G 2 as the stabiliser of ϕ in GL(7, R): The three-form ϕ induces an inner product · , · and an orientation on R 7 . The two objects in turn induce a Hodge star operator * on R 7 , so we have the four-form * ϕ = e 4567 + e 2367 + e 2345 + e 1357 − e 1346 − e 1256 − e 1247 .
Raising an indicex of ϕ gives a G 2 -cross product P : It is well known that G 2 acts transitively on the six-sphere S 6 ⊂ R 7 ∼ = V and that the isotropy group of (1, 0, . . . , 0) ∈ S 6 is isomorphic to the special unitary group SU(3). This implies S 6 is diffeomorphic to G 2 /SU(3). Let i : S 6 ֒→ R 7 be the standard immersion and denote by g the pullback of · , · by i. Call N the unit normal to the six-sphere and define J : R 7 → R 7 as JX := P (N, X). Let now p be a point in S 6 . From (3.2) it follows that J maps T p S 6 to itself. So one can view J as an endomorphism of each tangent space of S 6 , and we do so without changing our notations. Another easy consequence of (3.2) is that J is g-orthogonal if and only if J 2 = −Id. On the other hand, polarising the identity P (X, Y ) 2 = X 2 Y 2 − X, Y 2 yields the general formula P (X, Y ), P (X, Z) = X 2 Y, Z − X, Y X, Z , so for Y, Z tangent to the sphere Hence J is g-orthogonal and J 2 = −Id pointwise, i.e. (S 6 , g, J) is an almost Hermitian manifold.
Proof. One can perform the calculations on R 7 and then restrict the results to the sphere. Let (x k ) k=1,...,7 , be global coordinates on R 7 . Let the one-form dx k be the dual of the coordinate vector field ∂/∂x k for all k. The two-form J · , · turns out to have the following shape: A direct computation of its differential gives d J · , · = 3ϕ. Pulling back this identity to S 6 yields dσ = 3ψ + . Further, dψ − = −i * d(N * ϕ). By the expression of * ϕ it follows that d(N * ϕ) = 4 * ϕ and again the restrictions to S 6 are equal. Thus the claim is 4i * * ϕ = 2σ ∧ σ. Up to a rotation in G 2 mapping p to E 7 , and so N to ∂/∂x 7 , we have and this proves our claim.
Now let a two-torus act on R 7 ∼ = C 3 ⊕ R as follows. Take the maximal torus T 2 inside SU(3) given by matrices of the form A ϑ,φ := diag e iϑ , e iφ , e −i(ϑ+φ) , and let A ϑ,φ act effectively on the left on Because of the convention chosen for (3.1) we set , the fundamental vector fields have the form Plugging the two vectors in the two-form (3.3) and restricting to the six-sphere, one finds the multimoment map Using complex coordinates the T 2 -invariance is evident: Proposition 3.2. There are three two-dimensional spheres in S 6 where ν S 6 and dν S 6 vanish. These intersect at two common points. Further, there are two T 2 -orbits of critical points where ν S 6 attains its extrema.
To see where P (U, V ) is proportional to N = x 1 ∂/∂x 1 +· · ·+x 7 ∂/∂x 7 we need to find points (x 1 , . . . , x 7 ) such that the following equations hold for some real proportionality factor λ: The case λ = 0 gives points p where P (U, V ) = 0, i.e. U p and V p are linearly dependent over R. By comparing the expressions of U p , V p one obtains three two-spheres of critical points where the multi-moment map vanishes: Note that the three spheres have the poles (x 1 , . . . , x 6 , x 7 ) = (0, . . . , 0, ±1) in common. Now let us switch to the case λ = 0. Assume we are at a critical point, so in particular x 7 = 0. Up to the action of U and V , we can assume x 1 = x 2 = 0 and x 5 , x 6 ≥ 0. The equations characterising critical points yield x 4 = 0 and vanishes, so do all the others, and we get a contradiction as we need solutions on the six-sphere. Therefore we can assume without loss of generality all of them non-zero, which gives (x i ) 2 = λ 2 for i = 3, 5, 6. We thus obtain two stationary T 2 -orbits where ν S 6 attains its maximum and minimum, and ν S 6 ( The three two-spheres found can be recovered by looking for points with non-trivial stabilisers according to Proposition 2.1. This is done by solving the equation poles fixed by all of T 2 , t 2 + |z i | 2 = 1, i = 1, 2, 3, two-spheres of points fixed by a circle. Note the three two-spheres correspond to those found in the proof of Proposition 3.2. Projecting the latter to the orbit space S 6 /T 2 gives a graph of two points and three edges. As |z i | → 0 the two-spheres collapse to the common poles. Moreover, the spheres do not intersect each other at any point but the poles, so the edges of the graph are disjoint (see Figure 1).

The flag manifold
Let F 1,2 (C 3 ) be the set of pairs (L, U ) of subspaces in C 3 , where L is a complex line contained in the complex plane U . Such pairs are called flags. The special unitary group SU(3) acts transitively on F 1,2 (C 3 ). Let F 1 , F 2 , F 3 be the standard basis of C 3 . It turns out that the isotropy group of the point We now equip SU(3)/T 2 with an almost Hermitian structure. A matrix p ∈ SU(3) acts on SU(3) by left translation and induces a pullback map (p −1 We can thus define a Riemannian metric g on SU(3) such that where g 0 is the Killing form on su(3) normalised as g 0 (X, Y ) := (1/2) Tr( t XY ). The metric g is bi-invariant for g 0 is. In particular g is invariant under the action of the maximal torus in SU(3) above, so descends to a metric on the flag manifold, which we still denote by g. To construct an almost complex structure J we follow Gray [8,Section 3]. Let A := diag(e 2πi/3 , e 4πi/3 , 1) and define the conjugation mapθ : SU(3) → SU(3) so thatθ(B) = ABA −1 . It is clear by this definition that ϑ 3 = Id (where the cubic exponential stands for composing three times) and thatθ fixes the maximal torus T 2 in SU (3) above. Soθ induces a map on the quotient ϑ : SU(3)/T 2 → SU(3)/T 2 that fixes the coset T 2 and satisfies ϑ 3 = Id. We define J 0 at the identity as follows: The map J 0 : su(3)/t 2 → su(3)/t 2 is well defined as A commutes with diagonal matrices. We now check that This amounts to say that 0 = Id + dϑ + dϑ 2 , or more explicitly that X + AXA −1 + A 2 XA −2 = 0. Therefore as we wanted. A similar computation shows J 0 is an isometry. We can move the operator J 0 to every . From the invariance of g and J it follows that (g, J) is an almost Hermitian structure on the flag manifold of C 3 .

This implies the point p ∈ SU(3)/T 2 is critical if and only if
Using the relations z 1 + w 1 = −3p 31 p 32 , z 2 + w 2 = −3p 31 p 33 , z 3 + w 3 = −3p 32 p 33 we find that p is critical precisely when  Our set-up is invariant under cyclic permutations of columns or rows of p up to a sign of ν F1,2(C 3 ) , so in order to work out stationary orbits we can distinguish the cases c = 0 and at least one between a and b is zero, or a, b, c = 0.
In the first case the system is easy to discuss and generates critical points where the multi-moment map vanishes. In the second case d cannot be 0, otherwise the criticality conditions would imply either p 31 = 0 or p 33 = 0, namely a = 0 or c = 0. Then our equations are bp 31 p 33 = ap 33 d, cp 31 = ap 33 , cp 31 d = bp 31 p 33 .
Comparing the arguments we find β ≡ α + 4π/3 (mod 2π) and γ ≡ α + 2π/3 (mod 2π). Comparing the radii instead we obtain a = b = c = 1/ √ 3. Imposing the condition det p = 1 one gets α ≡ 7π/6 (mod 2π), so This gives a T 2 -orbit of points of minimum, the value of the multi-moment map at p is − √ 3/2. The last case ϕ = 4π/3 can be discussed similarly, and the point we find turns out to be , by (4.10) the value of the multi-moment map is √ 3/2. Summing up, we got two stationary orbits giving extrema, and Im ν F1,2( The goal now is to find which pairs of subspaces (L, U ) are fixed by some non-trivial element T 2 and compute their stabilisers. It will turn out that the action is not effective, a copy of Z 3 in T 2 fixes all the flags. However, there is an isomorphism T 2 ∼ = T 2 /Z 3 given by (e iϑ , e iφ ) → (e 3iϑ , e i(ϑ−φ) ). In the case below where Z 3 appears as a discrete stabilizer of all the flags, we can use this isomorphism to argue that the action of T 2 /Z 3 ∼ = T 2 is effective and the discrete stabilizers are all trivial.
Take a non-zero z = (z 1 , z 2 , z 3 ) ∈ C 3 and assume that L := Span(z) is a T 2 -invariant onedimensional subspace of C 3 . The equation we want to solve is A ϑ,φ z = λ(ϑ, φ)z, with λ some complexvalued function of ϑ, φ. Explicitly The case e iϑ = λ, e iφ = e iϑ gives two subcases, either 3ϑ ≡ 0 (mod 2π) or z 3 = 0. In the former we have ϑ ∈ {0, 2π/3, 4π/3} (mod 2π) and z i = 0 for every i = 1, 2, 3, which gives a discrete stabilizer of L as ϑ ≡ φ (mod 2π). This is a copy of Z 3 and we can argue as above to conclude that the stabilizer is trivial. Denote by F 1 , F 2 , F 3 the standard basis of C 3 . A plain discussion of the remaining cases yields the solutions fixed by all of T 2 , CF 1 ⊕ CF 2 , CF 1 ⊕ CF 3 , CF 2 ⊕ CF 3 , fixed by a circle.
Since the T 2 -action preserves CF i and the angles between two vectors, we have that CF j ⊕ CF k is preserved by T 2 as well, for different i, j, k ∈ {1, 2, 3}. On the other hand, since S 1 preserves Cz, z ∈ Span{F i , F j }, i = j, the pairs (Cz, CF i ⊕ CF j ) and (CF i , CF j ⊕ Span{z}), are fixed by S 1 . Therefore, we have six points in F 1,2 (C 3 ) fixed by all of T 2 and nine edges corresponding to twodimensional subspaces of points fixed by S 1 . The six points are represented by the A α,βγ : The edges a i , i = 1 . . . , 9 are the following: Cz 1 ⊕ CF 3 ), a 6 = (Cz 2 , CF 2 ⊕ Cz 2 ), a 9 = (Cz 3 , CF 1 ⊕ Cz 3 ).
In order to figure out what the vertices of, say, a 1 are, one can take the limit z → F 1 (resp. z → F 2 ) and see that a 1 → A 1,12 (resp. a 1 → A 2,12 ), see Figure 2.

The complex projective space
We proceed as in the previous section, and thus skip some technicalities. The compact symplectic group Sp(2) acts on C 4 ∼ = H 2 , transitively on the projective space CP 3 . An element in p ∈ Sp(2), p = ( x y z w ), fixes a := [1 : 0 : 0 : 0] ∈ CP 3 when x is a combination of 1, i and the quaternions z, y vanish. Since x has unit length it must lie in a circle U(1). The isotropy group of a is then isomorphic to Sp(1)U(1), and CP 3 is diffeomorphic to Sp(2)/Sp(1)U(1). Write H := Sp(1)U(1) and G := Sp(2). We then identify H with a subgroup of G containing elements of the form diag(e iϑ , α), where α is a unit quaternion and ϑ an angle. We denote by g and h the Lie algebras of G and H respectively, so g splits as h ⊕ m.
On the Lie algebra g we define the Killing form as g 0 (X, Y ) := Tr( t XY ) = − Tr(XY ). This can be translated to any point p yielding an inner product g p := Re((p −1 Id ) * g 0 ) on every tangent space T p G, and descends to the quotient modulo H as it is bi-invariant. The construction of the almost complex structure J follows from the existence of a diffeomorphism of order three as in the case of F 1,2 (C 3 ). Consider A := diag(e 2πi/3 , 1) in G and define an almost complex structure J p at the point p by translating the endomorphism J 0 : m → m, J 0 X := (2/ √ 3)(AXA −1 + 1 2 X). The Lie algebra g is spanned by Note that the indices range from 0 to 9, and not from 1 to 10. This will help keep the notation shorter here, but we will get back to numbering from 1 to 10 when we construct the multi-moment map. One can check the metric and the almost complex structure have the familiar shapes as in (4.3), (4.4). descend to the quotient g/h and satisfy dσ 0 = 3ϕ 0 and dψ 0 = −2σ 0 ∧ σ 0 . Consequently, the differential forms on Sp(2)/Sp(1)U(1) given by σ p := g p (J p · , · ), ψ +|p := ϕ 0 (p −1 · , p −1 · , p −1 · ), and ψ −|p := ψ 0 (p −1 · , p −1 · , p −1 · ) define a nearly Kähler structure on CP 3 .
Proof. This follows from the expressions of the differentials of e k , k = 0, . . . , 5: A routine computation gives the result.
The Lie algebra h contains elements of the form ( ia 0 0 ib ) + ( 0 0 0 c ) j, with a, b real and c complex. The projections (p −1 U p) m , (p −1 V p) m must be of the form xj ρ −ρ 0 , for x a complex number and ρ a quaternion, so (p −1 U p) m = 0 α −α 0 + γj 0 0 0 , α = i(p 1 11 p 1 12 − p 2 11 p 2 12 ) + i(p 1 11 p 2 12 + p 2 11 p 1 12 )j, γ = 2ip 1 11 p 2 11 , We now write (p −1 U p) m , (p −1 V p) m in terms of the basis introduced above: note that we shift the indices so that E 0 → E 1 , . . . , E 5 → E 6 , the first convention we used is no longer needed. Then It is convenient to write which is indeed invariant under the torus action, because α, β, γ, δ are invariant.
Proposition 5.2. There are exactly two T 2 -orbits of critical points where the multi-moment map ν CP 3 does not vanish. Thus they give maximum and minimum of ν CP 3 .
In terms of the p k ij , the latter are respectively We combine the left action of T 2 and the right action of Sp(1)U(1) so that p 12 = c is a non-negative real number, p 11 = a+bj for a, b non-negative real numbers, and p 21 = d+ρj, where d is a non-negative real and ρ is complex. The system giving critical points then reduces to ac + σd + ρτ = 0, c 2 + |σ| 2 + |τ | 2 = 1.

The product of three-spheres
It is convenient to view S 3 × S 3 as Sp(1) × Sp(1) ⊂ H × H, and recall that Sp(1) ∼ = SU (2). A triple (h, k, l) ∈ SU(2) 3 acts on S 3 ×S 3 as ((h, k, l), (p, q)) → (hpl −1 , kql −1 ). This action is obviously transitive and the stabiliser of the point (1, 1) is given by the triples (h, h, h) ∈ SU(2) 3 . We denote this isotropy group by SU(2) ∆ . Therefore S 3 × S 3 has the structure of smooth manifold and is diffeomorphic to We follow [1] to construct an almost Hermitian structure. We define an almost complex structure J on S 3 × S 3 at the point (p, q) as The standard product metric · , · on S 3 × S 3 is not invariant under J, so we define a metric g as the average of · , · and J · , J · , and normalise it by a factor 1/3. At the point (p, q) its expression is g(X, Y ) := 1 6 ( X, Y + JX, JY ), X, Y ∈ T (p,q) (S 3 × S 3 ). Trivially, J is g-orthogonal and (g, J) is then an almost Hermitian structure.
Proof. The differentials of the duals e k of E k satisfy de i = 2e jk for (ijk) cyclic permutation of (123) and (456), whence the result.
Therefore, the equation ψ + (U, V, · ) |(p,q) = 0 is equivalent to the following system: (6.2) Remark 6.2. As it turns out, S 3 × S 3 is the only homogeneous example where saddle points appear and where extrema of the multi-moment map are not symmetric with respect to the origin. We first discuss system (6.2) and then illustrate the various situations assigning explicit values to our parameters b 1 , b 2 , b 3 .
The vectors x and y lie in S 2 ⊂ Im H. discrete stabiliser given by t = ±Id, meaning that the action is not effective. By the usual argument as in the two cases above we can thus ignore it.
This shows we have a two-torus whose points are fixed by S 1 . The other cases are similar and the resulting graph is given by four disjoint circles (cf. Figure 4, C).
For every T 2 in T 3 , the stabilizers are still zero-or one-dimensional as Stab T 2 (p) ⊂ Stab T 3 (p). Thus there are no vertices in our graph, we get only disjoint circles. Further, a T 2 ⊂ T 3 cannot contain all the circles (t, t, t), (t, t −1 , t), (t, t, t −1 ), (t, t −1 , t −1 ). There are three cases: the two-torus may contain none, one or two of the circles above. For example, the first case happens when T 2 is of the form (r, s, Id), r, s ∈ S 1 , so in this case we get an empty graph and the T 2 -action is free. If it contains triples (r, rs, rs 2 ), r, s ∈ S 1 , then it contains the circle (r, r, r), so the graph is a single circle. Thirdly, if it is of the form (r, s, r), then it contains the circles (t, t, t) and (t, t −1 , t), but does not include (t, t, t −1 ) and (t, t −1 , t −1 ), so we get two circles in our graph (in Figure 4 A, B, the non-trivial cases are shown).