$G_2$-metrics arising from non-integrable special Lagrangian fibrations

We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms, connection 1-forms and equivariant $3\times3$ positive-definite symmetric matrix-valued functions on principal $G$-bundles over 3-manifolds. As applications, we describe regular parts of $G_2$-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of $G=\mathrm{T}^3$ and $\mathrm{SO}(3)$.


Introduction
The geometry of G 2 -structures on 7-manifolds is closely related to that of SU(3)structures on 6-manifolds. For a one-parameter family (ω(t), ψ(t)) of SU(3)structures on a 6-manifold X, the 3-form ω(t) ∧ dt + ψ(t) is a G 2 -structure on X × (t 1 , t 2 ). Here ω(t) and ψ(t) denote the 2-and 3-form on X defining an SU(3)-structure for each t ∈ (t 1 , t 2 ). Conversely, any G 2 -structure on Y is locally described by one-parameter families of SU(3)-structures on 6-dimensional hypersurfaces in Y as above. This viewpoints has been studied by many authors [BCFG15,Bry06,CS02,Hit01].
In the present paper, we study torsion-free G 2 -structures given in terms of oneparameter families of SU(3)-structures on T 3 -or SO(3)-bundles; the fibrations are special Lagrangian in the sense that the 2-and 3-forms defining each SU(3)structure vanish along the fibers.
Let G be a connected 3-dimensional Lie group with Lie algebra g, and P the total space of a principal G-bundle over a 3-manifold M. Denote by (ω, ψ) Ginvariant 2-and 3-forms defining a special Lagrangian SU(3)-structure on P (in the sense above). We first prove that if G is unimodular then such an SU(3)-structure decomposes uniquely into a triple (e, a, S) of a solder 1-form e, a connection 1-form a and a G-equivariant 3 × 3 positive-definite symmetric matrix-valued function S on P (Theorem 3.5).
Using this decomposition, we describe locally T 3 -and SO(3)-invariant torsionfree G 2 -structures whose definite 3-forms vanish along the fibers as orbits of constrained dynamical systems on the space of the triples (e, a, S). Here a definite 3-form is the 3-form defining a G 2 -structure on a 7-manifold. Let (ω(t), ψ(t)) be a one-parameter family of such SU(3)-structures defined on (t 1 , t 2 ), and (e(t), a(t), S(t)) the triples corresponding to (ω(t), ψ(t)). Then the 3-form ω(t) ∧ (det S) 1 2 dt + ψ(t) is a G 2 -structure on P × (t 1 , t 2 ). From now, we omit the symbol of summation adopting Einstein's convention and denote by ǫ ijk the Levi-Civita symbol for the permutation of {1, 2, 3}. Besides, abbreviateê i = 1 2 ǫ ijk e j ∧ e k . For a basis {X 1 , X 2 , X 3 } of g, we write e = e i X i and a = a i X i . Our main results are as follows.
In the case of T 3 -fibrations over M, we prove Theorem 1.1. Let G = T 3 , and {X 1 , X 2 , X 3 } a basis of the Lie algebra t 3 . The G 2 -structure ω(t) ∧ (det S) 1 2 dt + ψ(t) is torsion-free if and only if the triple (e(t), a(t), S(t)) is an orbit of the following constrained dynamical system on the space of triples (e, a, S): de i = 0, Ω ij = Ω ji , S iα,α = 0 (constraint conditions); (1.1) for i = 1, 2, 3. Here I = (δ ij ), and d H denotes the covariant derivation for each connection a(t). Also S ij;k and Ω ij are defined by d H S ij = S ij;k e k and d H a = Ω ijê j Y i .

ψ(t) is torsion-free if and only if the triple (e(t), a(t), S(t)) is an orbit of the
Remark 1.5. The condition d H e = 0 says that the connection a is the Levi-Civita one of the metric on M given by the local orthonormal coframe e = e i Y i . Then Ω ij e i ⊗ e j coincides with the Einstein tensor.
By scaling, we obtain Theorem 1.6. Every torsion-free SO(3)-invariant G 2 -structure whose definite 3form vanishes along the fibers is locally given by some orbit of the constrained dynamical system in Theorem 1.4.
Remark 1.7. We can see that equations of motion preserve the constraint conditions in Theorem 1.1 and 1.4. This is immediate for G = T 3 . See ([Chi19], Proposition 7) for G = SO(3).
The present paper is organized as follows. In Section 2, we review definitions and some basic results of SU(3)-and G 2 -structures. In Section 3, we consider G-invariant special Lagrangian fibrations of SU(3)-manifolds and prove the decomposition (Theorem 3.5). We apply this theorem to G-invariant G 2 -structures whose definite 3-form vanishes along the fibers in Section 4. In Section 5 and 6, as applications of the above results, we describe locally T 3 -and SO(3)-invariant G 2manifolds whose definite 3-forms vanish along the fibers as orbits of constrained dynamical systems on the spaces of the triples (Theorem 1.1, 1.2, 1.4 and 1.6).
Conventions. We omit the symbol of summation adopting Einstein's convention, and often abbreviate a ∧ b = ab and c i ∧ c j = c ij . Also we use the Levi-Civita symbol ǫ ijk and writeĉ i = (1/2)ǫ ijk c jk for a triple of 1-forms {c 1 , c 2 , c 3 }. Denote byÃ the adjugate matrix of an n × n matrix A satisfyingÃA = det (A)I. Here I = (δ ij ) is the identity matrix. Let G be a connected 3-dimensional Lie group with Lie algebra g, and P → M a principal G-bundle over a 3-manifold M. An equivariant g-valued 1-form e with respect to the adjoint action on g is called a solder 1-form if e = e i X i satisfies e 123 = 0 at each u ∈ P for a basis {X 1 , X 2 , X 3 } of g.

SU(3)and G 2 -structures
In this section we review SU(3)-structures on 6-manifolds and G 2 -structures on 7-manifolds, emphasizing relations between two structures. Throughout this paper, we assume all objects are of class C ∞ .
2.1. SU(3)-structures. Let X be a 6-manifold, Fr(X) the frame bundle over X, which is a principal GL(6; R)-bundle over X. We have the natural inclusion SU(3) ⊂ GL(6; R) by the standard identification R 6 ∼ = C 3 , where z i = x i + √ −1y i for i = 1, 2, 3. A subbundle of Fr(X) is said to be an SU(3)-structure on X if the structure group is contained in SU(3). For an SU(3)-structure on X, we have the associated real 2-form ω and real 3-form ψ pointwisely isomorphic to ω 0 = 3 i=1 dx i ∧ dy i and ψ 0 = Im(dz 1 dz 2 dz 3 ) on C 3 , respectively. Here Im( * ) denotes the imaginary part of * . We can identify an SU(3)-structure with such a pair (ω, ψ). Using this identification, for an SU(3)-structure (ω, ψ), we define by ψ # the real 3-form on X corresponding to Re(dz 1 ∧ dz 2 ∧ dz 3 ). Here Re( * ) denotes the real part of * , and ψ # is the same as −ψ in [Hit01].
It is useful to compare general cases with the following basic example.
Example 2.1. Let X be R 6 ∼ = C 3 . We have where dy k = 1 2 i,j ǫ ijk dy i dy j and dx k = 1 2 i,j ǫ ijk dx i dx j for k = 1, 2, 3. Thus the forms ω 0 , ψ 0 and ψ # 0 associated with the standard SU(3)structure on R 6 are expressed as follows.
Remark 2.3. In general, a G-structure F on an n-manifold N is said to be torsion-free if the tautological 1-form θ ∈ Ω 1 (F ; R n ) satisfies d H θ := dθ + a∧θ = 0 for some connection 1-form a ∈ Ω 1 (F ; Lie(G)) ⊂ Ω 1 (F ; gl(n; R)). The torsion-free condition in Definition 2.2 is known to be equivalent to this general definition of the torsion-free condition for G = SU(3).
By Remark 2.3 and the Ambrose-Singer theorem, we see that an SU(3)-structure (ω, ψ) is torsion-free if and only if the associated metric h (ω,ψ) has the holonomy group contained in SU(3). Moreover, If an SU(3)-structure is torsion-free, then the associated metric on X is Ricci-flat. A Riemannian metric h on X is said to be holonomy SU(3) if the holonomy group coincides with SU(3).
2.2. G 2 -structures. Let Y be a 7-manifold, Fr(Y ) the frame bundle over Y , which is a principal GL(7; R)-bundle over Y . Let us define G 2 -structures on Y as in the above subsection. The Lie group G 2 is defined as the linear automorphism group of the standard definite 3-form φ 0 (presented in Example 2.4) on R 7 . It is known that this group coincides with the linear automorphism group of the cross product structure on ImO, where ImO denotes the 7-dimensional imaginary part of the octonion algebra O. A subbundle of Fr(Y ) is said to be a G 2 -structure on Y if the structure group is contained in G 2 . For a G 2 -structure on Y , we have the associated real 3-form φ on Y pointwisely isomorphic to φ 0 on R 7 . Such a 3-form is called a definite 3-form. Then we identify the definite 3-form φ with a G 2 -structure on Y as in the above subsection. Since G 2 ⊂ SO(7), we have the Riemannian metric g φ and orientation associated with a G 2 -structure φ on Y . Besides, we denote by ⋆ φ the Hodge star associated with φ, and simply write ⋆ in situations without confusion.
The following example is the model of G 2 -structures as in Example 2.1.
Definition 2.5. Let φ be a G 2 -structure on Y .
A G 2 -structure φ is torsion-free if and only if the associated Riemannian metric g φ has the holonomy group contained in G 2 . If a G 2 -structure is torsion-free then the associated metric on Y is Ricci-flat. A 7-manifold Y with a torsion-free G 2structure φ is called a G 2 -manifold, and a Riemannian metric g on Y is called holonomy G 2 if the holonomy group coincides with G 2 .
It is useful the following lemma for a normal form of a G 2 -structure at a point y ∈ Y . Let φ be a G 2 -structure on Y , and {V 1 , V 2 , V 3 } orthonormal tangent vectors at y ∈ Y . Also denote by {V 1 , V 2 , V 3 } the dual cotangent vectors with respect to the metric g φ , i.e., V i ( * ) = g φ (V i , * ) for i = 1, 2, 3. Let φ y and ⋆ φ φ y denote 3-and 4-forms at y ∈ Y , respectively.
Proof. We can directly check this for V 1 = ∂ ∂x 1 , V 2 = ∂ ∂x 2 and V 3 = ∂ ∂x 3 in the case of φ 0 in Example 2.4. This suffices to prove Lemma 2.6 since it is known that the Lie group G 2 acts transitively (and faithfully) on the set of triple vectors , p. 115, Proposition 1.10).

Relations between SU(3)-structures
and G 2 -structures. G 2 -structures are related to SU(3)-structures as stated below (See [Bry10,Hit01] for more details). These propositions can be proved by direct calculation. Let X be a 6manifold, and consider a one-parameter family (ω(t), ψ(t)) of SU(3)-structures on X defined on an interval (t 1 , t 2 ).
. Let φ be the G 2 -structure on X × (t 1 , t 2 ) in Proposition 2.7. The torsion-free condition for φ is interpreted as the following constrained dynamical system on the space of SU(3)-structures on X.

G-invariant non-integrable special Lagrangian fibrations
Let G be a connected 3-dimensional Lie group with Lie algebra g, and P the total space of a principal G-bundle π : P → M over a 3-manifold M. In this section we introduce our main objects, G-invariant non-integrable special Lagrangian fibered SU(3)-structures on P . Next, we present examples of such SU(3)-structures. Finally, we prove that if G is unimodular then every such SU(3)-structure is uniquely constructed by a triple (e, a, S) of a solder 1-form e, a connection 1-form a and an equivariant Sym + (3; R)-valued function S on P . Here denote by Sym(3; R) and Sym + (3; R) the spaces of 3 × 3 symmetric and positive-definite symmetric matrices.
3.1. Definition. Let us start with the definition.
Definition 3.1. An SU(3)-structure (ω, ψ) on P is said to be a G-invariant nonintegrable special Lagrangian fibered SU(3)-structure if it satisfies: (1) ω and ψ are invariant for the right action of G on P ; (2) the restrictions of ω and ψ to the fibers F m ⊂ P vanish: ω| Fm = 0 and ψ| Fm = 0 for every m ∈ M.
Next we present some examples. Let V be a vector space with a representation ρ : G → GL(V ). Then denote by Ω p (P ; V ) G and Ω p (P ; V ) G hor the space of Gequivariant V -valued p-forms and horizontal p-forms on P .
We can generalize this examples as follows.
Fix a basis {X 1 , X 2 , X 3 } of g. By this basis, g ∼ = R 3 and Ad(g) ∈ GL(3; R). Let G act on Sym + (3; R) by g·S = Ad(g)·S· t Ad(g) for g ∈ G and S ∈ Sym + (3; R), where t Ad(g) is the transverse matrix of Ad(g). Given a triple (e, a, S) of a solder 1-form e = e k X k ∈ Ω 1 (P ; g) G hor , a connection 1-form a = a k X k ∈ Ω 1 (P ; g) G and an equivariant Sym + (3; R)-valued function S = (S ij ) ∈ Ω 0 (P ; Sym + (3; R)) G , we can see that, in the same way as Example 3.2, the 2-form ω and 3-form ψ defined by (3.1) and (3.2) is a G-invariant sLag SU(3)-structure on P . See also the proof of Theorem 3.5.
3.2. Decomposition. Next let us start with a G-invariant sLag SU(3)-structure (ω, ψ) on P and decompose it into a triple (e, a, S). Fix a basis {X 1 , X 2 , X 3 } of g and denote by A * the infinitesimal vector field on P for A ∈ g.
First, since the Riemannian metric h (ω,ψ) associated with (ω, ψ) is invariant for the right action of G on P , we have a connection 1-form a = a k X k ∈ Ω 1 (P ; g) G by the orthogonal decomposition of the tangent bundle T P .
Proposition 3.4. There exists uniquely a Sym + (3; R)-valued function S on P such that

G2-METRICS ARISING FROM NON-INTEGRABLE SPECIAL LAGRANGIAN FIBRATIONS 9
Proof. Let U be an open neighborhood of any point m ∈ M such that π −1 (U) ∼ = U ×G. Denote by T F the fiberwise tangent bundle ker (dπ) on P . Let us consider the restriction that are orthonormal with respect to the metric h (ω,ψ) and contained in T F | π −1 (U ) . Using this vector fields and the infinitesimal vector fields In this notation, we have In what follows, we use the following identities for the cofactor matrixQ = (Q ij ) of Q. These identities hold for general regular 3 × 3 matrices.
In order to apply Lemma 2.6 for normal forms, let us consider the 3-form φ = ω ∧ dt + ψ on P × R. The 3-form φ is a G 2 -structure by Proposition 2.7. Since the orthonormal vector fields {V 1 , V 2 , V 3 } satisfy φ(V 1 , V 2 , V 3 ) = 0, we can apply Lemma 2.6 to φ. Using the same notation E k and Z in Lemma 2.6 and the above identities for the cofactor matrixQ, we have for k = 1, 2, 3, whereQ −1 = (Q ij ). Moreover, by (3.3), (3.4) and (3.5), we have Then, by these identities, we have consequently Thus the function S is our goal. Further by the definition of S, we can see that S is independent of the choice of the orthonormal vector fields {V 1 , V 2 , V 3 }.

By Example 3.3 and Proposition 3.4, we obtain
Theorem 3.5. Let G be unimodular. Then there exists a one-to-one correspondence between G-invariant sLag SU(3)-structures and triples (e, a, S) in Example 3.3.
Proof. Let (ω, ψ) be a G-invariant sLag SU(3)-structure on P and (e, a, S) the triple corresponding to (ω, ψ) in Proposition 3.4. Then by the constructions, we have the following: )SAd(g) for i = 1, 2, 3 and for every g ∈ G, where R g : P → P denotes the right action of g and Ad(g) = (g ij ) is the representation of the adjoint action on g by the fixed basis {X 1 , X 2 , X 3 }. Further |Ad(g)| denotes the determinant. By this, if G is unimodular, then the triple (e, a, S) is a triple satisfying conditions in Example 3.3. Moreover the converse is straightforward by the above identities.
In summary, in the cases where G is unimodular, a G-invariant sLag SU(3)structure (ω, ψ) on P uniquely corresponds to a triple ({e k }, {a k }, S) of a solder 1-form e, a connection 1-form a and an equivariant Sym + (3; R)-valued function S on P such that Moreover, by the proof of Proposition 3.4, we have (3.9) Then the following 1-forms j Q ij a j and det(Q) j Q ij e j f or i = 1, 2, 3 are an orthonormal coframe with respect to the metric h (ω,ψ) associated with (ω, ψ) on P . Here S = Q · t Q and Q −1 = (Q ij ) as above.

Reduction of G-invariant G 2 -manifolds
In this section, we prove a generalization of results in Section 3 to a class of G-invariant G 2 -structures. Let G be a 3-dimensional Lie group with Lie algebra g, and Q → N a principal G-bundle over a 4-manifold N. Let us define G 2 -structures that we consider.
(1) φ is invariant for the right action of G on Q, (2) the restriction of φ to the fiber F n vanishes: φ| Fn = 0 for all n ∈ N.
In what follows, we refer to such G 2 -structures as G-invariant Lag G 2 -structures. The following example is typical.
Let φ be a G-invariant Lag G 2 -structure on Q → N, and fix a basis {X 1 , X 2 , X 2 } of g. The following is a generalization of Theorem 3.5 to such G 2 -structures. Proposition 4.3. Suppose G is unimodular and the 1-form ι(X * 3 )(ι(X * 2 )(ι(X * 1 ) ⋆ φ)) is closed. Then, for each n ∈ N, there exists a triple of a 3-dimensional submanifold (n ∈)D of N, a one-parameter family of G-invariant sLag SU(3)structures on π −1 (D) and a G-invariant function f (u, t) on π −1 (D) × (t 1 , t 2 ) such that φ is isomorphic to ω(t) ∧ f dt + ψ(t) on some neighborhood of π −1 (D) in Q, where (t 1 , t 2 ) denotes an interval on which the one-parameter family is defined.
(2) If φ is coclosed, then the 1-form ι(X * 3 )(ι(X * 2 )(ι(X * 1 ) ⋆ φ)) is closed. The former follows from the following equation . The latter is also proved in the same way. Thus if the invariant G 2 -structure φ is torsion-free and the action has both of irregular and regular parts, then φ is a G-invariant Lag G 2 -structure on regular parts of Y .
Remark 4.6. Let Y be a 7-manifold with a G 2 -structure φ. 3-dimensional submanifolds L in Y satisfying the condition φ| L = 0 in Definition 4.1 are studied in [GS15]. The authors characterized the infinitesimal deformation space of the Lagrangian-type submanifolds in Y . These submanifolds are called maximally ⋆φ-like submanifolds in ( [HL82], II.6.).
5. Local structure of T 3 -invariant G 2 -manifolds 5.1. Torsion of SU(3)-structures in the case of G = T 3 . In this subsection, we first calculate the torsion of T 3 -invariant sLag SU(3)-structures on the total space of a principal T 3 -bundle π : P → M over a 3-manifold M. Next we prove some corollaries that follow from the derived expressions.
Let (ω, ψ) be a T 3 -invariant sLag SU(3)-structure on P , which has the triple (e, a, S) corresponding to (ω, ψ). Denote by T = {T ij } and Ω = {Ω ij } the torsion and curvature forms of (e, a): de i = T ijê j and da i = Ω ijê j for i = 1, 2, 3. Also for a matrix-valued function A ∈ Ω 0 (M; M(k; R)), denote the derivatives by dA ij = A ij,k e k . Then, by a simple calculation, we have the following: for i = 1, 2, 3. By (3.6)-(3.9) and (5.1), we obtain the following expressions for the torsion of (ω, ψ) in terms of (e, a, S). The proof is straightforward calculation of differential forms.
Proposition 5.1. Let (e, a, S) be the triple corresponding to (ω, ψ). Then we have We have the following corollaries to Proposition 5.1.
The following corollary is consistent with the well-known fact that a T 3 -invariant Calabi-Yau 3-fold whose orbits are special Lagrangian submanifolds is locally constructed by a solution of the real Monge-Ampére equation on a domain of R 3 .
Corollary 5.3. Let (ω, ψ) be torsion-free, and (e, a, S) the triple corresponding to (ω, ψ). Then, for each m ∈ M, there exists a coordinate neighborhood This corollary is proved by combining the fourth in Corollary 5.2 and Poincaré's lemma.
Proof of Lemma 5.4. Let us take 1-forms f i = h ij dx j for i = 1, 2, 3. Then, by assumption, the 1-forms are closed. Thus, by Poincaré's lemma, there exist func- Hence, using Poincaré's lemma again, we obtain a function f ∈ Ω 0 (U; R) such that h ij = ∂ 2 f ∂x i ∂x j for i, j = 1, 2, 3. The function f is our goal.
Proof of Corollary 5.3. The existence of (e, a, F ) satisfying (5.2) is deduced form the fourth in Corollary 5.2. A function ρ satisfying (5.3) is constructed by combining the fourth in Corollary 5.2 and Lemma 5.4.
In [Bar10], Baraglia established a generalization of the Monge-Ampére equation to T 4 -invariant G 2 -manifolds whose orbits are coassociative submanifolds. In the sequel, we study generalizations of Corollary 5.3 to T 3 -and SO(3)-invariant Lagrangian-type fibered G 2 -manifolds. 5.2. Local structure of T 3 -invariant G 2 -manifolds. In this subsection, we prove that all torsion-free T 3 -invariant Lag G 2 -structures are locally described by orbits of constrained dynamical systems on the spaces of the triples (e, a, S).
Let (e(t), a(t), S(t), f (t)) be a one-parameter family defined on an interval (t 1 , t 2 ) of the triples corresponding to (ω, ψ) and T 3 -invariant positive functions f on P . we use the notation in Subsection 5.1.
In the proof of Proposition 5.5, we use the following lemmas. Let A be an n × n matrix-valued function defined on an interval andÃ the cofactor matrix of A, which is (det A)A −1 when A is regular. The proof of Lemma 5.6 is straightforward.
Proof of Theorem 1.1 and 1.2. By setting f = (det S) 1 2 in Proposition 5.5, we obtain Theorem 1.1. Moreover, by scaling of the parameter t, we can deduce Theorem 1.2 from Proposition 4.3 and 5.5.
Remark 6.1. Note that if there exists an SO(3)-invariant sLag SU(3)-structure then P is a trivial SO(3)-bundle over M. This is because P is isomorphic to the orthonormal frame bundle over M by the solder 1-form e, and any orientable 3-manifolds are parallelizable.
Let (ω, ψ) be an SO(3)-invariant SU(3)-structure on P , and (e, a, S) the triple corresponding to (ω, ψ). Denote by T = (T ij ) and Ω = (Ω ij ) the torsion and curvature forms of (e, a): Also for an equivariant matrix-valued function A = (A ij ) ∈ Ω 0 (P ; M(k; R)) SO(3) , denote by A ij;k the covariant derivative: We use the following formulas. The proof is straightforward.
Remark 6.4. The last terms in (6.1) and (6.2) are the only terms different from Proposition 5.1 in the case of G = T 3 . The non-commutativity of SO(3) appears at these points.
We have the following corollaries to Proposition 6.3.
(2) If dψ = dψ # = 0 holds, then the Riemannian metric on M given by the solder 1-form e is constant negative curvature.
Remark 6.6. If d H e = T ijê j Y i = 0, then the connection a is the Levi-Civita connection of the solder 1-form e. Then the Bianchi identity [d H a ∧ e] = d H d H e = 0 holds. Thus Ω ij = Ω ji holds for i, j = 1, 2, 3. Also Ω ij coincides with the orthonormal representation of the Einstein-tensor of the Riemannian metric defined by the local coframe {e 1 , e 2 , e 3 }.
Remark 6.7. The third statement of Corollary 6.5 is immediately derived from the Liouville -Arnold theorem, which implies compact fibers of Lagrangian fibrations of symplectic manifolds are tori.
6.2. Local structure of SO(3)-invariant G 2 -manifolds. In this subsection, we prove that all torsion-free SO(3)-invariant Lag G 2 -structures are locally described by orbits of constrained dynamical systems on the spaces of triples (e, a, S).
Remark 6.9. Since d H e = T ijê j Y i = 0, the behavior of a(t) is determined by (6.4). See the lemma below.
Lemma 6.10. Suppose that d H e = 0, ∂e i ∂t = p ik e k and p ij = p ji for i, j = 1, 2, 3.
Note that Lemma 5.6 and 5.7 hold by replacing the derivatives with covariant derivatives.
Proof of Proposition 6.8. Most of the proof is the same as that of Proposition 5.5. Setė i = p ij e j andȧ i = q ij e j for i = 1, 2, 3.
Proof of Theorem 1.4 and 1.6. By setting f = (det S) 1 2 in Proposition 6.8, we obtain Theorem 1.4. Moreover, by scaling of the parameter t, we can deduce Theorem 1.6 from Proposition 4.3 and 6.8.
Remark 6.12. The equations of motion preserve the constraint conditions in Theorem 1.4. This is proved by direct calculation in ([Chi19], Proposition 7).
In the paper, we also gave a Hamiltonian formulation of Theorem 1.4 and some observations on Bryant-Salamon's examples [BS89].
Example 6.13. Let M = SU(2), i.e., P = M × SO(3). Fix a global section of P , and by the pull-back, regard a triple (e(t), a(t), S(t)) in Theorem 1.4 as 1forms and functions on M. Assume (e(t), a(t), S(t)) is left-invariant for the group structure on M. The equations in Theorem 1.4 are reduced to the following ordinary differential system. This situation is contained in that of [MS13]. Let θ 1 , θ 2 , θ 3 be left-invariant 1-forms on M satisfying dθ i = −θ i for i = 1, 2, 3. Using A(t) = (A(t) ij ), B(t) = (B(t) ij ) and C(t) = (C(t) ij ) ∈ M(3; R), put e i = A ij e j , a i = B ij e j and de i = C ijê j for i = 1, 2, 3. Then C = − det(A) · A · t A, and by simple computation, we can see that (1.4) in Theorem 1.4 is equivalent to the