The constructions of general connections on second jet prolongation

A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXVIII, NO. 1, 2014 SECTIO A 67–89

MARIUSZ PLASZCZYK

The constructions of general connections on second jet prolongation

Abstract. We determine all natural operators D transforming general con- nectionsΓ on fibred manifolds Y → M and torsion free classical linear con- nections ∇ on M into general connections D(Γ, ∇) on the second order jet prolongationJ²Y → M of Y → M.

1. Introduction. The concept of r-th order connections was firstly intro- duced on groupoids by C. Ehresmann in [2] and next by I. Kol´aˇr in [3] for arbitrary fibred manifolds.

Let us recall that an r-th order connection on a fibred manifold p : Y → M is a section Θ : Y → J^rY of the r-jet prolongation β : J^rY → Y of p : Y → M . A general connection on p : Y → M is a first order connection Γ : Y → J¹Y or (equivalently) a lifting map

Γ : Y ×_M T M → T Y.

By Con(Y → M ) we denote the set of all general connections on a fibred manifold p : Y → M .

If p : Y → M is a vector bundle and an r-th order connection Θ : Y → J^rY is a vector bundle morphism, then Θ is called an r-th order linear connection on p : Y → M .

2000 Mathematics Subject Classification. 58A05, 58A20, 58A32.

Key words and phrases. General connection, classical linear connection, higher order jet prolongation, bundle functor, natural operator.

An r-th order linear connection on M is an r-th order linear connection Λ : T M → J^rT M on the tangent bundle π_M: T M → M of M . By Q^r(M ) we denote the set of all r-th order linear connections on M .

A classical linear connection on M is a first order linear connection

∇: T M → J¹T M or (equivalently) a covariant derivative ∇: X(M) × X(M) → X(M). A classical linear connection ∇ on M is called torsion free if its torsion tensor T (X, Y ) = ∇_XY − ∇_YX− [X, Y ] is equal to zero.

By Q_τ(M ) we denote the set of all torsion free classical linear connections on M .

Let FM denote the category of fibred manifolds and their fibred maps and let FM_m,n ⊂ FM be the (sub)category of fibred manifolds with m- dimensional bases and n-dimensional fibres and their local fibred diffeomor- phisms. Let Mf_m denote the category of m-dimensional manifolds and their local diffeomorphisms. Let F : FM_m,n → FM be a bundle functor on FMm,n of order r in the sense of [4]. Let Γ : Y ×M T M → T Y be the lifting map of a general connection on an object p : Y → M of FM_m,n. Let Λ : T M → J^rT M be an r-th order linear connection on M . The flow operator F of F transforming projectable vector fields η on p: Y → M into vector fields Fη := _{∂t |t=0}^∂ F (F l_t^η) on F Y is of order r. In other words, the value Fη(u) at every u ∈ FyY, y ∈ Y depends only on j_y^rη. There- fore, we have the corresponding flow morphism ˜F : F Y ×Y J^rT Y → T F Y , which is linear with respect to J^rT Y . Moreover, ˜F(u, j_y^rη) = Fη(u), where u∈ F_yY, y∈ Y . Let X^Γbe the Γ-lift of a vector field X on M to Y , i.e. X^Γis a projectable vector field on p : Y → M defined by X^Γ(y) = Γ(y, X(x)), y ∈ Y_x, x = p(y) ∈ M . Then the connection Γ can be extended to a morphism Γ : Y ט _M J^rT M → J^rT Y by the following formula ˜Γ(y, j_x^rX) = j_y^r(X^Γ).

By applyingF, we obtain a map F(˜Γ): F Y ×M J^rT M → T F Y defined by F(˜Γ)(u, j_x^rX) = ˜F(u, j_y^r(X^Γ)) = FX^Γ(u). Further the composition

F(Γ, Λ) := F(˜Γ) ◦ (id_{F Y} × Λ): F Y ×_M T M → T F Y

is the lifting map of a general connection on F Y → M. The connection F(Γ, Λ) is called F -prolongation of Γ with respect to Λ and was discovered by I. Kol´aˇr [5].

Let∇ be a torsion free classical linear connection on M. For every x ∈ M, the connection∇ determines the exponential map exp^∇_x : TxM → M (of ∇ in x), which is diffeomorphism of some neighbourhood of the zero vector at x onto some neighbourhood of x. Every vector v∈ TxM can be extended to a vector field ˜v on a vector space T_xM by ˜v(w) = _{∂t |t=0}^∂ [w+tv]. Then we can construct an r-th order linear connection E_r(∇) : T M → J^rT M , which is given by E_r(∇)(v) = j_x^r((exp^∇_x)∗˜v). This connection is called an exponential extension of∇ and was presented by W. Mikulski in [9]. Another equivalent definition (for corresponding principal connections in the r-frame bundles)

of the exponential extension was independently introduced by I. Kol´aˇr in [6].

Hence given a general connection Γ on Y → M and a torsion free classical linear connection∇ on M, we have the general connection

F(Γ, ∇) := F(Γ, E_r(∇)) : F Y ×_M T M → T F Y.

The canonical character of construction of this connection can be de- scribed by means of the concept of natural operators. The general concept of natural operators can be found in [4]. In particular, we have the following definitions.

Definition 1. Let F : FM_m,n→ FM be a bundle functor of order r on a category FM_m,n. AnFM_m,n-natural operator D : J¹× Q_τ(B)  J¹(F → B) transforming general connections Γ on fibred manifolds p: Y → M and torsion free classical linear connections ∇ on M into general con- nections D(Γ, ∇) : F Y → J¹F Y on F Y → M is a system of regular operators D_Y : Con(Y → M ) × Q_τ(M ) → Con(F Y → M ), (p : Y → M ) ∈ Obj(FM_m,n) satisfying the FM_m,n-invariance condition: for any Γ ∈ Con(Y → M ), Γ1 ∈ Con(Y1 → M1), ∇ ∈ Qτ(M ) and ∇1 ∈ Qτ(M1) such that if Γ is f -related to Γ₁ by an FM_m,n-map f : Y → Y₁ cover- ing f: M → M₁ (i.e. J¹f ◦ Γ = Γ₁ ◦ f) and ∇ is f-related to ∇₁ (i.e.

J¹T f ◦ ∇ = ∇1 ◦ T f), then DY(Γ, ∇) is F f -related to D_Y1(Γ₁,∇1) (i.e.

J¹F f◦D_Y(Γ, ∇) = D_Y1(Γ₁,∇₁) ◦ F f ). Equivalently the FM_m,n-invariance means that for any Γ ∈ Con(Y → M ), Γ₁ ∈ Con(Y1 → M1), ∇ ∈ Q_τ(M ) and ∇₁ ∈ Q_τ(M₁) if diagrams

J¹Y ^{J1f //}J¹Y₁

Y

Γ

OO

f //Y₁

Γ1

OO J¹T M ^J

1T f//J¹T M₁

T M

∇

OO

T f //T M₁

∇1

OO

commute for a FM_m,n-map f : Y → Y₁ covering f : M → M₁, then the diagram

J¹F Y ^{J1F f//}J¹F Y₁

F Y

DY(Γ,∇)

OO

F f //F Y₁

D_Y1(Γ1,∇1)

OO

commutes. We say that the operator D_Y is regular if it transforms smoothly parametrized families of connections into smoothly parametrized ones.

Definition 2. A Mfm-natural operator A : Qτ  Q^r extending torsion free classical linear connections∇ on m-dimensional manifolds M into r-th order linear connections A(∇) : T M → J^rT M on M is a system of regular

operators A_M: Qτ(M ) → Q^r(M ), M ∈ Obj(Mfm) satisfying the Mfm- invariance condition: if ∇ ∈ Q_τ(M ) and ∇₁ ∈ Q_τ(M₁) are f -related by a Mfm-map f : M → M₁ (i.e. J¹T f◦ ∇ = ∇1◦ T f), then A(∇) and A(∇1) are f -related, too (i.e. J^rT f ◦ A(∇) = A(∇₁) ◦ T f ). In other words, the Mf_m-invariance means that if for any ∇ ∈ Q_τ(M ), ∇₁ ∈ Q_τ(M₁) the diagram

J¹T M ^{J1T f//}J¹T M₁

T M

∇

OO

T f //T M₁

∇1

OO

commutes for aMfm-map f : M → M1, then the following diagram J^rT M ^{JrT f//}J^rT M₁

T M

A(∇)

OO

T f //T M₁

A(∇1)

OO

commutes, too. The regularity means that every A_M transforms smoothly parametrized families of connections into smoothly parametrized ones.

Thus the constructionF(Γ, Λ) can be considered as the FM_m,n-natural operator F : J¹ × Q_τ(B)  J¹(F → B). Similarly, the correspondence E_r: Q_τ  Q^r is the Mfm-natural operator.

In [4], the authors described allFM_m,n-natural operators D : J¹×Q_τ(B)

 J¹(F → B) for a bundle functor F = J¹: FM_m,n → FM. They constructed an additional FM_m,n-natural operator P and proved that all FM_m,n-natural operators D : J¹ × Q_τ(B)  J¹(J¹ → B) form the one parameter family tP + (1 − t)J¹, t∈ R.

In this paper we determine allFMm,n-natural operators D : J¹× Qτ(B)

 J¹(J²→ B). We assume that all manifolds and maps are smooth, i.e. of class C^∞.

2. Quasi-normal fibred coordinate systems. Let Γ : Y → J¹Y be a general connection on a fibred manifold p : Y → M with dim(M ) = m and dim(Y ) = m + n, ∇ be a torsion free classical linear connection on M and y₀ ∈ Y be a point with x₀ = p(y₀) ∈ M .

In [8] W. Mikulski presented a concept of (Γ, ∇, y0, r)-quasi-normal fi- bred coordinate systems on Y for any r. For r = 3 this concept can be equivalently defined in the following way.

Definition 3. A (Γ, ∇, y0, 3)-quasi-normal fibred coordinate system on Y is a fibred chart ψ on Y with ψ(y₀) = (0, 0) ∈ R^m,n covering a chart ψ on M with centre x₀ if the map id_Rm is a ψ

∗∇-normal coordinate system with

centre 0 ∈ R^m and an element j_(0,0)² (ψ∗Γ) ∈ J_(0,0)² (J¹R^m,n → R^m,n) is of the form

j_(0,0)² (ψ_∗Γ) = j_(0,0)²

 Γ₀+

m i,j,k=1

n p=1

a^p_kijx^kxⁱdx^j⊗ ∂

∂y^p

+

m i,j=1

n p,q=1

b^p_qijy^qxⁱdx^j⊗ ∂

∂y^p +

m i,j=1

n p=1

c^p_ijxⁱdx^j⊗ ∂

∂y^p

 (1)

for some (uniquely determined) real numbers a^p_kij, b^p_qij and c^p_ij satisfying a^p_kij− a^p_ikj = 0

a^p_kij+ a^p_kji+ a^p_ikj+ a^p_ijk+ a^p_jik+ a^p_jki = 0 b^p_qij+ b^p_qji = 0

c^p_ij + c^p_ji = 0, (2)

where Γ₀ = _m

i=1dxⁱ⊗ _∂x^∂i is the trivial general connection on R^m,n and x¹, . . . , x^m, y¹, . . . , yⁿ are the usual fibred coordinates on R^m,n.

In [8] W. Mikulski proved an important proposition ([8], Proposition 2.2) concerning (Γ, ∇, y0, r)-quasi-normal fibred coordinate systems. Below we recall this result for r = 3. A fibred-fibred manifold version of Proposition 2.2 from [8] for r = 1 is presented in [7].

Proposition 1. Let Γ : Y → J¹Y be a general connection on a fibred mani- fold p : Y → M with dim(M ) = m and dim(Y ) = m + n, ∇ be a torsion free classical linear connection on M and y₀ ∈ Y be a point with x₀ = p(y₀) ∈ M . Then:

(i) There exists a (Γ, ∇, y₀, 3)-quasi-normal fibred coordinate system ψ on Y . (ii) If ψ¹ is another (Γ, ∇, y₀, 3)-quasi-normal fibred coordinate system, then (3) j_y³₀ψ¹= j_y³₀((B × H) ◦ ψ)

for a linear map B ∈ GL(m) and diffeomorphism H : Rⁿ → Rⁿ preser- ving 0.

From the proof of Proposition 2.2 from [8] it follows that (B × H) ◦ ψ is a (Γ, ∇, y0, 3)-quasi-normal fibred coordinate system for any B ∈ GL(m) and any diffeomorphism H : Rⁿ → Rⁿ preserving 0. In other words, the FMm,n-maps of the form B × H for B ∈ GL(m) and diffeomorphisms H : Rⁿ→ Rⁿ preserving 0 ∈ Rⁿ transform quasi-normal fibred coordinate systems into quasi-normal ones.

From now on we will usually work in (Γ, ∇, y0, 3)-quasi-normal fibred coordinates for considered Γ and ∇. If coordinates are not necessarily quasi- normal, the reader will be informed.

3. Constructions of connections. Let Γ : Y → J¹Y be a general con- nection on an FM_m,n-object p : Y → M and let ∇ : T M → J¹T M be a torsion free classical linear connection on M .

Example 1. Let A : Q_τ  Q² be a Mf_m-natural operator and let Λ = A(∇) : T M → J²T M be a second order linear connection on M canonically depending on ∇. Then from Introduction for a functor F = J², we have a general connection

(4) J_(A)² (Γ, ∇) := J²(Γ, A(∇)) : J²Y → J¹J²Y on J²Y → M canonically depending on Γ and ∇.

Because of the canonical character of the constructionJ_(A)² (Γ, ∇) we ob- tain the following proposition.

Proposition 2. The family J_(A)² : J¹× Q_τ(B)  J¹(J²→ B) of functions J_(A)² : Con(Y → M ) × Qτ(M ) → Con(J²Y → M)

for allFM_m,n-objects Y → M is an FM_m,n-natural operator.

Example 2. For every torsion free classical linear connection ∇ on a man- ifold M we have a canonical vector bundle isomorphism ψ_∇: J²T M →

⊕²_k=0S^kT^∗M⊗ T M given by a formula

ψ_∇(τ ) = ⊕²_k=0S^kT₀^∗ϕ⁻¹⊗ T0ϕ⁻¹(I(J²T ϕ(τ ))),

where τ ∈ J_x²T M, x ∈ M, ϕ is a ∇-normal coordinate system on M with centre x and I : J₀²T R^m → ⊕²_k=0S^kT₀^∗R^m⊗T0R^mis the usual identification.

In the main result of [9], W. Mikulski showed thatMf_m-natural operators A : Q_τ  Q² are in bijection with Mfm-natural operators A₀ ≡ 0: Qτ  T^∗⊗ T, A₁: Q_τ  T^∗⊗ T^∗ ⊗ T and A₂: Q_τ  T^∗⊗ S²T^∗⊗ T . In other words, the second order linear connections Λ = A(∇) : T M → J²T M on M canonically depending on ∇ are in bijection with the tensor fields A₀(∇) ≡ 0 : M → T^∗M⊗ T M, A₁(∇) : M → T^∗M⊗ T^∗M⊗ T M and A₂(∇) : M → T^∗M⊗ S²T^∗M ⊗ T M on M canonically depending on ∇.

Now by means of ψ_∇, A₁(∇) ≡ 0 and A₂(∇) we can define a second order linear connection A(∇) : T M → J²T M on M by

(5) A(∇)(v) = ψ_∇⁻¹(v, 0, < A₂(∇)(x), v >), v ∈ T_xM, x∈ M In particular, for A₂(∇) ≡ 0 : M → T^∗M⊗ S²T^∗M⊗ T M we obtain (6) A^exp₂ (∇)(v) = ψ_∇⁻¹(v, 0, 0) : T M → J²T M,

On the other hand, from [9] it follows that A^exp₂ (∇)(v) = E₂(∇)(v).

It means that A^exp₂ (∇) is the second order exponential extension of ∇.

Finally, in the accordance with Example 1 we have a general connection (7) J_(A^{2 exp}

2 )(Γ, ∇) := J²(Γ, A^exp₂ (∇)) : J²Y → J¹J²Y on J²Y → M canonically depending on Γ and ∇.

Example 3. Let ρ ∈ (J²Y )_y0, y₀ ∈ Y_x0, x₀ ∈ M and consider a (Γ, ∇, y₀, 3)- quasi-normal fibred coordinate system ψ on Y . Then

j_(0,0)² (ψ_∗Γ) = j_(0,0)²

 Γ₀+

m i,j,k=1

n p=1

a^p_kijx^kxⁱdx^j ⊗ ∂

∂y^p

+

m i,j=1

n p,q=1

b^p_qijy^qxⁱdx^j ⊗ ∂

∂y^p +

m i,j=1

n p=1

c^p_ijxⁱdx^j⊗ ∂

∂y^p



for unique real numbers a^p_kij, b^p_qij and c^p_ij satisfying (2). Denote Γ^[1]= Γ₀+

m i,j,k=1

n p=1

a^p_kijx^kxⁱdx^j ⊗ ∂

∂y^p,

Γ^[2]= Γ₀+

m i,j=1

n p,q=1

b^p_qijy^qxⁱdx^j⊗ ∂

∂y^p +

m i,j=1

n p=1

c^p_ijxⁱdx^j⊗ ∂

∂y^p. (8)

Now we define general connectionsJ_[1]²(Γ, ∇) : J²Y →J¹J²Y andJ_[2]²(Γ, ∇) : J²Y → J¹J²Y on J²Y → M by

J_[1]²(Γ, ∇)(ρ) := J¹J²(ψ⁻¹)(J_(A²exp)(Γ^[1],∇⁰)(J²ψ(ρ))), J_[2]²(Γ, ∇)(ρ) := J¹J²(ψ⁻¹)(J_(A²exp)(Γ^[2],∇⁰)(J²ψ(ρ))), (9)

where∇⁰ is the usual flat classical linear connection on R^m.

Because of the canonical character of the construction J_[i]²(Γ, ∇) for i = 1, 2 we have the following proposition.

Proposition 3. The family J_[i]²: J¹× Qτ(B)  J¹(J²→ B) of functions J_[i]²: Con(Y → M ) × Q_τ(M ) → Con(J²Y → M)

for allFM_m,n-objects Y → M is an FM_m,n-natural operator.

4. The main result. We can consider the first jet prolongation functor J¹ as an affine bundle functor on the categoryFMm,n. The corresponding vector bundle functor is T^∗B ⊗ V , where B : FM_m,n → Mf_m is a base functor and V is a vertical tangent functor. For this reason, for any fibred manifold p : Y → M from the category FM_m,n, the first jet prolongation J¹Y → Y is the affine bundle with the corresponding vector bundle T^∗M⊗ V Y . Therefore, J¹J²Y → J²Y is the affine bundle with corresponding vector bundle T^∗M⊗ V J²Y . Thus the set of all FMm,n-natural operators D : J¹× Qτ(B)  J¹(J² → B) possesses the affine space structure.

The following theorem classifies all FMm,n-natural operators D : J¹ × Q_τ(B)  J¹(J²→ B).

Theorem 1. Let D : J¹ × Q_τ(B)  J¹(J² → B) be an FM_m,n-natural operator transforming general connections Γ : Y → J¹Y on FM_m,n-objects Y → M and torsion free classical linear connections ∇ on M into general connections D(Γ, ∇) : J²Y → J¹J²Y on J²Y → M.

If m≥ 2, then there exist uniquely determined real numbers t0, t₁, t₂ with t₀ + t₁ + t₂ = 1 and Mf_m-natural operator A : Q_τ  Q² transforming torsion free classical linear connections ∇ on Mf_m-objects M into second order linear connections A(∇) : T M → J²T M on M such that

(10) D(Γ, ∇) = t₀J_(A)² (Γ, ∇) + t₁J_[1]²(Γ, ∇) + t₂J_[2]²(Γ, ∇)

for any FM_m,n-object Y → M, any general connection Γ on Y → M and any torsion free classical linear connection ∇ on M. Besides, if t₀ = 0, then A is uniquely determined (else A can be arbitrary).

In the case m = 1, D = J².

In the proof we use methods for finding natural operators presented in [4] and lemmas from [1].

Proof. Let xⁱ, y^p be the usual fibred coordinates on R^m,n, y_i^p = ∂y^p

∂xⁱ, y^p_ij = y_ji^p = ∂²y^p

∂xⁱ∂x^j be the additional coordinates on J²R^m,n and

Y^p = dy^p, Y_i^p = dy_i^p, Y_ij^p = Y_ji^p = dy^p_ij

be the essential coordinates on the vertical bundle V J²R^m,n of J²R^m,n → R^m, where i, j = 1, . . . , m and p = 1, . . . , n.

On J₀²(J¹R^m,n) we have the coordinates Γ^p_i, Γ^p_ij = ∂Γ^p_i

∂x^j, Γ^p_iq = ∂Γ^p_i

∂y^q, Γ^p_ijk= ∂²Γ^p_i

∂x^j∂x^k, Γ^p_iqr = ∂²Γ^p_i

∂y^q∂y^r, Γ^p_ijq = ∂²Γ^p_i

∂x^j∂y^q.

The standard coordinates on J₀¹(Q_τ(R^m)) are ∇ⁱ_jk = ∇ⁱ_kj and ∇ⁱ_jkl= ∇ⁱ_kjl, where i, j, k, l = 1, . . . , m.

Let ω_kbe the usual coordinates on T^∗R^m. Then the induced coordinates on the tensor product (T^∗R^m⊗ V J²R^m,n)₀ are

Z_k^p = Y^pω_k, Z_i;k^p = Y_i^pω_k, Z_ij;k^p = Y_ij^pω_k.

Let D : J¹× Qτ(B)  J¹(J²→ B) be an FMm,n-natural operator trans- forming general connections Γ : Y → J¹Y on FMm,n-objects Y → M and

torsion free classical linear connections ∇ on M into general connections D(Γ, ∇) : J²Y → J¹J²Y on J²Y → M.

Since J¹J²Y → J²Y is the affine bundle with the corresponding vector bundle T^∗M⊗ V J²Y , we have the correspondingFM_m,n-natural operator

Δ_D: J¹× Q_τ(B)  (J², T^∗B ⊗ V J²).

It transforms a general connection Γ : Y → J¹Y on anFM_m,n-object Y → M and a torsion free classical linear connection ∇ on M into a fibred map (11) Δ_D(Γ, ∇) := D(Γ, ∇) − J_(A^2exp

2 )(Γ, ∇) : J²Y → T^∗M⊗ V J²Y.

Of course, the operator D is fully determined by Δ_Das D(Γ, ∇) = Δ_D(Γ, ∇) +J_(A²exp

2 )(Γ, ∇) for every Γ ∈ Con(Y → M ), ∇ ∈ Q_τ(M ). In other words D = Δ_D + J_(A²exp

2 ), so it is sufficient to investigate the operator Δ_D. Using the invariance of Δ_D with respect to the homotheties ψ_t= tid_R^m,n covering ψ_t= tid_R^m for t > 0, we have the homogeneous conditions

(T^∗(tid_R^m) ⊗ V J²(tid_R^m,n))(Δ_D(Γ, ∇)(ρ))

= (Δ_D((tid_R^m,n)_∗Γ, (tid_R^m)_∗∇))(J²(tid_R^m,n)(ρ)) for any general connection Γ on R^m,n, any torsion free classical linear con- nection∇ on R^m and any ρ∈ (J²R^m,n)_(0,0). Using the general theory and the above local coordinates, the above condition can be written as the sys- tem of homogeneous conditions. Now, by the non-linear Peetre theorem [4]

we obtain that the operator Δ_D is of finite order r in Γ and of order s in

∇. Having the natural operator Δ_D of order r in Γ and of finite order s in

∇, we shall deduce that r = 2 and s = 1.

The operators Δ_D of order 2 in Γ and of order 1 in ∇ are in bijection with G³_m,n-invariant maps of standard fibres f : S₁ × Λ × S0 → Z over f = id_S0, where S₁ = J₀²(J¹R^m,n), Λ = J₀¹(Q_τ(R^m)), S₀ = J₀²R^m,n, Z = (T^∗R^m⊗ V J²R^m,n)0. This map is of the form

Z_k^p = f_k^p(Γ^p_i, Γ^p_ij, Γ^p_iq, Γ^p_ijk, Γ^p_iqr, Γ^p_ijq,∇ⁱ_jk,∇ⁱ_jkl, y_i^p, y_ij^p) Z_i;k^p = f_i;k^p (Γ^p_i, Γ^p_ij, Γ^p_iq, Γ^p_ijk, Γ^p_iqr, Γ^p_ijq,∇ⁱ_jk,∇ⁱ_jkl, y_i^p, y_ij^p) Z_ij;k^p = f_ij;k^p (Γ^p_i, Γ^p_ij, Γ^p_iq, Γ^p_ijk, Γ^p_iqr, Γ^p_ijq,∇ⁱ_jk,∇ⁱ_jkl, y_i^p, y_ij^p).

The group G³_m,n acts on the standard fibre S₀ in the form y^p_i = a^p_qy^q_j˜a^j_i + a^p_j˜a^j_i

y^p_ij = a^p_qy^q_kl˜a^k_i˜a^l_j+ a^p_qry_k^qy_l^r˜a^k_i˜a^l_j + a^p_qky_l^q˜a^k_i˜a_j^l + a^p_qly^q_k˜a^k_i˜a^l_j + a^p_qy^q_k˜a^k_ij+ a^p_k˜a^k_ij+ a^p_kl˜a^k_i˜a^l_j