The constructions of general connections on the fibred product of q copies of the first 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. LXXII, NO. 1, 2018 SECTIO A 77–90

MARIUSZ PLASZCZYK

The constructions of general connections on the fibred product of q copies

of the first jet prolongation

Abstract. We describe all natural operators A transforming general con- nections Γ on fibred manifolds Y → M and torsion-free classical linear con- nections Λ on M into general connections A(Γ, Λ) on the fibred product J^<q>Y → M of q copies of the first jet prolongation J¹Y → M .

1. Introduction. All manifolds are smooth, Hausdorff, finite dimensional and without boundaries. Maps are assumed to be smooth, i.e. of class C^∞. The concept of r-th order connections for arbitrary fibred manifolds was introduced by I. Kol´aˇr in [3].

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 .

2010 Mathematics Subject Classification. 58A05, 58A20, 58A32.

Key words and phrases. General connection, classical linear connection, first jet pro- longation, bundle functor, natural operator.

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 .

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 πM: 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 on M , which can be also (equivalently) considered as its corresponding 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 F M denote the category of fibred manifolds and their fibred maps and let F M_m,n ⊂ F M 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 : F M_m,n → F M be a bundle functor on F M_m,n of order r in the sense of [4]. Let Γ : Y ×_M T M → T Y be the lifting map of a gen- eral connection on an F M_m,n-object p : Y → M . 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 F η := _{∂t |t=0}^∂ F (F l^η_t) on F Y is of order r. In other words, the value F η(u) at every u ∈ F_yY, y ∈ Y depends only on j_y^rη. Therefore, we have the cor- responding 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) = F η(u), where u ∈ F_yY, y ∈ Y . Let X^Γ be the Γ-lift of a vector field X on M to Y , i.e. X^Γ is a pro- jectable vector field on p : Y → M defined by X^Γ(y) = Γ(y, X(x)), y ∈ Yx, 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 applying F , 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)) = F X^Γ(u). Further, the composition

F (Γ, Λ) := F (˜Γ) ◦ (idF 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 [2].

In particular, if F : F M_m,n → F M is a bundle functor on F M_m,n of order r = 1 and Γ is a general connection on an F M_m,n-object p : Y → M and ∇ is a torsion-free classical linear connection on M , then one can obtain the general connection F (Γ, ∇) as in [2].

In the paper [1] authors introduced some interesting constructions on connections using other methods.

2. Natural operators. The canonical character of construction of this connection can be described by means of the concept of natural operators.

The general concept of natural operators can be found in the fundamental monograph [4]. In particular, we have the following definitions.

Definition 1. Let F : F Mm,n → F M be a bundle functor of order r = 1 on the category F M_m,n and B : F M_m,n → Mf_m be a base functor. An F M_m,n-natural operator D : J¹× Q_τ(B) J¹(F → B) transforming gen- eral connections Γ on fibred manifolds Y → M and torsion-free classical linear connections ∇ on M into general connections 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(F Mm,n) satisfying the F Mm,n- invariance condition.

The F M_m,n-invariance means that for any connections Γ ∈ Con(Y → M ), Γ₁ ∈ Con(Y₁ → M₁), ∇ ∈ Q_τ(M ) and ∇₁ ∈ Q_τ(M₁) such that if Γ is f -related to Γ1 by an F M_m,n-map f : Y → Y₁ covering f : M → M₁ (i.e. J¹f ◦ Γ = Γ₁ ◦ f ) and ∇ is f -related to ∇₁ (i.e. J¹T f ◦ ∇ =

∇₁◦T f ), then D_Y(Γ, ∇) is F f -related to DY1(Γ1, ∇1) (i.e. J¹F f ◦ DY(Γ, ∇)

= D_Y1(Γ₁, ∇₁) ◦ F f ).

Equivalently the F M_m,n-invariance means that for any Γ ∈ Con(Y → M ), Γ1 ∈ Con(Y₁ → M₁), ∇ ∈ Qτ(M ) and ∇1∈ Q_τ(M1) if diagrams

J¹Y ^J

1f //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 an F M_m,n-map f : Y → Y₁ covering f : M → M₁, then the diagram

J¹F Y ^J

1F 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 parame- trized families of connections into smoothly parametrized ones.

Thus the construction F (Γ, ∇) can be considered as an F M_m,n-natural operator F : J¹× Q_τ(B) J¹(F → B).

3. Quasi-normal fibred coordinates. According to [6], let Φ_r: J₀^r−1(T^∗R^m⊗ Rⁿ) → J₀^r(R^m, Rⁿ)₀ be the usual symmetrization

r−1

M

q=0

S^qT₀^∗R^m⊗ T₀^∗R^m⊗ Rⁿ→

r−1

M

q=0

S^q+1T₀^∗R^m⊗ Rⁿ modulo the following GL(m)-invariant identifications:

J₀^r−1(T^∗R^m⊗ Rⁿ) =

r−1

M

q=0

S^qT₀^∗R^m⊗ T₀^∗R^m⊗ Rⁿ,

J₀^r(R^m, Rⁿ)₀ =

r−1

M

q=0

S^q+1T₀^∗R^m⊗ Rⁿ.

In other words, Φ_r: J₀^r−1(T^∗R^m ⊗ Rⁿ) → J₀^r(R^m, Rⁿ)0 is the linear map such that

Φ_r



j₀^r−1 (xⁱ¹. . . x^iqdx^j)e_k



= 1

q + 1j^r₀(xⁱ¹. . . x^iqx^je_k)

for any i₁, . . . , iq, j = 1, . . . , m, q = 0, . . . , r −1 and k = 1, . . . , n, where (ek) is the usual canonical basis in Rⁿand (x¹, . . . , x^m) are the usual coordinates on R^m. Then it holds

Φr j₀^r−1(dσ)
= j₀^r(σ)

for any σ : R^m → Rⁿ with σ(0) = 0. In addition, Φ_r is GL(m)-invariant and linear.

Let Γ : Y → J¹Y be a general connection on a fibred manifold p : Y → M , where dim(M ) = m, dim(Y ) = m+n. Let Λ be a torsion-free classical linear connection on M . Let y₀ ∈ Y be a point such that x₀ = p(y₀) ∈ M .

We present a concept of (Γ, Λ, y₀, r)-quasi-normal fibred coordinate sys- tem on Y , which was introduced by W. Mikulski, [6], [7].

Definition 2. A fibred chart ψ on Y with ψ(y₀) = (0, 0) ∈ R^m,n covering a Λ-normal coordinate system ψ on M with centre x0 is called a (Γ, Λ, y₀, r)- quasi-normal fibred coordinate system on Y , if the condition

Φr

 j₀^r−1

 X

|α|+|β|≤r−1 m

X

j=1 n

X

k=1

Γ^k_jαβx^αdx^j⊗ e_k

= 0

holds for any multiindex β ∈ (N ∪ {0})ⁿ such that |β| ≤ r − 1, where j₀^r−1

 m

X

i=1

dxⁱ⊗ ∂

∂xⁱ + X

|α|+|β|≤r−1 m

X

j=1 n

X

k=1

Γ^k_jαβx^αy^βdx^j ⊗ ∂

∂y^k



is the expression of an element j_(0,0)^r−1(ψ∗Γ) and (x¹, . . . , x^m, y¹, . . . , yⁿ) are the usual coordinates on the product R^m× Rⁿ.

In [6], W. Mikulski proved the following theorem.

Theorem 1. Let Γ : Y → J¹Y be a general connection on an F Mm,n- object p : Y → M such that dim(M ) = m, dim(Y ) = m + n and let Λ be a torsion-free classical linear connection on M and let y₀ ∈ Y be a point such that x₀= p(y₀) ∈ M . Then:

(i) There exists a (Γ, Λ, y₀, r)-quasi-normal fibred coordinate system ψ on Y . (ii) If ψ¹ is another (Γ, Λ, y₀, r)-quasi-normal fibred coordinate system on Y , then

j_y^r₀ψ¹= j_y^r₀ (B × H) ◦ ψ

for a map B ∈ GL(m) and a diffeomorphism H : Rⁿ → Rⁿ preserving 0 ∈ Rⁿ.

From the proof of this theorem it follows that (B ×H)◦ψ is a (Γ, Λ, y₀, r)- quasi-normal fibred coordinate system on Y for any B ∈ GL(m) and any diffeomorphism H : Rⁿ → Rⁿ preserving 0 ∈ Rⁿ. In other words, the F M_m,n-maps of the form B × H for B ∈ GL(m) and diffeomorphisms H : Rⁿ → Rⁿ preserving 0 ∈ Rⁿ transform (Γ, Λ, y₀, r)-quasi-normal fibred coordinate systems on Y into (Γ, Λ, y₀, r)-quasi-normal fibred coordinate systems.

The generalization of this theorem in the case r = 2 for fibred-fibred manifolds was proved by J. Kurek and W. Mikulski in [5].

4. The fibred product of q copies of the first jet prolongation.

In [4], the authors described all F M_m,n-natural operators D : J¹× Q_τ (B) J¹(F → B) for a bundle functor F = J¹: F M_m,n → F M. They constructed an additional F M_m,n-natural operator P and proved that all F M_m,n-natural operators D : J¹ × Q_τ(B) J¹(J¹ → B) form the one parameter family tP + (1 − t)J¹, t ∈ R.

In other words, they showed that any F M_m,n-natural operator C : J¹× Q_τ(B) J¹(J¹ → B)

transforming pairs (Γ, Λ) consisting of general connections Γ : Y → J¹Y on F M_m,n-objects p : Y → M and torsion-free classical linear connections Λ : T M → J¹T M on M into general connections CY(Γ, Λ) : J¹Y → J¹J¹Y on J¹Y → M is of the form

(1) C = t · P + (1 − t) · J¹, t ∈ R,

where P and J¹ are natural operators constructed in the monograph [4].

In [8], we generalized this result to the case F = J². In other words, we classified all F M_m,n-natural operators D : J¹× Q_τ(B) J¹(J² → B).

A pair (Γ, Λ) consisting of a general connection Γ : Y → J¹Y on a fibred manifold p : Y → M and a torsion-free classical linear connection Λ : T M → J¹T M on M is called an admissible pair on p : Y → M .

We can consider the first jet prolongation functor J¹ as an affine bundle functor on the category F M_m,n. The corresponding vector bundle functor is T^∗B⊗V , where V is a vertical tangent functor. For this reason, for any fibred manifold p : Y → M , the first jet prolongation J¹Y → Y is the affine bundle with the corresponding vector bundle T^∗M ⊗V Y . Therefore, J¹J¹Y → J¹Y is the affine bundle with corresponding vector bundle T^∗M ⊗ V J¹Y . Thus the set of all F M_m,n-natural operators

C : J¹× Q_τ(B) J¹(J¹ → B)

transforming admissible pairs (Γ, Λ) on fibred manifolds p : Y → M into general connections C_Y(Γ, Λ) : J¹Y → J¹J¹Y on J¹Y → M possesses the affine space structure.

Let

∆ := J˜ ¹− P : J¹× Q_τ(B) (J¹, T^∗B ⊗ V J¹)

be an F M_m,n-natural operator transforming admissible pairs (Γ, Λ) on p : Y → M into fibred maps ˜∆_Y(Γ, Λ) : J¹Y → T^∗M ⊗ V J¹Y covering the identity id_J1Y : J¹Y → J¹Y , where V J¹Y = V (J¹Y → M ) is the vertical bundle of J¹Y → M .

By theorems presented in the monograph [4] it follows that the F M_m,n- natural operator ˜∆ : J¹× Q_τ(B) (J¹, T^∗B ⊗ V J¹) is of finite order.

Then the equality (1) can be written in the following form

(2) J¹− C = t · (J¹− P ).

If we denote E := J¹− C, then we can interpret the equality (2) in the following way.

Any F M_m,n-natural operator

E : J¹× Q_τ(B) (J¹, T^∗B ⊗ V J¹)

transforming admissible pairs (Γ, Λ) on p : Y → M into fibred maps EY(Γ, Λ) : J¹Y → T^∗M ⊗ V J¹Y covering the identity id_J¹_Y : J¹Y → J¹Y is of the form

E = t · ˜∆.

Let

J^<q>:= J¹×_Mfm· · · ×_Mfm

| {z }

q-times

J¹:F M_m,n→ F M

be the bundle functor transforming F M_m,n-objects Y → M into fibred products

J^<q>Y := J¹Y ×_M· · · ×_M

| {z }

q-times

J¹Y

of q copies of J¹Y → M and F M_m,n-maps f : Y → Y₁covering f : M → M₁ into the induced fibred maps

J^<q>f := J¹f ×_f· · · ×_f

| {z }

q-times

J¹f : J^<q>Y → J^<q>Y1.

5. The classification of constructions of general connections on the fibred product of q copies of the first jet prolongation. We want to describe all F M_m,n-natural operators

A : J¹× Q_τ(B) J¹(J^<q>→ B)

transforming admissible pairs (Γ, Λ) on p : Y → M into general connections AY(Γ, Λ) : J^<q>Y → J¹(J^<q>Y ) on J^<q>Y → M .

An example of such A is the F M_m,n-natural operator J^<q> of finite order constructed by I. Kol´aˇr.

By theorems presented in the monograph [4] it follows that F M_m,n- natural operators A : J¹× Q_τ(B) J¹(J^<q> → B) are of finite order.

Next, J¹(J^<q>Y ) → J^<q>Y is the affine bundle with corresponding vec- tor bundle T^∗M ⊗ V J^<q>Y , where V J^<q>Y = V (J^<q>Y → M ) is the vertical bundle of J^<q>Y → M . Therefore, we obtain the following F M_m,n- natural operator

∆ : J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V J^<q>)

of finite order transforming admissible pairs (Γ, Λ) on p : Y → M into fibred maps ∆_Y(Γ, Λ) : J^<q>Y → T^∗M ⊗ V J^<q>Y covering the identity map of J^<q>Y given by

∆_Y(Γ, Λ) := A_Y(Γ, Λ) − J_Y^<q>(Γ, Λ).

The natural operator A is completely described by the natural operator

∆, because it holds

AY(Γ, Λ) = ∆Y(Γ, Λ) + J_Y^<q>(Γ, Λ)

for any admissible pair (Γ, Λ). In other words, it holds A = ∆ + J^<q>. Therefore, in order to determine all F M_m,n-natural operators A : J¹ × Q_τ(B) J¹(J^<q> → B) it is sufficient to describe all F M_m,n- natural operators ∆ : J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V J^<q>).

Using the following identifications V J^<q>Y = V J¹Y ×_M· · · ×_M

| {z }

q-times

V J¹Y,

T^∗M ⊗ V J^<q>Y = (T^∗M ⊗ V J¹Y ) ×_M· · · ×_M

| {z }

q-times

(T^∗M ⊗ V J¹Y ), we find out that any fibred map

τ : J^<q>Y → T^∗M ⊗ V J^<q>Y

is the system τ = (τ₁, . . . , τ_q) of fibred maps

τi := (id_T^∗_M ⊗ V p_i) ◦ τ : J^<q>Y → T^∗M ⊗ V J¹Y

covering the usual projection p_i: J^<q>Y → J¹Y onto an i-th factor of the fibred product J^<q>Y = J¹Y ×M· · · ×_M

| {z }

q-times

J¹Y , i = 1, . . . , q.

So, the F M_m,n-natural operators A : J¹× Q_τ(B) J¹(J^<q> → B) are in a bijective correspondence with systems (B¹, . . . , B^q) of F M_m,n-natural operators

Bⁱ: J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V J¹)

transforming admissible pairs (Γ, Λ) on Y → M into fibred maps B_Yⁱ (Γ, Λ) : J^<q>Y → T^∗M ⊗ V J¹Y given by

B_Yⁱ (Γ, Λ) = (idT^∗M ⊗ V p_i) ◦ ∆Y(Γ, Λ) covering p_i: J^<q>Y → J¹Y for i = 1, . . . , q.

By theorems presented in the monograph [4] it follows that F M_m,n- natural operators Bⁱ: J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V J¹) are of finite order.

Therefore, in order to determine all F M_m,n-natural operators A : J¹ × Q_τ(B) J¹(J^<q> → B) it is sufficient to describe all F M_m,n- natural operators Bⁱ: J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V J¹) of the same type for i = 1, . . . , q.

Therefore, in order to determine all F M_m,n-natural operators A : J¹ × Q_τ(B) J¹(J^<q> → B) it is sufficient to describe all F M_m,n- natural operators

B : J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V J¹)

transforming admissible pairs (Γ, Λ) on p : Y → M into fibred maps B_Y(Γ, Λ) : J^<q>Y → T^∗M ⊗ V J¹Y given by

BY(Γ, Λ) = (idT^∗M ⊗ V p₁) ◦ ∆Y(Γ, Λ) covering id_Y : Y → Y .

By theorems presented in the monograph [4] it follows that F M_m,n- natural operators B : J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V J¹) are of finite order.

Consider the map δ : J¹Y → J¹Y ×_M· · · ×_M

| {z }

q-times

J¹Y given by δ(u) = (u, . . . , u)

for any element u ∈ J_x¹Y , where x ∈ M . Then the F Mm,n-natural operator B defines an F M_m,n-natural operator

B ◦ δ : J¹× Q_τ(B) (J¹, T^∗B ⊗ V J¹)

of finite order transforming admissible pairs (Γ, Λ) on p : Y → M into fibred maps (B ◦ δ)_Y(Γ, Λ) : J¹Y → T^∗M ⊗ V J¹Y given by

(B ◦ δ)_Y(Γ, Λ) := B_Y(Γ, Λ) ◦ δ

covering the identity id_J1Y.

Hence we see that B ◦ δ = t · ˜∆ for the real number t, i.e.

B_Y(Γ, Λ)(u, . . . , u) = t · ˜∆_Y(Γ, Λ)(u)

for any admissible pair (Γ, Λ) on Y → M and for any element u ∈ J_x¹Y , where x ∈ M .

We have the projection V π₀¹: V J¹Y → V Y , where π₀¹: J¹Y → Y is the jet projection. Then the F M_m,n-natural operator B defines an F M_m,n- natural operator

D := (idT^∗M⊗ V π₀¹) ◦ B : J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V )

transforming admissible pairs (Γ, Λ) on p : Y → M into fibred maps DY(Γ, Λ) : J^<q>Y → T^∗M ⊗ V Y given by

D_Y(Γ, Λ) = (id_T^∗_M ⊗ V π₀¹) ◦ B_Y(Γ, Λ) covering the projection π₀¹◦ p₁.

By theorems presented in the monograph [4] it follows that the F M_m,n-natural operator D : J¹ × Q_τ(B) (J^<q>, T^∗B ⊗ V ) is of finite order.

Because of the invariance of D with respect to fibred manifold charts, the existence of (Γ, Λ, y₀, r)-quasi-normal fibred coordinate systems and the non-linear Peetre theorem (see [4]), we deduce that D is determined by the values

(3)

D_Y

 Γ₀+

n

X

k=1 m

X

j=1

X

|α|+|β|≤r−1

Γ^k_jαβx^αy^βdx^j⊗ ∂

∂y^k,

 X

1≤|γ|≤s

Λⁱ_i¹

2i3γx^γi1=1,...,m i2,i3=1,...,m

 (˜u)

from T₀^∗R^m ⊗ V_(0,0)R^m,n for all ˜u = (u1, . . . , uq) such that u1, . . . , uq ∈ (J¹R^m,n)_(0,0), all natural numbers r, s = 1, 2, . . ., all Λⁱ_i¹

2i3γ ∈ R and all Γ^k_jαβ ∈ R satisfying the condition

(4) Φ_r



j₀^r−1 X

|α|≤r−1 m

X

j=1 n

X

k=1

Γ^k_jαβx^αdx^j ⊗ e_k

= 0

for any multiindex β ∈ Nⁿsuch that |β| ≤ r − 1, where Γ₀ =Pm

i=1dxⁱ⊗_∂x^∂_i is a trivial general connection on the fibred manifold R^m,n.

Using the invariance of D with respect to the homotheties t · id_R^m,n for t > 0 (they preserve u₁, . . . , u_q) and next applying the homogeneous function theorem (we can apply it because of the condition (4)) and putting t → 0, we see that every value (3) is equal to

DY(Γ0, Λ⁰)(u1, . . . , uq) ∈ T₀^∗R^m⊗ V_(0,0)R^m,n,

where Λ⁰ is a flat torsion-free classical linear connection on R^m.

Consider a tangent vector ξ ∈ T₀R^m and elements u₁ = j₀¹(id_R^m, ρ) ∈ (J¹R^m,n)_(0,0), u₂, . . . , u_q ∈ (J¹R^m,n)_(0,0) for a map ρ = (ρ₁, . . . , ρ_n) : R^m → Rⁿ such that j₀¹(ρa) 6= 0 for a = 1, . . . , n.

Write

BY(Γ0, Λ⁰)(u1, . . . , uq)(ξ) := d

dt|t=0 j₀¹(id_R^m, ρ + tv)
for some function v = (v₁, . . . , vn) : R^m→ Rⁿ and

va(0) := v⁰_a for a = 1, . . . , n. Then

D_Y(Γ₀, Λ⁰)(u₁, . . . , u_q)(ξ) = d

dt|t=0(0, tv_a⁰).

The fibred map

xⁱ = xⁱ, y^k= y^k+ (y^k)²

preserves: the trivial general connection Γ₀, the flat torsion-free classical linear connection Λ⁰, the F M_m,n-natural operator B, elements u₁, . . . , u_q, the vector ξ and sends _{dt |t=0}^d (j₀¹(id_R^m, ρ + tv)) into _{dt |t=0}^d j₀¹(ρ + ρ²_a+ 2t(¹₂v + ρ_av⁰_a+¹₂t(v_a⁰)²))
. Then it holds v⁰_a= 0 for a = 1, . . . , n. Hence we have the equality

D_Y(Γ₀, Λ⁰)(u₁, . . . , u_q)(ξ) = 0.

Consequently,

D : J¹× Q_τ(B) (J^<q>, T^∗B ⊗ V ) is the zero operator.

Consider the well-known exact sequence

(5) 0 → T^∗M ⊗ V Y → V J¹Y → V Y → 0 over J¹Y . Next we obtain the following exact sequence

0 → T^∗M ⊗ T^∗M ⊗ V Y → T^∗M ⊗ V J¹Y → T^∗M ⊗ V Y → 0 over J¹Y .

Therefore, the F M_m,n-natural operator B can be interpreted as an F M_m,n-natural operator

B : J¹× Q_τ(B) (J^<q>, T^∗B ⊗ T^∗B ⊗ V )

of finite order transforming admissible pairs (Γ, Λ) on p : Y → M into fibred maps B_Y(Γ, Λ) : J^<q>Y → T^∗M ⊗T^∗M ⊗V Y covering the projection π₀¹◦p₁.

Using the invariance of the F M_m,n-natural operator B with respect to fibred manifold charts, the existence of (Γ, Λ, y₀, r)-quasi-normal fibred coor- dinate systems and non-linear Peetre theorem, we deduce that the F M_m,n- natural operator B is determined by the values

(6)

B_Y

 Γ₀+

n

X

k=1 m

X

j=1

X

|α|+|β|≤r−1

Γ^k_jαβx^αy^βdx^j⊗ ∂

∂y^k,

 X

1≤|γ|≤s

Λⁱ_i¹

2i3γx^γ

i1=1,...,m i2,i3=1,...,m

 (˜u)

from T₀^∗R^m⊗ T₀^∗R^m⊗ V_(0,0)R^m,n for all elements ˜u = (u₁, . . . , u_q) such that u1, . . . , uq ∈ (J¹R^m,n)_(0,0), all natural numbers r, s = 1, 2, . . ., all numbers Λⁱ_i¹_2i3γ ∈ R and all numbers Γ^k_jαβ ∈ R satisfying the condition (4) for any multiindex β ∈ Nⁿ such that |β| ≤ r − 1.

We use the following identifications (J¹R^m,n)_(0,0) ∼= R^m∗⊗ Rⁿ,

T₀^∗R^m⊗ T₀^∗R^m⊗ V_(0,0)R^m,n∼= R^m∗⊗ R^m∗⊗ Rⁿ.

Using the invariance of the F M_m,n-natural operator B with respect to the homotheties t · id_R^m,n for t > 0 (they preserve the elements u₁, . . . , u_q) and next applying the homogeneous function theorem (we can apply it because of the condition (4)), we observe that every value (6) is equal to

(7)

BY

 Γ0+

n

X

k,l=1 m

X

j=1

Γ^k_jly^ldx^j⊗ ∂

∂y^k +

n

X

k=1 m

X

i,j=1

Γ^k_jixⁱdx^j⊗ ∂

∂y^k, Λ⁰



(u₁, . . . , u_q), where Γ^k_jl := Γ^k_j(0)e

l ∈ R and Γ^k_ji := Γ^k_je

i(0) ∈ R. Of course, Γ^k_ji = −Γ^k_ij. Moreover, the value (7) is linear in Γ^k_jiand Γ^k_jlwith coefficients being smooth functions in (u₁, . . . , u_q).

In particular, it holds

(8) B_Y



Γ₀+ y^ldx^j⊗ ∂

∂y^k, Λ⁰



(u₁, . . . , u_q)

= B_Y(Γ₀+ dx^j⊗ ˜Y , Λ⁰)(u₁, . . . , u_q) for ˜Y = (y^l+ 1)_∂y^∂k. Since Y₀ := _∂y^∂_k

|06= 0, there is a local diffeomorphism H : Rⁿ→ Rⁿ such that an element j₀¹H = id and H∗Y =˜ _∂y^∂k near 0.

The map id_R^m × H preserves elements u₁, . . . , uq, the F M_m,n-natural operator B and the connection Λ⁰, and sends Γ₀+dx^j⊗ ˜Y into Γ0+dx^j⊗_∂y^∂_k and acts on T₀^∗R^m⊗ T₀^∗R^m⊗ V_(0,0)R^m,n as the identity map. Then using

the invariance of the F M_m,n-natural operator B with respect to id_R^m× H, we see that

B_Y(Γ₀+ dx^j⊗ ˜Y , Λ⁰)(u₁, . . . , u_q) = B_Y



Γ₀+ dx^j⊗ ∂

∂y^k, Λ⁰



(u₁, . . . , u_q).

From (8) we obtain BY



Γ0+ dx^j⊗ ∂

∂y^k, Λ⁰



(u1, . . . , uq) = BY(Γ0, Λ⁰)(u1, . . . , uq) = 0.

Therefore, it holds B_Y



Γ0+ y^ldx^j⊗ ∂

∂y^k, Λ⁰



(u1, . . . , uq) = 0.

Consequently, the values (6) are equal to Xf_k^ji(u1, . . . , uq)Γ^k_ji for some smooth functions f_k^ji.

Using the invariance of F M_m,n-natural operator B with respect to fibre homotheties id_Rm× t · id_Rn for t > 0, we get the homogeneous conditions

t · f_k^ji(tu₁, . . . , tu_q) = t · f_k^ji(u₁, . . . , u_q).

Cancelling both sides by t and putting t → 0, we see that functions f_k^ji are constants.

Thus the F M_m,n-natural operator B is determined by the values BY



Γ0+ xⁱdx^j ⊗ ∂

∂y^k − x^jdxⁱ⊗ ∂

∂y^k, Λ⁰



(0, . . . , 0)

for 1 ≤ i < j ≤ m and k = 1, . . . , n. In other words, we claim that the F M_m,n-natural operator B is determined by the F M_m,n-natural operator B ◦ δ, where the map δ : J¹Y → J^<q>Y is given by δ(u) = (u, . . . , u). As we observed earlier, the equality B◦δ = t· ˜∆ holds for some t ∈ R. It means that F M_m,n-natural operators B ◦ δ form 1-parameter family of operators. Of course, any ˜∆ ◦ p_i is an example of a such B for i = 1, . . . , q. In particular, B is proportional to ˜∆ ◦ p1 and similarly Bⁱ is proportional to ˜∆ ◦ pi for i = 1, . . . , q. Thus we proved the following classification theorem.

Theorem 2. The F Mm,n-natural operators

A : J¹× Q_τ(B) J¹(J^<q>→ B)

transforming admissible pairs (Γ, Λ) on F M_m,n-objects p : Y → M into general connections A_Y(Γ, Λ) : J^<q>Y → J¹(J^<q>Y ) on J^<q>Y → M form the q-parameter family

(9) J^<q>+ (t_i· ˜∆ ◦ p_i)_i=1,...,q for real numbers t_i.

Remark 1. The q-parameter family (9) can be written equivalently in the following form:

(10) 

J^<q>+ t₁· ˜∆, J^<q>+ t₂· ˜∆, . . . , J^<q>+ t_q· ˜∆ . The curvature

R_Y(Γ) : Y →

2

^T^∗M ⊗ V Y can be treated as the fibred map

R_Y(Γ) : J¹Y → T^∗M ⊗ T^∗M ⊗ V Y.

Moreover, by the exact sequence (5) the curvature can be treated as the fibred map

R_Y(Γ) : J¹Y → T^∗M ⊗ V J¹Y.

Thus we obtain an F M_m,n-natural operator

(11) R : J¹× Q_τ(B) (J¹, T^∗B ⊗ V J¹).

By theorems presented in the monograph [4] it follows that the F M_m,n-natural operator R : J¹ × Q_τ(B) (J¹, T^∗B ⊗ V J¹) is of finite order.

Clearly, we can use R instead of ˜∆ in Theorem 2. Because of Theorem 2 for q = 1 we conclude that R is proportional to ˜∆. Therefore, we can reformulate Theorem 2 in the following way.

Theorem 3. The F M_m,n-natural operators

A : J¹× Q_τ(B) J¹(J^<q>→ B)

transforming admissible pairs (Γ, Λ) on F M_m,n-objects p : Y → M into general connections A_Y(Γ, Λ) : J^<q>Y → J¹(J^<q>Y ) on J^<q>Y → M form the q-parameter family

J^<q>+ (t_i· R ◦ p_i)_i=1,...,q

for real numbers t_i, where R is the F M_m,n-natural operator from (11).

References

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[2] Kol´aˇr, I., Prolongations of generalized connections, in: Differential Geometry (Bu- dapest, 1979), Colloq. Math. Soc. J´anos Bolyai, 31, North-Holland, Amsterdam, 1982, 317–325.

[3] Kol´aˇr, I., Higher order absolute differentiation with respect to generalized connections, in: Differential Geometry (Warsaw, 1979), PWN – Polish Sci. Publ., Warszawa, 1984, 153–162.

[4] Kol´aˇr, I., Michor, P. W., Slov´ak J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.

[5] Kurek, J., Mikulski, W. M., On prolongations of projectable connections, Ann. Polon.

Math. 101 (3) (2011), 237–250.

[6] Mikulski, W. M., On “special” fibred coordinates for general and classical connections, Ann. Polon. Math. 99 (2010), 99–105.

[7] Mikulski, W. M., On prolongation of connections, Ann. Polon. Math. 97 (2) (2010), 101–121.

[8] Plaszczyk, M., The constructions of general connections on second jet prolongation, Ann. Univ. Mariae Curie-Skłodowska Sect. A 68 (1) (2014), 67–89.

Mariusz Plaszczyk Institute of Mathematics

Maria Curie-Skłodowska University pl. M. Curie-Skłodowskiej 1 20-031 Lublin

Poland

e-mail: mariusz.plaszczyk@poczta.umcs.lublin.pl Received April 25, 2018