The constructions of general connections on second jet prolongation

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


Introduction.
The concept of r-th order connections was firstly introduced on groupoids by C. Ehresmann in [2] and next by I. Kolář 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 r Y of the r-jet prolongation β : J r Y → Y of p : Y → M .A general connection on p : Y → M is a first order connection Γ : Y → J 1 Y or (equivalently) a lifting map 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 r Y 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 r T 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 → J 1 T 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 ) = ∇ X Y − ∇ Y X − [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 mdimensional bases and n-dimensional fibres and their local fibred diffeomorphisms.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 FM m,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 r T 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 η t ) on F Y is of order r.In other words, the value Fη(u) at every u ∈ F y Y, y ∈ Y depends only on j r y η.Therefore, we have the corresponding flow morphism F : F Y × Y J r T Y → T F Y , which is linear with respect to J r T Y .Moreover, F(u, j r y η) = Fη(u), where u ∈ F y Y, y ∈ Y .Let X Γ be the Γ-lift of a vector field X on M to Y , i.e.X Γ is 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 × M J r T M → J r T Y by the following formula Γ(y, j r x X) = j r y (X Γ ).By applying F, we obtain a map F( Γ) : F Y × M J r T M → T F Y defined by F( Γ)(u, j r x X) = F(u, j r y (X Γ )) = FX Γ (u).Further the composition 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. Kolář [5].
Let ∇ be a torsion free classical linear connection on M .For every x ∈ M , the connection ∇ determines the exponential map exp ∇ x : T x M → M (of ∇ in x), which is diffeomorphism of some neighbourhood of the zero vector at x onto some neighbourhood of x.Every vector v ∈ T x M can be extended to a vector field ṽ on a vector space T x M by ṽ(w) = ∂ ∂t |t=0 [w+tv].Then we can construct an r-th order linear connection E r (∇) : T M → J r T M, which is given by E r (∇)(v) = j r x ((exp ∇ x ) * ṽ).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. Kolář 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 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 [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 .An FM m,n -natural operator D : O O commutes.We say that the operator D Y is regular if it transforms smoothly parametrized families of connections into smoothly parametrized ones.
O O commutes, too.The regularity means that every A M transforms smoothly parametrized families of connections into smoothly parametrized ones.
Thus the construction F(Γ, Λ) can be considered as the FM m,n -natural operator F : In [4], the authors described all FM m,n -natural operators D : J 1 ×Q τ (B) J 1 (F → B) for a bundle functor F = J 1 : FM m,n → FM.They constructed an additional FM m,n -natural operator P and proved that all FM m,n -natural operators D : In this paper we determine all FM m,n -natural operators D : We assume that all manifolds and maps are smooth, i.e. of class C ∞ .

Quasi-normal fibred coordinate systems.
Let Γ : Y → J 1 Y 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 y 0 ∈ Y be a point with x 0 = p(y 0 ) ∈ M .
In [8] W. Mikulski presented a concept of (Γ, ∇, y 0 , r)-quasi-normal fibred coordinate systems on Y for any r.For r = 3 this concept can be equivalently defined in the following way.Definition 3. A (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinate system on Y is a fibred chart ψ on Y with ψ(y 0 ) = (0, 0) ∈ R m,n covering a chart ψ on M with centre x 0 if the map id R m is a ψ * ∇-normal coordinate system with centre 0 ∈ R m and an element j 2 (0,0 for some (uniquely determined) real numbers a p kij , b p qij and c p ij satisfying where Γ 0 = m i=1 dx i ⊗ ∂ ∂x i is the trivial general connection on R m,n and x 1 , . . ., x m , y 1 , . . ., y n are the usual fibred coordinates on R m,n .
Proposition 1.Let Γ : Y → J 1 Y 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 y 0 ∈ Y be a point with x 0 = p(y 0 ) ∈ M .Then: (i) There exists a (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinate system ψ on Y .(ii) If ψ 1 is another (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinate system, then From the proof of Proposition 2.2 from [8] it follows that (B × H) • ψ is a (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinate system for any B ∈ GL(m) and any diffeomorphism H : R n → R n preserving 0. In other words, the FM m,n -maps of the form B × H for B ∈ GL(m) and diffeomorphisms H : R n → R n preserving 0 ∈ R n transform quasi-normal fibred coordinate systems into quasi-normal ones.
From now on we will usually work in (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinates for considered Γ and ∇.If coordinates are not necessarily quasinormal, the reader will be informed.

Constructions of connections.
Let Γ : Y → J 1 Y be a general connection on an FM m,n -object p : Y → M and let ∇ : T M → J 1 T M be a torsion free classical linear connection on M .

Example 1. Let
a Mf m -natural operator and let Λ = A(∇) : T M → J 2 T M be a second order linear connection on M canonically depending on ∇.Then from Introduction for a functor F = J 2 , we have a general connection (4) Because of the canonical character of the construction J 2 (A) (Γ, ∇) we obtain the following proposition.

Proposition 2. The family J
Example 2. For every torsion free classical linear connection ∇ on a manifold M we have a canonical vector bundle isomorphism ψ ∇ : , where τ ∈ J 2 x T M, x ∈ M, ϕ is a ∇-normal coordinate system on M with centre x and I : In other words, the second order linear connections Λ = A(∇) : T M → J 2 T M on M canonically depending on ∇ are in bijection with the tensor fields T M, On the other hand, from [9] it follows that Finally, in the accordance with Example 1 we have a general connection for unique real numbers a p kij , b p qij and c p ij satisfying (2).Denote Now we define general connections where ∇ 0 is the usual flat classical linear connection on R m .
Because of the canonical character of the construction J 2 [i] (Γ, ∇) for i = 1, 2 we have the following proposition.

The main result.
We can consider the first jet prolongation functor J 1 as an affine bundle functor on the category FM m,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 J 1 Y → Y is the affine bundle with the corresponding vector bundle J 1 (J 2 → B) possesses the affine space structure.
The following theorem classifies all FM m,n -natural operators D : If m ≥ 2, then there exist uniquely determined real numbers t 0 , t 1 , t 2 with any general connection Γ on Y → M and any torsion free classical linear connection ∇ on M .Besides, if t 0 = 0, then A is uniquely determined (else A can be arbitrary).
In the case m = 1, D = J 2 .
In the proof we use methods for finding natural operators presented in [4] and lemmas from [1].
Proof.Let x i , y p be the usual fibred coordinates on R m,n , be the additional coordinates on J 2 R m,n and , where i, j = 1, . . ., m and p = 1, . . ., n.On J 2 0 (J 1 R m,n ) we have the coordinates The standard coordinates on Let ω k be the usual coordinates on T * R m .Then the induced coordinates on the tensor product is the affine bundle with the corresponding vector bundle T * M ⊗ V J 2 Y , we have the corresponding FM m,n -natural operator It transforms a general connection Γ : Y → J 1 Y on an FM m,n -object Y → M and a torsion free classical linear connection ∇ on M into a fibred map , 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 for any general connection Γ on R m,n , any torsion free classical linear connection ∇ on R m and any ρ ∈ (J 2 R m,n ) (0,0) .Using the general theory and the above local coordinates, the above condition can be written as the system 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 G 3 m,n -invariant maps of standard fibres f : The group G 3 m,n acts on the standard fibre S 0 in the form and on the fibre S 1 by the formula The action on Λ is Finally, the group G 3 m,n acts on Z in the form Now we want to show that every FM m,n -natural operator ) is of order 2 in Γ and of order 1 in ∇.Using the general theory, the operators in question are in bijection with G q m,n -invariant maps where q = max{rank(J r J 1 ), rank(J s Q τ ), rank(J 2 ), rank(T * ), rank(V J 2 )} = max{r + 1, s + 2, 2, 1, 3} = max{r + 1, s + 2, 3} ≥ 3. We shall investigate these maps.Let α and γ be multi indices in x i and β be a multi index in y p .This associated map of our operator has the form , where |α| + |β| ≤ r and |γ| ≤ s.
Using the homotheties ãi j = tδ i j , ãp q = δ p q , a p i = 0, a p qr = 0, a p qi = 0, ãk ij = 0, a p ij = 0, a p qri = 0, a p qrs = 0, a p qij = 0, a p ijk = 0, ãl ijk = 0, we obtain From the homogeneous function theorem we deduce that f p k is linear in (Γ p i ) β , ∇ i jk , y p i and is independent of y p ij and of the variables with |α| > 0 or |γ| > 0. Therefore, (12) Considering invariance of (12) with respect to the homotheties ãi j = δ i j , a p q = tδ p q , a p i = 0, a p qr = 0, a p qi = 0, ãk ij = 0, a p ij = 0, a p qri = 0, a p qrs = 0, a p qij = 0, a p ijk = 0, ãl ijk = 0, we get the condition Using again the homogeneous function theorem, we see that The homotheties ãi j = δ i j , a p q = tδ p q , a p i = 0, a p qr = 0, a p qi = 0, ãk ij = 0, a p ij = 0, a p qri = 0, a p qrs = 0, a p qij = 0, a p ijk = 0, ãl ijk = 0, imply tf p i;k = f p i;k (t Hence the associated map of our operator is independent of (Γ p i ) αβ for |α| + |β| > 2 and (∇ i jk ) γ for |γ| > 1.This completes the proof of the fact that FM m,n -natural operator ) is of order 2 in Γ and of order 1 in ∇.In other words it means that the value Δ D (Γ, ∇)(ρ) is determined by j 2 (0,0) Γ and j 1 0 (∇) and ρ for any In the rest of the proof, we shall use (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinate systems,only.Consider the case m ≥ 2.
Since Δ D is invariant with respect to (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinate systems, Δ D is determined by the contractions all general connections Γ on R m,n and all torsion free classical linear connections ∇ on R m such that ψ = id R m,n is a (Γ, ∇, (0, 0), 3)-quasi-normal fibred coordinate system on R m,n over ψ = id R m .
For vector bundles E → M we have the standard identification V E = E × M E which is a vector bundle isomorphism.As R m,n is a vector bundle and J 2 R m,n is a vector bundle we can write that Next we use the usual GL(m) × GL(n)-invariant identification Using the polarization formula from linear algebra, we have that every symmetric bilinear form on a vector space is uniquely determined by the corresponding quadratic form.Therefore, for k = 2 the values ψ 2 Γ,∇ (ρ, v) are determined by the contractions ψ 2 Γ,∇ (ρ, v), (w w) ⊗ u for all v, u, w, Γ, ∇ as above, where denotes the symmetric tensor product.Then by the density argument and m ≥ 2, we can assume that v and w are linearly independent and u = 0.
Using the GL(m) × GL(n)-invariance of Δ D and Proposition 1, we can assume v = e 1 , w = e 2 , u = E 1 , where (e i ) is the standard basis in R m , (E p ) is the standard basis in R n and (E p ) is the dual basis in R n * .So we get that the operator Δ D is uniquely determined by the values In other words, Δ D is uniquely determined by the values for all ρ ∈ (J 2 R m,n ) (0,0) , all general connections Γ on R m,n and all torsion free classical linear connections ∇ on R m such that ψ = id R m,n is a (Γ, ∇, (0, 0), 3)-quasi-normal fibred coordinate system on R m,n over ψ = id R m .Consider locally defined FM m,n -maps ψ 2 : R m,n → R m,n , ψ 3 : R m,n → R m,n given by ψ 2 (x, y) = x, y 1 + (y 1 ) 2 , y 2 , . . ., y n ψ 3 (x, y) = x, y 1 + (y 1 ) 3 , y 2 , . . ., y n for x ∈ R n and y = (y 1 , y 2 , . . ., y n ) ∈ R n .They preserve ∂ ∂x 1 | 0 and can be written in the form ψ a (x, y) = (id R m (x), H a (y)), where H a (y) = (y 1 + (y 1 ) a , y 2 , . . ., y n ) and a = 2, 3.So ψ a = id R m × H a for H a : R n → R n being a diffeomorphism preserving 0. Hence by Proposition 1 these FM m,n -maps ψ a : R m,n → R m,n for a = 2, 3 transform quasi-normal fibred coordinate systems into quasi-normal ones.Using the invariance of Δ D with respect to ψ a : R m,n → R m,n for a = 2, 3 and the density argument, we show that the values all Γ, ∇, ρ as above.Using the action of the group G 3 m,n on S 0 for a = 2, we obtain y 1 22 = y 1  22 + 2y 1 y 1 22 + 2(y 1 2 ) 2 and then (15) Y n (i.e. for y 1 = 0).Similarly, for a = 3 we get ỹ1 22 = y 1  22 + 3(y 1 ) 2 y 1 22 + 6y 1 (y 1 2 ) 2 and then ( 16) By formula (16) for y 1 2 (ρ) = 0, we have all Γ, ∇, ρ as above.Then analogously from ( 15) and ( 17), we see that 2 ) 3 and therefore, the values all Γ, ∇, ρ as above.Summing up, we obtain that the operator Δ D is uniquely determined by the values for unique real numbers a p kij , b p qij and c p ij satisfying (2) and all torsion free classical linear connections ∇ such that the identity map id R m is a ∇-normal coordinate system with center zero (then for real numbers g p i , h p ij = h p ji .So, it is sufficient to study the values (18) for Γ, ∇, ρ as above.
Equivalently, in terms of G 3 m,n -invariant maps between the standard fibres we obtain that values of functions f 1 1 and f 1 2;1 are determined by values of functions f 1 22;1 .So we will study the values The invariance of f p ij;k with respect to the homotheties Then the homogeneous function theorem implies that with the coefficients being smooth functions in the coefficients g p i of ρ.Then using the invariance of f p ij;k with respect to the homotheties ãi j = δ i j , a p q = tδ p q , a p i = 0, a p qr = 0, a p qi = 0, ãk ij = 0, a p ij = 0, a p qri = 0, a p qrs = 0, a p qij = 0, a p ijk = 0, ãl ijk = 0, for t > 0 and the homogeneous function theorem, we observe that the coefficients on a p kij are constant, the coefficients on b p qij and ∇ l ij;k are linear and the coefficients on other terms from (22) are zero.
Considering the invariance of Δ D with respect to the maps id R m × H for diffeomorphisms H : R n → R n preserving 0, we get that n p,q=1 b p q12 y q ∂ ∂y p is near 0 equal to zero modulo some diffeomorphism H : R n → R n preserving 0. Hence we have that Δ D is determined by the real number α, the bilinear map g and the values (25) for all a ∈ R and all ρ ∈ (J 2 R m,n ) (0,0) .Next using the invariance of Δ D with respect to the homotheties ãi j = δ i j , a p q = tδ p q , a p i = 0, a p qr = 0, a p qi = 0, ãk ij = 0, a p ij = 0, a p qri = 0, a p qrs = 0, a p qij = 0, a p ijk = 0, ãl ijk = 0, from the homogeneous function theorem, it follows that (25) depends linearly in (a, ρ).This implies that Δ D is determined by the real number α, the bilinear map g and the values n is a (Γ 0 , ∇ 0 , (0, 0), 3)quasi-normal fibred coordinate system on R m,n over ψ = id R m .Since the FM m,n -maps of the form B × H (in question) preserve the trivial general connection Γ 0 and the flat torsion free classical linear connection ∇ 0 then we deduce that the values Δ D (Γ 0 , ∇ 0 )(ρ) are determined by the values But using the formula (23), we see that the last values are equal to zero.Therefore, (26) Δ D (Γ 0 , ∇ 0 )(ρ) = 0 for any ρ ∈ (J 2 R m,n ) (0,0) .This gives that Δ D is determined by the real number α, the bilinear map g and the values (27) The value ( 27) is determined by the evaluations for all p = 1, . . ., n and all i, j, k = 1, . . ., m.Since (25) depends linearly on a, using the invariance of Δ D with respect to the homotheties ãi j = δ i j , ãp q = tδ p q , a p i = 0, a p qr = 0, a p qi = 0, ãk ij = 0, a p ij = 0, a p qri = 0, a p qrs = 0, a p qij = 0, a p ijk = 0, ãl ijk = 0, we see that Therefore, Δ D is determined by the evaluations Then using the invariance of Δ D with respect to a t : R m,n → R m,n by a t (x, y) = (x, ty 1 , y 2 , . . ., y n ) for t > 0, we may assume p = 1, i.e.Δ D is determined by the evaluations Then using the invariance of Δ D with respect to b t : R m,n → R m,n by b t (x, y) = (t 1 x 1 , . . ., t m x m , y 1 , . . ., y n ), we see that the values (30) are all zero except the values Because of the invariance of Δ D with respect to exchanging x 1 and x 2 (i.e. with respect to the map c : R m,n → R m,n given by c(x 1 , x 2 , . . ., x m , y) = (x 2 , x 1 , . . ., x m , y)), we get Consequently, the vector space of all possible values (27) is of dimension ≤ 1.So, the vector space of all possible Δ D is of dimension ≤ 2 + K, where K is the dimension of the vector space of all possible g.
for any ρ ∈ (J 2 R m,n ) (0,0) and any torsion free classical linear connection ∇ ∈ Q τ (R m ) such that id R m is a ∇-normal coordinate system with center 0. By the flow argument we see that J 2 (x 2 ) 2 ∂ ∂y 1 (j 2 0 0) ∼ = j 2 0 ((x 2 ) 2 )J 2 ∂ ∂y 1 (j 2 0 0), J 2 x 2 ∂ ∂y 1 (j 2 0 0) ∼ = j 2 0 (x 2 )J 2 ∂ ∂y 1 (j 2 0 0), and then they are linearly independent.Using the dimension argument and the formula (23), we deduce that there exist unique real numbers t 1 and t 2 and an FM m,n -natural operator be the second order connection corresponding to Ã. So, we have constructed an Mf m -natural operator A transforming torsion free classical linear connections ∇ on m-manifolds M into second order linear connections A(∇) : T M → J 2 T M on M .
We prove (35) as follows.Using the invariance of A − A exp 2 with respect to the homotheties and applying the homogeneous function theorem, we see that A(∇ 0 ) − A exp 2 (∇ 0 ) is the zero tensor field of type T * ⊗ S 2 T * ⊗ T .Therefore, we obtain (34) for Δ J (A) instead of Δ D 1 .Then using the condition (34), we get for any ρ ∈ (J 2 R m,n ) (0,0) , any general connection Γ on R m,n and any torsion free classical linear connection ∇ on R m such that the identity map ψ = id R m,n is a (Γ, ∇, (0, 0), 3)-quasi-normal coordinate system on R m,n , where Because of Mf m -invariance it is sufficient to show (39) in the case M = R m , x = 0 and the identity map ψ = id Rm,n is a (Γ, ∇, (0, 0), 3)-quasinormal fibred coordinate system on R m,n .It is not difficult.So, A 1 = A, i.e.A satisfying (35) is uniquely determined.The proof of Theorem 1 for m ≥ 2 is complete.
The proof of Theorem 1 is complete.

1 , 2
obtained by composing the values Δ D (Γ, ∇)(ρ), v with the respective projections.So we can write for ∇ l ij;k = ∇ l ji;k ∈ R satisfying some "classical" conditions, ρ is of the form ρ g p i , h p ij = h p ji and g is the bilinear map as in (23).Then we have (35) because any Δ D (and then any D) is determined by the values (18).IfD 1 = J 2 (A 1 )for another Mf m -natural operator A 1 (of the type as the one of A), then (39) Ã(∇) |x , ω = Ã1 (∇) |x , ω for any torsion free classical linear connection ∇ on M and any ω ∈ T * x M, x ∈ M , where Ã1 (∇) = A 1 (∇) − A exp 2 (∇) : M → T * M ⊗ S 2 T * M ⊗ T M is the tensor field corresponding to A 1 (∇) : T M → J 2 T M.