Natural affinorson the r-th order adapted frame bundle over fibered-fibered manifolds

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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. LXII, 2008 SECTIO A 55–60

AGNIESZKA CZARNOTA

Natural affinors

on the r-th order adapted frame bundle over fibered-fibered manifolds

Abstract. We describe all F²Mm₁,m₂,n₁,n₂-natural affinors on the r-th order adapted frame bundle PA^rY over (m1, m2, n1, n2)-dimensional fibered- fibered manifolds Y .

Manifolds and maps are assumed to be of class C^∞. Manifolds are assumed to be finite dimension and without boundaries.

A fibered-fibered manifold (or a fibered square) is a commutative diagram

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Y −−−−→ X^π

q



 y



 y^p N −−−−→ M^π⁰

where all maps π, π₀, p, q are surjective submersions and the induced map Y → X ×_M N , y 7→ (π(y), q(y)) is a surjective submersion, [3], [5]. A fibered-fibered manifold (or fibered square) (1) is denoted by Y for short.

A fibered-fibered manifold (1) has dimension (m₁, m₂, n₁, n₂), if dim Y = m1 + m2+ n1+ n2, dim X = m₁+ m2, dim N = m₁+ n1, dim M = m₁. Recall that for two (m₁, m₂, n₁, n₂)-dimensional fibered-fibered manifolds Y , Y₁, a local isomorphism f : Y → Y₁ is a quadruple of local diffeomorphisms f : Y → Y1, f₁: X → X1, f₂ : N → N1, f₀: M → M1 such that all squares

2000 Mathematics Subject Classification. 58A20.

Key words and phrases. Fibered-fibered manifold, r-th order adapted frame bundle, natural affinor.

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of corresponding cube are commutative. All fibered-fibered manifolds of dimension (m₁, m2, n1, n2) and their local isomorphisms form a category which we will denote by F²M_m₁_,m₂_,n₁_,n₂.

Every F²M_m₁_,m₂_,n₁_,n₂-object Y is locally isomorphic to the standard fibered-fibered manifold

R^m¹× R^m²× Rⁿ¹ × Rⁿ² −−−−→ R^m¹ × R^m²



 y



 y R^m¹ × Rⁿ¹ −−−−→ R^m¹

which we will denote by R^m¹^,m²^,n¹^,n², where arrows are projections.

For any fibered-fibered manifold Y of dimension (m₁, m2, n1, n2) we define the r-th order adapted frame bundle

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P_A^rY = {j_(0,0,0,0)^r ϕ | ϕ : R^m¹^,m²^,n¹^,n² → Y is

F²M_m₁_,m₂_,n₁_,n₂-morphism}

over Y with jet target projection β : P_A^rY → Y , β(j_(0,0,0,0)^r ϕ) = ϕ(0, 0, 0, 0).

Every F²M_m₁_,m₂_,n₁_,n₂-map ψ : Y → Y₁ induces a map P_A^rψ : P_A^rY → P_A^rY1

given by P_A^rψ(j^r_(0,0,0,0)ϕ) = j_(0,0,0,0)^r (ψ ◦ ϕ), [1].

Definition 1. A F²M_m₁_,m₂_,n₁_,n₂-natural affinor A on P_A^r is a family of F²M_m₁_,m₂_,n₁_,n₂-invariant affinors A = {A_Y} (tensor fields of type (1, 1)):

(3) A_Y : T P_A^rY → T P_A^rY, on P_A^rY for any F²M_m₁_,m₂_,n₁_,n₂-object Y , [4].

The invariance means that

A_Y₁ ◦ T P_A^rψ = T P_A^rψ ◦ A_Y for any F²M_m₁_,m₂_,n₁_,n₂-map ψ : Y → Y₁.

In this article we describe all F²M_m₁_,m₂_,n₁_,n₂-natural affinors on P_A^r. All Mf_m-natural affinors on P^r were described by Kurek and Mikulski in [4].

We have the following examples of F²M_m₁_,m₂_,n₁_,n₂-natural affinors on P_A^r. Example 1. The identity F²M_m₁_,m₂_,n₁_,n₂-natural affinor Id on P_A^r such that Id : T P_A^rY → T P_A^rY is the identity map for any F²M_m₁_,m₂_,n₁_,n₂- object Y .

Remark 1. A vector field W on a fibered-fibered manifold Y is projectable- projectable on F²M_m₁_,m₂_,n₁_,n₂-object Y , (1), if there exist vector fields W₁ on X and W₂ on N and W₀ on M such that W , W₁ are π-related and W , W₂ are q-related and W₂, W₀ are π₀-related and W₁, W₀ are p-related.

Clearly, a vector field W on a fibered-fibered manifold Y is projectable- projectable on F²M_m₁_,m₂_,n₁_,n₂-object Y , (1), if and only if the flow {Φ_t} of the vector field W is formed by local F²M_m₁_,m₂_,n₁_,n₂-maps, [5].

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We write X_proj-proj(Y ) for the space of all projectable-projectable vector fields on F²M_m₁_,m₂_,n₁_,n₂-object Y . It is a Lie subalgebra of the Lie algebra X (Y ) of all vector fields on Y .

For a projectable-projectable vector field W ∈ X_proj-proj(Y ) its flow lifting P_A^rW is a vector field on P_A^r(Y ) such that if {Φ_t} is the flow of W , then P_A^r(Φt) is the flow of P_A^rW .

To give another example of a natural affinor on P_A^r we will use the following lemma, [2].

Lemma 1. Assume that Y is a fibered-fibered manifold (1) of dimension (m1, m2, n1, n2). Then any vector w ∈ T_vP_A^r(Y ), where v ∈ (P_A^r(Y ))y, y ∈ Y , is of the form w = P_A^rW_v for some W ∈ X_proj-proj(Y ) and j_y^rW is uniquely determined, where P_A^rW is the flow lifting of W to P_A^r(Y ).

Proof. Clearly, we can assume that Y = R^m¹^,m²^,n¹^,n² and y = (0, 0, 0, 0) ∈ R^m¹^+m²⁺ⁿ¹⁺ⁿ². Since P_A^r(R^m¹^,m²^,n¹^,n²) is obviously a principal subbun- dle of the r-th order frame bundle P^r(R^m¹^+m²⁺ⁿ¹⁺ⁿ²) then by the well- known manifold version of Lemma 1, we find W ∈ X (R^m¹^+m²⁺ⁿ¹⁺ⁿ²) such that w = P^rW_v and j_(0,0,0,0)^r W is uniquely determined, where P^rW is a vector field on P^r(R^m¹^+m²⁺ⁿ¹⁺ⁿ²) being a flow lifting of vector field W and v ∈ P_A^r(R^m¹^,m²^,n¹^,n²). For a projectable-projectable vector field fW ∈ X_proj-proj(R^m¹^,m²^,n¹^,n²) the vector P^rfW_v ∈ T_vP^r(R^m¹^+m²⁺ⁿ¹⁺ⁿ²) is tangent to P_A^r(R^m¹^,m²^,n¹^,n²) at the point v. On the other hand, the dimension of the space P_A^r(R^m¹^,m²^,n¹^,n²) and the dimension of the space of r-jets j_(0,0,0,0)^r W off projectable-projectable vector fields fW ∈ X_proj-proj(R^m¹^,m²^,n¹^,n²) are equal.

Then Lemma 1 follows from the dimension equality, since the flow operator

is linear.

Example 2. Let

B : J_(0,0,0,0)^r−1 T_proj-projR^m¹^,m²^,n¹^,n² →

J_(0,0,0,0)^r T_proj-projR^m¹^,m²^,n¹^,n²

0

be a linear map, where

J_(0,0,0,0)^r−1 Tproj-projR^m¹^,m²^,n¹^,n² = n

j_(0,0,0,0)^r−1 V | V ∈ Xproj-proj(R^m¹^,m²^,n¹^,n²) o and

J_(0,0,0,0)^r Tproj-projR^m¹^,m²^,n¹^,n²

0

=n

j_(0,0,0,0)^r V | V ∈ X_proj-proj(R^m¹^,m²^,n¹^,n²), V_(0,0,0,0)= 0o are vector spaces and X_proj-proj(Y ) is the vector space of all projectable- projectable vector fields on Y . Given a F²M_m₁_,m₂_,n₁_,n₂-object Y we define a vertical affinor A^B_Y : T P_A^rY → V P_A^rY ⊂ T P_A^rY by

(4) A^B_Y(v) = V P_A^rϕ((P_A^r˜v)θ), v ∈ Tj_(0,0,0,0)^r ϕP_A^rY, j_(0,0,0,0)^r ϕ ∈ P_A^rY,

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where v = (P_A^r¯v)_j^r

(0,0,0,0)ϕ, ˜v ∈ X_proj-proj(R^m¹^,m²^,n¹^,n²) is a projectable-projectable vector field on R^m¹^,m²^,n¹^,n² with j^r_(0,0,0,0)(˜v) = B(j_(0,0,0,0)^r−1 (ϕ⁻¹_? v))¯ and θ = j_(0,0,0,0)^r (id_Rm1+m2+n1+n2) ∈ P_A^rR^m¹^,m²^,n¹^,n². Here P_A^rV ∈ X (P_A^rY ) denotes the flow lifting of a projectable-projectable vector field V on Y to P_A^rY . We can show that A^B(v) is well defined. Precisely j_ϕ(0,0,0,0)^r ¯v is determined uniquely by v (see Lemma 1).

Then j_(0,0,0,0)^r−1 (ϕ⁻¹_? v) ∈ J¯ _(0,0,0,0)^r−1 (Tproj-projR^m¹^,m²^,n¹^,n²) is determined uniquely by v and j_(0,0,0,0)^r (˜v) ∈ J_(0,0,0,0)^r T_proj-projR^m¹^,m²^,n¹^,n²

0 is determined by v. Then (P^r˜v)θ is determined by v and it is a vertical vector.

Thus A^B_Y(v) is determined by v and it is a vertical vector. Using the linear- ity of the flow operator we obtain that A^B_Y : T P_A^rY → V P_A^rY ⊂ T P_A^rY is a vertical affinor.

It is easy to see that the family A^B = {A^B_Y} of affinors A^B_Y : T P_A^rY → T P_A^rY for any F²M_m₁_,m₂_,n₁_,n₂-object Y is a F²M_m₁_,m₂_,n₁_,n₂-natural affinor on P_A^r.

The main result of the present note is the following classification theorem:

Theorem 1. Any F²M_m₁_,m₂_,n₁_,n₂-natural affinor A on P_A^r is of the form

(5) A = λId + A^B,

for a (uniquely determined by A) real number λ and a (uniquely determined by A) linear map

(6) B : J_(0,0,0,0)^r−1 T_proj-projR^m¹^,m²^,n¹^,n² → J_(0,0,0,0)^r T_proj-projR^m¹^,m²^,n¹^,n²

0. In the proof of Theorem 1 we use the following fact.

Lemma 2. Let W1, W2 ∈ X_proj-proj(Y ) be projectable-projectable vector fields on an F²M_m₁_,m₂_,n₁_,n₂-object Y and let y ∈ Y . Let us assume that j_y^r−1W₁ = j_y^r−1W₂ and W₁(y) is not vertical with respect to the composition of the projections π : Y → X and p : X → M . Then there exists a local F²M_m₁_,m₂_,n₁_,n₂-morphism Φ : Y → Y such that j_y^r(Φ) = j_y^r(id_Y) and Φ?W1 = W2 near y.

Proof. The proof is a simple modification of the proof of Lemma 42.4 in

[2].

Proof of Theorem 1. Let θ = j_(0,0,0,0)^r (id_Rm1+m2+n1+n2) ∈ P_A^rR^m¹^,m²^,n¹^,n². Suppose that A((P_A^rV )θ) = (P_A^rV )˜ θ and V (0, 0, 0, 0) 6= µ ˜V (0, 0, 0, 0) for all µ and ˜V (0, 0, 0, 0) 6= 0. Then there exists an F²M_m₁_,m₂_,n₁_,n₂-map ψ : R^m¹^,m²^,n¹^,n² → R^m¹^,m²^,n¹^,n² preserving θ with J^rT ψ(j_(0,0,0,0)^r V ) = j_(0,0,0,0)^r V and J^rT ψ(j_(0,0,0,0)^r V ) 6= j˜ _(0,0,0,0)^r V . Then˜

(7) A((P_A^rV )_θ) = (P_A^r(ψ_?V ))˜ _θ 6= (P_A^rV )˜ _θ= A((P_A^rV )_θ)

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and it is a contradiction. Then

(8) T β^r◦ A((P_A^rV )θ) = λ(j_(0,0,0,0)^r V )V_(0,0,0,0),

for some (not necessarily unique and necessarily smooth) function λ : J_(0,0,0,0)^r T_proj-projR^m¹^,m²^,n¹^,n² → R,

where β^r : P_A^rR^m¹^,m²^,n¹^,n² → R^m¹^+m²⁺ⁿ¹⁺ⁿ² is the usual projection.

We prove that λ can be chosen from smooth functions. Let λ be such that (8) holds. Since the left side of (8) depends smoothly on j_(0,0,0,0)^r V then the function Φ : J_(0,0,0,0)^r (T_proj-projR^m¹^,m²^,n¹^,n²) → R given by

(9) Φ(j_(0,0,0,0)^r V ) = λ(j_(0,0,0,0)^r V )V¹(0) , 0 ∈ R^m¹^,m²^,n¹^,n² is smooth and Φ(j_(0,0,0,0)^r V ) = 0 if V¹(0) = 0 where

(10) V_(0,0,0,0)=

m1

X

i=1

Vⁱ(0) ∂

∂xⁱ|_(0,0,0,0)+ . . . .

Then (it is the well-known fact from mathematical analysis) there is a smooth map ψ : J_(0,0,0,0)^r (T_proj-projR^m¹^,m²^,n¹^,n²) → R such that

(11) Φ(j_(0,0,0,0)^r V ) = ψ(j_(0,0,0,0)^r V )V¹(0).

Then we can put λ = ψ and (8) holds.

Since the left hand side of (8) depends linearly on j_(0,0,0,0)^r V we have λ = const. Replacing A by A − λId we see that A(v) is vertical for any v ∈ T_θP_A^rR^m¹^,m²^,n¹^,n².

We define a linear map

(12) B : J_(0,0,0,0)^r−1 T_proj-projR^m¹^,m²^,n¹^,n² → J_(0,0,0,0)^r T_proj-projR^m¹^,m²^,n¹^,n²

0

by

(13) B(j_(0,0,0,0)^r−1 V ) = j_(0,0,0,0)^r V ,˜

where ¯V is a unique projectable-projectable vector field on R^m¹^,m²^,n¹^,n² with coefficients being polynomials of degree ≤ r−1 (with respect to the canonical basis of vector field on R^m¹^,m²^,n¹^,n²) such that j_(0,0,0,0,)^r−1 V = j¯ _(0,0,0,0)^r−1 V , and P_A^r( ˜V )_θ = A((P_A^rV )¯ _θ).

We will show that A = A^B. Clearly B is well defined. (For, j_(0,0,0,0)^r V˜ is determined by (P_A^rV )˜ θ = A((P_A^rV )¯ θ) and P_A^r( ¯V )θ is determined by j_(0,0,0,0)^r V (see Lemma 1) and j¯ _(0,0,0,0)^r V is determined by j¯ _(0,0,0,0)^r−1 V (by the definition of ¯V )). Moreover, since A is of vertical type then ˜V (0, 0, 0, 0) = 0.

That is why B is well defined. Then (by the definition of B) we see that A((P_A^rV )θ) = A^B((P_A^rV )θ) for all projectable-projectable vector fields V on R^m¹^,m²^,n¹^,n² with coefficients being polynomials of degree ≤ r − 1 (with

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respect to the canonical basis of vector fields on R^m¹^+m²⁺ⁿ¹⁺ⁿ²). But (by Lemma 2) any projectable-projectable vector fields W on R^m¹^,m²^,n¹^,n² with non-vanishing projection on R^m¹ is ψ-related (near (0, 0, 0, 0)) to some projectable-projectable vector field V with coefficients being polynomials of degree ≤ r−1 for some θ-preserving F²M_m₁_,m₂_,n₁_,n₂-map ψ : R^m¹^,m²^,n¹^,n²→ R^m¹^,m²^,n¹^,n². Consequently A(v) = A^B(v) for any v ∈ TθP_A^rR^m¹^,m²^,n¹^,n². Then A = A^B because of the F²M_m₁_,m₂_,n₁_,n₂-invariance of A and A^B and the fact that P_A^rY is the F²M_m₁_,m₂_,n₁_,n₂-orbit of θ.

If A^B = A^B⁰ then B = B⁰. If λ⁰ i B⁰ are such that A = λ⁰Id + A^B⁰, then λ = T β^r◦ A((P_A^r_∂x^∂₁)_θ) = λ⁰ and B = B⁰. Remark 2. Natural affinors on P_A^rY can be used to define a generalized torsion of connections on P_A^rY . Any natural affinor A : T P_A^rY → T P_A^rY defines a torsion τ^A(Γ) := [A, Γ] of a principal r-th order connection Γ : T P_A^rY → T P_A^rY on fibered-fibered manifold Y , where the bracket means the Fr¨olicher–Nijenhuis bracket.

A principal r-th order connection Γ on P_A^rY → Y is a right invariant section Γ : P_A^rY → J¹P_A^rY of the first jet prolongation J¹P_A^rY → P_A^rY of P_A^rY → Y . Equivalently, Γ can be treated as the corresponding lifting map Γ : T Y ×_Y P_A^rY → T P_A^rY , [2].

References

[1] Doupovec, M., Mikulski, W. M., Gauge natural constructions on higher order principal prolongations, Ann. Polon. Math. 92 (2007), no. 1, 87–97.

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

[3] Kol´aˇr, I., Connections on fibered squares, Ann. Univ. Mariae Curie-Skłodowska Sect.

A 59 (2005), 67–76.

[4] Kurek, J., Mikulski, W. M., The natural affinors on the r-th order frame bundle, Demonstratio Math. 41 (2008), 701–704.

[5] Mikulski, W. M., The jet prolongations of fibered-fibered manifolds and the flow oper- ator, Publ. Math. Debrecen 59 (3-4) (2001), 441–458.

Agnieszka Czarnota Institute of Mathematics M. Curie-Skłodowska University pl. Marii Curie-Skłodowskiej 1 20-031 Lublin, Poland

e-mail: czarnota.agnieszka@interia.pl Received June 12, 2008