Canonical vector valued 1-forms on higher order adapted frame bundles over category of fibered squares

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 31–36

ANNA BEDNARSKA

Canonical vector valued 1-forms on higher order adapted frame bundles

over category of fibered squares

Abstract. Let Y be a fibered square of dimension (m¹, m2, n1, n2). Let V be a finite dimensional vector space over R. We describe all F²Mm₁,m₂,n₁,n₂- canonical V -valued 1-form Θ : T P_A^r(Y ) → V on the r-th order adapted frame bundle P_A^r(Y ).

A fibered square (or fibered-fibered manifold) is any commutative dia- gram

(1)

Y −−−−→ X^π

q



 y



 y

p

N −−−−→ M^π0

where maps π, π₀, q, p are surjective submersions and induced map Y → X ×M N , y 7→ (π(y), q(y)) is a surjective submersion. We will denote a fibered square (1) by Y in short, [3], [5].

A fibered square (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. For two fibered squares Y₁, Y2 of the same dimension (m₁, m2, n1, n2), a fibered squares morphism f : Y₁ → Y₂ is quadruple of local diffeomorphisms f : Y₁ → Y₂,

2000 Mathematics Subject Classification. 58A20, 58A32.

Key words and phrases. Fibered square, projectable-projectable vector field, r-th order adapted frame bundle, canonical 1-forms.

f₁: X₁ → X₂, f₂: N₁ → N₂, f₀: M₁ → M₂ such that all squares of the cube in question are commutative.

All fibered squares of given dimension (m₁, m2, n1, n2) and their mor- phisms form a category which we denote by F²M_m1,m2,n1,n2.

Every object from the category F²M_m1,m2,n1,n2 is locally isomorphic to the standard fibered square

(2)

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



 y



 y R^m1 × Rⁿ¹ −−−−→ R^m1

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

Let Y be a fibered square (1) of dimension (m₁, m₂, n₁, n₂). We define the r-th order adapted frame bundle

(3)

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

F²M_m1,m2,n1,n2 - morphism}

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

The adapted frame bundle P_A^r(Y ) is a principal bundle with Lie group G^r_m1,m2,n1,n2 = P_A^r(R^m1,m2,n1,n2)_(0,0,0,0) (with multiplication given by the composition of jets) acting on the right on P_A^r(Y ) by composition of jets.

Every F²M_m1,m2,n1,n2-morphism Φ : Y₁ → Y₂ induces a local diffeomor- phism P_A^rΦ : P_A^r(Y₁) → P_A^r(Y₂) given by P_A^rΦ(j_(0,0,0,0)^r ϕ) = j_(0,0,0,0)^r (Φ ◦ ϕ), [1], [4].

Definition 1. Let V be a finite dimensional vector space over R. A F²M_m1,m2,n1,n2-canonical V -valued 1-form Θ on P_A^r is any F²M_m1,m2,n1,n2- invariant family Θ = {Θ_Y} of V -valued 1-forms Θ_Y: T P_A^r(Y ) → V on P_A^r(Y ) for any F²M_m1,m2,n1,n2-object Y , [2], [4].

The invariance of canonical 1-form Θ means that two V -valued forms Θ_Y1 and Θ_Y2 are P_A^rΦ-related (that is P_A^rΦ^∗Θ_Y2 = Θ_Y1, where P_A^rΦ^∗Θ_Y2 = Θ_Y2 ◦ T P_A^rΦ) for any F²M_m1,m2,n1,n2-morphism Φ : Y₁ → Y₂.

Example 1. For every F²M_m1,m2,n1,n2-object Y we define R^m1+m2+n1+n2- valued 1-form θ_Y on P_A¹(Y ) as follows. Consider the projection β : P_A¹(Y ) → Y given by β(j_(0,0,0,0)¹ ϕ) = ϕ(0, 0, 0, 0), an element u = j_(0,0,0,0)¹ ψ ∈ P_A¹(Y ) and a tangent vector W = j₀¹c ∈ TuP_A¹(Y ). We define the form θ_Y by (4) θY(W ) = u⁻¹◦ T β(W )

= j₀¹(ψ⁻¹◦ β ◦ c) ∈ T_(0,0,0,0)R^m1+m2+n1+n2 = R^m1+m2+n1+n2. Obviously, θ = {θ_Y} is a F²M_m1,m2,n1,n2-canonical 1-form on P_A¹.

A vector field W on Y is projectable-projectable on F²M_m1,m2,n1,n2- object (1), if there exists 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 W₁, W₀ are p-related and W2, W0 are π₀-related, [5].

We therefore see that vector field W on Y is projectable-projectable on F²M_m1,m2,n1,n2-object (1) if and only if the flow {Φ_t} of vector field W is formed by local F²M_m1,m2,n1,n2-maps.

The space of all projectable-projectable vector fields on F²M_m1,m2,n1,n2- object Y will be denoted by X_proj-proj(Y ). It is Lie subalgebra of Lie algebra X(Y ) of all vector fields on Y .

For projectable-projectable vector field W ∈ X_proj-proj(Y ) the flow lifting P_A^rW is vector field on P_A^r(Y ) such that if {Φt} is the flow of field W , then {P_A^r(Φt)} is the flow of field P_A^rW . (Since Φt are F²M_m1,m2,n1,n2-maps, we can apply functor P_A^r to Φ_t).

To present a general example of a F²M_m1,m2,n1,n2-canonical V -valued 1- form on P_A^r we need the following lemma, which is an obvious modification of the known fact for usual manifolds.

Lemma 1. Let Y be a fibered square (1) from the category F²M_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 any projectable-projectable vector field W ∈ X_proj-proj(Y ), where P_A^rW ∈ X(P_A^r(Y )) is the flow lifting of field W to P_A^r(Y ). Moreover j_y^rW is uniquely determined.

Proof. 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), by the well-known manifolds 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 vec- tor 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 ∈ Xproj-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 space P_A^r(R^m1,m2,n1,n2) and the dimension of space of r-jets j_(0,0,0,0)^r W of projectable-projectable vectorf fields fW ∈ Xproj-proj(R^m1,m2,n1,n2) are equal. Then Lemma 1 follows from dimension equality, since flow operators are linear.  Example 2. Let

(5) λ : J_(0,0,0,0)^r−1 (T_proj-proj(R^m1,m2,n1,n2)) → V

be a R-linear map, where J_(0,0,0,0)^r−1 (T_proj-proj(R^m1,m2,n1,n2)) is the vector space of all (r − 1)-jets j_(0,0,0,0)^r−1 W at point (0, 0, 0, 0) ∈ R^m1+m2+n1+n2 of projectable-projectable vector fields W ∈ X_proj-proj(R^m1,m2,n1,n2). Given

a fibered square Y , (1), from the category F²M_m1,m2,n1,n2 we define V - valued 1-form Θ^λ_Y : T P_A^r(Y ) → V on P_A^r(Y ) as follows. Let w ∈ TvP_A^r(Y ), where v = j_(0,0,0,0)^r ϕ ∈ (P_A^r(Y ))_y, y ∈ Y . By Lemma 1, we have w = P_A^rW_v for some projectable-projectable vector field W ∈ X_proj-proj(Y ) and j_y^rW is uniquely determined. Then is uniquely determined the (r − 1)-jet j_(0,0,0,0)^r−1 ((ϕ⁻¹)∗W ), where (ϕ⁻¹)∗W = T ϕ⁻¹◦ W ◦ ϕ. We define

(6) Θ^λ_Y(w) := λ(j_(0,0,0,0)^r−1 ((ϕ⁻¹)∗W )).

Obviously, Θ^λ = {Θ^λ_Y} is F²M_m1,m2,n1,n2-canonical V -valued 1-form on P_A^r.

The main result of this note is the following classification theorem.

Theorem 1. Any F²M_m1,m2,n1,n2-canonical V -valued 1-form on P_A^r is of the form Θ^λ for some uniquely determined R-linear map

λ : J_(0,0,0,0)^r−1 (T_proj-proj(R^m1,m2,n1,n2)) → V.

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 F²M_m1,m2,n1,n2-object Y and let y ∈ Y be a point. We sup- pose that j_y^r−1W₁= j_y^r−1W₂ and W₁(y) is not vertical vector with respect to composition of projections π : Y → X and p : X → M . Then there exists a (locally defined) F²M_m1,m2,n1,n2-map Φ : Y → Y such that j_y^r(Φ) = j_y^r(id_Y) and Φ_∗W1 = W2 near y.

Proof. It is a direct modification of the proof of Lemma 42.4 in [2].  Proof of Theorem 1. Let Θ be F²M_m1,m2,n1,n2-canonical V -valued 1- form on P_A^r. We must define λ : J_(0,0,0,0)^r−1 (T_proj-proj(R^m1,m2,n1,n2)) → V by (7) λ(ξ) := (Θ_Rm1,m2,n1,n2)(P^rWf_j^r

(0,0,0,0)(id_Rm1,m2,n1,n2))

for all ξ ∈ J_(0,0,0,0)^r−1 (T_proj-proj(R^m1,m2,n1,n2)), where fW is a unique (germ at (0, 0, 0, 0)) of projectable-projectable vector field on R^m1,m2,n1,n2 such that j_(0,0,0,0)^r−1 W = ξ and coefficients of ff W with respect to the basis of space X_proj-proj(R^m1,m2,n1,n2) composed of canonical vector fields are polynomials of degree ≤ r − 1. We are going to show that Θ = Θ^λ. Because of the F²M_m1,m2,n1,n2-invariance of Θ and Θ^λ it remains to show that

(8) (Θ_Rm1,m2,n1,n2)(w) = (Θ^λ_Rm1,m2,n1,n2)(w) for any w ∈ T_jr

(0,0,0,0)(id_Rm1,m2,n1,n2)P_A^r(R^m1,m2,n1,n2).

By the definition of λ and Θ^λ we have (8) for any w of the form w = P_A^rWf_j^r

(0,0,0,0)(id_Rm1,m2,n1,n2),

where fW ∈ Xproj-proj(R^m1,m2,n1,n2) is a projectable-projectable vector field such that coefficients fW with respect to the above mentioned basis of the space X_proj-proj(R^m1,m2,n1,n2) are polynomials of degree ≤ r − 1.

Now, let w ∈ T_j^r

(0,0,0,0)(id_Rm1,m2,n1,n2)P_A^r(R^m1,m2,n1,n2). Then by Lemma 1, w is of the form w = P_A^rW_j^r

(0,0,0,0)(id_Rm1,m2,n1,n2) for some projectable-pro- jectable vector field W ∈ X_proj-proj(R^m1,m2,n1,n2) and j_(0,0,0,0)^r W is uniquely determined. We can additionally assume that W (0, 0, 0, 0) is not verti- cal vector with respect to projection R^m1+m2+n1+n2 → R^m1. Let fW ∈ X_proj-proj(R^m1,m2,n1,n2) be projectable-projectable vector field such that j_(0,0,0,0)^r−1 W = jf _(0,0,0,0)^r−1 W and coefficients of field fW with respect to the basis of constant vector fields on R^m1,m2,n1,n2 are polynomials of degree

≤ r − 1. Let w = Pe _A^rWf_j^r

(0,0,0,0)(id_Rm1,m2,n1,n2). Then (see above) it holds (Θ_Rm1,m2,n1,n2)(w) = (Θe ^λ_Rm1,m2,n1,n2)(w).e

On the other hand by Lemma 2 there exists a F²M_m1,m2,n1,n2-map Φ : R^m1,m2,n1,n2→ R^m1,m2,n1,n2 such that j_(0,0,0,0)^r Φ = j_(0,0,0,0)^r (id_Rm1,m2,n1,n2) and Φ_∗W = W near (0, 0, 0, 0) ∈ Rf ^m1+m2+n1+n2. Since j_(0,0,0,0)^r Φ = id, then Φ preserves j_(0,0,0,0)^r (id_Rm1,m2,n1,n2). Then since Φ∗W = W , so Φ sendsf we into w. Then because of invariance of Θ and Θ^λ with respect to Φ, we obtain

(Θ_Rm1,m2,n1,n2)(w) = (Θ_Rm1,m2,n1,n2)(w) = (Θe ^λ_Rm1,m2,n1,n2)(w)e

= (Θ^λ_Rm1,m2,n1,n2)(w). 

For r = 1 we have J_(0,0,0,0)⁰ (Tproj-projR^m1+m2+n1+n2) ∼= R^m1+m2+n1+n2. Then by Theorem 1, the vector space of F²M_m1,m2,n1,n2-canonical V -valued 1-forms is of dimension (m₁+ m₂+ n₁+ n₂) dim V . Then we have:

Corollary 1. Any F²M_m1,m2,n1,n2-canonical 1-form Θ = {Θ_Y} on P_A¹ is of the form

(9) ΘY = λ ◦ θY : T P_A¹(Y ) → V

for some unique linear map λ : R^m1+m2+n1+n2 → V , where θ = {θ_Y} is a canonical R^m1+m2+n1+n2-valued 1-form on P_A¹ from Example 1.

Example 3. Notice that it holds

(10) J_(0,0,0,0)^r−1 (Tproj-projR^m1+m2+n1+n2) ∼= R^m1+m2+n1+n2 ⊕ g^r−1_m

1,m2,n1,n2 , where g^r−1_m

1,m2,n1,n2 = Lie(G_m^r−1_1,m2,n1,n2).

In this way for λ = id

Rm1+m2+n1+n2⊕g^r−1m1,m2,n1,n2 we have F²M_m1,m2,n1,n2- canonical 1-form

(11) θ^r_Y := Θ^idRm1+m2+n1+n2 ⊕gr−1m1,m2,n1,n2:

T P_A^r(Y ) → R^m1+m2+n1+n2 ⊕ g^r−1_m

1,m2,n1,n2

on P_A^r (see Example 2). For r = 1, we have θ¹ = θ as in Example 1.

Analogously as in Corollary 1 we have

Corollary 2. Any F²M_m1,m2,n1,n2-canonical V -valued 1-form Θ = {Θ_Y} on P_A^r is of the form:

(12) Θ_Y = λ ◦ θ^r_Y : T P_A^r(Y ) → V

for some uniquely determined linear map λ : R^m1+m2+n1+n2⊕g^r−1_m1,m2,n1,n2 → V , where θ^r is from Example 3.

Remark 1. A notion of fibered square is a generalization of a fibered man- ifold. The theory of projectable natural bundles over fibered manifolds is essentially related with the idea of fibered square, [2], [3], [5].

References

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[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., Canonical vector valued 1-forms on higher order adapted frame bundles, Arch. Math. (Brno) 44 (2008), 115–118.

[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.

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

e-mail: bednarska@hektor.umcs.lublin.pl Received April 7, 2008