DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 231–237

BUNDLE FUNCTORS OF THE JET TYPE

Ivan Kolář

Abstract. We introduce the general concept of jet functor on the product category Mf_{m} × Mf. We describe the extensions of jet functors to the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphism as base map.

Let Mf be the category of all manifolds and all smooth maps, Mf_{m} be the category of m-dimensional manifolds and their local diffeomorphisms and FM_{m} be the category of fibered manifolds over m-dimensional bases and their morphisms with the base map in Mf_{m}. In [4], W. Mikulski and the author described all fiber product preserving bundle functors on the category FM_{m} in terms of Weil algebras and their homomorphisms. In the present paper, we apply this point of view to various subbundles of the non-holonomic jet bundles by C. Ehresmann, [1], [5]. In §1 we introduce the general concept of jet functor on the category Mf_{m} × Mf and we characterize it by means of Weil algebras.

Then we discuss jet functors on the category FM_{m}. The construction of product fibered manifolds and product morphisms defines an injection i : Mf_{m} × Mf → FM_{m}. In general, a bundle functor E on FM_{m} is said to be extension of a bundle functor F on Mf_{m} × Mf, if E ∘ i = F. In §2 we point out that every jet functor on Mf_{m} × Mf determines two extensions, the horizontal one and the vertical one. Using the general description of constructing extensions from [4], we deduce in §3 that these two extensions are the only ones in the case of the functor of holonomic r-jets, r ≥ 2. In §4, we determine all extensions of the functor of nonholonomic r-jets. In the last paragraph we discuss an appropriate generalization of semiholonomic jets.

All manifolds and maps are assumed to be infinitely differentiable. Unless otherwise specified, we use the terminology and notation from [3].

1991 Mathematics Subject Classification. 58A20.

Key words and phrases. Jet bundle, non-holonomic jet, Weil bundle.

The author was supported by a grant of the GA ČR No. 201/96/0079.

This paper is in final form and no version of it will be submitted for publication elsewhere.

1. Jet functors on Mf_{m} × Mf. Every two manifolds M and N determine the bundle J^{r}(M, N) of all r-jets of M into N. In [3] we pointed out that this construction is to be interpreted as a functor on the product category Mf_{m} ×Mf, m = dim M. Indeed, every local diffeomorphism f : M → \overline{M} and every map g : N → \overline{N} induce a map

J^{r}(f, g) : J^{r}(M, N) → J^{r}(\overline{M}, \overline{N})

by the jet composition

(1)  J^{r}(f, g)(X) = (j_{y}^{r}g) ∘ X ∘ (j_{x}^{r}f)^{−1} ,  X ∈ J_{x}^{r}(M, N)_{y} .

In [1], C. Ehresmann introduced the bundle J̃^{r}(M, N) ⊃ J^{r}(M, N) of nonholonomic r-jets of M into N and defined the composition

(2)  Y ∘ X ∈ J̃_{x}^{r}(M, P)_{z}

for every X ∈ J̃_{x}^{r}(M, N)_{y} and Y ∈ J̃_{y}^{r}(N, P)_{z}. If X and Y are the classical holonomic r-jets, then (2) coincides with the classical composition. Hence J̃^{r} can also be interpreted as a functor on Mf_{m} × Mf, if one sets

(3)  J̃^{r}(f, g)(X) = (j_{y}^{r}g) ∘ X ∘ (j_{x}^{r}f)^{−1} ,  X ∈ J̃_{x}^{r}(M, N)_{y}

with the composition of non-holonomic jets.

There are several special classes of non-holonomic jets, see e.g. [5]. That’s why it seems to be reasonable to formulate the general concept of jet functor on Mf_{m} × Mf. The following definition is based on our results from [4].

Definition 1. A bundle functor F on Mf_{m} × Mf is called an r-th order jet functor, if J^{r} ⊂ F ⊂ J̃^{r} and F preserves products in the second factor, i.e.

(4)  F (M, N × P) = F (M, N) ×_{M} F (M, P) .

Thus we have J^{r}(M, N) ⊂ F (M, N). The inclusion F ⊂ J̃^{r} implies

(5)  F (f, g)(X) = (j_{y}^{r}g) ∘ X ∘ (j_{x}^{r}f)^{−1} ,  X ∈ F_{x}(M, N)_{y}

with the composition of non-holonomic jets.

We need to review some results from [4]. Every bundle functor F on Mf_{m}×Mf induces a bundle functor G^{F} on Mf by

G^{F} M = F_{0}(ℝ^{m}, N) , G^{F} f = F_{0}(id_{ℝ^{m}} × f) : G^{F} N → G^{F} P

for every manifold N and every smooth map f : N → P. We say that F is of order r in the first factor, if j_{x}^{r}f_{1} = j_{x}^{r}f_{2} implies that the restricted maps to the fiber of F (M, N) over x ∈ M coincide, i.e.

F_{x}(f_{1}, g) = F_{x}(f_{2}, g) : F_{x}(M, N) → F_{y}(\overline{M}, \overline{N}) , y = f_{1}(x) = f_{2}(x)

for every two local diffeomorphisms f_{1}, f_{2} : M → N and every smooth map g : N → \overline{N}. Such a functor induces a smooth group homomorphism H^{F} : G_{m}^{r} → N E(G^{F}), where G_{m}^{r} is the r-th jet group in dimension m and N E(G^{F}) is the group of all natural equivalences of the functor G^{F} into itself. Conversely, for every bundle functor G on Mf and every smooth group homomorphism H : G_{m}^{r} → N E(G) one constructs a bundle functor (G, H) on Mf_{m} × Mf by the associated bundle construction

(G, H)(M, N) = P^{r}M[GN, H] , (G, H)(f, g) = P^{r}f[Gg] ,

where P^{r}M is the r-th order frame bundle of M, [3]. In this way, we reconstruct F = (G^{F} , H^{F}). If G preserves products, then G = T^{A} is a Weil functor. Hence N E(G) is the Lie group Aut A of all algebra automorphisms of A and H : G_{m}^{r} → Aut A is a Lie group homomorphism.

In the case of J^{r}, G^{J^{r}} = T_{m}^{r} is the functor of (m, r)-velocities and H is the canonical action of G_{m}^{r} on T_{m}^{r}N by the jet composition on the right. In the case of J̃^{r}, GJ̃^{r} = T̃_{m}^{r} is the functor of non-holonomic m^{r}-velocities. The action of G_{m}^{r} on T̃_{m}^{r}N = J̃_{0}^{r}(ℝ^{m}, N) is given by the jet composition on the right.

The Weil algebra of T_{m}^{r} is 𝔻_{m}^{r} = ℝ[x_{1}, …, x_{m}]/⟨x_{1}, …, x_{m}⟩^{r+1}. The relation Aut 𝔻_{m}^{r} = G_{m}^{r} is well known, [4]. Write 𝔻̃_{m}^{r} for the Weil algebra of T̃_{m}^{r}. Since every element of G_{m}^{r} determines a natural equivalence T̃_{m}^{r} → T̃_{m}^{r} by the jet composition on the right, we have G_{m}^{r} ⊂ Aut 𝔻̃_{m}^{r}. There is a natural equivalence ε_{r} : T̃_{m}^{r} → T_{m}^{1} ∘ ⋯ ∘ T_{m}^{1} defined by the following induction. For r = 1, ε_{1} is the identity. Let t_{u} denote the translation on ℝ^{m} transforming 0 into u. If σ : ℝ^{m} → J̃^{r−1}(ℝ^{m}, N) is a section, then σ(u) ∘ j_{0}^{r−1}(t_{u}) ∈ T̃_{m}^{r−1}N. Hence

ε_{r−1}(σ(u) ∘ j_{0}^{r−1}t_{u}) ∈ \underbrace{T_{m}^{1} … T_{m}^{1}}_{(r-1)-times} N and we define ε_{r}(j_{0}^{1}σ) = j_{0}^{1}ε_{r−1}(σ(u) ∘ j_{0}^{r−1}t_{u}).

By the general theory, ε_{r} corresponds to an algebra isomorphism

(6)  𝔻̃_{m}^{r} ≈ 𝔻_{m}^{1} ⊗ … ⊗ 𝔻_{m}^{1} .

Since an r-th order jet functor F on Mf_{m} × Mf preserves products in the second factor, G^{F} is a Weil functor T^{A}. The natural injections J^{r} ⊂ F and F ⊂ J̃^{r} define two injective algebra homomorphisms i : 𝔻_{m}^{r} → A and j : A → 𝔻̃_{m}^{r}. Moreover, j defines a group homomorphism G_{m}^{r} → Aut A.

Proposition 1. The jet functors on Mf_{m} ×Mf are in bijection with quadruples (A, H, i, j), where A is a Weil algebra, H : G_{m}^{r} → Aut A is a group homomorphism and i : 𝔻_{m}^{r} → A and j : A → 𝔻̃_{m}^{r} are two G_{m}^{r}-equivariant injective algebra homomorphisms.

Proof. Both i and j are equivariant by Proposition 5 of [4]. Conversely, if we have an algebra A with a group homomorphisms H : G_{m}^{r} → Aut A and two equivariant injective algebra homomorphisms i : 𝔻_{m}^{r} → A and j : A → 𝔻̃_{m}^{r}, then F = (T^{A}, H) is a bundle functor on Mf_{m} × Mf and i or j induces a natural inclusion J^{r} ⊂ F or F ⊂ J̃^{r}, respectively. □

2. Horizontal and vertical extensions. Unless otherwise specified, M × N means the product fibered manifold M × N → M. The canonical injection i : Mf_{m} × Mf → FM_{m} is defined by i(M, N) = M × N and i(f, g) = f × g.

Definition 2. A bundle functor E on FM_{m} is called an extension of a bundle functor F on Mf_{m} × Mf, if F = E ∘ i.

From now on we assume that the bundle functor G on Mf has the so-called point property, i.e. G(pt) = pt, where pt denotes one-element set. (This is always true if G preserves products.) In [4] we clarified that the extensions of (G, H) are in bijection with the invariant diagonals in (G, H)(M, M). For a fibered manifold p : Y → M, the value of the extension D(G, H) determined by an invariant diagonal D_{M} is given by

D(G, H)(Y) = {X ∈ (G, H)(M, Y); (G, H)(id_{M}, p)(X) ∈ D_{M}} .

The effect of D(G, H) on an FM_{m}-morphism f is defined by the canonical restriction and corestriction of (G, H)(\underline{f}, f), where \underline{f} is the base map of f, [4].

The construction of r-th jet prolongations of fibered manifolds and FM_{m}-mor-phisms is a bundle functor on FM_{m}, which is usually also denoted by J^{r}. In this theoretical research, we shall use the symbol J_{h}^{r} for the latter functor on FM_{m} to distinguish it from the former functor J^{r} on Mf_{m} × Mf.

Definition 3. For every r-th order jet functor F on Mf_{m} × Mf, we define its horizontal extension F_{h} by

F_{h}Y = {X ∈ F (M, Y); F (id_{M}, p)(X) = j_{x}^{r}id_{M}}

and its vertical extension F_{v} by

F_{v}Y = {X ∈ F (M, Y); F (id_{M}, p)(X) = j_{x}^{r}x̂} ,

where x̂ denotes the constant map of M into x ∈ M. The effect of both F_{h} and F_{v} on FM_{m}-morphisms is defined by the canonical restriction and corestriction.

By a standard property of non-holonomic jets, F_{v}Y can also be expressed in the form

F_{v}Y = ⋃_{x∈M} F_{x}(M, Y_{x}) .

3. Extensions of J^{r}. We deduce a complete description of all extensions of the holonomic r-th jet functor J^{r}.

Proposition 2. For r ≥ 2, J_{h}^{r} and J_{v}^{r} are the only two extensions of J^{r}.

Proof. By invariancy, it is sufficient to find all invariant elements in L_{m,m}^{r} = J_{0}^{r}(ℝ^{m}, ℝ^{m})_{0}. So we are looking for all Y ∈ L_{m,m}^{r} satisfying

Y = X^{−1} ∘ Y ∘ X for all X ∈ G_{m}^{r} .

Let Y = (y_{j}^{i}, y_{j_{1}j_{2}}^{i}, …, y_{j_{1}...j_{r}}^{i}). There is a canonical injection of GL(m, ℝ) = G_{m}^{1} into G_{m}^{r}. The action of GL(m, ℝ) on each component of Y is the classical tensorial one, so that each y_{j_{1}...j_{k}}^{i} is an invariant tensor. By the invariant tensor theorem, [3], p. 214, y_{j_{1}...j_{k}}^{i} = 0 for k ≥ 2 and

(7)  y_{j}^{i} = cδ_{j}^{i} ,  c ∈ ℝ .

Consider the underlying 2-jet Y_{2} = (cδ_{j}^{i}, 0). An element X of the kernel of the canonical projection G_{m}^{2} → G_{m}^{1} is of the form X = (δ_{j}^{i}, b_{jk}^{i}). Then X^{−1} = (δ_{j}^{i}, −b_{jk}^{i}) and the jet composition yields

X^{−1} ∘ Y_{2} ∘ X = (cδ_{j}^{i}, −c^{2}b_{jk}^{i} + cb_{jk}^{i}) .

The second term implies

(8)  c = c^{2} ,  i.e. c = 0 or c = 1 .

One verifies directly that c = 0 corresponds to J_{v}^{r} and c = 1 corresponds to J_{h}^{r}. □

For r = 1, we have to stop at (7). Geometrically, J^{1}(M, N) = T^{∗}M ⊗ T N is a vector bundle over M × N. In particular, we can define an invariant diagonal

D_{c}^{1}M = {cj_{x}^{1}id_{M}, x ∈ M} , 0 ≠ c ∈ ℝ ,

and the corresponding extension

J_{h,c}^{1}Y = {X ∈ J^{1}(M, Y) , J^{1}(id_{M}, p)(X) ∈ D_{c}^{1}M} , c ≠ 0 .

(The case c = 0 corresponds to J_{v}^{1}.) Thus, we have proved

Proposition 3. All extensions of J^{1} are J_{v}^{1} and J_{h,c}^{1}, 0 ≠ c ∈ ℝ.

4. Extensions of J̃^{r}. Write L̃_{m,m}^{r} = J̃_{0}^{r}(ℝ^{m}, ℝ^{m})_{0}. The canonical coordinates of an element Y ∈ L̃_{m,m}^{r} are

y_{k_{1}...k_{r}}^{i} , k_{1}, …, k_{r} = 0, 1, …, m

with at least one non-zero subscript, [2], [6]. Each coordinate with s non-zero subscripts is called an s-th order coordinate, 1 ≤ s ≤ r. By [6] or [2], we have r geometric projections p_{s} : J̃^{r}(M, N) → J^{1}(M, N), whose restrictions to L̃_{m,m}^{r} are

p_{s}(Y) = (y_{0...j_{s}...0}^{i}) , j_{s} = 1, …, m .

Moreover, there are \binom{r}{2} geometric projections q_{st} : J̃^{r}(M, N) → J̃^{2}(M, N), s < t, whose restrictions to L̃_{m,m}^{r} are

q_{st}(Y) = (y_{0...j_{s}...0}^{i}, y_{0...j_{t}...0}^{i}, y_{0...j_{s}...j_{t}...0}^{i}) .

Lemma 1. The invariant elements of L̃_{m,m}^{r} are of the form

(9)  y_{j...0}^{i} = c_{1}δ_{j}^{i}, …, y_{0...j}^{i} = c_{r}δ_{j}^{i} ,  c_{1}, …, c_{r} ∈ ℝ

satisfying

(10)  c_{s}c_{t} = c_{s} ,  s < t ,

and with all zero coordinates in every order r ≥ 2.

Proof. The invariancy condition for Y ∈ L̃_{m,m}^{r} reads

Y = X^{−1} ∘ Y ∘ X for all X ∈ G_{m}^{r} .

Analogously to the proof of Proposition 2, if we consider the image of GL(m, ℝ) into G_{m}^{r}, then all coordinates of the order ≥ 2 vanish by the invariant tensor theorem and (9) holds. If we consider the invariancy of the underlying non-holonomic 2-jet q_{st}(Y), we find the condition −c_{s}c_{t}b_{jk}^{i} + c_{s}b_{jk}^{i} = 0, s < t. This is equivalent to (10). □

For r = 2, Lemma 1 yields the following assertion (in which J̃_{h}^{2} has the form J_{h}^{1} ∘ J_{h,1}^{1}).

Proposition 4. All extensions of J̃^{2} are

(11)  J̃_{v}^{2} , J_{h,c}^{1} ∘ J_{v}^{1}, J_{h}^{1} ∘ J_{h,c}^{1} ,  0 ≠ c ∈ ℝ .

Proof. The condition c_{1}c_{2} = c_{1} is fulfiled for c_{1} ≠ 0, c_{2} = 1 or c_{1} = 0, c_{2} arbitrary. The type of the extension is determined by the diagonal in question. In this way we find that for c_{1} = 0, c_{2} = 0 we obtain J̃_{v}^{2}. The case c_{1} = 0, c_{2} ≠ 0 corresponds to J_{h,c_{2}}^{1} ∘ J_{v}^{1}. For c_{1} ≠ 0, c_{2} = 1 we obtain J_{h}^{1} ∘ J_{h,c_{1}}^{1}. □

For arbitrary r, we deduce a recurrence formula.

Proposition 5. All extensions of J̃^{r} are

(12)  I ∘ J_{v}^{1} , J̃_{h}^{r−1} ∘ J_{h,c}^{1} ,  0 ≠ c ∈ ℝ ,

where I is any extension of J̃^{r−1}.

Proof. We have to discuss all conditions (10). If c_{1} ≠ 0, then c_{2} = ⋯ = c_{r} = 1. Such a diagonal corresponds to J̃_{h}^{r−1} ∘ J_{h,c}^{1}. If c_{1} = 0, then c_{2}, …, c_{r} satisfy the conditions from the order r − 1. This yields I ∘ J_{v}^{1}. □

5. One-semiholonomic jets. We introduce an appropriate generalization of the concept of semiholonomic r-jet, [1].

Definition 4. An r-th order jet functor F on Mf_{m} × Mf, r ≥ 2, is called one-semiholonomic, if

p_{1}(X) = ⋯ = p_{r}(X) ∈ J^{1}(M, N)

for every X ∈ F (M, N).

In particular, the classical functor \overline{J}^{r} of semiholonomic r-jets is one-semiho-lonomic. In the second order, p_{1}(X) = p_{2}(X) is the standard semiholonomicity condition.

Proposition 6. The only two extensions of a one-semiholonomic jet functor F are F_{h} and F_{v}.

Proof. By one-semiholonomicity, we have c_{1} = ⋯ = c_{r} in (10). Hence 0 and 1 are the only two possibilities. □

References

[1] Ehresmann, C., Extension du calcul des jets aux jets non holonomes, CRAS Paris 239(1954), 1762–1764.

[2] Kolář, I., Some higher order operations with connections, Czechoslovak Math. J., 24(1977), 311–330.

[3] Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer Verlag, 1993.

[4] Kolář, I., Mikulski, W. M., On the fiber product preserving bundle functors, to appear in Differential Geometry and Its Applications.

[5] Libermann, P., Introduction to the theory of semi-holonomic jets, Archivum Math.(Brno), 33(1997), 173–189.

[6] Virsik, G., Total connections in Lie groupoids, Archivum Math.(Brno), 31(1995), 183–200.

Department of Algebra and Geometry, Masaryk University Janáčkovo nám. 2a, 662 95 Brno, CZECH REPUBLIC E-mail: kolar@math.muni.cz