DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 215–229

BUNCH CONNECTIONS

George Virsik

Abstract. A path connection in a principal bundle P (M, G) is a lifting of paths in M onto paths in P. It is a well known fact that such liftings which are independent of reparametrisation, equivariant and infinitesimal of order one are in a one-to-one correspondence with ordinary connections in P. Even if r-th order jets are used, a path connection can be reasonably associated only with an r-th order connection which is the Ehresmann prolongation of a single first order connection. In order to make a similar relation hold for arbitrary (non-holonomic) r-th order connections, one needs to lift r-bunches rather than paths. Here an r-bunch in M is a map I^{r} → M, not necessarily continuous, with a certain semi-smoothness condition. In particular, the bunch connection associated with an r-th order connection that is the product of r first order connections is investigated.

1. Path connections in principal bundles. All manifolds are assumed smooth and finite dimensional. If M is such a manifold denote by Λ(M) the set of piecewise smooth paths in M, i.e. maps λ : I → M, where I = [0, 1]. If P (M, G) is a principal bundle with projection p : P → M and structure group G write Λ(M) ×_{M} P = {(λ, h) : λ ∈ Λ(M), h ∈ P, ph = λ(0)}.

A path connection in P (M, G) ([4], [5]) is a map

(1.1)  Σ : Λ(M) ×_{M} P → Λ(P)
(λ, h) ↦ [t ↦ Σ_{λ}(h; t)]

satisfying

(1.2)  pΣ_{λ}(h; t) = λ(t), t ∈ I  (lifting)
(1.3)  Σ_{λ}(h; 0) = h  (initial condition)

(1.4) if ψ : [0, 1] → [a, b] ⊆ [0, 1] is a piecewise smooth diffeomorphism then Σ_{λ}(h; ψ(s)) =

Σ_{λ∘ψ}(Σ_{λ}(h; ψ(0)); s), s ∈ I  (reparametrisation)
(1.5)  Σ_{λ}(hg; t) = Σ_{λ}(h; t)g, g ∈ G, t ∈ I  (equivariance)

1991 Mathematics Subject Classification. 53 C 05, 58 A 20. Key words and phrases. Path connection, semi-smooth, r-bunch, bunch connection, bloop. This paper is in final form and no version of it will be submitted for publication elsewhere.

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If h ∈ P (M, G) is fixed and Ω(M) ⊆ Λ(M) denotes the set of loops at x = ph, then the map H_{h} : Ω(M) → G defined by Σ_{λ}(h; 1) = hH_{h}(λ) is known as the holonomy map of the path connection Σ at h.

The condition (1.5) guarantees that a path connection defines an equivariant one-to one correspondence between the fibres P_{λ(0)} and P_{λ(t)} for any t ∈ I. Thus the lift (1.1) of λ ∈ Λ(M) is also uniquely determined by some a ∈ I and h_{a} ∈ P, where h_{a} = Σ_{λ}(h; a). We shall write Σ_{λ}(h_{a}; .) as well for such a lift. Thus Σ_{λ}(h_{a}; s) = Σ_{λ}(h_{0}; s), where h_{0} ∈ P_{λ(0)} is given by the condition Σ_{λ}(h_{0}; a) = h_{a}. Therefore (1.3) and (1.4) can be generalised to Σ_{λ}(h_{a}; a) = h_{a}, where ph_{a} = λ(a), and to Σ_{λ}(h; ψ(s)) = Σ_{λ∘ψ}(h; s), s ∈ I, respectively.

An infinitesimal non-holonomic r-th order connection in P (M, G) is a bundle map Γ : P → J̃_{r}P over M which satisfies π_{0}^{r} ∘ Γ = id_{P} and Γ(hg) = Γ(h) · j_{0}^{r}(t ↦ g) for any g ∈ G. Here J̃_{r}P is the bundle of r-th order non-holonomic jets of sections of P (M, G) → M and · denotes the jet prolongation of the action of G on P (c.f. [1], [2] and [3]).

We shal say that a path connection (1.1) is infinitesimal of order r ≧ 1 if there exists a smooth map Γ : P → J̃_{r}P such that

(1.6)  j_{0}^{r}Σ_{λ}(h; .) = Γ(h) ∘ j_{0}^{r}λ for any λ ∈ Λ(M) and any h ∈ P_{λ(0)}.

Theorem 1.1. The relation (1.6) with r = 1 defines a one-to-one correspondence between first order connections in P and path connections in P that are infinitesimal of order r = 1.

This is a well known result. For a general r we first observe that if a path connection satisfies (1.6) then it satisfies also

j_{t}^{r}Σ_{λ}(h; .) = Γ(h(t)) ∘ j_{t}^{r}λ, (1.7)

where h(t) = Σ_{λ}(h; t), for any h ∈ P_{λ(0)}, , t ∈ I.

In fact, fix t ∈ I and apply (1.4) to ψ_{t} : [0, 1] → [t, 1], defined as ψ_{t}(s) = t+(1−t)s, to obtain Σ_{λ}(h; ψ_{t}(s)) = Σ_{λ∘ψ_{t}}(Σ_{λ}(h; t); s). The r-jet of the left hand side of this gives j_{0}^{r}(Σ_{λ}(h; .) ∘ ψ_{t}) = j_{t}^{r}(Σ_{λ}(h; .) ∘ j_{0}^{r}(ψ_{t}), whereas the right hand side gives by (1.6) j_{0}^{r}(s ↦ Σ_{λ∘ψ_{t}}(Σ_{λ}(h; t); s) = Γ(Σ_{λ}(h; t)) ∘ j_{0}^{r}(λ ∘ ψ_{t}) = Γ(Σ_{λ}(h; t)) ∘ j_{t}^{r}(λ) ∘ j_{0}^{r}(ψ_{t}), whence we get (1.7) since j_{0}^{r}(ψ_{t}) is invertible.

Theorem 1.2. If (1.1) is a path connection which satisfies (1.6) for some r ≧ 1 and some r-th order connection Γ : P → J̃_{r}P then it also satisfies

j_{0}^{r+1}Σ_{λ}(h; .) = (Γ ⋆ ξ)(h) ∘ j_{0}^{r+1}λ for any λ ∈ Λ(M) and any h ∈ P_{λ(0)},

where ξ = π_{1}^{r} ∘ Γ : P → J_{1}P and Γ ⋆ ξ = J(Γ) ∘ ξ : P → J̃_{r+1}P is the Ehresmann prolongation of Γ (c.f. [3] and [6] for notation).

Proof. We have (Γ⋆ξ)(h)∘j_{0}^{r+1}λ = J(Γ){ξ(h)}∘j_{0}^{r+1}λ, where ξ : P → J_{1}P satisfies j_{0}^{1}Σ_{λ}(h; .) = ξ(h) ∘ j_{0}^{1}λ. If ξ(h) = j_{0}^{1}η, x = λ(0) ∈ M, η : M ⇝_{x} P, then this says that

(1.8)  j_{0}^{1}Σ_{λ}(h; .) = j_{0}^{1}(η ∘ λ),

and (Γ ⋆ ξ)(h) ∘ j_{0}^{r+1}λ can be written asj_{x}^{1}(u ↦ Γ(η(u))) ∘ j_{0}^{1}(t ↦ j_{t}^{r}λ) = j_{0}^{1}(t ↦ Γ(η(t)))∘j_{t}^{r}λ. This last expression depends only on first order derivatives of η∘λ and so by (1.8) it equals j_{0}^{1}(t ↦ Γ(Σ_{λ}(h; t))∘j_{t}^{r}λ) which by (1.7) is j_{0}^{1}(t ↦ j_{t}^{r}Σ_{λ}(h; .)) = j_{0}^{r+1}Σ_{λ}(h; ) as required.

As an immediate corollary we have a result that says that, as far as path connections are concerned, only infinitesimality of firts order is involved.

Theorem 1.3. If (1.1) is a path connection which satisfies (1.6) for some r ≧ 1 and some r-th order connection Γ : P → J̃_{r}P then it also satisfies

j_{0}^{r}Σ_{λ}(h; .) = (ξ ⋆ ξ ⋆ … ⋆ ξ)(h) ∘ j_{0}^{r}λ for any λ ∈ Λ(M) and any h ∈ P_{λ(0)},

where ξ = π_{1}^{r} ∘ Γ : P → J_{1}P.

In order to have a formula like (1.6) for general non-holonomic connections, ie. which are not only powers of first order connections, we need to replace holonomic jets of paths by non-holonomic jets of what we shall call bunches. The generalisation of (1.6) will be given by (3.12) below.

2. Non-holonomic jets. Let M and N be smooth manifolds. A typical element of J̃^{r}(M, N)_{x} can be written as

(2.1) j̃_{x}^{r}f =

j_{x}^{1}(u_{1} ↦ j_{u_{1}}^{1}(u_{2} ↦ j_{u_{2}}^{1}(u_{3} ↦ … ↦ j_{u_{r−1}}^{1}(u_{r} ↦ f(u_{1}, u_{2}, …, u_{r})) …),

or briefly, as j̃_{x}^{r}f = j̃_{x}^{r}(↦ f(u_{1}, u_{2}, …, u_{r})), where f : M × … × M ⇝ N is defined in a neighbourhood of (x, …, x) and has the appropriate derivatives at (x, …, x). Note that f need not be even continuous at (x, …, x).

In particular, j̃_{x}^{r}f ∈ J̃^{r}(ℝ, ℝ)_{0} is uniquely determined by the 2^{r} derivatives

(2.2) λ_{ι_{1},... ,ι_{r}} = (D_{1}^{ι_{1}} ∘ δ_{1} ∘ D_{2}^{ι_{2}} ∘ δ_{2} ∘ … ∘ D_{r}^{ι_{r}})λ(0, …, 0), ι_{1}, …, ι_{r} = 0 or 1,

where D_{i}^{ι} is the identity operator if ι = 0 and the partial derivative D_{i} if ι = 1. The operator

δ_{s} : Map(ℝ^{s+1}, ℝ) → Map(ℝ^{s}, ℝ), s = 1, …, r − 1,

assigns to the function g the function δ_{s}[g] : (u_{1}, …, u_{s}) ↦ g(u_{1}, …, u_{s}, u_{s}). In particular, (δ_{1} ∘ … ∘ δ_{r})(λ) = λ ∘ Δ : u ↦ (u, …, u).

Recall that Z = j̃_{x}^{r}f ∈ J̃^{r}(M, N)_{x} is semi-holonomic iff (c.f. (9) in [6] including notation)

π_{q}^{r}(Z) = J(π_{q−1}^{k−1})π_{k}^{r}(Z) ∈ J̃^{q}(M, N)_{x} whenever 1 ≦ q < k ≦ r,

ie. iff

(2.3) j̃_{x}^{q}(↦ f(x, …, x, u_{r−q+1}, u_{r−q+2}, …, u_{r})) =

j̃_{x}^{q}(↦ f(x, …, x, u_{r−k+1}, u_{r−k+1}, …, , u_{r−k+1}, u_{r−q+2}, …, u_{r}))
whenever 1 ≦ q < k ≦ r.

Here on the left hand side x is repeated (r − q) -times, on the right hand side x is repeated (r − k)-times and u_{r−k+1} is repeated (k − q + 1)-times. Also, the left hand side is a special case of the right hand side with k = q. In coordinates, j̃_{0}^{r}λ ∈ J̃^{r}(ℝ, ℝ)_{0} is semi-holonomic iff the coordinates (2.2) depend only on the number of zeros and ones among ι_{1}, …, ι_{r} but not on their position.

In particular, (2.3) gives for q = 1 the r maps

(2.4) J(π_{0}^{k−1})π_{k}^{r} : J̃^{r}(M, N)_{x} → J^{1}(M, N)_{x}

j̃_{x}^{r}f ↦ j_{x}^{1}(u ↦ f(x, …, x, u, …, u)) for k = 1, … r,

where x is repeated (r − k)-times and u_{r−k+1} = u is repeated k-times. The jet j̃_{x}^{r}f is holonomic if f can be chosen so that f(u_{1}, …, u_{r}) depends only on u_{r}. An inspection of (2.3) reconfirms the fact that holonomic jets are necessarily semiholonomic. In particular, if M = ℝ then there is no distinction between holonomic and semi-holonomic jets.

Recall also the rule for the composition of non-holonomic jets (c.f. [2]) defined recurrently as follows. If Z = j_{x}^{1}ρ ∈ J̃^{r}(M, N)_{x} and W = j_{y}^{1}σ ∈ J̃^{r}(N, Q)_{x}, where ρ : M ⇝ J̃^{r−1}(M, N) and σ : N ⇝ J̃^{r−1}(N, Q) are local sections in a neighbourhood of x ∈ M and y = π_{0}^{r}Z ∈ N respectively, then their composition W ∘ Z is given by (c.f.[1])

(2.5)  W ∘ Z = j_{x}^{1}(u ↦ σ(π_{0}^{r−1}ρ(u)) ∘ ρ(u)).

Finally, let us quote without proof the following result, easily verified in coordinates.

Lemma 2.1. If Z_{1}, Z_{2} ∈ J̃^{r}(M, N)_{x} are two non-holonomic jets such that Z_{1} ∘ j̃_{0}^{r}λ = Z_{2} ∘ j̃_{0}^{r}λ for any j̃_{0}^{r}λ ∈ J̃^{r}(ℝ, M)_{0}^{x} then Z_{1} = Z_{2}.

Note that the non-holonomity of j̃_{0}^{r}λ ∈ J̃^{r}(ℝ, M)_{0}^{x} is essential, ie. that Z_{1} ∘j_{0}^{r}λ = Z_{2} ∘ j_{0}^{r}λ for all λ : ℝ ⇝_{0} M, λ(0) = x, does not imply Z_{1} = Z_{2} (unless Z_{1} and Z_{2}

are also holonomic).

3. Bunch connections. If M is a smooth manifold, r ≧ 1 an integer then a (not necessarily continuous) map λ : I^{r} → M is called semi-smooth if the map

(3.1a)  λ ∘ Δ : I → M,  t ↦ λ(t, …, t)

as well as the maps

(3.1b)  λ(s_{1}, …, s_{i−1}, .) : I → M,  t ↦ λ(s_{1}, …, s_{i−1}, t, …, t))

are smooth for any s_{1}, …, s_{i−1} ∈ I^{i−1}, i = 2, …, r. The maps (3.1a) and (3.1b) are called the internal paths of λ.

A semi-smooth map λ : I^{r} → M is called an r-bunch if also the map

(3.2) j̃^{r}λ : I → J̃^{r}(ℝ, M)

t ↦ j̃_{t}^{r}λ = j_{t}^{1}(s_{1} ↦ j_{s_{1}}^{1}(s_{2} ↦ j_{s_{2}}^{1}(s_{3} ↦ … ↦ j_{s_{r−1}}^{1}(s_{r} ↦ λ(s_{1}, s_{2}, …, s_{r})) …) is smooth. Since λ ∘ Δ = π_{0}^{r}j̃^{r}λ this implies the smoothness of (3.1a) but not of (3.1b). Note that any r-jet Z ∈ J̃^{r}(ℝ, M)_{0} can be written as Z = j̃_{0}^{r}λ = j̃_{0}^{r}(↦ λ(s_{1}, s_{2}, …, s_{r})), where λ : I^{r} → M is an r-bunch.

An r-bunch need not be continuous. For instance, the function λ : I^{2} → ℝ, given by λ(x, y) = χ(x)(y − x)^{2}, where χ is the Dirichlet function, defines a map which is a two-bunch – ie. for any s_{1} ∈ I the functions t ↦ λ(t, t), t ↦ λ(s_{1}, t) as well as the function t ↦ \frac{d}{dt}D_{2}λ(t, t) are smooth – but λ : I^{2} → ℝ is obviously not continuous. Neither does every semi-smooth λ : I^{r} → M have to be a bunch ie. make also (3.2) smooth. In fact, if λ(x, y) = χ(x)(y−x) then λ is clearly again semismooth, however D_{2}λ(x, y) = χ(x), so t ↦ \frac{d}{dt}D_{2}λ(t, t) is not smooth. Of course, for r = 1, a smooth path, a semi-smooth path and a one-bunch are all the one and the same thing. We shall say that an r-bunch λ : I^{r−1} × I → M is stationary on the boundary if λ(u̇, t) is independent of t whenever u̇ ∈ İ^{r−1}. Note that λ is stationary on the boundary iff the internal paths λ(s_{1}, …, s_{r−1}, .) : I → M are constant for any (s_{1}, …, s_{r−1}) ∈ İ^{r−1}. If the preceding example is modified to λ(x, y) = (χ(x) − 1)(y − x)^{2} we obtain a two-bunch which is stationary on the boundary, ie. λ(0, y) and λ(1, y) are independent of y.

We shall say that an r-bunch λ : I^{r} → M is piecewise smooth if λ∘θ : I → M is a piecewise smooth path for any piecewise smooth θ : I → I^{r}. If λ : I^{r} → M is semismooth then for any (s_{1}, …, s_{r}) ∈ I^{r} there is a unique path starting at λ(0, …, 0), ending at λ(s_{1}, …, s_{r}), and consisting of a sequence of smooth internal paths of λ. This path is

(3.3)  λ⟨s_{1}, …, s_{r}⟩ = λ[s_{1}, …, s_{r}] ∗ … ∗ λ[s_{1}, s_{2}] ∗ λ[s_{1}] : I → M,

where (with a suitable reparametrisation to I → M) these paths are defined as (c.f. (3.1a), (3.1b))

(3.4)

λ[s_{1}] is the path λ ∘ Δ : t ↦ λ(t …, t), for 0 ≤ t ≤ s_{1}
λ[s_{1}, s_{2}] is the path λ(s_{1}, .) : t ↦ λ(s_{1}, t …, t),
for t from s_{1} to s_{2}
…
λ[s_{1}, …, s_{i}] is the path λ(s_{1}, …, s_{i−1}, .) : t ↦ λ(s_{1}, …, s_{i−1}, t …, t),
for t from s_{i−1} to s_{i}
…

λ[s_{1}, …, s_{r}] is the path λ(s_{1}, …, s_{r−1}, .) : t ↦ λ(s_{1}, …, s_{r−1}, t …, t),

for t from s_{r−1} to s_{r}.

Let Λ^{r}(M) denote the set of r-bunches λ : I^{r} → M. If P (M, G) is a principal bundle write again Λ^{r}(M) ×_{M} P = {(λ, h) : λ ∈ Λ^{r}(M), h ∈ P_{λ(0,... ,0)}}.

An r-th order bunch connection in P (M, G) is a map

(3.5)  Σ : Λ^{r}(M) ×_{M} P → Λ^{r}(P)
(λ, h) ↦ [(s_{1}, …, s_{r}) ↦ Σ_{λ}(h; s_{1}, …, s_{r})]

satisfying

pΣ_{λ}(h; s_{1}, …, s_{r}) = λ(s_{1}, …, s_{r}) (3.6)

for all (s_{1}, …, s_{r}) ∈ I^{r}  (lifting)
(3.7) Σ_{λ}(h; 0, …, 0) = h  (initial condition)

Σ_{λ}(hg; s_{1}, …, s_{r}) = Σ_{λ}(h; s_{1}, …, s_{r})g (3.8)

for all (s_{1} …, s_{r}) ∈ I^{r}, g ∈ G  (equivariance)

(3.9) If one of the internal paths (3.1a) or (3.1b)

is constant then so is the corresponding

internal path of Σ_{λ}(h; .) : I^{r} → P.  (stationary property)

Note that (3.9) implies that a bunch connection lifts bunches stationary on the boundary to bunches stationary on the boundary. Also, since every λ(s_{1}, …, s_{r}) ∈ I^{r} can be reached from λ(0, …, 0) by a succession of internal paths, namely by λ⟨s_{1}, …, s_{r}⟩, it follows from (3.9) that if λ =const then so is Σ_{λ}(h; .), ie. Σ_{λ}(h; .) = h.

As in the case of path connections, the condition (3.8) guarantees that a bunch connection defines an equivariant one-to one correspondence between the fibres P_{λ(0,... ,0)} and P_{λ(s_{1},... ,s_{r})} for any (s_{1}, …, s_{r}) ∈ I^{r}. Thus the lift (3.5) is uniquely determined by λ ∈ Λ^{r}(M) and some h_{a} ∈ P, where h_{a} = Σ_{λ}(h_{0}; a_{1}, …, a_{r}). We shall write again Σ_{λ}(h_{a}; .) for such a bunch in P as well. Thus Σ_{λ}(h_{a}; s_{1}, …, s_{r}) = Σ_{λ}(h_{0}; s_{1}, …, s_{r}), where h_{0} ∈ P_{λ(0,... ,0)} is given by Σ_{λ}(h_{0}; a_{1}, …, a_{r}) = h_{a}. In this sense (3.7) can be generalised to

Σ_{λ}(h_{a}; a_{1}, …, a_{r}) = h_{a}, where ph_{a} = λ(a_{1}, …, a_{r}).

The bunch connection (3.5) is called piecewise smooth (or continuous) if it lifts a piecewise smooth (or continuous) r-bunch λ into a piecewise smooth (or continuous) r-bunch Σ_{λ}(h; .) respectively, for any h ∈ P_{λ(0,... ,0)}. We shall see later that for r > 1 a bunch connections is usually not continuous. A bunch connection (3.5) of order r ≥ 1 is infinitesimal if there exists a smooth map Γ : P → J̃_{r}P such that

j̃_{t}^{r}Σ_{λ}(h; .) = Γ(Σ_{λ}(h; Δ(t))) ∘ j̃_{t}^{r}λ (3.11)

for any (λ, h) ∈ Λ^{r}(M) ×_{M} P, t ∈ I,

in particular (c.f. (1.6)),

(3.12)  j̃_{0}^{r}Σ_{λ}(h; .) = Γ(h) ∘ j̃_{0}^{r}λ for any (λ, h) ∈ Λ^{r}(M) ×_{M} P.

Since any j̃_{0}^{r}λ ∈ J̃^{r}(ℝ, M)_{0}^{x} can be expressed by means of an r-bunch, it follows from Lemma 2.1 that if a Γ exists which satisfies (3.11) then it is unique. It follows also easily from (3.7) and (3.12) that h = π_{0}^{r}Γ(h).

On the other hand, the left hand side of j̃_{0}^{r}Σ_{λ}(hg; .) = Γ(hg)∘j̃_{0}^{r}λ equals by (3.8)

j̃_{0}^{r}(↦ Σ_{λ}(hg; s_{1}, …, s_{r})) = (3.13)

j̃_{0}^{r}(↦ Σ_{λ}(h; s_{1}, …, s_{r})) · j̃_{0}^{r}(↦ g),

where j̃_{0}^{r}(↦ g) = j_{0}^{r}[g] is the r-jet of the constant g. Thus j̃_{0}^{r}Σ_{λ}(h; .) · j_{0}^{r}[g] = Γ(hg)∘j̃_{0}^{r}λ, and substitution from (3.12) gives (Γ(h)∘j̃_{0}^{r}λ)·j_{0}^{r}[g] = Γ(hg)∘j̃_{0}^{r}λ. Since j_{0}^{r}[g] = j_{x}^{r}[g] · j̃_{0}^{r}λ, where x = ph = λ(0, …, 0), we have derived that Γ(hg) ∘ j̃_{0}^{r}λ = {Γ(h)·j_{x}^{r}[g]}∘j̃_{0}^{r}λ for any j̃_{0}^{r}λ ∈ J̃^{r}(ℝ, M)_{0}. Again in view of Lemma 2.1 this implies

(3.14)  Γ(hg) = Γ(h) · j_{x}^{r}[g]

Thus if a bunch connection of order r admits a Γ : P → J̃_{r}P satisfying (3.12) for each (λ, h) ∈ Λ^{r}(M) ×_{M} P then this Γ is unique and it is necessarily an r-th order connection in P (M, G). On the other hand, an r-th order connection Γ will not necessarily determine a bunch connection satisfying (3.13). And if, this bunch connection is not unique, since t ↦ j̃_{t}^{r}λ determines the bunch λ : I^{r} → M only in a neighbourhood of the diagonal.

Note that a one-bunch is always piecewise smooth and it is the same thing as a smooth path. A path connection is the same thing as a first order piecewise smooth bunch connection, which is also parameter independent since in this case the infinitesimality property implies parameter independence.

The fact that a holonomic r-jet j_{x}^{r}f is a special case of (2.1), where f(u_{1}, …, u_{r}) = f(u_{r}), leads us to call an r-bunch λ : I^{r} → M holonomic if it is given by a smooth path I → M, ie. if λ(s_{1}, …, s_{r}) = λ(u_{r}). Clearly, any holonomic r-bunch is piecewise smooth. The infinitesimality condition (3.12) now has the same right hand side as (1.6) so we get from Theorem 1.3 directly

Theorem 3.1. An infinitesimal bunch connection (3.5) lifts holonomic bunches to holonomic bunches iff it is associated with the Ehresmann prolongation ξ ⋆ … ⋆ ξ of a first order connectionξ in P (M, G), ie. iff in (3.12) one can replace Γ(h) by (ξ ⋆ … ⋆ ξ)(h), ξ = π_{1}^{r} ∘ Γ, for any holonomic λ ∈ Λ^{r}(M) – in fact, λ ∈ Λ(M) – and h ∈ p^{−1}(λ(0)).

Let ξ_{k} = J(π_{0}^{k−1})π_{k}^{r} ∘ Γ : P → J_{1}P, k = 1, …, r, be the r first order connections determined by Γ. Applying J(π_{0}^{k−1})π_{k}^{r} to (3.11) and taking into account (2.4) we obtain

(3.15)  j_{t}^{1}(s ↦ Σ_{λ}(h; t, …, t, s, …, s)) = ξ_{k}(h_{t}) ∘ j_{t}^{1}(λ(t, …, t, s, …, s))

where t is repeated (r − k)-times, s is repeated k-times and h_{t} = Σ_{λ}(h; Δ(t)). Thus the one jet at t ∈ I of the internal path of Σ_{λ}(h; t, …, t, .) is the lift of the one jet at t ∈ I of the internal path λ(t, …, t, .) via the connection ξ_{k} : P → J_{1}P as in the case of infinitesimal path connections.

This suggests a bunch connection associated with an r-tuple ξ_{1}, ξ_{2}, …, ξ_{r} of first order connections in P (M, G) defined by means of lifts along internal paths. This bunch connection will be related to Γ = ξ_{1} ⋆ ξ_{2} ⋆ … ⋆ ξ_{r} in the sense of (3.11). With a slight abuse of language we shall refer to this bunch connection – to be defined explicitly in the next Section – as the bunch connection associated with ξ_{1} ⋆ ξ_{2} ⋆ … ⋆ ξ_{r}, though it does not follow from (3.11) that it is the only one satisfying the infinitesimality condition (3.11) with Γ = ξ_{1} ⋆ ξ_{2} ⋆ … ⋆ ξ_{r}.

4. The bunch connection associated with the connection ξ_{1}⋆ξ_{2}⋆…⋆ξ_{r}. Let now Γ = ξ_{1} ⋆ ξ_{2} ⋆ … ⋆ ξ_{r} : P → J̃_{r}P, where ξ_{1}, ξ_{2}, …, ξ_{r} are first order connections. Recall that ξ_{1} ⋆ ξ_{2} ⋆ … ⋆ ξ_{r} = J(ξ_{1} ⋆ … ⋆ ξ_{r−1}) ∘ ξ_{r}. We have also ξ_{k} = J(π_{0}^{k−1})π_{k}^{r}Γ : P → J_{1}P, for k = 1, …, r.

Denote by Σ^{k} : Λ(M) ×_{M} P → Λ(P) the path connection corresponding to ξ_{k} for k = 1, …, r. Thus by (1.7)

(4.1) j_{t}^{1}Σ_{λ}^{k}(h; .) = ξ_{k}(Σ_{λ}^{k}(h; t)) ∘ j_{t}^{1}λ

for any h ∈ P_{λ(0)}, t ∈ I; in particular, j_{0}^{1}Σ_{λ}^{k}(h; .) = ξ_{k}(h) ∘ j_{0}^{1}λ.

Recall also that Σ_{λ}^{k}(h; .) is well defined for any h such that ph = λ(s) even if s ≠ 0. Define the bunch connection Σ^{12...r} corresponding to ξ_{1} ⋆ ξ_{2} ⋆ … ⋆ ξ_{r} recurrently by

Σ_{λ}^{12}(h; s_{1}, s_{2}) := Σ_{λ(s_{1},.)}^{1}(Σ_{λ∘Δ}^{2}(h; s_{1}); s_{2}) and (4.2)

Σ_{λ}^{12...r}(h; s_{1} …, s_{r}) :=
Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(h; s_{1}); s_{2}, …, s_{r}),

where Δ : I → I^{r} is the diagonal map s ↦ (s, …, s).

We need to verify that (4.2) take semi-smooth maps into semi-smooth maps, and that (3.6), (3.7), (3.8) and (3.9), as well as (3.11), are satisfied.

Let us start with (3.9), which is certainly true for r = 1. Assume it holds for ξ_{1} ⋆ … ⋆ ξ_{r−1}, and let first λ ∘ Δ be constant. Then Σ_{λ}^{12...r}(h; s_{1} …, s_{1}) = Σ_{λ(s_{1},.)}^{12...r−1}(h; s_{1} …, s_{1}) = h since ph = λ(s_{1}, s_{1}, …, s_{1}). This shows that if (3.1a) is constant then so is the corresponding internal path of Σ_{λ}^{12...r}(h; .). As for (3.1b), let λ(s_{1}, …, s_{i−1}, t, …, t) = λ(s_{1}, …, s_{i−1}, 0, …, 0) for some 2 ≤ i ≤ r and all t ∈ I. This means that the internal path (3.1b) of λ(s_{1}, .) corresponding to Δ (if i = 2) or s_{2}, …, s_{i−1} (if i > 2) is constant. By the induction assumption, also the corresponding internal path of Σ_{λ(s_{1},.)}^{12...r−1}(h; .) is constant. Thus

Σ_{λ}^{12...r}(h; s_{1} …, s_{i−1}, t, …, t) =
Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(h; s_{1}); s_{2}, …, s_{i−1}, t, …, t) =
Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(h; s_{1}); s_{2}, …, s_{i−1}, 0, …, 0) =
Σ_{λ}^{12...r}(h; s_{1} …, s_{i−1}, 0 …, 0),

ie. also the corresponding internal path of Σ_{λ}^{12...r}(h; .) is constant as required. This proves (3.9).

To show that (4.2) takes r-bunches into r-bunches, we need to show that the internal paths (3.1a), (3.1b) as well as (3.2) corresponding to Σ_{λ}^{12...r}(h; .) are smooth if the ones corresponding to λ are smooth. We shall first verify (3.1a) and (3.1b) starting with r = 2. Thus we need to show that t ↦ Σ_{λ(s_{1},.)}^{1}(Σ_{λ∘Δ}^{2}(h; s_{1}); t) as well as t ↦ Σ_{λ(t,.)}^{1}(Σ_{λ∘Δ}^{2}(h; t); t) are smooth. The first map is smooth because it is the lift correspoonding to ξ_{1} of a smooth path t ↦ λ(s_{1}, t). The second map is by the generalised form of (1.3) equal to t ↦ Σ_{λ∘Δ}^{2}(h; t) which is the lift corresponding to ξ_{2} of the smooth t ↦ λ(t, t). Proceed now by induction, ie. assume that t ↦ Σ_{λ(s_{1},.)}^{12...r−1}(h; s_{2}, …, s_{i−1}, t, …, t) are smooth for i = 2, …, r. In particular

t ↦ Σ_{λ}^{12...r}(h; s_{1} …, s_{i−1}, t, …, t) = Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(h; s_{1}); s_{2}, …, s_{i−1}, t, …, t)

are smooth for i = 2, …, r. It only remains to show that also t ↦ Σ_{λ}^{12...r}(h; t, …, t) is smooth. The last expression equals t ↦ Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(h; t); t, …, t) which is t ↦ Σ_{λ∘Δ}^{r}(h; t) by (3.10) and thus smooth as well. This verifies (3.1a) and (3.1b).

As for (3.2), assume again first r = 2. Then we have j̃_{t}^{2}(↦ Σ_{λ}^{12}(h; s_{1}, s_{2})) = j_{t}^{1}(s_{1} ↦ j_{s_{1}}(s_{2} ↦ Σ_{λ∘Δ}(h; s_{1}); s_{2})) which by (4.1) is

j_{t}^{1}(s_{1} ↦ ξ_{1}(Σ_{λ(s_{1},.)}^{1}(Σ_{λ∘Δ}^{2}(h; s_{1}); s_{1})) ∘ j_{s_{1}}^{1}λ(s_{1}, .))

and this by the generalised form of (1.3) gives

(4.3)  j̃_{t}^{2}(↦ Σ_{λ}^{12}(h; s_{1}, s_{2})) = j_{t}^{1}(s_{1} ↦ ξ_{1}(Σ_{λ∘Δ}^{2}(h; s_{1})) ∘ j_{s_{1}}^{1}λ(s_{1}, .)).

Since Σ_{λ∘Δ}^{2}(h; t) ∈ P_{λ(t,t)}, we have ξ_{2}(Σ_{λ∘Δ}^{2}(h; t)) = j_{λ(t,t)}^{1}η for some section η : M ⇝ P, and by (4.1) alsoξ_{2}(Σ_{λ∘Δ}^{2}(h; t)) ∘ j_{t}^{1}(λ ∘ Δ) = j_{t}^{1}Σ_{λ∘Δ}^{2}(h; .). We conclude that j_{λ(t,t)}^{1}η ∘ j_{t}^{1}(λ ∘ Δ) = Σ_{λ∘Δ}^{2}(h; .), ie. that j_{t}^{1}(s_{1} ↦ η(λ(s_{1}, s_{1})) = j_{t}^{1}(s_{1} ↦ Σ_{λ∘Δ}^{2}(h; s_{1})). Since the expression in (4.3) depends only on the one-jet at t of s_{1} ↦ Σ_{λ∘Δ}^{2}(h; s_{1}), we can write it as j_{t}^{1}(s_{1} ↦ ξ_{1}(η(λ(s_{1}, s_{1}))) ∘ j_{s_{1}}^{1}λ(s_{1}, .)) and this by (2.5) is j_{λ(t,t)}^{1}(ξ_{1} ∘ η) ∘ j̃_{t}^{2}λ.

Thus we have proved for t ∈ I that

j̃_{t}^{2}(↦ Σ_{λ}^{12}(h; s_{1}, s_{2})) = (4.4)

J(ξ_{1}){ξ_{2}(Σ_{λ∘Δ}^{2}(h; t))} ∘ j̃_{t}^{2}λ = (ξ_{1} ⋆ ξ_{2})(Σ_{λ∘Δ}^{2}(h; t)) ∘ j̃_{t}^{2}λ,

whence the smoothness follows from that of Σ_{λ∘Δ}^{2}(h; .) and of (3.2) for λ. Assume now that a generalisation of (4.4) holds for r − 1, ie. assume that

j̃_{s_{1}}^{r−1}(↦ Σ_{λ}^{12...r−1}(h; s_{2}, …, s_{r})) =
(ξ_{1} ⋆ … ⋆ ξ_{r−1})(Σ_{λ∘Δ}^{r−1}(h; s_{1})) ∘ j̃_{s_{1}}^{r−1}λ

for all s_{1} ∈ I and that this map I → P is smooth. Then for each t ∈ I we have

(4.5) j̃_{t}^{r}(↦ Σ_{λ}^{12...r}(h; s_{1}, …, s_{r})) =
j_{t}^{1}(s_{1} ↦ j̃_{s_{1}}^{r−1}(↦ Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}(h; s_{1}); s_{2}, …, s_{r}))) =
j_{t}^{1}(s_{1} ↦ (ξ_{1} ⋆ … ⋆ ξ_{r−1})(Σ_{λ∘Δ}^{r−1}(Σ_{λ∘Δ}^{r}(h; s_{1}); s_{1})) ∘ j̃_{s_{1}}^{r−1}λ(s_{1}, .)) =
j_{t}^{1}(s_{1} ↦ (ξ_{1} ⋆ … ⋆ ξ_{r−1})(Σ_{λ∘Δ}^{r}(h; s_{1}) ∘ j̃_{s_{1}}^{r−1}λ(s_{1}, .)).

Since Σ_{λ∘Δ}^{r}(h; t) ∈ P_{λ(t,... ,t)}, we have ξ_{r}(Σ_{λ∘Δ}^{r}(h; t)) = j_{λ(t,... ,t)}^{1}η, for some section η : M ⇝ P, and by (4.1) also ξ_{r}(Σ_{λ∘Δ}^{r}(h; t))∘j_{t}^{1}(λ∘Δ) = j_{t}^{1}Σ_{λ∘Δ}^{r}(h; .). We conclude that j_{λ(t,... ,t)}^{1}η ∘ j_{t}^{1}(λ ∘ Δ) = Σ_{λ∘Δ}^{r}(h; .), ie. that j_{t}^{1}(s_{1} ↦ η(λ(s_{1}, …, s_{1})) = j_{t}^{1}(s_{1} ↦ Σ_{λ∘Δ}^{r}(h; s_{1})). Since the expression in (4.5) depends only on the one-jet at t of s_{1} ↦ Σ_{λ∘Δ}^{r}(h; s_{1}), we can write it as j_{t}^{1}(s_{1} ↦ (ξ_{1}⋆…⋆ξ_{r−1})(η(λ(s_{1}, …, s_{1})))∘j_{s_{1}}^{1}λ(s_{1}, .)) and this by (2.5) is j_{λ(t,... ,t)}^{1}((ξ_{1} ⋆…⋆ξ_{r−1})∘η)∘j̃_{t}^{r}λ. Thus we have proved for t ∈ I that

j̃_{t}^{r}(↦ Σ_{λ}^{12...r}(h; s_{1}, …, s_{r})) = (4.6)

J(ξ_{1} ⋆ … ⋆ ξ_{r−1})[ξ_{r}(Σ_{λ∘Δ}^{r}(h; t))] ∘ j̃_{t}^{r}λ =
(ξ_{1} ⋆ … ⋆ ξ_{r})(Σ_{λ∘Δ}^{r}(h; t)) ∘ j̃_{t}^{r}λ,

whence the smoothness follows from that of Σ_{λ∘Δ}^{r}(h; .) and of (3.2) for λ. This proves completely the semi-smoothness of (4.2).

By (4.2), Σ_{λ}^{12...r}(h; s_{1} …, s_{1}) = Σ_{λ∘Δ}^{r}(h; s_{1}) since pΣ_{λ∘Δ}^{r}(h; s_{1}) = λ(s_{1}, …, s_{1}). Thus (4.6) can also be written as

(4.7)  j̃_{t}^{r}Σ_{λ}^{12...r}(h; .) = (ξ_{1} ⋆ … ⋆ ξ_{r})(Σ_{λ}^{12...r}(h; t, …, t)) ∘ j̃_{t}^{r}λ,

which is the required infinitesimality condition, ie. the relation (3.11) between Σ_{λ}^{12...r}(h; .) and ξ_{1} ⋆ … ⋆ ξ_{r}.

The remaining conditions are easy. We have pΣ_{λ}^{12}(h; s_{1}, s_{2}) = λ(s_{1}, s_{2}) and Σ_{λ}^{12}(h; 0, 0) = Σ_{λ∘Δ}^{2}(h; 0) = h. This verifies (3.6) and (3.7) for r = 2 and similarly for r > 2 by induction. As for (3.8), assuming it for Σ^{12...r−1} one easily concludes that Σ_{λ}^{12...r}(hg; s_{1} …, s_{r}) = Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(hg; s_{1}); s_{2}, …, s_{r}) = Σ_{λ}^{12...r}(h; s_{1} …, s_{r})g. Thus we have proved

Theorem 4.1. If ξ_{1}, ξ_{2} …, ξ_{r} are first order connections in a principal bundle P (M, G) then the recurrent formula (4.2) defines an infinitesimal r-th order bunch connection. The associated r-th order connection Γ : P → J̃_{r}P is the product Γ = ξ_{1} ⋆ … ⋆ ξ_{r}, ie. (4.7) holds.

Observe that (4.2) says that the value Σ_{λ}^{12...r}(h; s_{1}, …, s_{r}) is obtained by transporting h along λ⟨s_{1}, …, s_{r}⟩, c.f. (3.3). More precisely, first ξ_{r} transports h along λ[s_{1}], then ξ_{r−1} transports the result along λ[s_{1}, s_{2}], then ξ_{r−2} continues along λ[s_{1}, s_{2}, s_{3}], etc., until ξ_{1} ends the transport along λ[s_{1}, …, s_{r}].

Thus Σ_{λ}^{12...r}(h; s_{1}, …, s_{r}) = h_{r} is defined recurrently by

h_{0} = h (4.8)

h_{k} = Σ_{λ}^{12...r}(h; s_{1}, s_{2}, …, s_{k}, …, s_{k}) = Σ_{λ[s_{1},... ,s_{k}]}^{r−k+1}(h_{k−1}; s_{k}),
for 1 ≤ k ≤ r.

Consider now the special case of a holonomic r-bunch, ie. when λ(s_{1}, …, s_{r}) = λ(s_{r}). In this case λ ∘ Δ = λ, the internal paths (3.1b) also coincide with λ, and the paths (3.4) are just suitable restrictions of λ, e.g. λ[s_{1}, …, s_{i}] is the portion of λ between t = s_{i−1} and t = s_{i}. It is therefore easily seen from the above that in case of a holonomic λ, the value Σ_{λ}^{12...r}(h; s_{1}, …, s_{r}) is obtained by transporting h along λ to λ(s_{1}) using ξ_{r}, then continuing along λ to λ(s_{2)} using ξ_{r−1}, etc. until ξ_{1} ends the transport along λ from the point λ(s_{r−1}) to the point λ(s_{r}). This agrees with Theorem 3.1 which says in particular that (4.2) will lift holonomic bunches to holonomic bunches if all the connections ξ_{1}, …, ξ_{r} coincide.

5. Bunch loops or bloops. An r-bunch (continuous or piecewise smooth) λ : I^{r} → M is called a (continuous or piecewise smooth) bunch loop or bloop, if it is constant on the boundary of I^{r}, i.e. if λ(s_{1}, …, s_{r}) = λ(0, …, 0) whenever (s_{1}, …, s_{r}) ∈ İ^{r}, ie. whenever one of the (s_{1}, …, s_{r}) is 0 or 1. Note that a bloop is necessarily stationary on the boundary and that the only holonomic bloop is the constant bloop. If λ(0, …, 0) = x we say that λ is a bloop at x. For example, λ(x, y) = [χ(x)−1](y−x)^{2}(1−y)ydefines a bloop λ : I^{2} → ℝ which is not continuous. Note that the homotopy classes rel İ^{r} of all continuous r-bloops at x form the r-th homotopy group of M at x.

Let x ∈ M and r ≥ 1 be fixed, and let Σ be an r-th order bunch connection in P (M, G). If λ is an r-bloop at x then by (3.6) Σ_{λ}(h; s_{1}, …, s_{r}) ∈ P for each h ∈ P_{x} and each (s_{1}, …, s_{r}) ∈ İ^{r}. Thus we get a map g_{λ}(h; .) : İ^{r} → G defined by

Σ_{λ}(h; s_{1}, …, s_{r}) = hg_{λ}(h; s_{1}, …, s_{r}) for (s_{1}, …, s_{r}) ∈ İ^{r}

satisfying

g_{λ}(hγ; s_{1}, …, s_{r}) = γ^{−1}g_{λ}(h; s_{1}, …, s_{r})γ for any γ ∈ G.

We shall call this map g_{λ}(h; , .) : İ^{r} → G the holonomy map of the bloop λ (corresponding to the given bunch connection and h ∈ P). It is continuous or piecewise smooth iff λ is. For r = 1 we have İ = {0, 1} with g_{λ}(h; 0) = e and g_{λ}(h; 1) = H_{h}(λ), where H_{h} : Ω(M) → G is the holonomy map of the path connection at h. Thus the holonomy map of a loop is in fact constant and equals the value at this loop of the standard holonomy map associated with the path connection.

Let us calculate the values of the holonomy map g_{λ}^{12...r}(h; , .) : İ^{r} → G of the bunch connection Σ_{λ}^{12...r}(h; .) that we associated with ξ_{1} ⋆ … ⋆ ξ_{r} in the preceding Section. Starting with r = 2, in the formula Σ_{λ}^{12}(h; s_{1}, s_{2}) = Σ_{λ(s_{1},.)}^{1}(Σ_{λ∘Δ}^{2}(h; s_{1}); s_{2}) we have to consider four cases:

(i) s_{1} = 0 : Σ_{λ}^{12}(h; 0, s_{2}) = Σ_{λ(0,.)}^{1}(Σ_{λ∘Δ}^{2}(h; 0); s_{2}) = Σ_{λ∘Δ}^{2}(h; 0) since λ(0, .) is a constant loop. Thus by (3.7) this is h, hence g_{λ}^{12}(h; 0, s_{2}) = e.

(ii) s_{1} = 1 : Σ_{λ}^{12}(h; 1, s_{2}) = Σ_{λ(1,.)}^{1}(Σ_{λ∘Δ}^{2}(h; 1); s_{2}) = Σ_{λ∘Δ}^{2}(h; 1) since λ(1, .) is a constant loop. Thus g_{λ}^{12}(h; 1, s_{2}) = H_{h/}^{2}(λ ∘ Δ), where H_{h}^{2} : Ω(M) → G is the holonomy map of ξ_{2}.

(iii) s_{2} = 0 : Σ_{λ}^{12}(h; s_{1}, 0) = h g_{λ}^{12}(h; s_{1}, 0). (iv) s_{2} = 1 : Σ_{λ}^{12}(h; s_{1}, 1) = h g_{λ}^{12}(h; s_{1}, 1). This element is obtained by trans-

porting = h g_{λ}^{12}(h; s_{1}, 0) along the loop λ(s_{1}, .) from s_{2} = 0 to s_{2} = 1. Thus it equals Σ_{λ(s_{1},.)}^{1}(h g_{λ}^{12}(h; s_{1}, 0); 1) = Σ_{λ(s_{1},.)}^{1}(h, 1) g_{λ}^{12}(h; s_{1}, 0).

Therefore

(5.1)  g_{λ}^{12}(h; s_{1}, 1) = H_{h}^{1}(t ↦ λ(s_{1}, t)) g_{λ}^{12}(h; s_{1}, 0).

So for r = 2 the holonomy map of the bloop is uniquely determined by the holonomy maps of the connections ξ_{1}, ξ_{2} and the values of g_{λ}^{12}(h; ., 0) : I → G.

For a general r ≥ 2 we obtain similarly

(5.2)  g_{λ}^{12...r}(h; 0, s_{2}, …, s_{r}) = e and g_{λ}^{12...r}(h; 1, s_{2}, …, s_{r}) = H_{h}^{r}(λ ∘ Δ).

Because of (3.10) one can replace in (4.2) the expression Σ_{λ∘Δ}^{r}(h; s_{1}), which is the value of Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(h; s_{1}); t, …, t) for t = s_{1}, by its value for t = 0, ie. by Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ∘Δ}^{r}(h; s_{1}); 0, …, 0) = Σ_{λ}^{12...r}(h; s_{1}, 0, …, 0). Thus we get

Σ_{λ}^{12...r}(h; s_{1}, …, s_{r}) = Σ_{λ(s_{1},.)}^{12...r−1}(Σ_{λ}^{12...r}(h; s_{1}, 0, …, 0); s_{2}, …, s_{r}).

If (s_{2}, …, s_{r}) ∈ İ^{r−1} then this can be written as h g_{λ}^{12...r}(h; s_{1}, …, s_{r}) =

Σ_{λ(s_{1},.)}^{12...r−1}(h g_{λ}^{12...r}(h; s_{1}, 0, …, 0); s_{2}, …, s_{r}) =
h g_{λ(s_{1},.)}^{12...r−1}(h; s_{2}, …, s_{r}) g_{λ}^{12...r}(h; s_{1}, 0, …, 0).

Thus

(5.3)  g_{λ}^{12...r}(h; s_{1}, …, s_{r}) = g_{λ(s_{1},.)}^{12...r−1}(h; s_{2}, …, s_{r}) g_{λ}^{12...r}(h; s_{1}, 0, …, 0)

for (s_{2}, …, s_{r}) ∈ İ^{r−1} and s_{1} ∈ I. It is easily seen that (5.2) are a special case of (5.3): If (s_{2}, …, s_{r}) ∈ I^{r−1} and s_{1} = 0 then λ(s_{1}, .) is the constant bloop and so g_{λ(0,.)}^{12...r−1}(h; s_{2}, …, s_{r}) = e, as well as e = g_{λ}^{12...r}(h; 0, 0, …, 0). If, instead, s_{1} = 1 then λ(s_{1}, .) is again the constant bloop, so g_{λ(1,.)}^{12...r−1}(h; s_{2}, …, s_{r}) = e, whereas g_{λ}^{12...r}(h; 1, 0, …, 0) = H_{h}^{r}(λ ∘ Δ). This proves the recurrence formula (5.3) for any (s_{1}, …, s_{r}) ∈ İ^{r} and leads easily to the following explicit formulae.

Theorem 5.1. If s_{i+1} is the first 0 in the sequence s_{1}, …, s_{r}, (i ≥ 1) then

(5.4)  g_{λ}^{12...r}(s_{1}, …, s_{r}) = ∏_{k=1}^{i} g_{λ(s_{1},... ,s_{k−1},.)}^{12...r−k+1}(s_{k}, 0, …, 0).

If s_{i+1} is the first 1 in the sequence s_{1}, …, s_{r}, (i ≥ 1) then g_{λ}^{12...r}(s_{1}, …, s_{r}) = (5.5)

H_{h}^{r−i}(t ↦ λ(s_{1}, …, s_{i}, t, …, t)) ∏_{k=1}^{i} g_{λ(s_{1},... ,s_{k−1},.)}^{12...r−k+1}(s_{k}, 0, …, 0).

Here λ(s_{1}, …, s_{k−1}, .) = λ if k = 1. The formulae (5.4) and (5.5) include i = 0, ie. (5.2), if the empty product is interpreted as e ∈ G. Also note that (5.4) and (5.5) do not contradict each other since λ(s_{1}, …, s_{k−1}, .) is the constant bloop if one of s_{1}, …, s_{k−1} is 0 or 1.

Thus the holonomy map of an r-bloop is completely determined by the holonomy maps of the connections ξ_{1}, …, ξ_{r} and the values g_{λ(s_{1},... ,s_{k−1},.)}^{12...r−k+1}(s_{k}, 0, …, 0) for k = 1, …, r − 1.

If Σ_{γ}^{k}(h; .) = Σ_{γ}^{1}(h; .) Δ_{γ}^{k}(h; .), where γ : I → M and Δ_{γ}^{k}(h; .) : I → G, k = 1, …, r, then clearly Δ_{γ}^{k}(ha; .) = ad(a^{−1})Δ_{γ}^{k}(h; .) for any a ∈ G. In (4.8) we get then recurrently for k = 1, …, r

(5.6)  h_{k} = Σ_{λ[s_{1},... ,s_{k}]}^{1}(h_{k−1}; s_{k}) Δ_{λ[s_{1},... ,s_{k}]}^{r−k+1}(h_{k−1}; s_{k}).

This could be used to express g_{λ(s_{1},... ,s_{k−1},.)}^{12...r−k+1}(s_{k}, 0, …, 0) by means of Σ_{γ}^{1}(h; .) and Δ_{γ}^{q}(h; .) for q = 2, …, r and various loops γ. We shall do this only for r = 2 and k = 1 to obtain easily the formula

(5.7.)  g_{λ}^{12}(h; s_{1}, 0) = H_{h}^{1}(λ⟨s_{1}, 0⟩) Δ_{λ∘Δ}^{2}(h; s_{1}).

Theorem 5.2. Let r > 1, let M be simply connected, and let (4.2) be the bunch connection associated with connections ξ_{1}, …, ξ_{r}. Consider the three conditions:

(a) ξ_{1} = … = ξ_{r} = ξ and ξ is curvature-free.

(b) The bunch connection (4.2) is piecewise smooth.

(c) The holonomy map g_{λ}^{12...r}(; .) : İ^{r} → G is trivial (ie. the constant e ∈ G) at any x ∈ M.

Then (a) implies (b) and (b) implies (c).

Proof. (b)⇒(c): If λ is a piecewise smooth r-bloop at x then for each (s_{1}, …, s_{r}) ∈ İ^{r} there is a unique piecewise smooth loop at x consisting of a sequence of internal paths (3.3), namely the loop λ⟨s_{1}, …, s_{r}⟩. If r > 1 then each one of these loops associated with a piecewise smooth λ is nullhomotopic (ie. contractible to the constant loop). So if (4.2) defines a piecewise smooth bunch connection then the lift of λ⟨s_{1}, …, s_{r}⟩ must also be a piecewise smooth loop, ie. h transported along λ⟨s_{1}, …, s_{r}⟩ remains h, thus g_{λ}^{12...r}(h; s_{1}, …, s_{r}) = e for any (s_{1}, …, s_{r}) ∈ İ^{r}.

(a)⇒(b) If ξ_{1} = … = ξ_{r} = ξ then, as observed in (4.8), if λ is an r-bloop at x then the value Σ_{λ}^{12...r}(h; s_{1}, …, s_{r}) is simply the result of the transport of h along λ⟨s_{1}, …, s_{r}⟩. Since ξ is curvature-free and M is simply connected this will also be the result of the transport of h along the path λ ∘ θ, where θ : I → I^{r} is any piecewise smooth path starting at (0, …, 0) and ending at (s_{1}, …, s_{r}) ∈ I^{r}. In paricular, Σ_{λ}^{12...r}(h; .) ∘ θ : I → P will be piecewise smooth for any piecewise smooth θ : I → I^{r} proving that (4.2) defines a piecewise smooth bunch connection.

Recall that a space is called n-connected, if it is path connected and all its homotopy groups of orders 1, …, n are trivial.

Theorem 5.3. If M is (r − 1)-connected, (r > 1), then the three conditions of Theorem 5.2 are equivalent.

Proof. It remains to show that (c)⇒(a). First assume r = 2. Since g_{λ}^{12}(h; 1, s_{2}) = H_{h}^{2}(λ ∘ Δ) by (ii), the assumption g_{λ}^{12}(h; 1, s_{2}) = e will imply that ξ_{2} is curvature free if we can find for any smooth loop γ : I → M a 2-bloop λ such that λ ∘ Δ = γ. Since M is simply connected, there is a homotopy F : γ ≃ const_{x} rel{0, 1}, which we can assume to be smooth, ie.

(5.8)  F : I × I → M

satisfying

F (s, 0) = γ(s), F (s, 1) = F (0, t) = F (1, t) = x for all s, t ∈ I.

Now it is easily verified, that λ : I^{2} → M defined as

(5.9)  λ(s_{1}, s_{2}) =F (s_{1}, 1 − \frac{s_{2}}{s_{1}} )  if s_{2} ≤ s_{1}
F (s_{1}, \frac{s_{2} − s_{1}}{1 − s_{1}} )  if s_{1} ≤ s_{2}

is a piecewise smooth 2-bloop such that λ(t, t) = γ(t). Thus ξ_{2} is curvature free. On the other hand, it follows from (5.1) that e.g. H_{h}^{1}(↦ λ(\frac{1}{2}, t)) = e. This will imply that also ξ_{1} is curvature free if we can find for any smooth loop γ : I → M a 2-bloop λ such that λ(\frac{1}{2}, t) = γ(t). Using (5.8) again, we see that

(5.10)  λ(s_{1}, s_{2}) =F (s_{2}, 1 − 2s_{1})  if s_{1} ≤ \frac{1}{2}
F (s_{2}, 2s_{1} − 1)  if s_{1} ≥ \frac{1}{2}

defines the required 2-bloop. It remains to show that ξ_{1} = ξ_{2}. If Σ_{γ}^{2}(h; .) = Σ_{γ}^{1}(h; ,)∘ Δ_{λ}^{2}(h; , .) then formula (5.7) shows that Δ_{λ∘Δ}^{2}(h; s_{1}) = e for any 2-bloop λ and any s_{1} ∈ I. Since, as we have seen, any loop γ can be represented as some λ ∘ Δ, we conclude that Σ_{γ}^{2}(h; .) = Σ_{γ}^{1}(h; ,) for any loop at x whence ξ_{1} = ξ_{2}. This completes the proof of (c)⇒(a) for r = 2.

Proceed by induction, ie. assume (c)⇒(a) for some r−1. By (5.3) our assumption implies that g_{λ(s_{1},.)}^{12...r−1}(h; s_{2}, …, s_{r}) = e for any r-bloop λ. Since M is (r − 1)- connected, any (r − 1)-bloop γ can be seen as λ(\frac{1}{2}, .), where λ is an r-bloop, in a similar manner as (5.8) was used to construct (5.10). Explicitly, given an (r − 1)- bloop γ we have now a smooth

(5.11)  F : I^{r−1} × I → M

satisfying

F (s, 0) = γ(s), F (s, 1) = x for all s ∈ I^{r−1} and
F (s, t) = x for all t ∈ I and s ∈ İ^{r−1}

which gives rise to the r-bloop

λ(s_{1}, …, s_{r}) =F (s_{2}, …, s_{r}, 1 − 2s_{1})  if s_{1} ≤ \frac{1}{2}
F (s_{2}, …, s_{r}, 2s_{1} − 1)  if s_{1} ≥ \frac{1}{2}

such that λ(\frac{1}{2}, s_{2}, …, s_{r}) = F (s_{2}, …, s_{r}, 0) = γ(s_{2}, …, s_{r}). We conclude that also g_{γ}^{12...r−1}(h; .) : İ^{r−1} ∈ G is trivial for any (r − 1)-bloop and so by the induction assumption, ξ_{1} = … = ξ_{r−1} = ξ is one and the same curvature free connection.

Next we need to show that for any loop γ : I → M there is an r-bloop λ : I^{r} → M such that γ = λ ∘ Δ. Since M is (r − 1)-connected there are smooth maps F_{q} : I^{q−1} × I → M, q = 2, …, r (c.f. (5.11)) which are homotopies from (q − 1)- bloops to the constant bloop rel{0, 1} constructed as follows: F_{2} is the hotopy (5.8) from γ = λ_{1} to const_{x} rel{0, 1} and recurrently, F_{q} is the homotopy from the bloop λ_{q−1} : I^{q−1} → M to the constant bloop at x rel İ^{q−1} for q = 3, …, r. Explicitly, we have for q = 2, …, r

F_{q} : I^{q−1} × I → M,

satisfying

F_{q}(s, 0) = λ_{q−1}(s), F_{q}(s, 1) = x for all s ∈ I^{q−1} and
F_{q}(s, t) = x for all t ∈ I and s ∈ İ^{q−1}

where λ_{1} = γ, λ_{2} is given by (5.9) and recurrently for q = 3, …, r

λ_{q}(s_{1}, …, s_{q}) =F_{q}(s_{1}, …, s_{q−1}, 1 − \frac{s_{q}}{s_{q−1}} )  if s_{q} ≤ s_{q−1}

F_{q}(s_{1}, …, s_{q−1}, \frac{s_{q} − s_{q−1}}{1 − s_{q−1}} )  if s_{q−1} ≤ s_{q}.

It is easily verified that each λ_{q} is a piecewise smooth bloop. Especially λ = λ_{r} is an r-bloop such that γ = λ ∘ Δ. Thus we can conclude again from (5.2) that ξ_{r} is curvature free.

It remains to show that ξ_{r} = ξ. Since now ξ_{1} = … = ξ_{r−1} = ξ we have also Δ_{λ[s_{1},... ,s_{k}]}^{r−k+1}(h_{k−1}; .) = e for k = 2, …, r, and so the recurrence formula (5.6) will give for Σ_{λ}^{12...r}(s_{1}, …, s_{r}) = h_{r} the recurrence formulae

h_{r} = Σ_{λ[s_{1},... ,s_{r}]}(h_{r−1}; s_{r}
h_{r−1} = Σ_{λ[s_{1},... ,s_{r−1}]}(h_{r−2}; s_{r−1}
…
h_{2} = Σ_{λ[s_{1},s_{2}]}(h_{1}; s_{2})
h_{1} = Σ_{λ[s_{1}]}(h, s_{1}) Δ_{λ[s_{1}]}(h, s_{1}).

Denoting by λ∗ the composition of paths λ[s_{1}, …, s_{r}] ∗ … ∗ λ[s_{1}, s_{2}] and observing that λ[s_{1}] = λ ∘ Δ (c.f. (3.4)), we get

Σ_{λ}^{12...r}(h; s_{1}, …, s_{r}) = Σ_{λ∗}^{1}(h_{1}; s_{r}), where h_{1} = Σ_{λ∘Δ}^{1}(h; s_{1}) Δ_{λ∘Δ}^{r}(h; s_{1}).

Thus denoting by < λ > the path in (3.3), we get

Σ_{λ}^{12...r}(h; s_{1}, …, s_{r}) = Σ_{<λ>}^{1}(h; s_{r}) Δ_{λ∘Δ}^{r}(h; s_{1}).

If s_{r} = 0, this says h g_{λ}^{12...r}(h; s_{1} …, s_{r−1}, 0) = h Δ_{λ∘Δ}^{r}(h; s_{1}) because < λ > is a loop and ξ_{1} = ξ is curvature free. Since the holonomy map of λ was assumed trivial we conclude that Δ_{λ∘Δ}^{r}(h; s_{1}) = e for all s_{1} ∈ I, and since any loop γ can be seen as λ ∘ Δ for some r-bloop λ, we conclude further that ξ_{r} = ξ as required. This completes the proof of Theorem 5.3.

References

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

[2] Ehresmann C., Sur les connexions d’ordre supérieur, Atti V^{0} Cong. Un. Mat. Italiana, Pavia-Torino, 1956, pp.326–328.

[3] Kolář I., On some operations with connections, Math. Nachrichten 69(1975), 297-306.

[4] Telemann C., Connection and bundles I, Proc. Koninkljike Nederlandse Akad. van Wettenschappen (Ser A) 72(1969), 89-112.

[5] Virsik J. (George), On the holonomity of higher order connections, Cahiers Topol. Géom. Diff. 12(1971), 197–212.

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

Department of Mathematics, Monash University Clayton,Victoria 3168, Australia