Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 257–264

ON STRUCTURE EQUATIONS FOR SECOND ORDER CONNECTIONS

E. Aguirre-Dabán and I. Sánchez-Rodríguez ^{∗}

Abstract The basic facts concerning the structure equations of connections in the principal bundle of second order (holonomic) frames F^{2}M on a differentiable manifold M are presented here in a self contained and synthetic way. Advantage is taken from the embedding of F^{2}M as a subset of F (F M) and from viewing the group G^{2}(n) as a semidirect product.

Keywords. Connections, frame bundles.

MS classification. 53C.

1 Introduction

There is a great paralellism between the linear (or first order) frame bundle, LM ≡ F^{1}M, and the second order (holonomic) frame bundle, F^{2}M, on a n-dimensional manifold M. As standard references see e.g. [1],[3]. On both principal bundles there exist canonical 1-forms: in the linear case it is the well known IR^{n}-valued 1-form, denoted by θ^{L}; in the second order case, the canonical 1-form, denoted simply by θ, is valued into the vector space IR^{n} + 𝔤𝔩(n, IR).

On both bundles we can define connection 1-forms, denoted by λ or ω, valued in the Lie algebras, 𝔤𝔩(n, IR) or 𝔤^{2}(n), of their respective structure groups, GL(n, IR) or G^{2}(n). The so called first and second structure equations of a connection λ in LM are

D^{λ} θ^{L} = dθ^{L} + λ ˙∧θ^{L}

D^{λ} λ = dλ + \frac{1}{2}[λ, λ]

where D^{λ} means the covariant differentiation with respect to the connection λ.

We could expect to obtain similar expressions for the second order case. The second structure equation, giving an expression for the covariant derivative of ω with respect to ω itself, is well known as a general equation of a connection in any principal bundle:

D^{ω} ω = dω + \frac{1}{2}[ω, ω].

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

The first structure equation, giving an expression of D^{ω} θ in terms of θ and ω, was first obtained by Yuen [5]; it could also be deduced from the work of Kolář [4] in the more abstract framework of prolongations of general principal bundles.

We want to stress two previous steps, undoubtly well known, but having not been fully exploited before. The first one is to see G^{2}(n) as a semidirect product of GL(n, IR) and S^{2}(n), where S^{2}(n) means the abelian Lie group of symmetric bilinear maps of IR^{n} × IR^{n} into IR^{n}, and also its abelian Lie algebra. The second step is to see F^{2}M as the first prolongation (LM)_{1} of LM, in the sense of [3] (thus, as a subset of L(LM)).

2 The Lie group G^{2}(n) and its Lie algebra 𝔤^{2}(n)

The Lie group G^{2}(n) is the set of all 2-jets j^{2}(g), with source 0 ∈ IR^{n}, of diffeomorphisms between open neighbourhoods of 0 in IR^{n}, with g(0) = 0, and with product law given by the 2-jet of the composition map.

The following two descriptions of G^{2}(n) by global charts are interesting:

I : G^{2}(n) ⟶^{≈} (GL(n, IR) × S^{2}(n), ⋆)

II : G^{2}(n) ⟶^{≈} (GL(n, IR) × S^{2}(n), •)

where \{ \begin{matrix}(a, τ) ⋆ (a′, τ′) := (aa′, aτ′ + τ(a′, a′)) \\ (a, t) • (a′, t′) := (aa′, t + at′(a^{−1}, a^{−1})) \end{matrix}

and \{ \begin{matrix}I : j^{2}(g) ⟼ (a, τ) \\ II : j^{2}(g) ⟼ (a, t) \end{matrix} with \{ \begin{matrix}a := Dg(0) ≡ ( \frac{∂g^{i}}{∂u^{j}} |_{0}) \\ τ := D^{2}g(0) ≡ ( \frac{∂^{2}g^{i}}{∂u^{j}∂u^{k}} |_{0}) \\ t := D^{2}g(0)(Dg(0)^{−1}, Dg(0)^{−1}) , \end{matrix}

(u^{j}) being the standard coordinates in IR^{n}.

Description I is the standard one with partial derivatives, and the product law ⋆ expresses nothing else but the chain rule.

Description II is very useful because it describes G^{2}(n) as a semidirect product, GL(n, IR) [figure: ×]S^{2}(n), associated to Q ∈ Hom(GL(n, IR), Aut S^{2}(n)) given by

Q(a) : t ⟼ t_{a} ≡ a t(a^{−1}, a^{−1}).

We will use only II.

The Lie algebra 𝔤^{2}(n) of G^{2}(n) can be identified –in both descriptions I or II– with the semidirect sum 𝔤𝔩(n, IR) [figure]S^{2}(n), associated to q := Q_{∗}|_{Id} ∈ Hom(𝔤𝔩(n, IR), End S^{2}(n)), explicitly

q(α) : t ⟼ t^{α} ≡ αt − t(α, ·) − t(·, α).

The adjoint representations are given by

Ad : GL(n, IR) [figure: ×]S^{2}(n) ⟶  Aut(𝔤𝔩(n, IR) [figure]S^{2}(n)) (a, t)  ⟼ ((β, s) ↦ (aβa^{−1}, s_{a} − t^{aβa^{−1}})) ad : 𝔤𝔩(n, IR) [figure]S^{2}(n) ⟶  End(𝔤𝔩(n, IR) [figure]S^{2}(n)) (α, t)  ⟼ ((β, s) ↦ ([α, β], s^{α} − t^{β})) .

It is well known that there exists a representation of G^{2}(n) into IR^{n} + 𝔤𝔩(n, IR). We can see it in the following way:

Let g : U ⟶ V be a diffeomorphism between neighbourhoods of 0 ∈ IR^{n}, with g(0) = 0. Let us define

\begin{aligned}I_{g} : & LIR^{n}|_{U} & ⟶ & LIR^{n}|_{V} \\ & j^{1}(ψ) & ⟼ & j^{1}(g ∘ ψ ∘ g^{−1}) ; \end{aligned}

it thus follows that

I_{g_{∗}}|_{j^{1}(id)} : T_{j^{1}(id)}LIR^{n} ⟶ T_{j^{1}(g)}LIR^{n},

id being the identity map in IR^{n}, depends only on j^{2}(g).

Then, identifying LIR^{n} with the product manifold IR^{n} × GL(n, IR) and T_{j^{1}(g)}LIR^{n} with IR^{n} + 𝔤𝔩(n, IR), we get a representation of G^{2}(n) into this vector space. Explicitly:

A : GL(n, IR) [figure: ×]S^{2}(n) ⟶  Aut(IR^{n} + 𝔤𝔩(n, IR))
(a, t)  ⟼ ((v, β) ↦ (av, t(av) + aβa^{−1}))

and the corresponding representation of the Lie algebra

𝔞 := A_{∗}|_{j^{2}(id)} : 𝔤𝔩(n, IR) [figure]S^{2}(n) ⟶  End(IR^{n} + 𝔤𝔩(n, IR))
(α, t)  ⟼ ((v, β) ↦ (αv, t(v) + [α, β])).

This representation plays an important role, as we will see.

3 The Second Order Frame Bundle, F^{2}M

The second order frame bundle F^{2}M, is the set of all 2-frames on M, i.e. 2-jets, j^{2}(ϕ), with source 0 ∈ IR^{n}, of local diffeomorphisms ϕ of open neighbourhoods of 0 ∈ IR^{n} into M.

It has a structure of G^{2}(n)-principal bundle, with right action j^{2}(ϕ)·j^{2}(g) := j^{2}(ϕ∘g) and projection

π_{2} : F^{2}M ⟶ M, j^{2}(ϕ) ⟼ ϕ(0) .

There exists a canonical embedding of F^{2}M into L(LM), obtained by the following process:

Given a local diffeomorphism ϕ : U ⊂ IR^{n} → M with 0 ∈ U, we define

\begin{aligned}ϕ̃ : & LIR^{n}|_{U} & ⟶ & LM|_{ϕ(U)} \\ & j^{1}(ψ) & ⟼ & j^{1}(ϕ ∘ ψ) \end{aligned}

then, it can be proved that its tangent map at j^{1}(id)

ϕ̃_{∗}|_{j^{1}(id)} : IR^{n} + 𝔤𝔩(n, IR) ≈ T_{j^{1}(id)}LIR^{n} ⟶ T_{j^{1}(ϕ)}LM

depends only on j^{2}(ϕ) ; and so we obtain the embbeding

\begin{aligned}ι : & F^{2}M & ⟶ & L(LM) \\ & j^{2}(ϕ) & ⟼ & ϕ̃_{∗}|_{j^{1}(id)} ≡ z \end{aligned} .

The isomorphisms z : IR^{n} + 𝔤𝔩(n, IR) → T_{j^{1}(ϕ)}LM corresponding to 2-frames j^{2}(ϕ) are characterized as follows:

Lemma 1 z ∈ L(LM) belongs to ι(F^{2}M) if and only if:

(a) The vertical part is determined by the condition z(0, α) = α^{∗}|_{l}, α^{∗} being the corresponding fundamental vectorfield on LM, ∀α ∈ 𝔤𝔩(n, IR).

(b) Defining z(IR^{n}, 0) =: H^{z}, the isomorphism IR^{n} → H^{z}, v ↦ z(v, 0), is the inverse of the canonical form acting on H^{z}; i.e. θ^{L}(z(v, 0) = v, ∀v ∈ IR^{n}.

(c) The subspaces H^{z} have no torsion; i.e. dθ^{L}|_{H^{z}×H^{z}} = 0.

Then we have a one-to-one correspondence between 2-frames at points of M and horizontal torsionless n-subspaces in T LM.

On the other hand, it can be proved that, given z ≡ ι(j^{2}(ϕ) ∈ L(LM), every connection λ in LM having H^{z} as horizontal plane at l ≡ j^{1}(ϕ) yields a normal chart x_{λ}, centered at m ≡ ϕ(0) and corresponding to the linear frame l, such that j^{2}(ϕ) = j^{2}(x_{λ}^{−1}). We denote by C_{z} the set of linear connections having H^{z} as horizontal plane. The isomorphism z satisfies z(v, α) = B_{λ}^{L}(v)|_{l} + α^{∗}|_{l}, B_{λ}^{L}(v) being the standard λhorizontal vectorfield corresponding to v.

It turns out that the properties a, b, and c define what Kobayashi calls (see [3]) first prolongation (LM)_{1} of the linear frame bundle LM . Initially, (LM)_{1} is a S^{2}(n)- principal bundle over LM with projection Π : (LM)_{1} → LM given by the restriction of the projection L(LM) → LM, and right action

zt := z ∘ A_{(I,t)} .

But there is also a natural GL(n, IR)-action on (LM)_{1} given by

za := R_{a ⋆}^{L} ∘ z ∘ A_{(a,0)}

(R^{L} is the right action on LM), which may be characterized by the condition R_{a ∗}^{L}(H^{z}) = H^{za}, or C_{za} = C_{z}.

Mixing in appropriate way the last two actions we obtain:

Lemma 2

(a) The bundle π^{L} ∘ Π : (LM)_{1} ⟶ M is a GL(n, IR) [figure: ×]S^{2}(n)-principal bundle over M (denoting π^{L} : LM → M), with action

\begin{matrix}R : (LM)_{1} × (GL(n, IR) [figure: ×]S^{2}(n)) & ⟶ & LM_{1} \\ (z, (a, t)) & ⟼ & z(a, t) := (zt)a = R_{a ∗}^{L} ∘ z ∘ A_{(a,t)}. \end{matrix}

(b) The bijection ι : F^{2}M ⟶ (LM)_{1}, becomes an isomorphism of principal bundles over M corresponding to the identity map in GL(n, IR) [figure: ×]S^{2}(n).

We resume the situation in the following diagram:

[figure: F^{2}M; LM_{1}; L(LM); LM; M; ι; ⊂; π_{2,1}; π_{2}; Π; π^{L}]

In what follows we shall identify F^{2}M ≡ LM_{1}, π_{2,1} ≡ Π, π_{2} ≡ π^{L} ∘ Π.

Let us now fix our conventions concerning exterior products of forms on F^{2}M : Let Ψ ∈ Λ^{j}(F^{2}M, 𝔤𝔩(n, IR) [figure]S^{2}(n)) and Φ ∈ Λ^{k}(F^{2}M, IR^{n} + 𝔤𝔩(n, IR)) be two pseudotensorial forms of type Ad and A, respectively. We define Ψ ˙∧Φ ∈ Λ^{j+k}(F^{2}M, IR^{n} + 𝔤𝔩(n, IR)) by

Ψ ˙∧Φ(X_{1},…, X_{j+k}) := \frac{1}{j!k!} ∑_{σ}(−1)^{σ}𝔞_{Ψ(X_{σ(1)},...,X_{σ(j)})}Φ(X_{σ(j+1)},…, X_{σ(j+k)}),

where the sum is taken over the set of all permutations σ of {1,…, j + k} and (−1)^{σ} = ±1 is the sign of σ.

Also, if Ψ ∈ Λ^{j}(F^{2}M, 𝔤𝔩(n, IR) [figure]S^{2}(n)) and Ψ′ ∈ Λ^{k}(F^{2}M, 𝔤𝔩(n, IR) [figure]S^{2}(n)), we define [Ψ, Ψ′] ∈ Λ^{j+k}(F^{2}M, 𝔤𝔩(n, IR) [figure]S^{2}(n)) by

[Ψ, Ψ′](X_{1},…, X_{j+k}) := \frac{1}{j!k!} ∑_{σ}(−1)^{σ}[Ψ(X_{σ(1)},…, X_{σ(j)}), Ψ′(X_{σ(j+1)},…, X_{σ(j+k)})].

4 Structure Equations of a Connection in F^{2}M

We proceed to define the main objects appearing in the structure equations, and to give their most important properties.

The fundamental vectorfield (α, s)^{∗} on F^{2}M is defined as usual in any principal bundle, ∀(α, s) ∈ 𝔤𝔩(n, IR) [figure]S^{2}(n).

The canonical 1-form θ ≡ (θ_{−1}, θ_{0}) ∈ Λ^{1}(F^{2}M, IR^{n} + 𝔤𝔩(n, IR)) is defined by θ(X) := z^{−1}(Π_{∗}X), ∀z ∈ F^{2}M, ∀X ∈ T_{z}F^{2}M.

Lemma 3 It holds:

(a) θ is pseudotensorial of A type: i.e. R_{(a,t)}^{∗}θ = A_{(a,t)^{−1}}θ. It is not tensorial (contrarily to the linear case), because θ((α, s)^{∗}|_{z}) = (0, α).

(b) The IR^{n}-and 𝔤𝔩(n, IR)-valued components of θ verify:

θ_{−1} = Π^{∗}θ^{L} and θ_{0}|_{z} = (Π^{∗}λ)|_{z}, ∀λ ∈ C_{z}.

(c) dθ_{−1} = −θ_{0} ˙∧θ_{−1} (as a direct application of the first structure equation of a symmetric linear connection dθ_{−1} = −λ ˙∧θ^{L}).

A second order connection ω ≡ (ω_{0}, ω_{1}) ∈ Λ^{1}(F^{2}M, 𝔤𝔩(n, IR) [figure]S^{2}(n)) is a connection form in F^{2}M defined as usual. From ω we obtain a linear connection λ^{ω} by the condition Π^{∗}λ^{ω} = ω_{0} (because Π is a morphism of principal fiber bundles).

We define the standard ω-horizontal vectorfield corresponding to v ∈ IR^{n} as the unique vectorfield B_{ω}(v) on F^{2}M satisfying ω(B_{ω}(v)) = 0 and θ_{−1}(B_{ω}(v)) = v.

Lemma 4

(a) Π_{∗}(B_{ω}(v)|_{z}) = B_{λ^{ω}}^{L}(v)|_{l} with B_{λ^{ω}}^{L}(v) the corresponding standard λ^{ω}-horizontal vectorfield on LM.

(b) [(α, t)^{∗}, B_{ω}(v)] = B_{ω}(αv).

Lemma 5 Given a second order connection ω and v ∈ IR^{n}, the C^{∞}-function Υ_{ω}(v) := θ_{0}(B_{ω}(v)) : F^{2}M → 𝔤𝔩(n, IR), z ↦ θ_{0}(B_{ω}(v)|_{z}) satisfies (α, t)^{∗}Υ_{ω}(v) = Υ_{ω}(αv) − [α, Υ_{ω}(v)] − t(v).

Proof: Let exp and Exp be the exponential maps in the Lie groups GL(n, IR) and GL(n, IR) [figure: ×]S^{2}(n), respectively. Note that the GL(n, IR)-component of Exp(α, t) is expα. Then we have (∀z ∈ F^{2}M and ∀(α, t) ∈ 𝔤𝔩(n, IR) [figure]S^{2}(n) ) :

(α, t)_{z}^{∗}Υ_{ω}(v) = \frac{d}{dϵ}|_{ϵ=0} θ_{0}(B_{ω}(v)|_{z·Exp ϵ(α,t)}) = (Lemma 4b)

\frac{d}{dϵ}|_{ϵ=0} θ_{0}(R_{Exp ϵ(α,t) ∗}B_{ω}(expϵα v)|_{z}) = (Lemma 3a)

\frac{d}{dϵ}|_{ϵ=0} (A_{Exp −ϵ(α,t)}θ)_{0} (B_{ω}(expϵα v)|_{z}) = (𝔞_{−(α,t)}θ)_{0} (B_{ω}(v)|_{z}) + θ_{0}(B_{ω}(α v)|_{z}) =

(−[α, θ_{0}] − t ∘ θ_{−1}) (B_{ω}(v)|_{z}) + θ_{0}(B_{ω}(α v)|_{z}) = −[α, Υ_{ω}(v) (z)] − t(v) + Υ_{ω}(α v) (z)

and the result follows. □

We remark that, given X ≡ B_{ω}(v)|_{z} +(α, t)^{∗}|_{z} ∈ T_{z}(F^{2}M), it holds: (θ_{0} −ω_{0})(X) = θ_{0}(B_{ω}(v)|_{z}).

Finally, we could define, by analogy to the linear case, the ’torsion’ of second order (see [5]) as the two form

Θ = (Θ_{−1}, Θ_{0}) := D^{ω}θ ∈ Λ^{2}(F^{2}M, IR^{n} + 𝔤𝔩(n, IR)),

which is a tensorial 2-form of A type, and the curvature of second order as

Ω = (Ω_{0}, Ω_{1}) := D^{ω}ω ∈ Λ^{2}(F^{2}M, 𝔤𝔩(n, IR) [figure]S^{2}(n)),

which is a tensorial 2-form of Ad-type.

Theorem (Structure equations). Let ω be a second order connection. It holds:

(a) dθ = −ω ˙∧θ + \frac{1}{2} [ω_{0}, ω_{0}] + Θ

Structure equations for second order conections 263

(b) dω = −\frac{1}{2} [ω, ω] + Ω Proof: (a) By linearity, it suffices to compute both members on pairs X, Y ∈ T_{z}(F^{2}M) in the following three cases:

• Both X and Y are ω-horizontal. The equation follows trivially.

• Both X and Y are vertical, thus X = (α, t)^{∗}|_{z} and Y = (β, s)^{∗}|_{z} for some α, β ∈ 𝔤𝔩(n, IR) and t, s ∈ S^{2}(n). Then:

\{ \begin{matrix}dθ(X, Y) = X(θ(β, s)^{∗}) − Y (θ(α, t)^{∗}) − θ|_{z} [(α, t)^{∗}, (β, s)^{∗}] = (Lem. 3a) \\ −(0, [α, β]) \\ ω ˙∧θ (X, Y) = 𝔞_{(α,t)}(θ|_{z}(β, s)^{∗}) − 𝔞_{(β,s)}(θ|_{z}(α, t)^{∗}) = 2(0, [α, β]) \\ [ω_{0}, ω_{0}] (X, Y) = 2 [ω_{0}(α, t)^{∗}|_{z}, ω_{0}(β, s)^{∗}|_{z}] = 2(0, [α, β]) \\ Θ(X, Y) = 0 \end{matrix}

and the result follows.

• X is vertical and Y is ω-horizontal, thus X = (α, t)^{∗}|_{z} and Y = B_{ω}(v)|_{z} for some

α ∈ 𝔤𝔩(n, IR), t ∈ S^{2}(n) and v ∈ IR^{n}. Then:

dθ(X, Y) = X(θB_{ω}(v)) − Y (θ(α, t)^{∗}) − θ|_{z} [(α, t)^{∗}, B_{ω}(v)] = (Lem. 5 and 4b) (0, Υ_{ω}(α v) − [α, Υ_{ω}(v)] − t(v)) − θ|_{z} B_{ω}(α v) = (−α v, −[α, Υ_{ω}(v)] − t(v))

ω ˙∧θ (X, Y) = 𝔞_{(α,t)}(θ|_{z} B_{ω}(v)) = (α v, t(v) + [α, Υ_{ω}(v)])

[ω_{0}, ω_{0}] (X, Y) = 0

Θ(X, Y) = 0 and the result follows.

(b) This is a general result for connections in any principal bundle. □ We remark that the equation (a) decomposes into:

• dθ_{−1} = −ω_{0} ˙∧θ_{−1} + Θ_{−1} (IR^{n}-valued component)

• dθ_{0} = −ω_{1} ˙∧θ_{−1} − [ω_{0}, θ_{0}] + \frac{1}{2}[ω_{0}, ω_{0}] + Θ_{0} (𝔤𝔩(n, IR)-valued component)

Taking covariant derivatives in the equations of the Theorem, we obtain the following (Bianchi) identities: Corollary

(a) D^{ω}Θ = Ω ˙∧θ − [Ω_{0}, ω_{0}]

(b) D^{ω}Ω = 0.

References

[1] L.A.Cordero, C.T.J.Dodson, M. de León, Differential Geometry of Frame Bundles, Kluwer Academic Publishers, Dordrecht 1989

[2] S.Kobayashi, Canonical forms on frame bundles of higher order contact, Proceedings of Symposia in Pure Math., vol III, A.M.S.(1961),186-193 [3] S.Kobayashi, Transformation Groups in Differential Geometry, Springer Verlag, Heidel-

berg 1972

[4] I.Kolář, Canonical forms on the prolongations of principal fibre bundles, Rev. Roumaine Math. Pures et Appl., \underline{16} (1971), 1091-1106; A generalization of the torsion form, Časopis pro pěstování matematiky, Praha, roč.\underline{100} (1975), 284-290

[5] P.C.Yuen, Higher order frames and linear connections, Cahiers de topologie et géométrie diffèréntielle, \underline{12} (1971), 337-371 .

E. Aguirre-Dabán, Dep. de Geometría y Topología, Fac. de Matemáticas, Univ. Complutense, 28040 Madrid, Spain.

I. Sánchez-Rodríguez, Dep. de Economía Financiera y Contabilidad I, Fac. de CC.EE., Univ. Complutense, Campus de Somosaguas, 28023 Madrid, Spain.

Received 5 November 1992