Proceedings Ist International Meeting on Geometry and Topology Braga (Portugal)

Public. Centro de Matemática da Universidade do Minho p. 191–205, 1998

Explicit Formulas for the 3-Jet Lift of a Matrix Group. Apllications to Conformal Geometry

E. Aguirre-Dab and \underline{I. Sánchez-Rodríguez}

Abstract:

The 3-jet lift G^{3} of a matrix group G is isomorphic, via a map that we give explicitly, to a semidirect product of G itself and a nilpotent group builded up from the first two prolongations of its Lie algebra. Using this isomorphism, we write down the formulas for the most natural representations of G^{3}, as well as for one additional representation of the 2-jet lift G^{2} appearing when G is of finite type 2. We apply these results to the case of the (linear) conformal group and we point out the geometric implications of these representations.

1 Introduction

The r-jet lift G^{r} (with r ∈ N, r > 0) of a Lie subgroup G of the general linear group GL(n, R) is the Lie group of r-jets at 0 ∈ R^{n} of (local, 0-preserving) diffeomorphisms of R^{n} which preserve, in an obvious sense and up to order (r−1), the canonical G-structure on R^{n}.

The r-jet lift of GL(n, R) itself is the structure group of the principal r-frame bundle F^{r}(M) over any n-dimensional manifold M, and some of its properties have been studied in this context [5,8,10]. The second order frame bundle F^{2}(M) has been extensively studied and explicit expressions concerning the 2-jet lift of GL(n, R) are well known; see e.g. [3]. A proper use of such expressions leads [1] to a new proof of the first structure equation —first found in [10]— for connections in F^{2}(M).

192 E. Aguirre-Dab and I. Sánchez-Rodríguez

For a proper Lie subgroup G of GL(n, R), the group G^{r} comes into play when one deals with “(r − 1)-integrable” G-structures (“uniformly (r − 1)-flat” in the sense of [4]), i.e. G-structures P ⊂ LM having, over each point of M, contact of order (r − 1) with coordinate sections of the linear frame bundle LM —let us remember that 1-integrable G-structures are those admitting symmetric (linear) connections [2,4]. For each (r − 1)-integrable G-structure P, one can define its r-jet lift P^{r} as the set of r-jets at 0 ∈ R^{n} of (local) diffeomorphisms of R^{n} into M yielding such order of contact, and it turns out that P^{r}(M, G^{r}) becomes a principal subbundle of F^{r}(M).

Consider the Lie algebra g of G. It may happen that the so called “r-prolonga-tion” g_{r} vanishes. Remember that when g_{r} = 0 and g_{r−1} ≠ 0, one says that G is of finite type r. In the case g_{r} vanishes, G^{r+1} becomes isomorphic to G^{r}, and G^{r} inherits some additional properties from the (r + 1) order, mainly an additional natural representation that it is in the background of the appearance, in some cases [4], of Cartan connections on the r-jet lifts P^{r} of (r − 1)-integrable G-structures P.

The (pseudo) orthogonal conformal group CO(n, R) is (for n ≥ 3) of finite type 2 and every CO(n, R)-structure is 1-integrable. Both CO(n, R) and its 2-jet lift CO(n, R)^{2} can be considered subgroups of the full conformal group O(n+2, R)/Z_{2}. A lot of work has been done in the study of 2-jet lifts of conformal structures in dimension ≥ 3, see e.g. [6,7], mainly by specific methods of the conformal case. We are interested in a more systematic treatment of connections on 2-jet lifts P^{2}(M, G^{2}) of 1-integrable G-structures P when G is of finite type 2, and their relationships with Cartan connections when they exist. Such treatment needs an appropriate handling of the 3-jet lift of a general matrix group. To our knowledge, no explicit formulas are up to now available.

In this paper we give (Lemma) a map which shows the isomorphism between G^{3} and a semidirect product of G itself and a nilpotent group builded up from the first two prolongations of the Lie algebra g of G. This map allows us to write down (Theorem) explicit formulas for the most natural representations of G^{3} and for the additional representation of G^{2} appearing when G is of finite type 2. When G is the (linear) conformal group CO(n, R), one can check that this additional representation is the restriction to CO(n, R)^{2} of the adjoint representation of the full conformal group O(n + 2, R)/Z_{2}. This last representation of CO(n, R)^{2} controls the behaviour of conformal Cartan connections on the 2-jet lift of a CO(n, R)-structure.

2 Basic definitions

For each v ∈ R^{n}, let D_{0,v}(R^{n}) be the set of diffeomorphisms φ between open subsets of R^{n}, containing 0 and v respectively, such that φ(0) = v. The r-jet j_{0}^{r}(φ) is the equivalence class of φ in D_{0,v}(R^{n}) induced by the equivalence relation

φ ∼^{r} φ′ ⟺ D^{s}φ|_{0} = D^{s}φ′|_{0} , 1 ≤ s ≤ r.

Explicit Formulas for the 3-Jet Lift of a Matrix Group 193

The set G^{r}(n) := D_{0,0}(R^{n})/ ∼^{r} = {j_{0}^{r}(g) : g ∈ D_{0,0}(R^{n})} is a Lie group, the group of r-jets, with product law

j_{0}^{r}(g) · j_{0}^{r}(g′) := j_{0}^{r}(g ∘ g′)

and differentiable structure such that the bijection

\begin{aligned}(u_{0}, u_{1}, …, u_{r−1}) : & G^{r}(n) & ⟶ & GL(n, R) × S^{2}(n) × ⋯ × S^{r}(n) \\ & j_{0}^{r}(g) & ⟼ & (Dg|_{0} , D^{2}g|_{0} , …, D^{r}g|_{0}) \end{aligned}

becomes a global chart; we denote by S^{r}(n) the subspace of symmetric r-linear maps in L^{r}(n) ≡ L(R^{n}× ⋯^{r} ×R^{n}, R^{n}). This bijection induces a product law on GL(n, R) × S^{2}(n) × ⋯ × S^{r}(n) which can be explicitly computed by applying the chain rule to the derivatives of the composition of functions. The canonical Lie group isomorphism G^{1}(n) → GL(n, R), j_{0}^{1}(g) ↦ Dg|_{0}, will be understood in what follows.

The set F^{r}(R^{n}) := {j_{0}^{r}(φ) : φ ∈ D_{0,v}(R^{n}), v ∈ R^{n}} is a (trivial) G^{r}(n)- principal bundle over R^{n} [5], the bundle of r-frames over R^{n}, with projection π^{r} : F^{r}(R^{n}) → R^{n}, j_{0}^{r}(φ) ↦ φ(0), G^{r}(n)-right action R_{j_{0}^{r}(g)}^{r}j_{0}^{r}(φ) ≡ j_{0}^{r}(φ) · j_{0}^{r}(g) := j_{0}^{r}(φ ∘ g), and (global) trivialization

Ψ^{r} : F^{r}(R^{n}) → R^{n} × G^{r}(n), j_{0}^{r}(φ) ↦ (φ(0) , j_{0}^{r}(T_{−φ(0)} ∘ φ)), T_{v} being the translation by v ∈ R^{n}. The canonical principal fibre bundle isomorphism F^{1}(R^{n}) → LR^{n}, j_{0}^{1}(φ) ↦ (φ_{∗}|_{0} : R^{n} → T_{φ(0)}R^{n}), will be understood in what follows.

3 The r-jet lift of a matrix group

Let G be a Lie subgroup of GL(n, R). The canonical G-structure GR^{n} is the subset of the linear frame bundle LR^{n} obtained by moving with G the global section σ_{I} of π^{1} : LR^{n} → R^{n} induced by the identity chart I ≡ (u^{1}, u^{2}, …, u^{n}) of R^{n}

GR^{n} := (Im σ_{I}) · G.

Every diffeomorphism φ : U ⊂ R^{n} → V ⊂ R^{n} raises a principal bundle isomorphism φ̃ : LR^{n}|_{U} → LR^{n}|_{V}, j_{0}^{1}(ψ) ≡ l ↦ j_{0}^{1}(φ ∘ ψ) ≡ φ_{∗}|_{ψ(0)} ∘ l.

The r-jet lift G^{r} of G is defined as the subset of G^{r}(n) induced by those diffeomorphisms g ∈ D_{0,0}(R^{n}) whose associated g̃ takes σ_{I} into a (local) section of LR^{n} having (r − 1)-order contact over 0 ∈ R^{n} with some section of GR^{n}

G^{r} := {j_{0}^{r}(g) ∈ G^{r}(n) : ∃ σ ∈ Sec(GR^{n}) with j_{0}^{r−1}(g̃ ∘ σ_{I}) = j_{0}^{r−1}(σ ∘ g)}.

It can be described equivalently by

G^{r} = {j_{0}^{r}(g) ∈ G^{r}(n) : ∃ κ ∈ F(R^{n}, G) with Dg|_{0} = κ(0)
and D^{s+1}g|_{0} = D^{s}κ|_{0} (1 ≤ s ≤ r − 1)};

we have identified D^{s}κ|_{0} ∈ L(R^{n}× ⋯^{s} ×R^{n}, gl(n, R)) with the element of L^{s+1}(n) given by D^{s}κ|_{0}(v_{1}, …, v_{s+1}) ≡ D^{s}κ|_{0}(v_{1}, …, v_{s})(v_{s+1}). We have that (GL(n, R))^{r} = G^{r}(n) and G^{1} is canonically isomorphic to G.

Let us show up some algebraic properties of G^{r}:

(i) For every 0 < r′ < r, the projection β^{r,r′} : G^{r} ⟶ G^{r′}, j_{0}^{r}(g) ⟼ j_{0}^{r′}(g), is a Lie group homomorphism and the invariant Lie subgroup G^{r,r′} ≡ Ker β^{r,r′} is nilpotent. In particular, G^{r,r−1} (r > 1) is abelian.

(ii) For r > 1, one sees very easily the existence of the following splitting exact sequence

1 ⟶ G^{r,1} ↪ G^{r} ⟶^{β} ; \overleftarrow{γ} G ⟶ 1,

with ↪ the inclusion (G^{r,1} := {j_{0}^{r}(g) ∈ G^{r} : Dg|_{0} = I} = {j_{0}^{r}(Dg|_{0}^{−1} ∘ g) : j_{0}^{r}(g) ∈ G^{r}}), β(j_{0}^{r}(g)) := Dg|_{0} and γ(a) := j_{0}^{r}(a).

Thus G^{r} is canonically isomorphic with the semidirect product G× ⊃ G^{r,1}, associated to the homomorphism of G into Aut(G^{r,1}) given by

a ⟼ (j_{0}^{r}(g) ↦ j_{0}^{r}(a ∘ g ∘ a^{−1})),

and this isomorphism is defined by the map

\begin{aligned}χ : & G^{r} & ⟶ & G× ⊃ G^{r,1} \\ & j_{0}^{r}(g) & ⟼ & (Dg|_{0} , j_{0}^{r}(Dg|_{0}^{−1} ∘ g)). \end{aligned}

Remark The following sequence is also exact

1 ⟶ G^{r,r−1} ↪ G^{r} ⟶ G^{r−1} ⟶ 1;

however it does not split, unless r = 2 (in which case both sequences are the same) or G^{r,r−1} = {0} (trivial case); thus, in general, one cannot write G^{r} ≃ G^{r−1} × ⊃ G^{r,r−1} (the group G^{r−1} has no natural structure as a subgroup of G^{r}).

By looking at curves in G^{r} through the identity j_{0}^{r}(I), it is easy to see [4] that the Lie algebra g^{r} of G^{r} must be isomorphic —as a linear space— to the direct sum ∑_{s=0}^{r−1} g_{s}, g_{s} being the s-prolongation of g, defined as g_{0} := g and, for s > 0, as the vector space

g_{s} := {t ∈ S^{s+1}(n) : t(v_{1}, …, v_{s}, ·) ∈ g, ∀ v_{1}, …, v_{s} ∈ R^{n}} = L(R^{n}, g_{s−1})∩S^{s+1}(n)

(obviously (gl(n, R))_{s} = S^{s+1}(n); moreover: g_{s} = 0 ⇒ g_{s+1} = 0).

One can also prove [4,6,8] that the Lie algebra g^{r} of G^{r} is isomorphic to the Lie algebra (∑_{s=0}^{r−1} g_{s}, [ , ]_{0}^{r}), with the Lie bracket [ , ]_{0}^{r} defined as follows

[t_{p}, t′_{q}]_{0}^{r} := \frac{(p + q + 1)!}{p!(q + 1)!}S(t_{p}(t′_{q}(·, …^{q+1}, ·), …^{p}, ·))−\frac{(p + q + 1)!}{(p + 1)!q!}S(t′_{q}(t_{p}(·, …^{p+1}, ·), …^{q}, ·)),

if p + q ≤ r − 1

[t_{p}, t′_{q}]_{0}^{r} := 0, if p + q > r − 1,

where t_{p} ∈ g_{p}, t′_{q} ∈ g_{q}, p, q ≥ 0 (thus [t_{p}, t′_{q}]_{0}^{r} ∈ g_{p+q}), and the symmetrizer S is given by (∀ t ∈ L^{s}(n) and ∀ v_{1}, …, v_{s} ∈ R^{n}):

S(t)(v_{1}, …, v_{s}) := \frac{1}{s!} ∑_{σ∈P_{s}} t(v_{σ(1)}, …, v_{σ(s)}).

It follows that the abelian group G^{r,r−1} (r > 1) is isomorphic to the vector group g_{r−1}. If g_{r−1} = 0, the homomorphism β^{r,r−1} : G^{r} → G^{r−1} is a Lie group isomorphism.

Remarks.

(i) For r > 1, consider the Lie algebra (∑_{s=1}^{r−1} g_{s}, [ , ]_{1}^{r}), with [ , ]_{1}^{r} being simply the restriction of [ , ]_{0}^{r}. One always has

(∑_{s=0}^{r−1} g_{s}, [ , ]_{0}^{r}) = g+ ⊃ (∑_{s=1}^{r−1} g_{s}, [ , ]_{1}^{r}) (semidirectsum).

Moreover —and corresponding to the above Remark on groups— (∑_{s=0}^{r−2} g_{s}, [ , ]_{0}^{r−1}) closes as a Lie subalgebra of (∑_{s=0}^{r−1} g_{s}, [ , ]_{0}^{r}) if and only if either r = 2 or g_{r−1} = 0 (trivial case).

(ii) The Lie bracket [ , ]_{0}^{r} on g^{r} can be extended in a natural way to some kind of “bracket” [ , ]_{−1}^{r} on R^{n} + g^{r}, defined as follows

[ ,]_{−1}^{r}|_{g^{r}×g^{r}} := [ , ]_{0}^{r}
[v, v′]_{−1}^{r} := 0
[t_{p}, v]_{−1}^{r} := t_{p}(v) =: −[v, t_{p}]_{−1}^{r},

where v, v′ ∈ R^{n}, t_{p} ∈ g_{p}, p ≥ 0. Certainly R^{n} + g^{r} always closes under [ , ]_{−1}^{r}; however, the Jacobi-identity is fulfilled —only then it becomes possible to define the Lie algebra R^{n} + g^{r} ≡ (R^{n} + g^{r}, [ , ]_{−1}^{r})— if and only if either r = 1 (we get the affine algebra R^{n} ⊂ +g) or the prolongation g_{r} vanishes.

Besides the adjoint representation, G^{r} has another natural representation; let us show how this one comes out.

The r-jet lift G^{r}(R^{n}) of GR^{n} is defined as the subset of F^{r}(R^{n}) induced by those diffeomorphisms φ ∈ D_{0,v}(R^{n}), v ∈ R^{n}, whose associated φ̃ takes σ_{I} into a (local) section of LR^{n} having (r − 1)-order contact over φ(0) ∈ R^{n} with some section of GR^{n}

G^{r}(R^{n}) := {j_{0}^{r}(φ)∈F^{r}(R^{n}) : ∃ σ ∈ Sec(GR^{n}) with j_{0}^{r−1}(φ̃ ∘ σ_{I})=j_{0}^{r−1}(σ ∘ φ)}.

One sees very easily that G^{r}(R^{n}) is a (trivial) G^{r}-principal bundle over R^{n}, in fact a reduction of F^{r}(R^{n}) (see section 2).

The canonical isomorphism G^{1} ≃ G yields a natural representation of G^{1} in R^{n}. For r > 1, the above construction of G^{r}(R^{n}) leads to a natural representation A^{G^{r}} of G^{r} in the vector space R^{n} + g^{r−1}, as follows:

(i) Every diffeomorphism g : U → V in D_{0,0}(R^{n}) raises a diffeomorphism I_{g}^{r−1} : F^{r−1}(R^{n})|_{U} → F^{r−1}(R^{n})|_{V}, j_{0}^{r−1}(ψ) ↦ j_{0}^{r−1}(g ∘ ψ ∘ g^{−1}) (it is not a principal bundle isomorphism!). One can prove [4] that the tangent map I_{g ∗}^{r−1}|_{j_{0}^{r−1}(I)}, which depends in fact on j_{0}^{r}(g), preserves the subspace T_{j_{0}^{r−1}(I)}(G^{r−1}(R^{n})) of T_{j_{0}^{r−1}(I)}(F^{r−1}(R^{n})); thus we can write

I_{g ∗}^{r−1}|_{j_{0}^{r−1}(I)} : T_{j_{0}^{r−1}(I)}(G^{r−1}(R^{n})) ⟶ T_{j_{0}^{r−1}(I)}(G^{r−1}(R^{n})).

(ii) The global trivialization (see section 2) Ψ^{r−1} : G^{r−1}(R^{n}) → R^{n} × G^{r−1} and the associated canonical identification T_{j_{0}^{r−1}(I)}G^{r−1}(R^{n}) ≃ R^{n} + g^{r−1} lead to the (injective) representation

A^{G^{r}} : G^{r} ⟶  GL(R^{n} + g^{r−1})
j_{0}^{r}(g) ⟼ (I_{g ∗}^{r−1}|_{j_{0}^{r−1}(I)} : R^{n} + g^{r−1} → R^{n} + g^{r−1}).

This representation of G^{r} induces a representation a^{g^{r}} of the Lie algebra g^{r}, given by

a^{g^{r}} ≡ A_{∗}^{G^{r}} |_{j_{0}^{r}(I)} : g^{r} ⟶ gl(R^{n} + g^{r−1}).

Let us consider what happens if g_{r−1} = 0. In that case, the above cited Lie group isomorphism β^{r,r−1} : G^{r} → G^{r−1} gives an obvious representation

\overline{A}^{G^{r−1}} := A^{G^{r}} ∘ (β^{r,r−1})^{−1} : G^{r−1} → GL(R^{n} + g^{r−1}).

4 The 2-jet lift of G

We now give a short review of the 2-jet lift G^{2} of a matrix group G. The material of this section is fairly well known, although our description shows up the semidirect product structure of G^{2}(n), so the outcoming formulas are not the usual ones

As the Abelian group G^{2,1}(n) = {j_{0}^{2}(Dg|_{0}^{−1} ∘ g) : j_{0}^{2}(g) ∈ G^{2}(n)} is isomorphic —via the map u_{1}— to the vector group S^{2}(n), we get the isomorphism

\begin{aligned}(u_{0}, ū_{1}) := (I, u_{1}) ∘ χ : & G^{2}(n) & ⟶^{≃} & GL(n, R)× ⊃ S^{2}(n) \\ & j_{0}^{2}(g) & ⟼ & (Dg|_{0} , Dg|_{0}^{−1} D^{2}g|_{0}), \end{aligned}

where the product law in GL(n, R)× ⊃ S^{2}(n) is given by

(a, t) · (a′, t′) := (aa′, a′^{−1}t(a′, a′) + t′).

Restricting (u_{0}, ū_{1}) to

G^{2} = {j_{0}^{2}(g) ∈ G^{2}(n) : ∃ κ ∈ F(R^{n}, G) with Dg|_{0} = κ(0) and D^{2}g|_{0} = Dκ|_{0}}

we get, just because Dg|_{0}^{−1} D^{2}g|_{0} = (κ^{−1}Dκ)|_{0} ∈ S^{2}(n) ∩ L(R^{n}, g), the isomorphism (which will be understood in what follows)

\begin{aligned}(u_{0}, ū_{1}) : & G^{2} & ⟶ & G× ⊃ g_{1} \\ & j_{0}^{2}(g) & ⟼ & (Dg|_{0} , Dg|_{0}^{−1} D^{2}g|_{0}). \end{aligned}

Before looking into the representation Ad^{G^{2}} and A^{G^{2}} of G^{2}, we want to introduce the following useful notations: ∀ t ∈ L^{r}(n) ≃ L(R^{n}, L^{r−1}(n)), ∀ a ∈ GL(n, R) and ∀ α ∈ gl(n, R), we write

t_{a} ≡ at(a^{−1}, …, a^{−1}) ∈ L^{r}(n)t^{α}
L^{r}(n)t^{α} ≡ αt − t(α ·, ·, …, ·) − … − t(·, …, ·, α ·) ∈ L^{r}(n).

Note that if we call Q ∈ Hom(GL(n, R), GL(L^{r}(n))), Q(a) : t → t_{a}, and q ∈ Hom(gl(n, R), gl(L^{r}(n))), q(α):t→t^{α}, it is very easy to see that q = Q_{∗}|_{I}.

In the case t ∈ S^{2}(n) ⊂ L^{2}(n) ≃ L(R^{n}, gl(n, R)), it is easy to check the following properties:

(i) Both maps t ↦ t_{a} and t ↦ t^{α} belong to gl(S^{2}(n)),

(ii) t_{ab} = (t_{b})_{a} and t^{[α,β]} = (t^{β})^{α} − (t^{α})^{β},

(iii) t_{a}(v) = at(a^{−1}v)a^{−1} and t^{α}(v) = [α, t(v)] − t(αv), ∀ v ∈ R^{n},

(iv) t^{t(v)}(w) − t^{t(w)}(v) = 2[t(v), t(w)], ∀ v, w ∈ R^{n},

(v) (t^{α})_{a} = (t_{a})^{aαa^{−1}}.

Remark. We can rewrite the product law in GL(n, R)× ⊃ S^{2}(n) as follows:

(a, t) · (a′, t′) = (aa′, t_{a′}^{−1} + t′), thus (a, t)^{−1} = (a^{−1}, −t_{a}).

With this notations, the representations Ad^{G^{2}} (adjoint) and A^{G^{2}} —and the corresponding representations ad^{g^{2}} and a^{g^{2}}— have the form

\begin{aligned}Ad^{G^{2}} : & G^{2} ≃ G× ⊃ g_{1} & ⟶ & Aut(g+ ⊃ g_{1}) \\ & (a, t) & ⟼ & ( \begin{matrix}Ad_{a}^{G} & 0 \\ −(t^{(·)})_{a} & (·)_{a} \end{matrix} ) , \end{aligned}

or Ad_{(a,t)}^{G^{2}}(β, s) = (aβa^{−1}, s_{a} − (t^{β})_{a});

\begin{aligned}ad^{g^{2}} : & g^{2} ≃ g+ ⊃ g_{1} & ⟶ & Der(g+ ⊃ g_{1}) \\ & (α, t) & ⟼ & ( \begin{matrix}ad_{α}^{g} & 0 \\ −t^{(·)} & (·)^{α} \end{matrix} ) , \end{aligned}

or ad_{(α,t)}^{g^{2}}(β, s) = ([α, β], s^{α} − t^{β}) = [(α, t), (β, s)]_{0}^{2};

\begin{aligned}A^{G^{2}} : & G^{2} ≃ G× ⊃ g_{1} & ⟶ & GL(R^{n} + g) \\ & (a, t) & ⟼ & ( \begin{matrix}a & 0 \\ Ad_{a}^{G}t(·) & Ad_{a}^{G} \end{matrix} ) , \end{aligned}

or A_{(a,t)}^{G^{2}}(v, β) = (av, a(t(v) + β)a^{−1});

\begin{aligned}a^{g^{2}} : & g^{2} ≃ g+ ⊃ g_{1} & ⟶ & gl(R^{n} + g) \\ & (α, t) & ⟼ & ( \begin{matrix}α & 0 \\ t(·) & ad_{α}^{g} \end{matrix} ) , \end{aligned}

or a_{(α,t)}^{g^{2}}(v, β) = (αv, t(v) + [α, β]).

The case g_{1} = 0. In that case, one has the isomorphism

\begin{matrix}G & ⟶ & G^{2} ≃ G× ⊃ \{0\} \\ a & ⟼ & (a, 0) \end{matrix}

and \begin{aligned}\overline{A}^{G} : & G & ⟶ & Aut(R^{n} ⊂ +g) \\ & a & ⟼ & ( \begin{matrix}a & 0 \\ 0 & Ad_{a}^{G} \end{matrix} ) , \end{aligned}

or \overline{A}_{a}^{G}(v, β) = (av, aβa^{−1}) (we write \overline{A}_{a}^{G} ∈ Aut(R^{n} ⊂ +g) because \overline{A}^{G} preserves the bracket in the Lie algebra R^{n} ⊂ +g).

5 The 3-jet lift of G

In analogous way as we did for G^{2}, we first describe G^{3} as a semidirect product of G itself and a (now non-abelian) nilpotent group builded up on the first two prolongations g_{1} and g_{2} of its Lie algebra g.

The nilpotent group G^{3,1}(n) = {j_{0}^{3}(g) ∈ G^{3}(n) : Dg|_{0} = I} is isomorphic —via (u_{1}, u_{2})— to the group (S^{2}(n) × S^{3}(n), ⋆), ⋆ being the product law induced by applying the chain rule to the derivatives of the composition of functions

(t, T) ⋆ (t′, T′) := (t + t′, T + 3S(t(·)t′(·, ·)) + T′).

Now, instead of using the chart (u_{1}, u_{2}), it will be more convenient (see the next lemma) to use the chart (u_{1}, u_{2} − \frac{3}{2}S(u_{1}(·)u_{1}(·, ·))), which establishes an isomorphism between G^{3,1}(n) and the group (S^{2}(n) × S^{3}(n), ⋄), where ⋄ stands for the product law

(t, T) ⋄ (t′, T′) := (t + t′, T + \frac{1}{2}[t, t′]_{0}^{3} + T′),

and the bracket [t, t′]_{0}^{3} := 3S(t(·)t′(·, ·)) − 3S(t′(·)t(·, ·)) was given in section 3. In that way we get the isomorphism

(u_{0}, ū_{1}, ū_{2}) := (I, u_{1}, u_{2} − \frac{3}{2} S(u_{1}(·)u_{1}(·, ·))) ∘ χ : G^{3}(n) ⟶
⟶ GL(n, R)× ⊃ (S^{2}(n) × S^{3}(n), ⋄)
j_{0}^{3}(g) ⟼ (Dg|_{0} , Dg|_{0}^{−1} D^{2}g|_{0} , Dg|_{0}^{−1} D^{3}g|_{0} −

−\frac{3}{2}S(Dg|_{0}^{−1} D^{2}g|_{0} (·) Dg|_{0}^{−1} D^{2}g|_{0} (·, ·))),

where the product law in GL(n, R)× ⊃ (S^{2}(n) × S^{3}(n), ⋄) is given by

\begin{aligned}(a, t, T) · (a′, t′, T′) & := & (aa′ , (t_{a′}^{−1} , T_{a′}^{−1}) ⋄ (t′ , T′)) = \\ & = & (aa′ , t_{a′}^{−1} + t′ , T_{a′}^{−1} + \frac{1}{2}[t_{a′}^{−1}, t′]_{0}^{3} + T′) \end{aligned}

(what implies (a, t, T)^{−1} = (a^{−1}, −t_{a}, −T_{a})).

Remark. Note that the term \frac{1}{2}[t_{a′}^{−1}, t′]_{0}^{3} prevents this product to be a semidirect product of the form G^{2}(n)× ⊃ S^{3}(n); the group G^{2}(n) has no natural structure as a subgroup of G^{3}(n) (see the first remark in section 3).

As the following lemma establishes, the above chart (u_{0}, ū_{1}, ū_{2}) provides, by restriction, an isomorphic image of G^{3} in terms of G and the first two prolongations of g, and this is a very interesting fact for computations.

Lemma. The group G^{3} is isomorphic —via (u_{0}, ū_{1}, ū_{2})— to the group G×⊃ (g_{1} × g_{2}, ⋄).

Proof. As we said above, it holds: j_{0}^{3}(g) ∈ G^{3} ⇔ ∃ κ ∈ F(R^{n}, G) such that (1) Dg|_{0} = κ(0) ∈ G,

(2) D^{2}g|_{0} = Dκ|_{0}(·) ∈ S^{2}(n),

(3) D^{3}g|_{0} = D^{2}κ|_{0}(·, ·) ∈ S^{3}(n).

As for G^{2}, conditions (1) and (2) lead to

ū_{1}(j_{0}^{3}(g)) ∈ g_{1}.

Now we consider condition (3). Let us call κ_{1}(·) ≡ κ^{−1}Dκ ∈ F(R^{n}, L(R^{n}, g)) and κ_{2}(·, ·) ≡ κ^{−1}D^{2}κ ∈ F(R^{n}, L_{sim}(R^{n} × R^{n}, gl(n, R))). Note that although κ_{1}|_{0}(·) ∈ L(R^{n}, g), in general κ_{2}|_{0}(·, ·) does not belong to L(R^{n} × R^{n}, g); so conditions (1) and (3) do not lead to Dg|_{0}^{−1} D^{3}g|_{0} ∈ g_{2}.

From D(κ^{−1})(·) = −κ^{−1}Dκ(·)κ^{−1}, it immediately follows

κ_{2}(·, ·) = Dκ_{1}(·, ·) + κ_{1}(·)κ_{1}(·);

then, at the origin of R^{n},

κ_{2}|_{0}(·, ·) ∈ κ_{1}|_{0}(·)κ_{1}|_{0}(·) + L(R^{n} × R^{n}, g).

Thus conditions (1) and (3) lead to

(3′) κ_{2}|_{0}(·, ·) ∈ (κ_{1}|_{0}(·)κ_{1}|_{0}(·) + L(R^{n} × R^{n}, g)) ∩ S^{3}(n).

Now observe that, given any s ∈ g_{1} := L(R^{n}, g) ∩ S^{2}(n), the identity

s(·)s(·, ·) ≡ \frac{3}{2}S(s(·)s(·, ·)) + \frac{1}{2}s^{s(·)}(·, ·)

holds with s^{s(·)} ∈ L(R^{n}, g_{1}) ⊂ L(R^{n} × R^{n}, g); applying this identity to s = κ_{1}|_{0}, we get from (3’)

κ_{2}|_{0}(·, ·) − \frac{3}{2} S(κ_{1}|_{0}(·)κ_{1}|_{0}(·, ·)) ∈ L(R^{n} × R^{n}, g) ∩ S^{3}(n) = g_{2};

thus again (1) and (3) lead to

Dg|_{0}^{−1} D^{3}g|_{0} − \frac{3}{2}S(Dg|_{0}^{−1} D^{2}g|_{0} (·) Dg|_{0}^{−1} D^{2}g|_{0} (·, ·)) ∈ g_{2},

or, in other words, ū_{2}(j_{0}^{3}(g)) ∈ g_{2}.

Once we know that (u_{0}, ū_{1}, ū_{2})(G^{3}) ⊂ G × g_{1} × g_{2}, the assert of the lemma follows immediately.

□

Corollary. If g_{2} = 0, every j_{0}^{3}(g) ∈ G^{3} satisfies

Dg|_{0}^{−1} D^{3}g|_{0} − \frac{3}{2}S(Dg|_{0}^{−1} D^{2}g|_{0} (·) Dg|_{0}^{−1} D^{2}g|_{0} (·, ·)) = 0.

Remark.

Thus the algebraic condition for a matrix group G of being of finite type 2 analytic property: every GR^{n}-preserving diffeomorphism φ ∈ D_{0,v}(R^{n}), v ∈ R^{n}, must satisfy

Dφ|_{0}^{−1} D^{3}φ|_{0} − \frac{3}{2}S(Dφ|_{0}^{−1} D^{2}φ|_{0} (·) Dφ|_{0}^{−1} D^{2}φ|_{0} (·, ·)) = 0.

In particular, let CO(n, R) (with n ≥ 3) be the linear conformal group with respect to some (pseudo) euclidean scalar product η in R^{n}; as it is well known, CO(n, R) is of finite type 2. It thus follows that every conformal (local) diffeomorphism in (R^{n}, η) satisfies this condition, as one immediately checks.

Using the above isomorphism (u_{0}, ū_{1}, ū_{2}), we get the following

Theorem. The representations Ad^{G^{3}} (adjoint) and A^{G^{3}} —and the corresponding representations ad^{g^{3}} and a^{g^{3}}— are given by

\begin{matrix}Ad^{G^{3}} : & G^{3} ≃ G× ⊃ (g_{1} × g_{2}, ⋄) & → & Aut(g + g_{1} + g_{2}, [·, ·]_{0}^{3}) \\ (a, t, T) & ↦ & ( \begin{matrix}Ad_{a}^{G} & 0 & 0 \\ −(t^{(·)})_{a} & (·)_{a} & 0 \\ −(T^{(·)} − \frac{1}{2}[t^{(·)}, t]_{0}^{3})_{a} & ([t, ·]_{0}^{3})_{a} & (·)_{a} \end{matrix} ) , \end{matrix}

or Ad_{(a,t,T)}^{G^{3}}(β, s, S) = (aβa^{−1}, s_{a} − (t^{β})_{a}, (S − T^{β} + [t, s]_{0}^{3} + \frac{1}{2}[t^{β}, t]_{0}^{3})_{a});

\begin{aligned}ad^{g^{3}} : & g^{3} ≃ G× ⊃ (g_{1} × g_{2}, ⋄) & → & Der(g + g_{1} + g_{2}, [·, ·]_{0}^{3}) \\ & (α, t, T) & ↦ & ( \begin{matrix}ad_{α}^{g} & 0 & 0 \\ −t^{(·)} & (·)^{α} & 0 \\ −T^{(·)} & [t, ·]_{0}^{3} & (·)^{α} \end{matrix} ) , \end{aligned}

or ad_{(α,t,T)}^{g^{3}}(β, s, S)=([α, β], s^{α}−t^{β}, S^{α}−T^{β} + [t, s]_{0}^{3})≡[(α, t, T), (β, s, S)]_{0}^{3});

\begin{aligned}A^{G^{3}} : & G^{3} ≃ G× ⊃ (g_{1} × g_{2}, ⋄) & → & GL(R^{n} + g + g_{1}) \\ & (a, t, T) & ↦ & ( \begin{matrix}a & 0 & 0 \\ Ad_{a}^{G}t(·) & Ad_{a}^{G} & 0 \\ (T (·) − \frac{1}{2}t^{t(·)})_{a} & −(t^{(·)})_{a} & (·)_{a} \end{matrix} ) , \end{aligned}

or A_{(a,t,T)}^{G^{3}}(v, β, s) = (av , a(t(v) + β)a^{−1} , (T (v) − \frac{1}{2}t^{t(v)} − t^{β} + s)_{a});

\begin{aligned}a^{g^{3}} : & g^{3} ≃ G× ⊃ (g_{1} × g_{2}, ⋄) & → & gl(R^{n} + g + g_{1}) \\ & (α, t, T) & ↦ & ( \begin{matrix}α & 0 & 0 \\ t(·) & ad_{α}^{g} & 0 \\ T (·) & −t^{(·)} & (·)^{α} \end{matrix} ) , \end{aligned}

or a_{(α,t,T)}^{g^{3}}(v, β, s) = (αv, t(v) + [α, β], T (v) + s^{α} − t^{β}).

Proof. A careful computation is needed; for more details see [9]. The expressions for Ad_{(a,t,T)}^{G^{3}} := Int_{(a,t,T) ∗}^{G^{3}}|_{(I,0,0)} and A_{j_{0}^{3}(g)≃(a,t,T)}^{G^{3}} := I_{g ∗}^{2}|_{j_{0}^{2}(I)≃(0,I,0)} follow from the following expressions of the inner automorphisms of G^{3}

\begin{matrix}Int_{(a,t,T)}^{G^{3}} : G× ⊃ (g_{1} × g_{2}, ⋄) & → G× ⊃ (g_{1} × g_{2}, ⋄) \\ (u_{0}, ū_{1}, ū_{2}) & ↦(a, t, T)(u_{0}, ū_{1}, ū_{2})(a, t, T)^{−1} = \end{matrix}

= (au_{0}a^{−1} , (t_{u_{0}^{−1}} + ū_{1} − t)_{a}, (T_{u_{0}^{−1}} + ū_{2} + \frac{1}{2}([t_{u_{0}^{−1}}, ū_{1}]_{0}^{3} + [t, ū_{1}]_{0}^{3} + [t, t_{u_{0}^{−1}}]_{0}^{3}))_{a}),

and of the diffeomorphism

I_{g}^{2} : F^{2}(R^{n})|_{U} ≃ U × GL(n, R) × S^{2}(n) → F^{2}(R^{n})|_{V}
(u, u_{0}, u_{1})  ↦
↦ (g(u) , Dg|_{u} u_{0} Dg^{−1}|_{0} , D^{2}g|_{u} (u_{0} Dg^{−1}|_{0} , u_{0} Dg^{−1}|_{0}) +
+ Dg|_{u} u_{1}(Dg^{−1}|_{0} , Dg^{−1}|_{0}) + Dg|_{u} u_{0} D^{2}g^{−1}|_{0}),

respectively.

□

The case g_{2} = 0. In that case, the map

\begin{matrix}G^{2} ≅ G× ⊃ g_{1} & ⟶ & G^{3} ≅ G× ⊃ (g_{1} × \{0\}, ⋄) \\ (a, t) & ⟼ & (a, t, 0) \end{matrix}

gives an isomorphism G^{2} ≃ G^{3} and a representation \overline{A}^{G^{2}} which preserves the bracket in the Lie algebra (R^{n} + g + g_{1}, [·, ·]_{−1}^{2})

\begin{aligned}\overline{A}^{G^{2}} : & G^{2} ≃ G× ⊃ g_{1} & ⟶ & Aut(R^{n} + g + g_{1}, [·, ·]_{−1}^{2}) \\ & (a, t) & ⟼ & ( \begin{matrix}a & 0 & 0 \\ Ad_{a}^{G}t(·) & Ad_{a}^{G} & 0 \\ −\frac{1}{2}(t^{t(·)})_{a} & −(t^{(·)})_{a} & (·)_{a} \end{matrix} ) , \end{aligned}

or \overline{A}_{(a,t)}^{G^{2}}(v, β, s) = (av , a(t(v) + β)a^{−1} , (−\frac{1}{2}t^{t(v)} − t^{β} + s)_{a}).

Remarks.

(i) In the case g_{2} = 0, observe that: \overline{A}_{(a,t)}^{G^{2}}|_{g+g_{1}} = Ad_{(a,t)}^{G^{2}} and \overline{a}_{(α,t)}^{g^{2}} = ad_{(0,α,t)}^{R^{n}+g^{2}}.

(ii) When G = CO(n, R), the linear conformal group, it is well known [6] that CO(n, R)^{2} is isomorphic to a Lie subgroup L_{0} of the full conformal group L ≡ O(n + 2)/Z_{2}, as follows:

CO(n, R)^{2} ≃ CO(n, R)× ⊃ co(n, R)_{1} ⟶^{≃} L_{0}

(kb, −μ̃) ⟼ A_{(k,b,μ)} ≡ ( \begin{matrix}k & 0 & 0 \\ bμ^{♯} & b & 0 \\ (2k)^{−1}η(μ^{♯}, μ^{♯}) & k^{−1}μ & k^{−1} \end{matrix} ) ,

with k ∈ R^{+}, b ∈ O(n, R), μ ∈ R^{n ∗}, μ̃ ∈ co(n, R)_{1}, μ^{♯} ∈ R^{n} (we have used the isomorphisms: R^{n ∗} ⟶ R^{n}, μ ⟼ μ^{♯}, determined by η(μ^{♯}, ·) = μ, with η a standard scalar product of arbitrary signature, and R^{n ∗} ⟶ co(n, R)_{1}, μ ⟼ μ̃, where

μ̃(v, w) := μ(v)w + μ(w)v − η(v, w)μ^{♯}).

This induces the following isomorphism between the Lie algebras

co(n, R)^{2} ≃ co(n, R)+ ⊃ co(n, R)_{1} ⟶^{≃} l_{0}

(κI + β, −μ̃) ⟼ a_{(κ,b,μ)} ≡ ( \begin{matrix}κ & 0 & 0 \\ μ^{♯} & β & 0 \\ 0 & μ & −κ \end{matrix} )

with κ ∈ R, β ∈ o(n, R), μ ∈ R^{n ∗}.

Moreover one sees very easily the isomorphism

(R^{n} + co(n, R) + co(n, R)_{1}, [ , ]_{−1}^{2}) ⟶^{≃} l

(w, κI + β, −μ̃) ⟼ ( \begin{matrix}κ & w^{♮} & 0 \\ μ^{♯} & β & w \\ 0 & μ & −κ \end{matrix} )

(being w^{♮} = η(w, ·)) with

[w, w′]_{−1}^{2} = 0,
[β, w′]_{−1}^{2} = βw′,  [β, β′]_{−1}^{2} = ββ′ − β′β,
[μ̃, w′]_{−1}^{2} = μ̃(w′),  [μ̃, β′]_{−1}^{2} = −μ̃^{β′} = μ̃β′,  [μ̃, μ̃′]_{−1}^{2} = 0.

The last isomorphism yields (after a straightforward computation) the correspondence

\overline{A}_{(kb,−μ̃)}^{2}(w, κI + β, −ν̃) ⟼ Ad_{A_{(k,b,μ)}}^{L}a_{(κ,β,ν)}.

Thus we conclude that

\overline{A}_{j_{0}^{2}(g)}^{2} = Ad_{j_{0}^{2}(g)}^{L}, ∀ j_{0}^{2}(g) ∈ CO(n, R)^{2};

for details see [9].

6 Final comment

As we mentioned in the introduction we are interested in the relatonships between true connections and Cartan connections on 2-jet lifts P^{2}(M, G^{2}) of 1-integrable G-structures when P is of finite type 2. The behaviour of such geometrical objects under the principal action is controlled by the above representations Ad^{G^{2}} and \overline{A}^{G^{2}}, respectively. Using such representations we have been able to prove that, given a Cartan connection on P^{2} and a symmetric connection on P, a distinguished connection on P^{2} arises which is not the trivial prolongation of the latter one; in the conformal case, the geodesics of that second order connection are closely related to the so called “conformal circles”.

References

[1] E. Aguirre-Dabán and I. Sánchez-Rodríguez, ‘On structure equations for second order connections’, Proc. Int. Conf. on Diff. Geometry and its Applications (Opava, Czechoslovakia, 92) (eds. O. Kowalski and D. Krupka, Silesian University, Opava, 1993), pp. 257-264.

[2] D. Bernard, ‘Sur la géometrie différentielle des G-structures’, Ann. Inst. Fourier of Grenoble, 10 (1960) 151-270.

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

[4] V. Guillemin, ‘The integrability problem for G-structures’, Trans. Amer. Math. Soc., 116 (1965) 544-560.

[5] S. Kobayashi, ‘Canonical forms on frame bundles of higher order contact’, Proc. Symp. Pure Math. 3 (American Mathematical Society, 1961), pp. 186-193.

[6] S. Kobayashi, Transformation Groups in Differential Geometry (Springer Verlag, Heidelberg, 1972).

[7] K. Ogiue, Theory of conformal connections, Kodai Math. Sem. Rep., 19 (1967) 193-224.

[8] R. S. Palais and C. L. Terng, ‘Natural bundles have finite order’, Topology, 16 (1977) 271-277.

[9] I. Sánchez-Rodríguez, ‘Conexiones en el fibrado de referencias de segundo orden. Conexiones conformes’, Thesis (Department of Geometry and Topology, Universidad Complutense, Madrid, 1994).

[10] P. C. Yuen, ‘Higher order frames and linear connections’, Cahiers de Topologie et Géometrie Différentielle, 12 (1971) 337-371.

Eduardo Aguirre-Dabán Departamento de Geometría y Topología Facultad de Matemáticas Universidad Complutense 28040 Madrid Spain

Email: Eduardo_{−}Aguirre@@Mat.UCM.Es

Ignacio Sánchez-Rodríguez Departamento de Geometría y Topología Facultad de Ciencias Universidad de Granada 18071 Granada Spain

Email: ignacios@@goliat.ugr.es