© Hindawi Publishing Corp. MULTILOCAL INVARIANTS FOR THE CLASSICAL GROUPS

Multilocal higher-order invariants, which are higher-order 
invariants defined at distinct points of representation space, 
for the classical groups are derived in a systematic way. The 
basic invariants for the classical groups are the well-known 
polynomial or rational invariants as derived from the Capelli 
identities. Higher-order invariants are then constructed from the 
former ones by means of total derivatives. At each order, it 
appears that the invariants obtained in this way do not generate 
all invariants. The necessary additional invariants are 
constructed from the invariant polynomials on the Lie algebra of 
the Lie transformation groups.


Introduction.
The study of invariants stands in contrast with the more structural approaches in differential geometry as, for example, formulated in the books of Kobayashi and Nomizu [13].One of the main reasons for this is that within this structural approach, invariants appear with a precise geometrical meaning inherited from the structure.When not considered in this context, invariants are, by definition, members of invariant rings of functions which, if certain regularity conditions are imposed, are nothing else but rings generated by a set of defining coordinates of the orbits of the group action on a given manifold.Moreover, when considered as member of the coordinate ring of the orbits, subspaces of a jet bundle, a constant function is not necessarily an absolute invariant.The function may depend on the coordinates with respect to which the partial derivatives are calculated and hence is not necessarily invariant under general reparameterization.For this reason the distinction is made between relative and absolute invariants [26].But, in geometry, we are, in particular, interested in invariants of a certain weight, which then determine invariant symmetric covariant tensorfields, which are coordinate independent quantities.
We start with a good reason to undertake the study of invariants.Consider the following well-known problem: "Given the action of a group G on a manifold M and two sets of points in M, when does an element exist in G which transforms the first set into the second one?" This is the classical problem of invariants.We prefer to tackle the question in a more general form.Consider two sets of points in M and/or tangent vectors and/or higher-order tangent vectors at a distinct points.Say 1 = {ξ i ,...,X| ξ j ,...,Y Y ...Y | ξ k ,...} and 2 = {η i ,...,X| η j ,...,Y Y ...Y | η k ,...}.The question then is when does there exist an element g ∈ G such that g( 1 ) = 2 and what are the general criteria?The problem, though interesting in itself, appears in problems related to the existence of invariant kernels [7] and partial differential equations and, in particular, is raised among others in computer vision [3,16,20,24] and in numerical algorithms related to symmetry [18].
One way to answer this type of question is to study invariants for the extended action of G on higher-order tangent vectors located at distinct points of the manifold.Such invariants will be called multilocal differential invariants because they are defined at distinct points of the manifold and also depend on the higher-order differential structure of curves in M through the distinct points.The natural setting for these problems seems to be multispace and multijet space.Such invariants appear under different names in the literature, in particular, as semidifferential invariants [24] and as joint invariants [19].
In this paper, we restrict attention to the local problem and to invariants which are typically curve invariants.This implies that the invariants are meant to be evaluated locally along curves.The development of the same ideas to submanifolds is not considered here.Nonetheless, it turns out that the invariants yielding the key information are tensors on jet bundles and hence, the results are suitable for generalization to the theory of submanifolds.
For each of the groups, we start with "point invariants," namely, invariant functions defined on the zero-order jet space over the manifold.For the linear transformation groups on R n , these invariants are derived from the Capelli identities and are found in the classical literature on the subject [9,25].The point invariants are, as one expects, orbit invariants on multispace, which is an r -fold product of the representation space with itself.Invariants, depending on higher-order tangent vectors, are then defined on appropriate jet bundles over these products.
The simplest example is given by the standard action of SO(3) on V = E 3 .Let Ᏹ 2 = V 1 × V 2 be the multispace.With x as variable on the first factor and y as variable on the second, the functions ᏽ(x, x), ᏽ(x, y), and ᏽ(y, y) are the well-known invariants of this action.ᏽ stands for the Euclidean quadratic form on V .The rank of multispace equals 2, which is the smallest number of factors needed for the orbits to have maximal dimension.Taking the firstorder jet bundle over the first factor gives the space J 1 V 1 × V 2 .The invariants, together with their total derivatives, yield the set {ᏽ(x, x), ᏽ(x, y), ᏽ(y, y), ᏽ( ẋ, x), ᏽ( ẋ, y)}.The set thus obtained is not maximal, one lacks an invariant function in order to have a complete set.The missing function is the wellknown Euclidean metric function on the first-order jet bundle, namely ᏽ( ẋ, ẋ).There are several ways to derive this function.(1) Because the action is linear and extends linearly to the tangent space, we may reapply the Capelli identities to the variables (x, y, ẋ).(2) We may construct the space J 1 V 1 × V 2 × J 1 V 3 equipped with the variables x, ẋ, y, z, ż.The total derivative with respect to the third factor of the invariant ᏽ( ẋ, z) yields ᏽ( ẋ, ż).The equivariant embedding  : 3 given by z = x, ż = ẋ then yields the desired invariant namely  * ᏽ( ẋ, ż).(3) Because the orbits of SO(3) are diffeomorphic to the group, they carry the invariant metric determined by the translation of the Killing form on the orbits.An invariant Riemannian metric on Ᏹ 2 is found by means of the construction of an invariant bundle transversal to the orbits.The metric results from the Killing form on the orbits and the choice of an Euclidean metric in the transversal bundle which is taken orthogonal to the orbits.Consider the generating vector fields and the invariant transversal vector fields Defining these vector fields as orthonormal determines an invariant metric.This metric, taken at the invariant subspace y = 0, restricted to the tangent bundle of the first component of the multispace Ᏹ 2 , and considered as function on J 1 V , is given by φ = (x 2 1 + x 2 2 + x 2 3 )[( ẋ1 ) is invariant, we find the extra invariant ᏽ( ẋ, ẋ).In this paper, we show that this happens at each level for all the classical groups.The same approach is then applied.
The results of the present paper can be summarized as follows.On the firstorder bundle, we find, besides the zero-order invariants and their first-order prolongations, an additional set of functionally independent invariants which, together with the former ones, generate the sheaf of invariant functions.We show that, if formulated properly, these additional invariants are related to the invariant polynomes on the Lie algebra of the transformation group.Hence, they are symmetric tensors on the orbits of the group in the zero-order jet bundle.In this sense, they relate directly to the geometry of the orbits of the group in zero-order space.The same procedure then repeats at each order.The dimension of the space upon which the group acts is crucial at each step.For each of the groups, we follow the same procedure and present some aspects of the geometry involved.
For each of the groups from (1) to (5), we examine the action of their linear part and construct the invariant sheaf.Then, by means of a homogenization procedure, the sheaf of invariants for the transitive groups is found.Remark that the following groups are semisimple: SO(n), O(n), Sp(n), Sl(n), Pl(n), and CO (1) (n).Furthermore, the results for CO(n) are derived from SO(n) and those for Gl(n) from Sl(n) by projectivization of a set of generators of the invariant sheaf.
In the first section, we review the basic results on invariants for the classical linear groups such as given in the classical book of Weyl on the subject [25] or more recently by Fulton and Harris [9].The subsequent sections then treat the higher-order multilocal invariants for the different groups.The Capelli construction of the invariants is a well-debated subject, and we refer to the literature [12,23].A different approach has been given by Olver [17,19], and Fels [8] in terms of moving coframes.The construction of the invariant metric in multispace, which we use here, is very close to this approach, although for the point invariants, we use the classical theorems.
Besides the relative invariants, which depend on the parameterization of the curves, we have to find a parameterization which is invariant under the induced action of the group.Such parameters are called G-invariant parameters [16].Clearly, the construction of such parameters is not unique; there are several types of possible parameters.For example, the parameter may be determined by the integral of the kth root of an invariant function of weight k, determined on the jet bundle.But we may also determine a parameter by means of the normalization of the prolonged curve with respect to an invariant symmetric tensor field on the jet bundle.For example, in the Euclidean case, there exists a tensor field which lives on the zero-order jet bundle determining the Euclidean arc length, but, for example, in the case of a curve in one projective space, we will show that the Schwarzian derivative, which determines a projective parameter, is nothing but a normalization with respect to the Killing form on the orbit of the projective group in the second-order jet bundle.As said in former paragraphs, we will not go into a systematic examination of the existence of such invariant parameters but will give examples of such parameters.
The general setting for this paper is the C ∞ -category of manifolds and functions.Restriction to polynomials or rational functions would be more advisable in most cases, but, for our interest, the distinction is not very important.The language of sheaf theory is used, which makes the formulation easier and more elegant, but, as the reader will notice, we could also do without it at this stage.The sheafs here are sheafs of germs of C ∞ -real valued functions, which are fine sheafs, and their global analysis is outside the scope of this paper.For the Lie groups and their algebras used in this paper, we refer to Helgason [11].

Elements of the classical theory.
Let V be the standard representation space of Gl(n) and G a subgroup of Gl(n).Let (e i ) be a basis on V ; we denote by (ξ i ) the corresponding coordinates and use ξ = (ξ i ) to indicate a point in V .Classical invariant theory aims at constructing all polynomials P (ξ 1 ,ξ 2 ,...,ξ m ) in m variables on V which are invariant under the action of G.
It is indicated to define these polynomials on the product space Ᏹ r = r α=1 V α , where each V α is a copy of V .The action of the group G on Ᏹ r is the product action.Invariant polynomes or invariant functions in m variables are polynomes or functions on Ᏹ r , depending on m factors, with r ≥ m.
Results on the invariants for the classical linear subgroups of Gl(n) are consequences of the identities of Capelli [9,25].We summarize the results in the form of a theorem.Theorem 2.1 [9].(1) The only polynomial invariants for Gl(n) and CO(n) are the constants.

The projective group Pl(n)
Gl(n + 1)/ ± e λ Id and the conformal group CO (1) (n) do not act linearly on V and hence need a different treatment.
First, consider the projective group.Let ξ α 1 ,ξ α 2 ,...,ξ αm be m variables on V and ξ k an extra variable on V .We then define (2.1) Theorem 2.4.All rational functions in where the products are taken such that for each α r the number of factors containing the variable ξ αr in the nominator equals the number of factors containing ξ αr in the denominator, are invariant for the standard action of Pl(n) on V .
Proof.The above expression is clearly invariant for translations and the action of the general linear group.For this, we first remark that, because they are invariant under translations, we may put ξ k = 0. Invariance under the special projective transformations ξj = ξ j /1 + a i ξ i is easily checked, which proves the proposition.
The group of conformal transformations CO (1) (n) is the first prolongation of the group CO(n) [10,22].The group is generated by the elements of the similarity group Sim(n) and the inversions on the n sphere.As transformation on V , the conformal group is generated by the similarity group Sim(n), together with the special conformal transformations [1] Theorem 2.5.Let (ξ α i ) be m variables on V and d the distance function defined by the quadratic form ᏽ. All rational functions in Πd(ξ α i ξ α j )/Πd(ξ β i ξ β j ), with 1 ≤ α i < α j ≤ m and 1 ≤ β i < β j ≤ m and where the product is taken such that for each α r the degree of the variable ξ αr in the nominator equals the degree of this variable in the denominator, are invariant for the standard action of CO (1) (n) on V .
Proof.The expressions in the theorem are clearly invariant for the action of the translations and the group CO(n).We only have to prove invariance for the action of the special transformations (2.3).The proof then follows from the application of the following lemma to the above functions.Lemma 2.6 [1].Let ξ 1 , ξ 2 be two points in V .Then, (2.4)

3.
Multispace.Let V be as before and Ᏹ = r α=1 V α , the product of r copies of V .Ᏹ is called multispace constructed upon V .We denote by (ξ i α ) the coordinates on the αth copy in the product and call V α the α-layer of Ᏹ.The projection ρ α : Ᏹ → V α is the mapping onto the αth component.The identification of V with a specific layer is called the layer mapping, which is given by i α : V → V α , and the image projection π is defined as the superposition of the different layers upon V .In other words, π is the r -valued mapping defined by the set of r functions (ρ α ) of Ᏹ followed by the natural identification of each layer with V .Remark that πoi α = Id for each α.
The order of multispace Ᏹ equals the number of layers of Ᏹ.When necessary, we indicate the order of multispace explicitly and use Ᏹ r .
Calling kl : ξ k = ξ l a diagonal of Ᏹ r , we define the following operations: (1) the permutation mapping of two layers s kl : Ᏹ → Ᏹ, which permutes the layers V k and V l and (2) the diagonal mapping  kl : Ᏹ r −1 → Ᏹ r , which is the injection defined by (ξ α ) (ξ 1 ,...,ξ k ,...,ξ k ,...,ξ r −1 ) identifying Ᏹ r −1 with the diagonal kl .Any multispace possesses, in a natural way, sets of commuting vector fields which are useful in the construction of invariant.The following lemma is one example.
Lemma 3.1.Let (ξ i α ), α = 1,...,r , i = 1,...,n, be coordinates on Ᏹ r .Then, for all k, l, α, Proof.A simple calculation gives We call a point ξ in multispace Ᏹ r generic if the image projection π(ξ) of ξ consists of r distinct points in V .Let ξ o be a point in V , then there exists a canonical lift to Ᏹ r which consists of the unique point ξ o = (ξ o ,ξ o ,...,ξ o ).We call such a point ξ o ∈ Ᏹ r diagonal or canonical.
Remark that an arbitrary curve in Ᏹ r describes the evolution of r points in V .A generic curve in Ᏹ is, by definition, a curve containing a dense subset of generic points.A generic curve is locally projected upon r distinct curves in V by the image map.On the other hand, if a curve in V is given, many different lifts to Ᏹ r are possible.First of all, if ξ = γ(t) is a curve in V , then it is easy to see that there exists a unique curve γ(t) in Ᏹ r such that for each α, ρ α (γ(t)) = γα (t) = γ(t).We call such curves diagonal or canonical.It describes the evolution of r coinciding points in V .
In general, we would like to have a parameterization of the layer curve different from the one of the original curve in V .This brings us to the following definition.Definition 3.2.Let γ(t) be a curve in V and (ξ i ) a set of r points in V , all belonging to γ(t).Then, an α-layer curve γ, associated with γ(t), is a curve γ(t) : t (ξ 1 ,ξ 2 ,...,ξ α = γ(φ(t)),...,ξ r ), where φ is a reparameterization of γ.
The construction of a general lifting of a curve is, in fact, a kind of superposition of r layer curves.Hence, we have the following definition.Definition 3.3.Let γ(t) be a curve in V .Then, γ(t) in Ᏹ is a lift of γ(t) if the image of the image projection of γ(t) equals the image of γ(t).
Let (ξ i α ) be the coordinates on V α and ξ i = γ i (t) a curve in V .A lifted curve on Ᏹ is given by where φ is a reparameterization in the domain of the curve γ.We remark that parameterization may differ from layer to layer.Hence the lift, as a parameterized curve, is not unique; indeed, we accept the freedom to move the distinct points with a different speed along the curve.Hence, keeping the r − 1 points ξ β , with β ≠ α for a given α, fixed and ξ α as the running point determines a layer curve associated with γ(t).

Jets of curves in multispace.
Consider the set of germs of curves in V and denote this space by C(V ).Recall that germ equivalence of two curves is defined as γ 1 (t) ∼ t o γ 2 (t) if and only if both curves coincide on a neighborhood of t o .The set of germs is the sheaf of germ equivalent curves on R with values in V .
Jet equivalence is then introduced as an equivalence on C(V ).Let γ 1 (t) and γ 2 (t) be the representatives of two germs in C(V ), both are k jet equivalent if their derivatives up to order k coincide at t.This equivalence clearly does not depend upon the choice of the representatives.We generally indicate a germ by a representative and the k jet of a curve germ by j k t (γ).It is assumed that the germ is taken at the point t.The set of k jets of germs of curves in V is the space J k (R,V ).The set of infinite jets J(RV ), is defined as the inverse limit of the sequence (J k (R,V )) k .This space admits a ring of C ∞ -functions Ᏺ(J(R,V )), defined by the direct limit of the sets {Ᏺ(J k (R,V ))} with respect to the projections π k : J(R,V ) → J k (R,V ).In the sequel, we consider each set Ᏺ(J k (R,V )) as a natural subset of Ᏺ(J(R,V )).A section σ : R → J(R,V ) is called integrable if σ equals the jet of a curve at each of its source points.
Let α : J k (R,V ) → R be the source map defined as j k t γ t.The fibre α −1 (0) is a bundle over the target space V .We call this bundle J k (V ).This bundle is equipped with the projection map β : J k (V ) → V by j k 0 (γ) γ(0).We denote elements in J k (V ) by j k (γ).The bundle of infinite jets is denoted by J(V ).Again, we call a curve σ : R → J(V ) integrable if it is the infinite jet of a curve in V at each of the points of β(σ (t)).Let ∂ t be the derivative operator on R; the total lift of this operator, ∂ h t , as an operator on Ᏺ(J(R,V )), is given by Let X ∈ ᐄ(V ) be a vector field with flow φ t .Because φ t is a local diffeomorphism on V , composition with curve mappings defines the prolonged action of the local diffeomorphism on the jet space J k (V ).Let γ(s) be a curve, then φ t (γ(s)) is the composition with the local diffeomorphism for a fixed t and the action on J k (V ) is given by φ This determines what is called the complete lift X (k) , generator of φ (k) , on J k (V ).The following proposition, which is well known in jet bundle theory [5,14,15], gives a constructive definition.
Definition 4.1.Let X be a vector field on V .Then, the complete lift X (k)  is the unique vector field on The above definitions extend in a natural way to multijet space.Let γ(t) be a curve in V and γα (t) an α-layer curve corresponding to This operation is also defined on general curves γ in Ᏹ r .The jet extension is applied to the αth component of the curve.It means that we consider the curve as a layer curve keeping all other components fixed.Moreover, we may extend each component γα to a specific jet order k α .This brings us to products of jet bundles over V , which is a bundle over the base manifold Ᏹ r .This space takes the form of the product and is called multijet space.Points in this space are multijets of curves in V .The component J kα V α is the α-layer jet bundle.Let γ(t) be a curve in Ᏹ r , and let p be a generic point on γ(t).A multijet of γ(t) at p defines, via the image projection, r points together with tangents up to a given order of the curve γ(t) at each of the points.
The bundle (k 1 ,...,kr ) (Ᏹ r ) is a subbundle of J k (Ᏹ r ), where k = max{k 1 ,...,k r }, but the jet bundle operations are defined on each layer jet bundle.A section σ of (k 1 ,...,kr ) (Ᏹ r ) can be written as σ = (σ α ).The section is integrable if and only if each σ α is integrable.Each layer carries a total derivative operator, which we denote by T α .Let p = (σ 1 (t o ),...,σ r (t o )), we then have , where f is any function on multijet space.
The complete lift of a vector field on Ᏹ r on a jet bundle over a specific layer J kα (V α ) is defined as in Definition 4.1 where the total derivative is T α .

Let (ξ (lα)i α
) be the natural coordinates on (k 1 ,...,kr ) (Ᏹ r ) and X = A i α ∂ ξ i α a vector field on Ᏹ r , then is the total lift of the vector field.The operations which have been determined on multispaces extend to multijet spaces.Call J kα V α the jet bundle over the αth layer.
(1) Let the jet bundles over the layers V k and V l be of same order.The permutation mapping ι kl : (Ᏹ) r → (Ᏹ) r is a mapping which permutes the two layers together with their jet bundles J m V k and J m V l .
(2) A diagonal in multijet space kl is defined as follows.Let J m k V k and J m l V l be two jet bundles over different layers in (Ᏹ) r .Suppose that m k ≤ m l .Then, the diagonal subspace is given by kl : J m k V k = J m k V l .At kl , the m k jets over both components are identified.A diagonal mapping is an injection
Let g be the Lie algebra of G and (e a ), a = 1,...,dim g, a basis.The vector field κ : e a X a = Dτ(e a )| e , with e the identity in G, is a fundamental (or generating) vector field.Given the action of G on Ᏹ r , we define the singular subset Σ G = {ξ ∈ Ᏹ r | rk{X a } < dim g}, where X a is a complete set of fundamental vector fields.The regular subset is the complement Let M be a manifold and [20].The conditions for a set of functions to be functionally independent on a manifold are those of implicit function theorem asserting the existence of a local submanifold at p, whose tangent space is the kernel of the set of one forms {df i }.The rank k of equals the codimension of the submanifold.
Because the invariant sheaf is determined by its sections, we only need to give at each point a complete functionally independent set of germs of invariant C ∞ -functions.Such a set will be called a set of generators.We agree not to mention the passage from functions to germs.To simplify notation, we do not mention the base points of the germs either and will give a minimal set of functions needed in the construction of a maximal functionally independent set of sections of the sheaf at each point in the domain of the sheaf.Such a set of generators will be called complete.
Let X a be a fundamental vector field, then f ∈ Ꮽ(ᐃ) → X a (f ) = 0, for all a.The converse is not always true.It follows that, on a regular subset ᐃ, the rank of the sheaf of germs of C ∞ -invariant functions equals the codimension of the level surface of any set of generators.
Consider the projective equivariant system The orbit dimension is a nondecreasing function on the orbits.Let be the set of maximal orbits.We then define the order of stabilization of the action of G in the limiting system defined by Ᏹ r , for r large enough, as the rank k such that for all l > 0 : dimᏻ k+l = dim ᏻ k , for orbits in .Remark 5.2.(1) Let k be the order of stabilization of G in Ᏹ r .Then, as a consequence of the dimensional requirements, we have rkᏭ (2) Let P (ξ 1 ,ξ 2 ,...,ξ m ) be a generator in Ꮽ(ᐃ k ).Then, is a generator in Ꮽ(ᐃ m+1 ) which does not belong to Ꮽ(ᐃ m ).
The product action of a Lie group G on Ᏹ prolongs in a natural manner to any multijet bundle (Ᏹ).Using the natural prolongation of one-parameter local diffeomorphisms on the target space to the jet bundles, we find the prolongation of fundamental vector fields of a group action.Let X a be a fundamental vector field on V ; with respect to the coordinates we set 3) The corresponding fundamental vector field on Ᏹ r becomes where A i a,α = A i a (x i = ξ i α ).The prolongation of the fundamental vector field onto the multijet bundle (k 1 ,...,kr ) (Ᏹ r ) = α J kα V α is obtained from (4.2).

The first-order transformations groups
6.1.The linear subgroups of Gl(n)

Invariants for O(n)
. Let e = (e i ), i = 1,...,n, be an orthonormal basis, (ξ i ) the corresponding coordinates on V , and ᏽ the Euclidean quadratic form on V .

Zero-order invariants
The following properties are elementary.We present them without proof.
(2) The order of stabilization equals r = n − 1.Moreover, if r = n − 1, the polynomial invariants for SO(n) coincide with those for O(n) as a consequence of Theorem 2.1. ( It follows that the subset ᐃ in Ᏹ n−1 is foliated by orbits of SO(n), and the leaves are the level surfaces of Ᏽ ᐃ .Let Ꮽ(ᐃ) be the sheaf generated by Ᏽ ᐃ in Ᏹ n−1 .Remark 6.2.Because Ᏹ r is the zero-order level in jet space, a curve γ(t) in ᐃ is lying in an orbit if and only if the invariant sheaf Ꮽ(ᐃ) restricted to γ is constant.
Let ξ = {ξ 1 ,...,ξ n−1 } be a set of n − 1 points in V .The isotropy group at ξ n−1 is the group SO(n − 1), and, by recursion, the isotropy group leaving k points fixed, say {ξ n−k ,ξ n−k+1 ,...,ξ n−1 }, is the group SO(n − k) [2].From this, it follows that, if γ(t) is a nowhere constant layer curve such that the invariant sheaf restricted to γ(t) is constant, the image of γ(t) is locally the image of an orbit of SO (2).
Letbe a level surface of the invariant sheaf Ꮽ (1) (ᐃ ).Then,is foliated by orbits of SO(n).As a consequence of former Proposition 6.3, the codimension of the orbits inequals n − 1.This implies that we need n − 1 extra independent generators to construct the complete invariant sheaf.
In order to derive n − 1 extra invariants, we construct an invariant normal bundle N(ᐃ), as subbundle of T ᐃ.This bundle is a vector bundle whose sections are transversal to the leaves and which is invariant as a subbundle under the action of SO(n).
The normal bundle N(ᐃ) is the subbundle of the tangent bundle spanned by the vector fields Y a .The fibre of the bundle N(ᐃ) is isomorphic to R n(n−1)/2 .Because the action of SO(n) is effective on ᐃ, the group SO(n) acts trivially on N(ᐃ).Let L(ᐃ) be the subbundle of T ᐃ which is tangent to orbits and hence is spanned by the fundamental vector fields X a .We define an invariant metric on ᐃ as where K ab is the Killing form on the Lie algebra so(n).
The lift of the semi-Riemannian tensor field q2 is a function q2 on the tangent space T ᐃ.Using the natural embedding the function ı * q2 is an invariant function on ᐃ .
From the invariant metric, we construct invariant layer metrics.Let qα 2 be the restriction of the metric q2 to the fibre of the α layer The function qα 2 = ı * qα 2 will, by abuse of language, be called the α-layer metric on J (1,1,...,1) Ᏹ n−1 .
Proof.Because differentials of the layer metrics with respect to the coordinates ξα are linear in the fibre coordinates ξα , we find functional independence of the layer metrics with respect to the sheaf generators of Ꮽ (1) (ᐃ ).Moreover, they are functional independent among each other which proves the proposition.
Remark 6.11.The expression ᏽ( ξα , ξα ) is a tensor field on the α layer for each α and determines the well-known Euclidean arc length parameter along each layer curve.
Higher-order invariants.Before formulating the general case, we consider in detail the one-layer case and work by induction on the order.Let for 1 ≤ l < n− 1 and let ᐃ l = ᐃ (l,0,...,0) be the regular subset.The following theorem is a consequence of the identities of Capelli and the linear action of the prolonged action.

The rank of Ᏽ
(1) Using the unique extension of the sheaf on ᐃ l+1 , we find that one generator is missing for the construction of the invariant sheaf on ᐃ l+1 .For the construction of the extra generator, we follow the same procedure as in the first-order case.Because the group action of SO(n) extends linearly to the jet bundle, the same construction for the invariant transversal bundle can be applied at each order.From the invariant Killing metric on the orbits and an invariant metric in the normal bundle, we construct a layer metric on the first layer of multijet space (l+1,0,...,0) Ᏹ n−l−1 .The construction is analogous to the constructions in Propositions 6.4 and 6.5.
Restriction of this metric to the tangent space to the jet bundle over the first layer gives us a layer metric.This metric lifts as a function on ᐃ l+1 denoted by q1 2 .We then have the following theorem.
Top-order invariants.The above construction works as long as n − l − 1 > 1.Then, l = n − 2 determines what we call the top order.Proceeding as above, we have the following diagram: Notice that dim J n−2 V equals n(n − 1).Let ᐃ n−2 be the subset on which the fundamental vector fields are linear independent.The sheaf Ꮽ n−2 (ᐃ n−2 ) is constructed the same way as before.Again, prolongation of the sheaf to J n−1 V does not generate the invariant sheaf.An additional generator is constructed from the Killing form on the involutive distribution spanned by the fundamental vector fields, together with a metric in a normal invariant distribution spanned by a set of normal vector fields Y a as before.This determines an invariant metric q2 on ᐃ n−2 .Let ᐃ n−1 be the open set in J n−1 V on which the fundamental vector fields are linear independent.Let Ꮽ(ᐃ) n−1 be the invariant sheaf generated in this way.The following theorem shows that we are really dealing with the top invariants.Theorem 6.15.Let Ꮽ(ᐃ n−1 ) be the invariant sheaf as defined above.Then, the invariant sheaf on ᐃ n+l−1 is generated by the lth prolongation of Ꮽ(ᐃ) n−1 for l > 0.
The theorem is a consequence of the dimensions of the jet spaces involved.Remark there is no restriction here because multispace contains only one layer.

Invariants for CO(n)
. The connected component of the identity of the group CO(n) is generated by SO(n), together with the dilatations d λ : ξ e λ ξ with λ ∈ R. Let ᐃ ⊂ Ᏹ n−1 be the subset of all sets of n − 1 linear independent vectors in V , and ᏼ(ᐃ) the quotient space of ᐃ by the action of d λ .
Hence, the projectivization of the set is a set of generators of the invariant sheaf for CO(n).The same method determines generators for the top-order invariants.Their order equals n − 1.We omit their derivation.
Remark 6.16.The invariant ᏽ( ξ1 , ξ1 )/ᏽ(ξ 1 ,ξ 1 ) is an invariant tensorfield on the first layer determining an invariant parameterization along first-layer curves.Remark 6.17.Dimension n = 2 is exceptional.The sheaf of zero-order invariants is the constant sheaf.The group acts prehomogeneously on V , and the origin is the singular part.Because a set of generators, is given by with ᐃ = J 1 V \{π −1 (0)}, which generate the top-order invariants.Because the group CO( 2) is Abelian, the geometric structure is locally flat.

Invariants for Sp(n).
Let ᏽ be the skew-symmetric form on V determining the symplectic structure, and e a basis of V .The dimension of V is even and equals 2n.Recall that the dimension of the group equals n(2n + 1).
The level surfaces of Ᏽ ᐃ are submanifolds of constant dimension and, consequently, are disjoint unions of orbits.The generated sheaf is invariant and has rank n(2n−1).Each orbit is a pseudo-Riemannian manifold which inherits its geometric structure from the semi simple group Sp(n).
Higher-order invariants.We show how to construct first-order invariants; then, higher-order invariants are derived by repeating the construction the way it has been done for the orthogonal group.
The fundamental vector fields on ᐃ are given by (6.28) Define the following vector fields on (6.29) Proposition 6.21.The vector fields are linear independent on ᐃ.
Let q 2 be the pseudo-Riemannian metric which, in the level sets, is given by the Killing form on the Lie algebra sp(n) and, in the normal space, by an Euclidean metric.The restriction of this metric to the tangent space of the first layer defines the layer metric q 1  2 .The lift of this metric, q1 2 , as a function on (1,0,...,0) (ᐃ), is an invariant function which is functionally independent from the set (6.27).

Invariants for Sl(n)
Zero-order invariants.Let V be of dimension n, and consider the standard action of Sl(n) on V .The key invariant for Sl(n) is the determinant |ξ 1 ,...,ξ n |.
The following lemma shows that the order of stabilization equals n in multispace Ᏹ r .Lemma 6.24.Let r < n, and ᏻ an orbit of maximal dimension in Ᏹ r .Then, dim ᏻ < n 2 − 1. (6.33) Proof.The proof follows from dim Ᏹ r ≤ n(n − 1) for n > 1 and r < n.
The regular subset ᐃ of Ᏹ n is defined by the set of points ξ = (ξ 1 ,...,ξ n ) which is linear independent.Let Ꮽ(ᐃ) be the invariant sheaf generated by Ᏽ ᐃ = {|ξ 1 ,...,ξ n |}.All orbits in ᐃ are of maximal dimension.Let Γ be a connected component of a level set of Ᏽ ᐃ , then Γ is an orbit of Sl(n) and hence is a connected pseudo-Riemannian manifold.
Let Z a = {X 1 ,X 2 ,...,X n 2 −1 ,Y }, and set Z a = A αi a ∂ ξ i α .We define the one forms ω a = B a αi dξ i α , where B = (B a αi ) is the inverse matrix of A = (A αi a ).Using the chosen basis for the Lie algebra determines the canonical one form In particular, for any layer curve γ(t) in ᐃ with running point in the first layer, we find ω(γ(t)) ∈ sl (n)⊕R.Remark that γ(t) is a point in ᐃ for fixed t.

Invariants for Gl(n)
. The connected component of the identity of the general linear group is generated by Sl(n), together with the dilatations d λ : ξ e λ ξ with λ ∈ R. Let ᐃ ⊂ Ᏹ n be the subset of all sets of n linear independent vectors in V .The general linear group acts prehomogeneously on ᐃ.Because the open orbit has maximal, the stabilization order of Gl(n) equals n.Hence, the zero-order invariant sheaf is constant.

The transitive transformation groups.
We start with the Euclidean case and then indicate how the same constructions are applied to the other transitive groups.The central key in the construction is the introduction of an extra layer in a multispace and the homogenization of the invariants with respect to the zero-order coordinates in order to make them invariant under translations.
Proof.Let X a be the generating vector fields on Ᏹ.The matrix of the components in terms of the coordinates is (A i aα ), which takes the form where each A α is the matrix of the fundamental rotation vector fields in the layer V α .The proof follows from a rearrangement of the matrix such that

.46)
Zero-order invariants.Using the results for the orthogonal group on ᏺ, we find that the set Ᏽ ᐂ = ρ * Ᏽ ᐃ is functionally independent.Ᏽ ᐃ is the set given by Proposition 6.1.Let Ꮽ(ᐂ) be the sheaf generated by Ᏽ ᐃ .Because the level surfaces of Ᏽ ᐃ are orbits of SO(n), the inverse image of an orbit of SO(n) in ᐃ by ρ is an orbit of Eucl(n) and a level surface of Ᏽ ᐂ .Hence, the regular subset ᐂ is foliated by orbits of Eucl(n).
Top-order invariants.Consider next the case l = n − 1.For the space (l,0) (Ᏹ 2 ), we have the invariant sheaf Ꮽ(ᐂ), with ᐂ the regular subset of the jet space.Construction of the first layer prolongation yields Ꮽ (1) (ᐂ (1) ), with ᐂ (1)  the inverse image of ᐂ.Consider next the diagonal embedding  : J (n−1) V → J (n−1) V × V , which is equivariant.The pullback of the set of generators of Ꮽ (1) (ᐂ (1) ) by  generates the invariant sheaf on the regular subset in J (n−1) V .We omit the construction.

Invariants for the similarity transformation group.
The similarity group is generated by CO(n), together with the translations.We use the same methods as for the Euclidean group.At zero order, we consider sets ξ = {ξ 1 ,...,ξ n } of n points in V such that the sets = {ξ pulls back the generators (6.24).For the higher-order invariants, the same procedure applied to the set (6.25) gives the desired results.We omit the details.Remark 6.33.The invariants ᏽ( ξα , ξα )/ᏽ(ξ α − ξ β ,ξ α − ξ β ), for β ≠ α, determine invariant parameters along layer curves except for the top order where only one layer is available.

Invariants for the symplectic transformation group.
The homogenization construction in this case is identical with the one used for the Euclidean group.We omit their construction.

Invariants for the volume-preserving transformation group.
The volume-preserving group is generated by Sl(n), together with the translations.Via the map ρ : Ᏹ n+1 → Ᏹ n as η α = ξ α − ξ n+1 , the pullback of the sets of invariants given in Proposition 6.25 determines a set of invariant generators on ρ (−1) ᐃ.For the higher-order invariants, we follow the same construction using the generators formulated in Proposition 6.26.To construct the top-order invariants, consider the jet bundle J n V × V and the sheaf of invariants under the action of Sl(n).Let Ꮽ(ᐃ) be this sheaf.Construct the prolongation Ꮽ (1) (ᐃ (1) ) with ᐃ (1) = π (−1) ᐃ.The pullback along the following maps yields the desired result: , where  is the diagonal embedding and κ the homogenization map η = ξ 1 − ξ 2 .Homogenization of the invariant parameters for the Sl(n) action yield invariant parameters.The same remains true at the top order.

Invariants for the affine transformation group.
At the zero order, we take multispace Ᏹ n+1 of rank n + 1.Let ρ : Ᏹ n+1 → Ᏹ n be the projection map defined by η α = ξ α −ξ n+1 , and consider the preimage of ᐃ ⊂ Ᏹ n .Because ᐃ is an orbit of Gl(n), we find that ρ −1 ᐃ is an orbit of the affine group.The invariant sheaf is consequently the constant sheaf.
Proof.The set ∆(k 1 ,...,k n+1 ) = 0 is invariant under translations; hence, by means of a translation, we choose ξ n+1 = 0. Using the transformation formula we find which proves statement (a).To prove (b), we remark that the set of fundamental vector fields drops rank at ξ if and only if ξ ∈ Σ.
Proof.At each point of ᐃ, the set of fundamental vector fields span the tangent space.The singular subset Σ is closed in Ᏹ n+1 ; hence, ᐃ is an open subset.On the other hand, it is not hard to prove that, given two points ξ1 and ξ2 in ᐃ, there exists an element in g ∈ Pl(n) such that ξ2 = g • ξ1 .It suffices to show that the point (δ i 1 ,δ i 2 ,...,δ i n , n k=1 δ i k , 0) can be transformed into any (ξ 1 ,...,ξ n ,ξ n+1 , 0).Using the transformation (7.2), we find that for k = 1,...,n.Rewriting this system as we find that the determinant of this system is different from zero on ᐃ, and hence, the system yields an unique solution.
Because the subset ᐃ is an orbit of the projective group, it inherits all the properties of the group.The set of fundamental vector fields {X a } has rank n(n + 2) on ᐃ.Again, we define X a = A αi a ∂ ξ i α .Let B = (B a αi ) be the inverse matrix of A = (A αi a ).The one form ω = B a αi dξ i α ⊗ e a , (7.6) where (e a ) is a basis for the Lie algebra pl (n), determines an isomorphism ω : As a consequence, the n canonical ad-invariant forms {q 2 ,...q n+1 } on pl (n) define the invariant symmetric tensors for i = 2,...,n+ 1.The form Φ 2 defines a pseudo-Riemannian metric on ᐃ.
The construction applies to all higher orders.
The top-order invariants are obtained for l = n+2.Remark that, in this case, the tangent space to the first layer coincides with the tangent bundle of the jet bundle.The lifts of the ad-invariant forms are nothing else but their lifts to the tangent bundle of the group.Remark 7.7.In each case, the function j * Φ2 determines a semi-Riemannian metric on the appropriate jet bundle over the first layer.Normalization of a layer curve with running point in the first layer fixes a projective invariant parameter on the curve.In case n = 1, this parameter is determined by the Schwarzian derivative as will be shown in (7.14).
We present some results in low dimensions.
Projective transformations on E 1 [6].Consider the action of Sl(2) on E ≡ E 1 = R.Let (∂ x ,x∂ x ,x 2 ∂ x ) be fundamental fields on E.
(1) Consider Ᏹ 3 = E x × E y × E z .The fundamental vector fields on Ᏹ 3 are The singular subset is given by vanishing of the Vandermonden determinant Σ : (y − x)(z − x)(z − y) = 0. Consider a regular layer curve γ(t) with running point y(t) and lying in the regular subset ᐃ = Ᏹ 3 ⊂ Σ.Then, the norm of the tangent taken with respect to the layer metric g is given by from which it follows that the curve is always lying outside the null cone of the metric.
(2) Consider (Ᏹ 2 ) = E x × J 1 E y .The fundamental vector fields are The singular subset is given by Σ = (y − x) 2 y = 0. Again, we consider a regular layer curve γ(t) with running point y(t) lying outside Σ.Then, with g the layer metric.This curve is either lying outside the null cone or on the null cone of the metric.
(3) Finally, consider (Ᏹ 1 ) = J 3 E x .The fundamental vector fields are now and singular subset Σ : (x ) 3 = 0. Consider a regular curve γ(t) in E x .Then, which lies inside or outside the null cone of the metric.The last invariant equals minus 4 the well-known Schwarzian derivative in projective geometry [21,26].Notice that any regular curve in E is in the regular subset of the projective action.
Remark 7.8.In each of the cases, we have a parameter along the curve such that g(γ , γ ) = constant which determines a projective invariant parameter.
Projective transformations on E 2 .In this section, we study the projective motion of points in E ≡ E 2 = R 2 .Let x = (x 1 ,x 2 ) be the standard coordinates on E.
The one-parameter groups generated by the standard basis vectors in Pl(2) define through the action on E the set of fundamental vector fields on E given by (7.15) The invariant polynomials on the Lie algebra are given by the coefficients of the polynomial in λ .16)This yields the quadratic form q 2 = d 2 − de + e 2 + bf + ah + cg (7.17) and the cubic form The Killing form equals 12 times q 2 as follows from a direct calculation.The prehomogeneous spaces on which Pl(2) acts and which are used as reference spaces are (1) From the quadratic and the cubic forms constructed for the prehomogeneous spaces, we are able to derive invariant generators on appropriate multijet spaces using the equivariant lifting procedure given in the chapter on jet bundles.

Higher-order invariants.
In case n = 2, the conformal group acts prehomogeneously on the space Ᏹ 3 .Let ᐃ 1 be the regular subset of (1,0,0) Ᏹ 3 , and ω the canonical one form on ᐃ with values in the Lie algebra co (1) (2).Define Φ 1 = Tr ad(ω) 2 , Φ 2 = Tr ad(ω) 4 ; (7.26) then, the set where Φ1 1 , Φ1 2 are the functions which are the lifts of the restriction of the symmetric tensors Φ 1 and Φ 2 to the tangent space to the first layer, is a set of invariant generators.
Let n > 2. In this case, the regular subset of multispace Ᏹ n+1 is foliated by orbits of CO (1) (n).In order to construct all invariants, we need the following construction.
Let ᐃ be the regular subset in multispace ᐂ = n+1 α=1 E (1,n+1) and let (z i α ), with i = 0, 1,...,n + 1 and α = 1,...,n + 1, be the standard coordinates.The following set of vector fields are fundamental fields for the action of O + (1,n+ 1): Remark that the generators of the group action on Ᏹ n+1 are found by projection of the generators X ij and X i along the generators of the null cones taken over σ .The projection along the generators of the null cone may be seen as the projection along the z 0 α -axis followed by a radial projection on the tangent planes to the n spheres.The generators X ij for i, j = 1,...,n are the usual rotations in the layers of Ᏹ n+1 .The generators X n+1,i for i = 1,...,n are the translations.The generators X i for i = 1,...,n are the special conformal transformations and X n+1 are the dilatations.The generators X ij are lying in the plane P and are tangent to σ .Now, consider the fundamental vector fields over the subspace ᏺ + at the section z 0 α = 1 defined by σ .The vector fields X ij are tangent to σ by construction.We will construct the projection of the fields X i .Denote the projection by π .But the following set of vector fields commute with the fundamental fields and are transversal to the orbits in ᐃ : We omit the proof of following proposition.
The existence of these vector fields allows the construction of an invariant metric on σ .We proceed as in the orthogonal case and require the normal

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5} is a set of invariant generators. Proposition 7 . 10 .
Provided r ≥ 3, the number of generators in the set Ᏽ ᐃr equals r (r − 3)/2.Proof.The number of functions in the set Φ α equals α − 1.Hence, the number of functions in Ᏽ ᐃ n+1 equals r −2 i=2 i, which proves the proposition.

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