2 Jet prolongations of L

J´an Brajerˇc´ık

Abstract.The aim of this paper is to characterize all second order tensor- valued and scalar differential invariants of the bundle of linear framesF X over an n-dimensional manifold X. These differential invariants are ob- tained by factorization method and are described in terms of bases of invariants. Second order natural Lagrangians of frames have been charac- terized explicitly; ifn= 1,2,3,4, the number of functionally independent second order natural Lagrangians isN = 0,6,33,104, respectively.

M.S.C. 2010: 53A55, 58A10, 58A20, 58A32.

Key words: Frame; differential invariant; equivariant mapping; differential group;

natural Lagrangian.

1 Introduction

Throughout this paper, aleft G-manifoldis a smooth manifold endowed with a left action of a Lie groupG. A mapping between two leftG-manifolds transforming G- orbits into G-orbits is said to be G-equivariant. As usual, we denote byR the field of real numbers. Ther-th differential group L^r_n ofRⁿ is the Lie group of invertible r-jets with source and target at the origin 0 ∈ Rⁿ; the group multiplication in L^r_n is defined by the composition of jets. Note that L¹_n =GLn(R). For generalities on spaces of jets and their mappings, differential groups, their actions, etc., we refer to [6, 11, 13].

LetP andQbe two leftL^r_n-manifolds, andU be an open,L^r_n-invariant set inP. A smoothL^r_n-equivariant mapping F :U →Qis called adifferential invariant. IfQ is the real lineR, endowed with the trivial action ofL^r_n, an equivariant mappingF is called ascalar invariant.

LetX be ann-dimensional manifold. By anr-frameat a pointx∈X we mean an invertibler-jet with source 0∈Rⁿ and targetx. The set ofr-frames together with its natural structure of a principalL^r_n-bundle with baseX is denoted byF^rX, and is called thebundle of r-frames overX. For r= 1, we get thebundle of linear frames, and writeF¹X =F X. IfS is a leftL^r_n-manifold, then the bundle with type fibreS, associatedwithF^rX is denoted byF_S^rX.

Balkan Journal of Geometry and Its Applications, Vol.15, No.2, 2010, pp. 22-33 (electronic version);∗

pp. 14-25 (printed version).

°c Balkan Society of Geometers, Geometry Balkan Press 2010.

If S is a leftL¹_n-manifold, we denote by T_n^rS the manifold ofr-jets with source 0∈Rⁿ and target in S. For finding differential invariants of frames it is convenient to realizeF X as a bundle with type fibreL¹_n, associated with itself. Then, ther-jet prolongationJ^rF X ofF X can be considered as a fibre bundle with type fibreT_n^rL¹_n, associated withF^r+1X.

For characterizingnaturalLagrangians onJ^rF X, i.e. Lagrangians invariant with respect to alldiffeomorphisms ofX, it is sufficient to describe all differential invariants defined on the type fiberP =T_n^rL¹_nofJ^rF X. The aim of this paper is to give explicit characterization of second order natural Lagrangians.

Most of differential invariants with values inQappearing in differential geometry correspond with the case when Q is an L¹_n-manifold. These differential invariants can be described as follows. LetK_n^r,s be thekernelof the canonical group morphism π_n^r,s:L^r_n →L^s_n, wherer≥s. IfL^r_n acts onQ via its subgroupL¹_n, eachcontinuous, L^r_n-equivariant mappingF :U →Qhas the form F =f ◦π, whereπ:P →P/K_n^r,1 is the quotient projection,P/K_n^r,1is the space of K_n^r,1-orbits, andf :P/K_n^r,1→Qis a continuous,L¹_n-equivariant mapping. Indeed, in this scheme P/K_n^r,1 is considered with the quotient topology, but is not necessarily a smooth manifold. The quotient projectionπis continuous but not necessarily smooth. IfP/K_n^r,1has a smooth struc- ture such that πis a submersion, we call πthe basis of differential invariants on P (for more details of a basis, see [12]). The general concepts on equivariant mappings, related with a normal subgroup of a Lie group, and corresponding assertions with the proofs can be found in [4, 8].

A method based on this observation was first time applied to the problem of finding differential invariants of a linear connection in [8]. The initial problem was reduced to a more simple problem of the classical invariant theory (see e.g. [14, 15]) to describe allL¹_n-equivariant mappings from P/K_n^r,1 to L¹_n-manifolds. Our aim in this paper is to study invariants of linear frames by the same method, which allows us to simplify expressions of the action ofL^r_n onP.

In this paper, we first introduce the domain of second order differential invariants with values inL¹_n-manifolds, which is, according to the prolongation theory of mani- folds endowed with a Lie group action (see e.g. [6, 7, 11]), theL³_n-manifoldP =T_n²L¹_n. Then we describe the frameaction of L³_n on T_n²L¹_n. This action corresponds to the second jet prolongation of the frame lift of a diffeomorphism ofX. Using a tensor de- composition, we also construct the corresponding orbit space of the normal subgroup K_n^3,1ofL³_n. We show that this orbit space can be identified with Cartesian products of L¹_n with some tensor spaces over Rⁿ; in this way the differential invariants with values in L¹_n-manifolds are described in terms of their basis. Note that the second order differential invariants with values inL²_n-manifolds can be easily obtained by the same manner.

These results are subsequently used for extension of the theory of the first order differential invariants of frames in [5], studied in terms of integrals of canonical dif- ferential system, to the second order case. Applying factorization method, we give an explicit characterization of second orderscalar invariantsof frames andLagrangians, defined onJ²F X, invariant with respect to all diffeomorphisms on X. In Section 8 we also introduce the concept ofcanonical odd n-formon F X, where n= dimX, which gives an exact description of globally defined invariant object in the role of volume form. In [5], (ordinary)n-form was considered, which is only available over

orientable manifolds. We also extend the remarks of the authors on the use of differ- ential invariants as Lagrangians overnon-orientablemanifolds. For the theory of odd de Rham forms and odd base forms we refer to [9].

The calculations in this paper rely on the jet description of differential invariants.

This is based on the existence of a Lie group (the differential group) whose invariants are exactly the differential invariants. Such description implies, in particular, that the arising theory is comparatively simpler than other versions of the theory of differential invariants.

Using another left action of L¹_n on itself, called coframe action, it is possible to obtain the corresponding differential invariants of coframes. Note that they can be obtained by the same method. For differential invariants of coframes, see [3]; it represents an extension of Ph.D. thesis of the first author, devoted to the second order case, to the third order case.

There are several types of invariance of Lagrangians on frame bundles. One of them is invariance with respect to the canonical action of L¹_n on J^rF X. All L¹_n- invariant Lagrangians onJ^rF X are explicitly described in [2].

2 Jet prolongations of L

manifolds

In this section, the general prolongation theory of leftG-manifolds is applied to the case of the Lie groupG=L¹_n =GLn(R). We use the prolongation formula derived in [7], and the terminology and notation of the book [11].

Recall that the r-th differential group L^r_n of Rⁿ is the group of invertible r-jets with source and target at the origin 0∈Rⁿ. The group multiplication inL^r_nis defined by the composition of jets. LetJ₀^rα∈L^r_n, where α= (αⁱ) is a diffeomorphism of a neighborhoodU of the origin 0∈Rⁿ intoRⁿ such thatα(0) = 0. Thefirst canonical coordinatesaⁱ_j1,aⁱ_j1j2, . . . , aⁱ_j1j2...jr, where 1≤i≤n, 1≤j1≤j2 ≤. . .≤jr≤n, on L^r_n are defined by

(2.1) aⁱ_j1j2...jk(J₀^rα) =D_j1D_j2. . . D_jkαⁱ(0), 1≤k≤r.

We also define thesecond canonical coordinates bⁱ_j1, bⁱ_j1j2, . . . , bⁱ_j1j2...jr, onL^r_n by bⁱ_j1j2...jk(J₀^rα) =aⁱ_j1j2...jk(J₀^rα⁻¹), 1≤k≤r.

Indeed,aⁱ_jb^j_k=δⁱ_k (the Kronecker symbol).

Let us consider a leftL¹_n-manifoldS, and denote by T_n^rS the manifold of r-jets with source 0∈Rⁿand target inS. According to the general theory of prolongations of leftG-manifolds,T_n^rShas a (canonical) structure of a leftL^r+1_n -manifold. To define this structure, denote bytxthe translation of Rⁿ defined bytx(y) =y−x. Consider elementsq∈T_n^rS,q=J₀^rγ, anda∈L^r+1_n ,a=J₀^r+1α. Denoting ¯αx=tx◦α◦t−α⁻¹(x), and ¯α(x) =J₀¹α¯x we get an element of the groupL¹_n. Then formula

(2.2) a·q=J₀^r(¯α·(γ◦α⁻¹))

defines a left action of the differential groupL^r+1_n onT_n^rS. We usually call formula (2.2) theprolongation formula for the action of the group L¹_n on S. The leftL^r+1_n - manifoldT_n^rS is called ther-jet prolongationof the leftL¹_n-manifoldS.

3 Frames

LetX be an n-dimensional manifold. Recall that anr-frameat a pointx∈X is an invertibler-jet with source 0 ∈Rⁿ an target at x. The set of r-frames, denoted by F^rX, will be considered with its natural structure of a principal L^r_n-bundle over X.

We writeF X =F¹X;F X is the bundle oflinearframes.

For computing differential invariants of frame bundles it is important to realize F X as a fibre bundle with type fibre L¹_n, associatedwith itself. Thus, the structure group of F X is the group L¹_n = GLn(R), with canonical coordinates (aⁱ_j), defined by (2.1). If (pⁱ_j) are the canonical coordinates on the type fibre L¹_n of fibre bundle F X, then the left action of the structure group L¹_n of F X on the type fibre L¹_n is represented by the group multiplication L¹_n ×L¹_n 3 (J₀¹α, J₀¹η) → J₀¹(α◦η) ∈ L¹_n. In the canonical coordinates,pⁱ_j(J₀¹(α◦η)) =aⁱ_k(J₀¹α)p^k_j(J₀¹η), which can be written simply by

(3.1) p¯ⁱ_j =aⁱ_kp^k_j.

(3.1) is called theframe action ofL¹_n on itself.

J^rF X denotes the r-jet prolongation ofF X. It follows from the general theory of jet prolongations of fibre bundles thatJ^rF X can be considered as a fibre bundle overX with type fibreT_n^rL¹_n, associated withF^r+1X. Equations of the group action ofL^r+1_n onT_n^rL¹_n can be obtained from (2.2) and (3.1).

4 The second jet prolongation of the frame action

Now we derive an explicit expression for the action (2.2) of the groupL³_n onT_n²L¹_n, associated with (3.1).

LetU be a neighborhood of the origin 0∈Rⁿ. Letαbe a diffeomorphism ofU onto α(U)⊂Rⁿ such thatα(0) = 0. Then ¯α(x) =J₀¹α¯x, where ¯αx=tx◦α◦t_−α⁻¹_(x). Let γ:U →L¹_n be a mapping. For every x∈α(U) we denoteψ(x) = ¯α(x)·γ(α⁻¹(x)), and the dot on the right hand side means the multiplication in the group L¹_n. In coordinates,

(4.1) pⁱ_j(ψ(x)) =pⁱ_j(¯α(x)·γ(α⁻¹(x))) =aⁱ_s(¯α(x))p^s_j(γ(α⁻¹(x))).

Note that in this formula,

(4.2) aⁱ_s(¯α(x)) =Dsαⁱ(α⁻¹(x)).

Now the chart expression of the frame action is obtained by expressing the r-jet J₀^rψ=J₀^r+1α·J₀^rγ (2.2) in coordinates. Consider the caser= 2.

Lemma 1. The group action of L³_n on T_n²L¹_n induced by the frame action of L¹_n onL¹_n is defined by the equations

(4.3)

¯

pⁱ_j=p^s_jaⁱ_s,

¯

pⁱ_j,k=p^s_j,taⁱ_sb^t_k+p^s_jaⁱ_stb^t_k,

¯

pⁱ_j,kl=p^s_j,tuaⁱ_sb^u_lb^t_k+p^s_j,t(aⁱ_sub^u_kb^t_l+aⁱ_sub^u_lb^t_k+aⁱ_sb^t_kl) +p^s_j(aⁱ_stub^u_lb^t_k+aⁱ_stb^t_kl).

Proof. First equation is obtained from (4.1) and (4.2), to get the remaining equa- tions, we differentiate (4.1) twice, and then putx= 0. ¤

5 Differential invariants with values in L

-manifolds

In this part we are interested in differential invariants F : T_n²L¹_n → Q, where Q is arbitrary leftL¹_n-manifold. We define the homomorphism

π^3,1_n :L³_n→L¹_n, π^3,1_n (aⁱ_j, aⁱ_jk, aⁱ_jkl) = (aⁱ_j).

Notice that each differential invariantF with values inL¹_n-manifold satisfies F(a·q) =π^3,1_n (a)·F(q),

for eacha∈L³_n, q∈T_n²L¹_n.

LetK_n^3,1 denote the kernel of the canonical homomorphismπ^3,1_n ; K_n^3,1 is normal subgroup ofL³_nrepresented by elements in coordinates written as (δⁱ_j, aⁱ_jk, aⁱ_jkl). Now we restrict the action (4.3) to the subgroup K_n^3,1 of L³_n. The following result is fundamental for the discussion of the corresponding orbit spaces.

Lemma 2. The group action of K_n^3,1 onT_n²L¹_n induced by the frame action ofL¹_n onL¹_n is defined by the equations

(5.1)

¯ pⁱ_j =pⁱ_j,

¯

pⁱ_j,k=pⁱ_j,k+p^s_jaⁱ_sk,

¯

pⁱ_j,kl=pⁱ_j,kl+p^s_j,laⁱ_sk+p^s_j,kaⁱ_sl+ (pⁱ_j,t+p^s_jaⁱ_st)b^t_kl+p^s_jaⁱ_skl.

Proof. We takeaⁱ_j =bⁱ_j=δⁱ_j in (4.3). ¤ Corollary 1. The action(5.1)is free.

Now we describeorbitsof the group actions (5.1). Let us introduce some notation.

Using the second canonical coordinates onT_n²L¹_n, we denote byqⁱ_j the inverse matrix of the matrixpⁱ_j; thus,qⁱ_j:T_n²L¹_n→Rare functions such that q_sⁱp^s_j =δ_jⁱ.

We also use the special notation for symmetrization and antisymmetrization of indexed families of functions through selected indices. Symmetrization (resp. anti- symmetrization) in some indicesj, k, l, . . .is denoted by writing a bar (resp. a tilde) over these indices, i.e., by writing ¯j,k,¯ ¯l, . . .(resp. ˜j,˜k,˜l, . . .).

First, we state some auxiliary assertions on the Young decomposition of tensors of type (0,3). Let us have a tensor ∆ = ∆jkl and let n be the dimension of the underlying vector space. We define

(5.2)

S∆ = ¹₆(∆jkl+ ∆ljk+ ∆klj+ ∆jlk+ ∆lkj+ ∆kjl),

Q∆ = ¹₃(∆_jkl+ ∆_kjl−∆_lkj−∆_klj) +¹₃(∆_jkl+ ∆_lkj−∆_kjl−∆_ljk), A∆ = ¹₆(∆jkl+ ∆ljk+ ∆klj−∆jlk−∆lkj−∆kjl).

Note that (5.2) can be equivalently written by

(5.3) S∆ = ∆¯j¯k¯l, Q∆ = ¹₃(2∆jkl−∆ljk−∆klj), A∆ = ∆˜jk˜˜l. Lemma 3. (a)If n≥3, a tensor ∆ = ∆jkl has a unique decomposition

∆ =S∆ +Q∆ +A∆,

such that S(S∆) = S∆, Q(Q∆) = Q∆, A(A∆) = A∆, S(Q∆) = Q(S∆) = A(Q∆) =Q(A∆) =S(A∆) =A(S∆) = 0.

(b)If n= 2, a tensor∆ = ∆jkl has a unique decomposition

∆ =S∆ +Q∆,

such thatS(S∆) =S∆,Q(Q∆) =Q∆,S(Q∆) =Q(S∆) = 0.

Proof. These assertions can be verified by a direct computation. ¤ Corollary 2. For tensor∆ = ∆_jkl symmetric in the indicesk, l there is a unique decomposition

(5.4) ∆ =S∆ +Q∆,

such thatS(S∆) =S∆,Q(Q∆) =Q∆,S(Q∆) =Q(S∆) = 0.

Finally, we introduce the following functions onT_n²L¹_n:

(5.5)

I_jkⁱ (p^a_b, p^a_b,c, p^a_b,cd) =q^l_˜jpⁱ_l,˜k,

I_jklⁱ (p^a_b, p^a_b,c, p^a_b,cd) = 2q_j^spⁱ_s,kl−q_k^spⁱ_s,lj−q_l^spⁱ_s,jk

−3(q_˜^t_jp^m_t,˜lq^s_m¯pⁱ_s,¯k+q_˜j^tp^m_t,˜kq_m^s_¯pⁱ_s,¯l) +pⁱ_s,m(2q^s_jqk^t¯p^m_t,¯l−q^s_kq^t¯lp^m_t,¯j−q_l^sq¯j^tp^m_t,¯k).

It is obvious that the functionsI_jkⁱ are antisymmetric in indicesj, k, and the functions I_jklⁱ are symmetric in indices k, l, which gives us I_jkⁱ +I_kjⁱ = 0, I_jklⁱ −I_jlkⁱ = 0, respectively. Moreover, we have the identityI_jklⁱ +I_ljkⁱ +I_kljⁱ = 0.

Lemma 4. K_n^3,1-orbits in T_n²L¹_n induced by the frame action of L¹_n on L¹_n is defined by the equations

pⁱ_j =cⁱ_j, I_jkⁱ (p^a_b, p^a_b,c, p^a_b,cd) =cⁱ_jk, I_jklⁱ (p^a_b, p^a_b,c, p^a_b,cd) =cⁱ_jkl, wherecⁱ_j, cⁱ_jk, cⁱ_jkl∈Rare arbitrary constants such thatdetcⁱ_j6= 0.

Proof. Consider the action (5.1) ofK_n^3,1onT_n²L¹_n, induced by the frame action of L¹_n onL¹_n, in standard notation given by ¯pⁱ_j=aⁱ_kp^k_j. Rewrite this action in the form (5.6) p¯ⁱ_j=pⁱ_j, p¯ⁱ_j,k=pⁱ_j,k+p^s_jaⁱ_sk, p¯ⁱ_j,kl=pⁱ_j,kl+χⁱ_j,kl+p^s_jaⁱ_skl,

where the functions

(5.7) χⁱ_j,kl=p^s_j,laⁱ_sk+p^s_j,kaⁱ_sl+ (pⁱ_j,t+p^s_jaⁱ_st)b^t_kl

are symmetric in the indicesk, l. From (5.6) we get

(5.8) q¯_jⁱ =q_jⁱ, aⁱ_sk=q_s^j_¯(¯pⁱ_j,¯k−pⁱ_j,¯k), aⁱ_skl=q^j_¯s(¯pⁱ_j,k¯¯l−pⁱ_j,¯k¯l−χⁱ_j,¯k¯l).

Substituting the second equation of (5.8) to (5.6) we have q^j_s(¯pⁱ_j,k−pⁱ_j,k) =q_s^j_¯(¯pⁱ_j,¯k−pⁱ_j,¯k),

which means that we compare the tensor on the left hand side with its symmetric part. It gives us ¯q^l_˜jp¯ⁱ_l,˜k =q^l_˜jpⁱ_l,˜k. Thus, for the functionsI_jkⁱ , defined by (5.5), we have (5.9) I_jkⁱ (¯p^a_b,p¯^a_b,c,p¯^a_b,cd) =I_jkⁱ (p^a_b, p^a_b,c, p^a_b,cd),

and the functionsI_jkⁱ are invariant with respect to the action (5.1) ofK_n^3,1 onT_n²L¹_n. Again, substituting (5.8) to (5.6) we have

q_j^s(¯pⁱ_s,kl−pⁱ_s,kl−χⁱ_s,kl) =q¯j^s(¯pⁱ_s,¯k¯l−pⁱ_s,¯k¯l−χⁱ_s,¯k¯l).

It means that we compare the tensor

(5.10) ∆ⁱ_jkl=q_j^s(¯pⁱ_s,kl−pⁱ_s,kl−χⁱ_s,kl),

symmetric in the subscriptsk, l, on the left hand side, with its symmetric partS∆ =

∆ⁱ_¯j¯k¯l. Using decomposition (5.4) for the tensor ∆ given by (5.10), we get

(5.11) Q∆ = 0.

Applying (5.3) to (5.11), and using (5.7), and (5.8), after long calculation we obtain that (5.11) is equivalent to

(5.12) I_jklⁱ (¯p^a_b,p¯^a_b,c,p¯^a_b,cd) =I_jklⁱ (p^a_b, p^a_b,c, p^a_b,cd),

which means that the functions I_jklⁱ , defined by (5.5), are invariant with respect to

the action (5.1) ofK_n^3,1 onT_n²L¹_n. ¤

6 Basis of the second order invariants

Now, from the assertions on equivariant mappings of manifolds (see [4, 8]) we can obtain the exact characteristics of basis of differential invariants with values inL¹_n- manifolds.

First, let us denote by S_n⁰ the vector subspace of the tensor productN₂ R^n∗ = R^n∗⊗R^n∗, defined in the canonical coordinates onRby the equations

xjk+xkj= 0.

Similarly, S_n¹ denotes the vector subspace of the tensor product N₃

R^n∗ = R^n∗ ⊗ R^n∗⊗R^n∗, defined by the equations

xjkl−xjlk= 0, xjkl+xljk+xklj = 0.

We can summarize the discussion of Section 5 in the following theorem, describing differential invariants onT_n²L¹_n with values inL¹_n-manifolds.

Theorem 1. (a)The frame action defines onT_n²L¹_nthe structure of a left principal K_n^3,1-bundle.

(b) The quotient space T_n²L¹_n/K_n^3,1 is canonically isomorphic to the space L¹_n × (Rⁿ⊗S_n⁰)×(Rⁿ⊗S_n¹).

Proof. (a) Since we have already proved that the action (5.1) of K_n^3,1 onT_n²L¹_n is free (see Corollary 1), in order to show that T_n²L¹_n is a principal K_n^3,1-bundle it remains to show that the equivalence ”J₀²γ∼J₀²¯γif and only if there exists an element J₀³α∈K_n^3,1such thatJ₀²γ¯=J₀³α·J₀²γ” is a closed submanifold inT_n²L¹_n×T_n²L¹_n. But using (5.1) with help of (5.9) and (5.12), we see that this submanifold is defined by the equations

pⁱ_j(J₀²γ)¯ −pⁱ_j(J₀²γ) = 0, I_jkⁱ (J₀²¯γ)−I_jkⁱ (J₀²γ) = 0, I_jklⁱ (J₀²¯γ)−I_jklⁱ (J₀²γ) = 0, and is therefore closed.

(b) Let J₀²γ ∈T_n²L¹_n and let [J₀²γ] be the corresponding class of J₀²γ in quotient T_n²L¹_n/K_n^3,1. We set

(6.1)

pⁱ_j([J₀²γ]) =pⁱ_j(J₀²γ), I_jkⁱ ([J₀²γ]) =I_jkⁱ (J₀²γ), I_jklⁱ ([J₀²γ]) =I_jklⁱ (J₀²γ).

Relations (5.1), (5.9), and (5.12) imply that coordinates defined by (6.1) do not depend on a representant of given class, and for two different classes there are different sets of numbers. It means that factor projection π : T_n²L¹_n → T_n²L¹_n/K_n^3,1 can be expressed by

π= (pⁱ_j, I_jkⁱ , I_jklⁱ ).

Thus, canonical isomorphism of bundles maps the class fromT_n²L¹_n/K_n^3,1 with coor- dinates (pⁱ_j, I_jkⁱ , I_jklⁱ ), to the element of L¹_n×(Rⁿ⊗S_n⁰)×(Rⁿ⊗S_n¹) with the same

canonical coordinates. ¤

Theorems 1 says that every second order differential invariant of frames factorizes through the corresponding bundle projection. Consider the components of the isomor- phisms defined bypⁱ_j:T_n²L¹_n →L¹_n,I_jkⁱ :T_n²L¹_n→Rⁿ⊗S_n⁰, andI_jklⁱ :T_n²L¹_n→Rⁿ⊗S_n¹. We have the following results.

Corollary 3. The mappingspⁱ_j,I_jkⁱ ,I_jklⁱ represent a basis of second order invari- ants of frames with values in leftL¹_n-manifold.

7 Basis of scalar invariants

Note that ifGis a Lie group,K is its normal subgroup, andP is a G-manifold then the quotient manifold (P/K)/(G/K) is canonically isomorphic withP/G(see [1]).

This means that for finding scalar invariants of Lie groupG, we can equivalently factorizeP by normal subgroupKand subsequently by the factor groupG/K. Thus, to obtain scalar invariants ofL³_n on T_n²L¹_n it is sufficient to consider L¹_n-equivariant mappings defined onT_n²L¹_n/K_n^3,1.

Let us define some functionsI_jkⁱ ,I_jklⁱ , onT_n²L¹_n, by

(7.1)

I_jkⁱ =qⁱ_lp^s_˜kp^l_˜j,s,

I_jklⁱ =q_rⁱ(2p^s_kp^t_lp^r_j,st−p^s_lp^t_jp^r_k,st−p^s_jp^t_kp^r_l,st

−³₂q^u_mp^r_u,s(p^s_kp^t_˜lp_˜^m_j,t+p^s_lp^t_˜kp^m_˜j,t)−⁵₂p^m_j,sp^sk¯p^r¯l,m

+¹₂p^s_jp^rk,m¯ p^m¯l,s+ 2p^r_j,mp^s¯kp^m¯l,s).

We have the following

Theorem 2. The functions I_jkⁱ , I_jklⁱ represent a basis of second order scalar invariants of frames.

Proof. The group L¹_n ' L³_n/K_n^3,1 acts in factor space T_n²L¹_n/K_n^3,1, where the functionsI_jkⁱ andI_jklⁱ live, by

(7.2) I¯_jkⁱ =aⁱ_rb^s_jb^t_kI_st^r, I¯_jklⁱ =aⁱ_rb^s_jb^t_kb^u_lI_stu^r ,

respectively. Using relationsaⁱ_r=q^m_r p¯ⁱ_m, andb^s_j = ¯q_j^vp^s_v, obtained from (3.1), in (7.2), we have

¯

q_i^ap¯^j_bp¯^k_cI¯_jkⁱ =q^a_rp^s_bp^t_cI_st^r, q¯^a_ip¯^j_bp¯^k_cp¯^l_dI¯_jklⁱ =q^a_rp^s_bp^t_cp^u_dI_stu^r , which describesL¹_n-invariant objets inT_n²L¹_n/K_n^3,1. Using (5.5), we get

q^a_rp^s_bp^t_cI_st^r =I_bc^a, q_r^ap^s_bp^t_cp^u_dI_stu^r =I_bcd^a ,

whereI_bc^a,I_bcd^a are given by (7.1). ¤

Using factorization method, we are allowed to determine the number of indepen- dentL³_n-invariant functions on T_n²L¹_n as the dimensions of the corresponding factor spaces. Thus, the number of functionally independent invariantsI_jkⁱ ,I_jklⁱ is given by

dim(T_n¹L¹_n/L²_n) =¹₂n²(n−1), dim(T_n²L¹_n/L³_n) =¹₃n²(n²−1),

respectively. For instance ofn = 1,2,3,4, the number of independentL³_n-invariant functions onT_n²L¹_n isN= 0,6,33,104, respectively.

Let us consider a left action of the general linear groupL¹_n on the real lineRby L¹_n×R3(a, t)7→ |deta⁻¹| ·t∈R.

The real line, endowed with this action, is anL¹_n-manifold, denoted byR.e We also introduce the functionI0defined onT_n²L¹_n by

T_n²L¹_n3q7→ I0(q) =|detq_jⁱ(q)| ∈R.