J´
an Brajerˇ
c´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 byRthe field of real numbers. Ther-th differential group Lr
n ofRn is the Lie group of invertible r-jets with source and target at the origin 0 ∈ Rn; the group multiplication in Lr
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 leftLr
n-manifolds, andU be an open,Lrn-invariant set inP.
A smoothLr
n-equivariant mapping F :U →Qis called adifferential invariant. IfQ
is the real lineR, endowed with the trivial action ofLr
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 principalLr
n-bundle with baseX is denoted byFrX, and is
called thebundle of r-frames overX. For r= 1, we get thebundle of linear frames, and writeF1X =F X. IfS is a leftLr
n-manifold, then the bundle with type fibreS,
associatedwithFrX is denoted byFr SX.
∗
Balkan Journal of Geometry and Its Applications, Vol.15, No.2, 2010, pp. 22-33 (electronic version); pp. 14-25 (printed version).
c
If S is a leftL1
n-manifold, we denote by TnrS 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
prolongationJrF X ofF X can be considered as a fibre bundle with type fibreTr nL1n,
associated withFr+1X.
For characterizingnaturalLagrangians onJrF X, i.e. Lagrangians invariant with
respect to alldiffeomorphisms ofX, it is sufficient to describe all differential invariants defined on the type fiberP =Tr
nL1nofJrF 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. LetKr,s
n be thekernelof the canonical group morphism πr,s
n :Lrn →Lsn, wherer≥s. IfLrn acts onQ via its subgroupL1n, eachcontinuous, Lr
n-equivariant mappingF :U →Qhas the form F =f ◦π, whereπ:P →P/Knr,1
is the quotient projection,P/Kr,1
n is the space of Knr,1-orbits, andf :P/Knr,1→Qis
a continuous,L1
n-equivariant mapping. Indeed, in this scheme P/Knr,1 is considered
with the quotient topology, but is not necessarily a smooth manifold. The quotient projectionπis continuous but not necessarily smooth. IfP/Kr,1
n has 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/Knr,1 to L1n-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 ofLr
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 =Tn2L1n.
Then we describe the frameaction of L3
n on Tn2L1n. 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
K3,1
n ofL3n. 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
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 JrF X. All L1n
-invariant Lagrangians onJrF X are explicitly described in [2].
2
Jet prolongations of
L
1n
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 Lr
n of Rn is the group of invertible r-jets
with source and target at the origin 0∈Rn. The group multiplication inLr
nis defined
by the composition of jets. LetJr
0α∈Lrn, where α= (αi) is a diffeomorphism of a
neighborhoodU of the origin 0∈Rn intoRn such thatα(0) = 0. Thefirst canonical coordinatesai
j1,aij1j2, . . . , aij1j2...jr, where 1≤i≤n, 1≤j1≤j2 ≤. . .≤jr≤n, on
Lr
n are defined by
(2.1) ai
j1j2...jk(J
r
0α) =Dj1Dj2. . . Djkα
i(0), 1≤k≤r.
We also define thesecond canonical coordinates bi j1, b
i
j1j2, . . . , b i
j1j2...jr, onL
r n by bi
j1j2...jk(J
r
0α) =aij1j2...jk(J
r
0α−1), 1≤k≤r.
Indeed,ai jb
j k=δ
i
k (the Kronecker symbol).
Let us consider a leftL1
n-manifoldS, and denote by TnrS the manifold ofr-jets
with source 0∈Rnand target inS. According to the general theory of prolongations
of leftG-manifolds,Tr
nShas a (canonical) structure of a leftLrn+1-manifold. To define
this structure, denote bytxthe translation of Rn defined bytx(y) =y−x. Consider
elementsq∈Tr
nS,q=J0rγ, anda∈Lrn+1,a=J r+1
0 α. Denoting ¯αx=tx◦α◦t−α−1(x),
and ¯α(x) =J1
0α¯x we get an element of the groupL1n. Then formula
(2.2) a·q=Jr
0(¯α·(γ◦α−1))
defines a left action of the differential groupLr+1
n onTnrS. We usually call formula
(2.2) theprolongation formula for the action of the group L1
n on S. The leftLrn+1
-manifoldTr
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
FrX, will be considered with its natural structure of a principal Lr
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 L1n = GLn(R), with canonical coordinates (aij), defined
by (2.1). If (pi
j) are the canonical coordinates on the type fibre L1n of fibre bundle F X, then the left action of the structure group L1
n of F X on the type fibre L1n is
represented by the group multiplicationL1
n ×L1n ∋ (J01α, J01η) → J01(α◦η) ∈ L1n.
In the canonical coordinates,pi
j(J01(α◦η)) =aik(J01α)pkj(J01η), which can be written
(3.1) is called theframe action ofL1
n on itself.
JrF X denotes the r-jet prolongation ofF X. It follows from the general theory
of jet prolongations of fibre bundles thatJrF X can be considered as a fibre bundle
overX with type fibreTr
nL1n, associated withFr+1X. Equations of the group action
ofLr+1
n onTnrL1n 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 onTn2L1n,
and the dot on the right hand side means the multiplication in the group L1
n. In
Note that in this formula,
(4.2) ais(¯α(x)) =Dsαi(α−1(x)).
Now the chart expression of the frame action is obtained by expressing the r-jet
Jr
0ψ=J
r+1
0 α·J0rγ (2.2) in coordinates. Consider the caser= 2.
Lemma 1. The group action of L3
n on Tn2L1n induced by the frame action of L1n
onL1
n is defined by the equations
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
1n
-manifolds
In this part we are interested in differential invariants F : T2
nL1n → Q, where Q is
arbitrary leftL1
n-manifold. We define the homomorphism π3n,1:L3n→L1n, π3
,1
n (a i j, a
i jk, a
i
jkl) = (a i j).
Notice that each differential invariantF with values inL1
n-manifold satisfies F(a·q) =π3,1
n (a)·F(q),
for eacha∈L3
n,q∈Tn2L1n.
LetK3,1
n denote the kernel of the canonical homomorphismπ3n,1; Kn3,1 is normal
subgroup ofL3
nrepresented by elements in coordinates written as (δij, aijk, a i
jkl). Now
we restrict the action (4.3) to the subgroup K3,1
n of L3n. The following result is
fundamental for the discussion of the corresponding orbit spaces.
Lemma 2. The group action of K3,1
n onTn2L1n induced by the frame action ofL1n
onL1
n is defined by the equations
(5.1)
¯
pi j =p
i j,
¯
pi j,k=p
i j,k+p
s ja
i sk,
¯
pij,kl=p i j,kl+p
s j,la
i sk+p
s j,ka
i sl+ (p
i j,t+p
s ja
i st)b
t kl+p
s ja
i skl.
Proof. We takeai j =b
i j=δ
i
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 onT2
nL1n, we denote byqij the inverse matrix
of the matrixpi
j; thus,qij:Tn2L1n→Rare functions such that qsipsj =δji.
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∆ = 1
6(∆jkl+ ∆ljk+ ∆klj+ ∆jlk+ ∆lkj+ ∆kjl),
Q∆ = 13(∆jkl+ ∆kjl−∆lkj−∆klj) +13(∆jkl+ ∆lkj−∆kjl−∆ljk),
Note that (5.2) can be equivalently written by
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 onT2
nL1n:
It is obvious that the functionsIi
jkare antisymmetric in indicesj, k, and the functions Ii
jkl are symmetric in indices k, l, which gives us I i
respectively. Moreover, we have the identityIi jkl+I
defined by the equations
pi
jkl∈Rare arbitrary constants such thatdetc i
j. Rewrite this action in the form
are symmetric in the indicesk, l. From (5.6) we get
Substituting the second equation of (5.8) to (5.6) we have
qj
which means that we compare the tensor on the left hand side with its symmetric part. It gives us ¯ql
and the functionsIi
jk are invariant with respect to the action (5.1) ofK
3,1
n onTn2L1n.
Again, substituting (5.8) to (5.6) we have
qs
It means that we compare the tensor
(5.10) ∆ijkl=q
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) Ii
which means that the functions Ii
jkl, defined by (5.5), are invariant with respect to
the action (5.1) ofK3,1
n onTn2L1n. ¤
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 S0
n the vector subspace of the tensor product
N2
Rn∗ =
Rn∗⊗Rn∗, defined in the canonical coordinates onRby the equations
xjk+xkj= 0.
Similarly, S1
n denotes the vector subspace of the tensor product
N3
Rn∗ = Rn∗ ⊗
Rn∗⊗Rn∗, defined by the equations
We can summarize the discussion of Section 5 in the following theorem, describing differential invariants onT2
nL1n with values inL1n-manifolds. Theorem 1. (a)The frame action defines onT2
nL1nthe structure of a left principal K3,1
n -bundle.
(b) The quotient space T2
nL1n/Kn3,1 is canonically isomorphic to the space L1n ×
(Rn⊗S0
n)×(Rn⊗Sn1).
Proof. (a) Since we have already proved that the action (5.1) of K3,1
n onTn2L1n
is free (see Corollary 1), in order to show that T2
nL1n is a principal Kn3,1-bundle it
remains to show that the equivalence ”J2
0γ∼J02¯γif and only if there exists an element
J3
0α∈Kn3,1such thatJ02γ¯=J03α·J02γ” is a closed submanifold inTn2L1n×Tn2L1n. But
using (5.1) with help of (5.9) and (5.12), we see that this submanifold is defined by the equations
pi j(J
2
0γ¯)−pij(J
2 0γ) = 0,
Ii jk(J
2
0¯γ)−Ijki (J
2 0γ) = 0,
Ijkli (J02¯γ)−I
i
jkl(J02γ) = 0,
and is therefore closed. (b) Let J2
0γ ∈Tn2L1n and let [J02γ] be the corresponding class of J02γ in quotient
T2
nL1n/Kn3,1. We set
(6.1)
pi j([J
2
0γ]) =pij(J
2 0γ),
Ii
jk([J02γ]) =Ijki (J02γ),
Ijkli ([J02γ]) =I
i
jkl(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 : T2
nL1n → Tn2L1n/Kn3,1 can be
expressed by
π= (pi j, I
i jk, I
i jkl).
Thus, canonical isomorphism of bundles maps the class fromT2
nL1n/Kn3,1 with
coor-dinates (pi j, Ijki , I
i
jkl), to the element of L
1
n×(Rn⊗Sn0)×(Rn⊗Sn1) 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:Tn2L1n →L1n,I i
jk:Tn2L1n→R n⊗S0
n, andI i
jkl:Tn2L1n→R n⊗S1
n.
We have the following results.
Corollary 3. The mappingspi j,Ijki ,I
i
jklrepresent a basis of second order
invari-ants of frames with values in leftL1n-manifold.
7
Basis of scalar invariants
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 Tn2L1n it is sufficient to consider L1n-equivariant
mappings defined onT2
nL1n/K3 ,1
n .
Let us define some functionsIi
jk,Ijkli , onTn2L1n, by
We have the following
Theorem 2. The functions Ii
jk, Ijkli represent a basis of second order scalar
invariants of frames. Proof. The group L1
n ≃ L3n/Kn3,1 acts in factor space Tn2L1n/Kn3,1, where the
respectively. Using relationsai
r=qmr p¯im, andbsj = ¯qjvpsv, obtained from (3.1), in (7.2),
which describesL1
n-invariant objets inTn2L1n/Kn3,1. Using (5.5), we get qa
rpsbptcIstr =Ibca, qrapsbptcpduIstur =Ibcda ,
whereIa
bc,Ibcda are given by (7.1). ¤
Using factorization method, we are allowed to determine the number of indepen-dentL3
n-invariant functions on Tn2L1n as the dimensions of the corresponding factor
spaces. Thus, the number of functionally independent invariantsIi
jk,Ijkli is given by
functions onT2
nL1n isN= 0,6,33,104, respectively.
Let us consider a left action of the general linear groupL1
n on the real lineRby L1
n×R∋(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 onTn2L1n by
Lemma 5. The functionI0:Tn2L1n →Re is a differential invariant.
Proof. Obviously, the function I0 is smooth, and for every a ∈ L3n, and every q∈T2
nL1n we haveI0(a·q) =|deta−1| · I0(q). ¤
Corollary 4. Every differential invariantI defined onT2
nL1n, with values in Re,
is the product of some scalar invariant and the functionI0.
8
Canonical odd
n
-form on
F X
In order to compare our results with [5], we recall in this Section the concept of volume form needed for integration on not necessarily orientable manifold.
Any chart (U, ϕ),ϕ= (xi), onX, induces thefibredchart (V, ψ),ψ= (xi, xi j), on F X. For every frame Ξ∈V we have detxi
j(Ξ)6= 0, and we can define some other
coordinatesykj of Ξ by settingx i jy
j k =δ
i
k. We define a functionV ∋Ξ7→ |dety i j(Ξ)| ∈
R,associatedwith the chart (V, ψ). With a chart (V, ψ) we also associate the object (8.1) ω˜(V,ψ)=|detyji| ·ϕ˜⊗dx1∧dx2∧. . .∧dxn,
where ˜ϕis afield of odd scalarsonX, associated with (U, ϕ) (see [9]). It is easily seen that (8.1) represent a globally defined odd base form onF X; we denote this form by ˜
ω, and call itcanonical oddn-form on X. This form has the following properties:
1. For each frame fieldζ:W →F X, whereW is an open set onX, the pullback
ζ∗ω˜ is an odd volume form onW.
2. The construction of ˜ω does not depend on orientability of base manifoldX. In the case of orientable and oriented manifoldsX, ˜ωis equivalent to an (ordinary)
n-form onF X.
3. The form ˜ω is diff X-invariant, i.e. (F α)∗ω˜ = ˜ω for all diffeomorphisms αof
X, where F αis the canonical lift ofαtoF X.
It should be pointed out that oddn-formsζ∗ω˜ may be used as volume forms for
integration on the base manifoldX. In particular, these forms naturally appears as a components of Lagrangians for variational problems for frame fields. We discuss these questions in the subsequent Section.
9
Second order natural Lagrangians of frames
Our aim in this Section is to characterize all Lagrangians onJ2F X, invariant with
respect to all diffeomorphisms ofX. First we recall main concepts to this purpose. We present basic definitions in full generality (for odd base forms). If the under-lying manifoldX is orientable, odd base forms may be replaced by ordinary forms.
Let us denote by µ the bundle projection µ : F X → X. The canonical jet projectionµ2 :J2F X →X is, for everyJ2
xζ ∈J2F X, defined by µ2(Jx2ζ) =x. A
second order Lagrangian forF X is anyµ2-horizontaln-formλdefined on the second
jet prolongation J2F X of F X. In a chart (V, ψ), ψ = (xi, xi
associatedchart (V2, ψ2),ψ2 = (xi, xi
j, xij,k, xij,kl), on J2F X, a Lagrangianλhas an
expression
λ=L ·ϕ˜⊗ω0,
whereω0 =dx1∧dx2∧. . .∧dxn, ˜ϕis a field of odd scalars, andL:V2 →Ris the
component ofλwith respect to (V, ψ) (the Lagrange function associated with (V, ψ)). We say that a second order Lagrangianλ isnatural, if for every diffeomorphism
α: W →X, where W is an open set in X, the canonical lift F α of αto F X is an invariance transformation ofλ, i.e.,
(J2F α)∗λ=λ
on the corresponding open set inJ2F X.
The following theorem is an application of a general result to the structure we consider in this paper (see [10, 11]).
Theorem 3. Let X be an n-dimensional manifold. There exists a one-to-one correspondence between natural Lagrangians onJ2F X and differential invariantsI:
T2
nL1n →Re.
We denote byAdiffX the algebra of diff X-invariant functions on J2F X. Define
in any chart (V, ψ),ψ= (xi, xi
j), onF X, functionsLijk,Lijkl, by
(9.1)
Lijk=y i lx
s
˜
kx l
˜
j,s,
Lijkl=y i r(2x
s kx
t lx
r j,st−x
s lx
t jx
r k,st−x
s jx
t kx
r l,st
−32yumx r u,s(x
s kx
t
˜
lx m
˜
j,t+x s lx
t
˜
kx m
˜
j,t)−
5 2x
m j,sx
s
¯
kx r
¯
l,m
+1 2x
s jx
r
¯
k,mx m
¯
l,s+ 2x r j,mx
s
¯
kx m
¯
l,s)
(compare with (7.1)). The functionsLi
jk,Lijkl, in coordinates expressed by (9.1), are
globally defined functions onJ2F X.
Corollary 5. The functions Li jk,L
i
jkl ∈ AdiffX, and every function L ∈ AdiffX
can be locally written as a differentiable function of the functions Li
jk, Lijkl, defined
by(9.1).
The following theorem is an immediate consequence of the invariance theory.
Theorem 4. Every natural Lagrangian λonJ2F X is of the form
λ=Lω,˜
whereL ∈ AdiffX andω˜ is canonical oddn-form ofF X.
Acknowledgments
The author is grateful to the Slovak Research and Development Agency (Grants MVTS SK-CZ-0081-07 and SK-CZ-0006-09), and to the Czech Grant Agency (Grant No. 201/09/0981).
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Author’s address:
J´an Brajerˇc´ık
