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Differential Geometry and its Applications 3 (1993) 323–329

arXiv:math/9209219v1 [math.DG] 1 Sep 1992Differential Geometry and its Applications 3 (1993) 323–329North-HollandCHARACTERISTIC CLASSES FOR G-STRUCTURESD. V. AlekseevskyPeter W. MichorCenter ‘Sophus Lie’, MoscowInstitut f¨ur Mathematik, Universit¨at Wien, AustriaAugust 3, 2018Abstract.

Let G ⊂GL(V ) be a linear Lie group with Lie algebra g and let A(g)G bethe subalgebra of G-invariant elements of the associative supercommutative algebraA(g) = S(g∗) ⊗Λ(V ∗). To any G-structure π : P →M with a connection ω weassociate a homomorphism µω : A(g)G →Ω(M).The differential forms µω(f)for f ∈A(g)G which are associated to the G-structure π can be used to constructLagrangians.

If ω has no torsion the differential forms µω(f) are closed and definecharacteristic classes of a G-structure. The induced homomorphism µ′ω : A(g)G →H∗(M) does not depend on the choice of the torsionfree connection ω and it is thenatural generalization of the Chern Weil homomorphism.1.

G-structures1.1.G-structures. By a G-structure on a smooth finite dimensional manifoldM we mean a principal fiber bundle π : P →M together with a representationρ : G →GL(V ) of the structure group in a real vector space V of dimension dim Mand a 1-form σ (called the soldering form) on M with values in the associatedbundle P[V, ρ] = P ×G V which is fiber wise an isomorphism and identifies TxMwith P[V ]x for each x ∈M.

Then σ corresponds uniquely to a G-equivariant 1-formθ ∈Ω1hor(P; V )G which is strongly horizontal in the sense that its kernel is exactlythe vertical bundle V P.The form θ is called the displacement form of the G-structure. A G-structure is called 1-integrable if it admits torsionfree connections,see 1.4 below.We fix this setting ((P, p, M, G), (V, ρ), θ) from now on.1.2.

Invariant forms. We consider a multilinear form f ∈Nk V ∗= Lk(V ) whichis invariant in the sense that f ◦(Nk ρ(g)) = f for each g ∈G.

Let us denote by1991 Mathematics Subject Classification. 53C10, 57R20.Key words and phrases.

G-structures, characteristic classes.Supported by Project P 7724 PHY of ‘Fonds zur F¨orderung der wissenschaftlichen Forschung’.Typeset by AMS-TEX1

2D. V. ALEKSEEVSKY PETER W. MICHORLk(V )G the space of all these invariant forms.

For each f ∈Lk(V )G we have forany X ∈g, the Lie algebra of G,0 = ddt|0f(ρ(exp(tX))v1, . .

. , ρ(exp(tX))vk),=kXi=1f(v1, .

. ., ρ′(X)vi, .

. ., vk),where ρ′ = Teρ : g →gl(V ) is the differential of the representation ρ.1.3 Products of differential forms.

For ϕ ∈Ωp(P; g) and Ψ ∈Ωq(P; V ) let usdefine the form ρ′∧(ϕ)Ψ ∈Ωp+q(P; V ) by(ρ′∧(ϕ)Ψ)(X1, . .

. , Xp+q) ==1p!

q!Xσsign(σ)ρ′(ϕ(Xσ1, . .

. , Xσp))Ψ(Xσ(p+1), .

. .

, Xσ(p+q)).Then ρ′∧(ϕ) : Ω∗(P; V ) →Ω∗+p(P; V ) is a graded Ω(P)-module homomorphism ofdegree p. Recall also that Ω(P; g) is a graded Lie algebra with the bracket[ϕ, ψ]∧(X1, . .

. , Xp+q) ==1p!

q!Xσsignσ [ϕ(Xσ1, . .

. , Xσp), ψ(Xσ(p+1), .

. .

, Xσ(p+q))]g.One may easily check that for the graded commutator in End(Ω(P; V )) we haveρ′∧([ϕ, ψ]∧) = [ρ′∧(ϕ), ρ′∧(ψ)] = ρ′∧(ϕ) ◦ρ′∧(ψ) −(−1)pqρ′∧(ψ) ◦ρ′∧(ϕ)so that ρ′∧: Ω∗(P; g) →End∗(Ω(P; V )) is a homomorphism of graded Lie algebras.Let N V be the tensoralgebra generated by V . For Φ, Ψ ∈Ω(P; N V ) we willuse the associative bigraded product(Φ ⊗∧Ψ)(X1, .

. .

, Xp+q) ==1p! q!Xσsign(σ)Φ(Xσ1, .

. .

, Xσp) ⊗Ψ(Xσ(p+1), . .

. , Xσ(p+q))1.4.The covariant exterior derivative.

Let ω ∈Ω1(P; g)G be a principalconnection on the principal bundle (P, p, M, G). Let χ : TP →HP denote thecorresponding projection onto the horizontal bundle HP := ker ω.

The covariantexterior derivative dω : Ωk(P; V ) →Ωk+1hor (P; V ) is then given as usual by dωΨ =χ∗dΨ = (dΨ) ◦Λk+1(χ).Lemma. For Ψ ∈Ωhor(P; V )G the covariant exterior derivative is given by dωΨ =dΨ + ρ′∧(ω)Ψ.Proof.

If we insert one vertical vector field, say the fundamental vector field ζX forX ∈g, into dωΨ, we get 0 by definition. For the right hand side we use iζXΨ = 0 and

CHARACTERISTIC CLASSES FOR G-STRUCTURES3LζXΨ =∂∂t0 (FlζXt )∗Ψ =∂∂t0 Ψ ◦Λp(rexp tX) =∂∂t0 ρ(exp(−tX))Ψ = −ρ′(X)Ψto getiζX(dΨ + ρ′∧(ω)Ψ) = iζXdΨ + diζXΨ + ρ′∧(iζXω)Ψ −ρ′∧(ω)iζXΨ= LζXΨ + ρ′∧(X)Ψ = 0.Let now all vector fields ξi be horizontal, then we get(dωΨ)(ξ0, . .

. , ξk) = (χ∗dΨ)(ξ0, .

. .

, ξk) = dΨ(ξ0, . .

. , ξk),(dΨ + ρ′∧(ω)Ψ)(ξ0, .

. .

, ξk) = dΨ(ξ0, . .

. , ξk).□1.5.Definition.

If θ ∈Ω1hor(P; V )G is the displacement form of a G-structurethen the torsion of the connection ω with respect to the G-structure is τ := dωθ =dθ + ρ′∧(ω)θ.Recall that a G-structure is called 1-integrable if it admits a connection withouttorsion. This notion has also been investigated in [Kol´aˇr, Vadoviˇcov´a] where it wascalled prolongable.1.6.

Chern-Weil forms. For differential forms ψi ∈Ωpi(P; V ) and f ∈Lk(V ) =(Nk V )∗we can construct the following differential forms:ψ1 ⊗∧· · · ⊗∧ψk ∈Ωp1+···+pk(P; V ⊗· · · ⊗V ),f ψ1,...,ψk := f ◦(ψ1 ⊗∧· · · ⊗∧ψk) ∈Ωp1+···+pk(P).The exterior derivative of the latter one is clearly given byd(f ◦(ψ1 ⊗∧· · · ⊗∧ψk)) = f ◦d(ψ1 ⊗∧· · · ⊗∧ψk) == f ◦Pki=1(−1)p1+···+pi−1ψ1 ⊗∧· · · ⊗∧dψi ⊗∧· · · ⊗∧ψk.We also set f ψ := f ψ,...,ψ = alt f(ψ, .

. ., ψ) for ψ ∈Ωp(P; V ).

Note that the formf ψ1,...,ψk is G-invariant and horizontal if all ψi ∈Ωpihor(P; V )G and f ∈Lk(V )G. Itis then the pullback of a form on M.1.7. Lemma.

Let 0 ̸= ψ ∈Ωp(P; V ) and f ∈Lk(V ). Then we have:f ψ ̸= 0 ⇐⇒ alt f ̸= 0,if p is odd,sym f ̸= 0,if p is even,where alt and sym are the natural projections onto Λ(V ∗) and S(V ∗), respec-tively.□1.8.

Lemma. If f ∈Lk(V )G is invariant then we havef ◦Pki=1(−1)p1+···+pi−1ψ1 ⊗∧· · · ⊗∧ρ′∧(ω)ψi ⊗∧· · · ⊗∧ψk= 0.Proof.

This follows from the infinitesimal condition of invariance for f given in 1.2by applying the alternator.□

4D. V. ALEKSEEVSKY PETER W. MICHOR2.

Obstructions to 1-integrability of G-structures2.1.Proposition. Let π : P →M be a G-structure and let f ∈Lk(V )G bean invariant tensor.

For arbitrary G-equivariant horizontal V -valued forms ψi ∈Ωpihor(P; V )G we consider the (p1 + · · ·+ pk)-form f ψ1,...,ψk on M as above. If thereis a connection ω for the G-structure π such that dωψi = 0 for all i, then the formf ψ1,...,ψk is closed.Proof.

We use dωψi = dψi + ρ′∧(ω)ψi from lemma 1.4, and lemma 1.8, to obtaindf ψ1,...,ψk = f ◦Pki=1(−1)p1+···+pi−1ψ1 ⊗∧· · · ⊗∧dωψi ⊗∧· · · ⊗∧ψk= 0.□2.2. Corollary.

1. For a G-structure π : P →M with displacement form θ wehave a natural homomorphism of associative algebrasν : Λ(V ∗)G →Ω(M),f 7→f θ = f(θ, .

. ., θ).2.

If the G-structure is 1-integrable then the image of ν consists of closed formsand we get an induced homomorphismν∗: Λ(V ∗)G →H∗(M).If M and G are compact then ν∗is injective.Proof. If the G-structure is 1-integrable then there is a connection ω with vanishingtorsion τ = dωθ = 0.

Then the result follows from proposition 2.1.If G is compact, any torsionfree connection ω for π : P →M is the Levi-Civitaconnection for some Riemannian metric. Any form f θ, which is parallel with respectto to ω, is harmonic and can thus not be exact for compact M. So ν∗is injective.□Problem.

Is the homomorphism ν∗injective for compact M but noncompact G?2.3. Remark.

Given a principal connection ω on P there is the induced covariantexterior derivative ∇: Ωp(M; P[V ]) →Ωp+1(M; P[V ]) on the associated vectorbundle P[V ]. The soldering form (see 1.1) σ : TM →P[V ] is an isomorphismof vector bundles and we may consider the pull back covariand derivative σ∗∇onTM.

Next we consider the ‘combined’ covariant derivative Dσ∗∇,∇on the vectorbundle L(TM, P[V ]) given by Dσ∗∇,∇XA = ∇X ◦A −A ◦(σ∗∇)X. Obviously wehave Dσ∗∇,∇σ = 0.

Consequently for any f ∈Lk(V )G we have that f θ ∈Ωk(M)is parallel for the connection induced on ΛkT ∗M from σ∗∇on TM.3. The generalized Chern-Weil homomorphism for G-structures3.1.

The Chern-Weil homomorphism. Let ω be a connection for a G-structureπ : P →M with curvature form Ω∈Ω2hor(P, g).

Then the Bianchi identity dωΩ= 0holds. If we apply proposition 2.1 to ψi = Ωwe obtain a homomorphismγ : S(g∗)G →Ω(M),given by γ(f) = f Ω.

Since the image of γ consists of closed forms we have aninduced homomorphismγ′ : S(g∗)G →H∗(M).This is the well known Chern-Weil homomorphism.

CHARACTERISTIC CLASSES FOR G-STRUCTURES53.2. The algebra A(g, V ).

In oder to generalize the Chern Weil homomorphismwe associate to a Lie algebra g and a vector space V the associative graded com-mutative algebraA(g, V ) := S(g∗) ⊗Λ(V ∗),where the generators of the symmetric algebra S(g∗) have degree 2. We may alsoconsider A(g, V ) as a graded Lie algebra with the bracket[a ⊗ϕ, b ⊗ψ] := {a, b} ⊗ϕ ∧ψ,a, b ∈S(g∗), ϕ, ψ ∈Λ(V ∗),where {a, b} is the usual Poisson-Lie bracket in S(g∗).Let now g be the Lie algebra of the Lie group G and let ρ : G →GL(V ) bea representation.

Then G acts naturally on A(g, V ), and we denote A(g, V )G thesubalgebra of G-invariant elements in A(g, V ).3.3. Remark.

The associative algebra A(g, V )G contains the subalgebra S(g∗)G⊗Λ(V ∗)G, in general as a proper subalgebra. Actually, let G ⊂GL(V ) be the isotropygroup of an irreducible Riemannian symmetric space M. Then the curvature tensorof M defines an element of (g∗⊗Λ2V ∗)G ⊂A(g, V )G that does not belong to(g∗)G ⊗(Λ2V ∗)G = 03.4.

The generalized Chern-Weil homomorphism. Now we are in a positionto combine the constructions 2.2 and 3.1.Theorem.

Let π : P →M be a G-structure on M with displacement form θ. Anyconnection ω in π defines a homomorphism of associative algebrasµ : A(g, V )G →Ω(M)(Sp(g∗) ⊗ΛqV ∗)G ∋f 7→f Ω,θ = f(Ω, . .

., Ω|{z}p, θ, . .

., θ| {z }q)If the connection ω has no torsion then the image of µ consists of closed forms andµ induces a homomorphismµ′ : A(g, V )G →H∗(M),which is independent of the choice of the torsionfree connection.In other words, any G-invariant tensor f ∈Sp(g∗)⊗Λq(V ∗) defines a cohomologyclass [f Ω,θ] ∈H2p+q(M) which is an invariant of the 1-integrable G-structure. Wecall it a characteristic class of the 1-integrable G-structure π.Proof.

It just remains to show that the cohomology class [f Ω,θ] does not dependon the choice of the torsionfree connection for the G-structure π : P →M.So let ω0, ω1 be two torsionfree connections for the G-structure, let ϕ = ω1 −ω0,and denote by Ωt = dωtΩt the curvature form of the torsionfree connection ωt =ω0 + tϕ = (1 −t)ω0 + tω1. We claim that for f ∈(Sp(g∗) ⊗Λq(V ∗))G we havef Ω1,θ −f Ω0,θ = d(Tf),where(1)Tf = pZ 10f(ϕ, Ωt, .

. .

, Ωt, θ, . .

., θ) dt

6D. V. ALEKSEEVSKY PETER W. MICHORis the transgression form of f on P. The assertion is immediate from (1).

To proveit we compute ∂tf Ωt,θ using the identities ∂tΩt = dωtϕ (see [Kobayashi, Nomizu II,p. 296]), dωtΩt = 0, and dωtθ = 0.∂tf Ωt,θ = p f(∂tΩt, Ωt, .

. ., Ωt, θ, .

. ., θ)= p f(dωtϕ, Ωt, .

. ., Ωt, θ, .

. ., θ)= p dωtf(ϕ, Ωt, .

. ., Ωt, θ, .

. ., θ)= p d f(ϕ, Ωt, .

. ., Ωt, θ, .

. ., θ).□3.5.

Remarks about secondary characteristic classes. If the characteristicforms f Ω1,θ and f Ω0,θ associated with two torsionfree connections ω1 and ω0 vanishwe obtain a secondary characteristic class [Tf].

It is a natural generalization ofthe classical Chern-Simons characteristic class, see [Chern, Simons], [Kobayashi,Ochiai].Problem: study conditions when the secondary characteristic class [Tf] does notdepend on the choice of the torsionfree connections ω1 and ω0.3.6.Examples of characteristic classes. Assume that a linear group G ⊂GL(V ) preserves some pseudo Euclidean metric in V = Rn.

Then we may identifythe Lie algebra g = Lie(G) with a subspace g ⊂Λ2V . Suppose that there exists aG-invariant supplement d to g in Λ2V .

Then the G-equivariant projection Λ2V →galong d determines a G-invariant element q ∈g⊗Λ2V ∗∼= g∗⊗Λ2V ∗. The element qdefines a 4-form qΩ,θ on the base of any G-structure π : P →M with a connectionω and curvature Ω.

It may be written asqΩ,θ = q(Ω, θ, θ) = qabcdRbaefθc ∧θd ∧θe ∧θf,where (qabcd) is the coordinate expression of q in the standard basis (ea) of V = Rn,θ = ea ⊗θa, and Ω= Rabefθe ∧θf.If ω is torsionfree the 4-form qΩ,θ is closed and it defines a cohomology class[qΩ,θ] ∈H4(M) independently of the choice of ω.3.7. Remarks about the classification of characteristic classes.

The clas-sification of characterictic classes for G −structures with a given Lie group Greduces to the construction of generators of the associative algebra A(g, V )G =(S(g∗) ⊗Λ(V ∗))G. We may also use the bracket to multiply characteristic classes.It suffices to solve this problem for those Lie groups G which appear as holonomygroups of torsionfree connection. Only for such groups G there exist 1-integrablenon-flat G-structures.

Under the additional hypothesis of irreducibility, all suchgroups were classified by [Berger], up to some gaps which were filled by [Bryant]and [Alekseevsky, Graev].ReferencesAlekseevsky, D. V.; Graev M. M., Twistors and G-structures, Preprint, University of Roma,(1991).Berger, M., Sur les groupes d’holonomie des vari´et´es `a connexion affine et des vari´et´es rie-manni´ennes, Bull. Soc.

Math. France 83 155 (1955), 279–330.

CHARACTERISTIC CLASSES FOR G-STRUCTURES7Bryant, R., Holonomy groups, Lecture at the conference: Global Differential Geometry and GlobalAnalysis, Berlin 1990.Chern, S. S.; Simons, J., Some cohomology classes in principal fiber bundles and their applicationsto Riemannian geometry, Proc. Nat.

Acad. Sci.

(USA) 68 (1971), 791–794.Chern, S. S.; Simons, J., Charateristic forms and geometrical invariants, Annals of Math. 99(1974), 48–69.Dupont, Johan L., Curvature and characteristic classes, Lecture Notes in Mathematics 640,Springer-Verlag, Berlin, 1978.Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry.

Vol. II., J. Wiley-Interscience,1969.Kobayashi, Sh.

; Ochiai, T., G-structures of order two and transgression operators, J. Diff. Geo.

6(1971), 213–230.Kol´aˇr, I; Vadoviˇcov´a, I, On the structure functions of a G-structure, Math. Slovaca 35 (1985),277–282.D.

V. Alekseevsky: Center ‘Sophus Lie’, Krasnokazarmennaya 6, 111250 Moscow,USSRP. W. Michor:Institut f¨ur Mathematik, Universit¨at Wien, Strudlhofgasse 4,A-1090 Wien, AustriaE-mail address: MICHOR@AWIRAP.BITNET

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