The aim of this article is to proof a necessary and sufficient condition for
the existence of a Cartan connection on a principal bundle. After collecting
the essentially well known facts to fix the terminology, soldering forms and
geometrizable principal bundles are defined to finally prove the existence
criterion.Comment: 6 pages, corrected the principal typo :

Barakat, Mohamed

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M. Barakat

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The existence of Cartan connections and geometrizable principle bundles

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arXiv:math/0206136v2  [math.DG]  19 Jul 2007THE EXISTENCE OF CARTAN CONNECTIONSAND GEOMETRIZABLE PRINCIPAL BUNDLESMOHAMED BARAKATAbstract. The aim of this article is to give necessary and sufficient conditions for theexistence of Cartan connections on principal bundles. For this purpose we recall thedefinition of abstract soldering forms and introduce the notion of geometrizable principalbundles.1. Principal bundles and associated bundles.Throughout the paper let H → Ppi→ M denote a principal H-bundle over the basemanifold M , where P is a manifold, and H a Lie group with a smooth free proper rightaction on PR :{H → Diff(P );h 7→ Rh,and M := P/H the quotient manifold.Further let F be an H-manifold, i.e. a manifold with a smooth left H action. For p ∈ Pand ξ ∈ F denote bypξ := {(ph, h−1ξ)| h ∈ H}(1)the orbit of (p, ξ) under the natural right action of H on P ×F . Then the right orbit spaceE = E(P, F ) := P ×H F := (P × F )/H = {pξ| p ∈ P, ξ ∈ F}(2)is called the associated fiber bundle to P with typical fiber F , and denoted by F → EpiE→M .Every p ∈ P with pi(p) = x ∈M defines a diffeomorphismp :{F → Fx := pi−1E (x);ξ 7→ pξ.(3)Moreover we have: (ph)ξ = p(hξ) for every h ∈ H .Lemma 1.1. The following are in bijective correspondence:(i) A global section f : M → E.(ii) An H-equivariant map f˜ : P → F , i.e. f˜(ph) = h−1f˜(p) for all h ∈ H.Definition 1.2 (Reduction). Let K be a Lie subgroup of H . The principal H-bundleP is called K-reducible, if there exists a principal K-bundle Q ⊆ P over M , where theembedding ι : Q → P is K-equivariant, i.e. ι(qk) = ι(q)k for all q ∈ Q and k ∈ K. Q iscalled a K-reduction of P .I would like to thank Prof. E. Ruh and Prof. H. Baum for the fruitful discussions.12 MOHAMED BARAKATDefinition 1.3 (Symmetry breaking). Let K be a Lie subgroup of H . An H-equivariantmap f : P → H/K is called a K-symmetry breaking.Theorem 1.4. Let K be a closed Lie subgroup of H. The following are in pairwise bijectivecorrespondence:(i) A K-reduction Q ⊆ P .(ii) A global section f : M → P/K.(iii) A K-symmetry breaking f˜ : P → H/K.Proof. The equivalence (i) ⇔ (ii) is given by factoring out the free action of K onQ and P . The equivalence (ii)⇔ (iii) is established by the fact that P/K ∼= P ×HH/K =E(P,H/K) and Lemma 1.1. Definition 1.5 (H-Structure). Let M be an n-dimensional manifold and H be a Liesubgroup of GL(Rn). The principal bundle H → Ppi→ M is called an H-structure, if P isan H-reduction of the frame bundle GL(Rn)→ L(M)→M .Definition 1.6 (Pseudotensorial, Tensorial). Let V be anH-module via the representationρ : H → GL(V ). A V -valued r-Form φ : ∧rTP → V is called pseudotensorial of type (V, ρ)if it is H-equivariant, i.e. ifR∗hφ = ρ(h−1)φ, ∀h ∈ H.A pseudotensorial r-form is thus an element of fixed space Γ(∧rT ∗P ⊗ V )H .φ is called tensorial if it is pseudotensorial and horizontal1 meaning thatφ(X1, . . . , Xr) = 0whenever at least one of the Xi’s is vertical, i.e. tangent to the fiber. The set of tensorialr-forms is denoted by Γhor(∧rT ∗P ⊗ V )H .A tensorial r-form factors over the induced projection TPpi∗→ TM and we obtain thenatural identificationΓhor(∧rT ∗P ⊗ V )H ∼= Γ(∧rpi∗T ∗M ⊗ V )H .(4)Further if the action of H on V is trivial, then every tensorial form is the pullback to theprincipal bundle P of a V -valued form on the base2 manifold M , i.e. Γhor(∧rT ∗P ⊗ V )H ∼=pi∗Γ(∧rT ∗M ⊗ V ).Definition 1.7 (Bundle valued). Let P , V be as above and V → EpiE→ M the associatedvector bundle. Elements of Γ(∧rT ∗M ⊗E) are called bundle valued r-forms.Now we can state the key lemma used in proof of the main theorem:Lemma 1.8. Let P , V and E be as above. The following are in bijective correspondence:(i) A bundle valued r-form on M .1This notion is found in [KoNo], Chapter II.5. Sharpe [Sh] uses ‘semibasic’ instead.2This motivates the notion ‘basic’ for such tensorial forms, cf. [Sh], Definition 1.5.23 and Lemma 1.5.25.See also [KoNo], Example II.5.1.THE EXISTENCE OF CARTAN CONNECTIONS AND GEOMETRIZABLE PRINCIPAL BUNDLES 3(ii) A tensorial V -valued r-form on P .Thus we have the natural identification:Γ(∧rT ∗M ⊗ E) ∼= Γhor(∧rT ∗P ⊗ V )H .(5)Proof. See [KoNo], Example II.5.2. 2. ConnectionsFor the rest of the paper let h denote the Lie algebra of H .Definition 2.1 (Ehresmann connection). An h-valued 1-form γ : TP → h is called anEhresmann connection or simply a connection, if the following two conditions hold:(i) R∗hγ = Ad(h−1)γ for all h ∈ H .(ii) γ(X†) = X for all X ∈ h.By X† we denote the fundamental vector field of X induced by the infinitesimal action ofh on P induced from the action of H .Because of (i) a connection is pseudotensorial of type (h,Ad) and because of (ii) it isnot tensorial.Definition 2.2 (Soldering form). LetM be an n-dimensional manifold, H → Ppi→M andρ : H → GL(Rn) a representation. A surjective Rn-valued tensorial 1-form: θ : TP ։ Rnis called a soldering form3.Example 2.3 (Fundamental form). Every H-structure P has a natural soldering form,the so called fundamental form, defined byθu(X) := u−1pi∗(X) ∀u ∈ P and X ∈ TP,where u−1 denotes the inverse of the isomorphism u : Rn → Tpi(u)M defined in (3). ρ isgiven by the natural action of H ≤ GL(Rn) on Rn.The definition of the fundamental form relies on the fact, that the tangent bundle TMis associated to P . This directly motivates the next definition, which seems to be new.Definition 2.4 (Geometrizable principal bundle). Let M be an n-dimensional manifold.We call the principal bundle H → Ppi→ M geometrizable if the tangent bundle TM isassociated to P , i.e. if there exists a representation ρ : H → GL(Rn) turning Rn into anH-module, such that the associated bundle satisfies the soldering condition:P ×H Rn ∼= TM.(6)We call P first order geometrizable if ρ is faithful, otherwise higher order. Once we fix arepresentation fulfilling the soldering condition, we call P geometrical.Example 2.5.• P is an H-structure, iff P is a first order geometrizable principal H-bundle. (Cf.Theorem 3.2 and [Sh], Exercise 5.3.21.)3In [Ko] θ is called a form of soudure (cf. Theorem 2 and the above lines). See also [AM], 2.4, 5.1.4 MOHAMED BARAKAT• The trivial bundle S2 × SO(R2) is not geometrizable.• An n-dimensional manifold M (n ≥ 3) is spin, iff there exists a geometrizableprincipal Spinn(R)-bundle over M .It is worth mentioning, that, in contrast to relativistic gravitation theories, most of theprincipal bundles appearing in classical gauge theories are not geometrizable, unless TMis trivial.3. Cartan ConnectionsNow we come to the main object of this paper.Definition 3.1 (Cartan connection). Let g be a Lie algebra with h ≤ g and dim g = dimP .A g-valued 1-form ω : TP → g satisfying4(i) ω : TpP → g is an isomorphism for all p ∈ P . (trivialization of TP → P )(ii) R∗hω = Ad(h−1)ω for all h ∈ H . (pseudotensorial of type (g,Ad))(iii) ω(X†) = X for all X ∈ h.We call a Cartan connection reductive, if h has an H-invariant complement5 p in g: g =h⊕H p.In the definition of a Cartan connection we do not make use of the Lie algebra structureof g. We only use the H-module structure6 of g and the fact that h is an H-submodule ofg. The first place where we do need the Lie algebra structure on g is in the definition ofcurvature of a Cartan connection. This is exactly where the power and flexibility of thisnotion lies, cf. [Ruh] and [Sh].Theorem 3.2. The following statements are equivalent:(i) P has a Cartan connection for some g ≥ h.(ii) P has a reductive Cartan connection for some g ≥ h.(iii) P has a reductive Cartan connection for some g ≥ h with [p, p] = 0.(iv) P has a soldering form.(v) P is a geometrizable bundle.Proof. (v) ⇒ (iv): TM is associated to P via an H-module structure of Rn. Theimage of the identity section id ∈ T ∗M⊗TM ∼= EndTM under the bijective correspondence(5) for r = 1, E = TM and V = Rn (carrying the above H-module structure) is thedesired soldering form. (iv) ⇒ (iii): Let θ be the soldering form. Due to [KoNo], TheoremII.2.1 there exists a connection form γ. ω = γ + θ is a reductive Cartan connection withg := h⋉Rn, and Rn =: p viewed as an H-module (and h-module) via the representation ρappearing in the definition of the soldering form θ. (iii) ⇒ (ii): Trivial. (ii) ⇒ (i): Trivial.(i) ⇒ (v): [Sh], Theorem 5.3.15. 4For an alternative definition see [AM], 2.1, 2.2.5We do not require that p is a Lie subalgebra.6and the induced h-module structureTHE EXISTENCE OF CARTAN CONNECTIONS AND GEOMETRIZABLE PRINCIPAL BUNDLES 5Note, for a Cartan connection ω the composition ωg/h := TPω→ g ։ g/h is a solderingform, providing a direct proof for (i) ⇒ (iv). As mentioned in the proof, ωg/h is identifiedvia (5) with the identity section id ∈ EndTM . Cf. [Ko], Theorem 2, and [AM], 2.4, 5.1.References[AM] Dmitri V. Alekseevsky and Peter W. Michor. Differential geometry of Cartan connec-tions Publ. Math. Debrecen 47 (1995), 349-375. Preprint math.DG/9412232. 3, 4, 5[Ko] S. Kobayashi. On Connections of Cartan. Canad. J. Math. 8 (1956), 145-156. 3, 5[KoNo] S. Kobayashi and K. Nomizu. Foundations of differential geometry. Volume I. IntersciencePublishers, New York, 1963. 2, 3, 4[Ruh] E. Ruh. Almost flat manifolds. J. of Differential Geom. 17 (1982), 1-14. 4[Sh] R.W. Sharpe. Differential Geometry. Cartan’s Generalization of Klein’s Erlangen Program.Springer 1996. 2, 3, 4Lehrstuhl B fu¨r Mathematik, Technical University of Aachen, 52062 GermanyE-mail address : mohamed.barakat@rwth-aachen.de

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