Cartan connection in nLab

Contents

Contents

Idea

A Cartan connection is a principal connection on a smooth manifold equipped with a certain compatibility condition with the tangent bundle of the manifold. It combines the concept of G-structure with that of soldering form. This combination allows us to express various types of geometric structures on XX – such as notably (pseudo-)Riemannian geometry, conformal geometry and many more (see below) – in terms of connection data, i.e. in terms of nonabelian differential cohomology-data. In particular the first order formulation of gravity in terms of Cartan connections has been one of the historical motivations (Cartan 23).

In a little bit more detail, a Cartan connection on a manifold XX for a given subgroup inclusion H↪GH \hookrightarrow G is data that identifies all the tangent spaces T xXT_x X of XX with the tangent space 𝔤/𝔥=T eH(G/H)\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H) of the coset space Klein geometry G/HG/H, such that the choice of these identifications is transported along compatibly.

Therefore a manifold equipped with a Cartan connection is also called a Cartan geometry (see also there), a generalization (globalization) of the concept of Klein geometry.

In yet a little bit more detail, an (H↪G)(H \hookrightarrow G)-Cartan connection on XX is a GG-principal connection on XX equipped with a reduction of its structure group along H→GH \to G and such that the connection 1-form linearly identifies each tangent space T xXT_x X of XX with the tangent space 𝔤/𝔥=T eH(G/H)\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H) of the coset space.

History

The concept essentially originates around (Cartan 23), but the formulation in terms of principal connections and in fact the terminology “Cartan connection” is due to Charles Ehresmann who formulated principal connections as what, in turn, today are called Ehresmann connections (Ehresmann 50).

In (Ehresmann 50) Cartan’s ideas are formalized (see Marle 14, page 9, 10 for review) by saying that an (G↪H)(G \hookrightarrow H)-Cartan connection is a GG-Ehresmann connection on a GG-principal bundle PP equipped with an HH-principal subbundle QQ, such that the restriction of the connection form along this inclusion yields a form that determines an isomorphism of each tangent space of QQ with 𝔤\mathfrak{g}.

Definition

Traditional

Let GG be a Lie group and H↪GH \hookrightarrow G a sub-Lie group. (So that we may think of the coset space G/HG/H as a Klein geometry.) Write 𝔥↪𝔤\mathfrak{h} \hookrightarrow \mathfrak{g} for the corresponding Lie algebras.

There are various equivalent forms of the definition of Cartan connections. The following one characterizes it as a GG-principal connection equipped with extra structure and property.

See also Wikipedia – Cartan connection – As principal connections.

(Sharpe, section 5.3, below def. 3.1, Cap-Slovák 09, section 1.5.7, p. 85, Lott 01, section 3).

Synthetically in terms of differential cohesion

We discuss a synthetic formulation of Cartan connections in terms of differential cohesion.

under construction

Write

• Ω 1(−,𝔤)\Omega^1(-,\mathfrak{g}) for the sheaf (on the site of formal smooth manifolds) of Lie algebra valued differential forms, regarded as the smooth moduli space of 𝔤\mathfrak{g}-differential forms (as explained at geometry of physics in the chapter on differential forms)

• BG conn≃Ω 1(−,𝔤)//G\mathbf{B} G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G for the universal moduli stack of connections, which is equivalently the homotopy quotient of Ω 1(−,𝔤)\Omega^1(-,\mathfrak{g}) by the action of GG (regarded as a smooth group) by gauge transformations;

• Ω 1(−,𝔤)//H\Omega^1(-,\mathfrak{g})//H for the homotopy quotient by just the subgroup H↪GH \hookrightarrow G;

• J:Ω 1(−,𝔤)//H⟶Ω 1(−,𝔤)//G≃BG conn\mathbf{J} \;\colon\;\Omega^1(-,\mathfrak{g})//H \longrightarrow \Omega^1(-,\mathfrak{g})//G\simeq \mathbf{B}G_{conn} for the canonical morphism.

We need this and one more ingredient for synthetically formalizing Cartan connections:

The following is a synthetic formulation of Cartan connections, def. .

Weaker definitions (pre- and semi-Cartan geometry)

We discuss here some weakining of the above definition of Cartan connection that have their uses.

Pre-Cartan geometry

…(Kuranishi 95)…

Examples

Table of example

(pseudo-)Riemannian geometry

Let G=Iso(d,1)G = Iso(d,1) be the Poincare group and H⊂GH \subset G the orthogonal group O(d,1)O(d,1). Then the quotient

𝔦𝔰𝔬(d,1)/𝔰𝔬(d,1)≃ℝ d+1 \mathfrak{iso}(d,1)/\mathfrak{so}(d,1) \simeq \mathbb{R}^{d+1}

is Lorentzian spacetime. Therefore an (O(d,1)↪Iso(d,1))(O(d,1)\hookrightarrow Iso(d,1))-Cartan connection is equivalently an O(d,1)O(d,1)-connection on a manifold whose tangent spaces look like Minkowski spacetime: this is equivalently a pseudo-Riemannian manifold from the perspective discussed at first-order formulation of gravity:

the ℝ d+1\mathbb{R}^{d+1}-valued part of the connection is the vielbein.

GG-Structures

More generally, G-structures equipped with compatible principal connections are given by Cartan connections. (We will speak of “HH-structure” here, since the reduced structure will correspond to the group denoted HH above, while what is denoted GG above will be the semidirect product of HH with the translation group).

Let H→GL(ℝ n)H \to GL(\mathbb{R}^n) be a Lie group homomorphism, so that reduction of the structure group of the frame bundle of a manifold of dimension nn along this map is an H-structure on the manifold. Then write

G≔ℝ n⋊H G \coloneqq \mathbb{R}^n \rtimes H

for the semidirect product of HH with the translation group ℝ n\mathbb{R}^n, given via the induced action of HH on ℝ n\mathbb{R}^n via the canonical action of the general linear group GL(ℝ n)GL(\mathbb{R}^n).

With this an (H↪G)=(H↪ℝ n⋊H)(H \hookrightarrow G)= (H \hookrightarrow \mathbb{R}^n \rtimes H)-Cartan connection is equivalently an H-structure equipped with a vielbein field and with an HH-principal connection.

(CapSlovak 09, section 1.3.6 and 1.6.1)

With this identification the torsion of a Cartan connection maps into the torsion of a G-structure.

• connection on a bundle

  • parallel transport, holonomy

• principal connection

  • affine connection, Levi-Civita connection, Cartan connection

• connection on a 2-bundle

• connection on an ∞-bundle

  • ∞-Cartan connection

  • higher parallel transport

References

General

The idea originates in Élie Cartan‘s “method of moving frames” (cf. Cartan geometry)L

• Élie Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Annales scientifiques de l’École Normale Supérieure, Sér. 3 40 (1923) 325-412 [numdam:ASENS_1923_3_40__325_0]

Historical review:

• Erhard Scholz, E. Cartan’s attempt at bridge-building between Einstein and the Cosserats – or how translational curvature became to be known as “torsion”, The European Physics Journal H 44 (2019) 47-75 [doi:10.1140/epjh/e2018-90059-x]

The formalization in terms of principal connections (in their incarnation as Ehresmann connections) is due to

• Charles Ehresmann, Les connexions infinitesimales dans un espace fibre diff´erentiable, Colloque de topologie de Bruxelles, 1950, p. 29–55.

reviewed in

• Charles-Michel Marle, The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles (arxiv:1401.8272)

see also

• Shôshichi Kobayashi, On Connections Of Cartan, Canadian Journal of Mathematics 8 (1956) 145-156 [doi:10.4153/CJM-1956-018-8]

Introduction:

• Richard W. Sharpe, An introduction to Cartan Geometries, Proceedings of the 21st Winter School “Geometry and Physics”, Circolo Matematico di Palermo (2002) 61-75 [eudml:220395, dml:701688]

Monographs:

• Richard W. Sharpe, Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Graduade Texts in Mathematics 166, Springer (1997) [ISBN:9780387947327]

• Andreas Čap, Jan Slovák, chapter 1 of: Parabolic Geometries I – Background and General Theory, AMS (2009) [ISBN:978-1-4704-1381-1]

Discussion with an eye towards first-order formulation of gravity:

• Gabriel Catren, Geometrical Foundations of Cartan Gauge Gravity, Int. J. Geom. Methods in Modern Physics 12 04 (2015) 1530002 [arXiv:1407.7814, doi:10.1142/S0219887815300020]

• Kirill Krasnov, §3 in: Formulations of General Relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2020) [doi:10.1017/9781108674652, taster:pdf]

Discussion with an eye towards torsion constraints in supergravity:

• John Lott, Torsion Constraints in Supergravity, Comm. Math. Phys. 133 (1990) 563-615

• John Lott, The Geometry of Supergravity Torsion Constraints [arXiv:0108125]

See also

• Masatake Kuranishi, CR geometry and Cartan geometry, Forum mathematicum (1995) Volume: 7, Issue: 2, page 147-206 (EuDML page, page with link to pdf)

• Dmiti Alekseevesky, Peter Michor, Differential geometry of Cartan connections Publ. Math. Debrecen 47/3-4 (1995), 349-375 (pdf)

Further discussion of Cartan connections as models for the first order formulation of gravity is in

• Derek Wise, MacDowell-Mansouri gravity and Cartan geometry, Class.Quant.Grav.27:155010,2010 (arXiv:gr-qc/0611154)

• Gabriel Catren, Geometrical Foundations of Cartan Gauge Gravity (arXiv:1407.7814)

See also

• wikipedia Cartan connection