nLab torsion of a Cartan connection

for disambiguation see at torsion

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Definiton

Given an $(H \stackrel{i}{\to} G)$-_Cartan connection_ $\nabla$ on a manifold $X$, its torsion is the component of its curvature in the coimage of $i_\ast : \mathfrak{h}\to \mathfrak{g}$.

So if $U \to X$ is a cover over which the underlying $G$-principal bundle trivializes (for instance the total space of the $G$-principal bundle itself if one models the Cartan connection by an Ehresmann connection) and

$$\omega \in \Omega^1(U,\mathfrak{g})$$

is the Lie algebra valued curvature form, then the torsion form is

$$\tau \coloneqq coim(i)_\ast \omega \in \Omega^1(U,\mathfrak{g}/\mathfrak{h}) \,.$$

e.g. (Sharpe 97, section 5.3, below def 3.1, Cap-Slovák 09, section 1.5.7, p. 85)

Properties

Integrability of local charts

If the torsion of a Cartan connection vanishes, then it has flat $\mathfrak{g}/\mathfrak{h}$-valued parallel transport. In partciular then every point in the base manifold has a neighbourhood $U_x$ over which the given $G$-principal bundle trivializes and then this parallel trasport gives an identification $U_x \simeq U_{\sigma(x)}(G/H)$ with an open subset in the Klein geometry $G/H$.

Relation to integrability of $G$-structure

If the Cartan connection is regarded as providing, in particular, a G-structure, then the condition that its torsion vanishes is the integrability of G-structures.

Examples

(Pseudo-)Riemannian geometry

For $(H \to G)$ being the inclusion $O(d)\hookrightarrow Iso(\mathbb{R}^d)$ of the orthogonal group into the Euclidean group or the inclusion $O(d-1,1)\hookrightarrow Iso(\mathbb{R}^{d-1,1})$ of the Lorentz group into the Poincare group, then an $(H\to G)$-Cartan connection encodes a (pseudo-)Riemannian geometry with metric connection. Its torsion then is the torsion of a metric connection. See also at first-order formulation of gravity.

Related concepts

• torsion of a G-structure

• torsion of a metric connection

References

General

Historical origin of the notion in Cartan geometry:

• Élie Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces á torsion, C. R. Acad. Sci. 174 (1922) 593-595 .

• 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 [doi: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]

Monographs:

• Richard W. Sharpe, Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Graduare 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 of torsion in gravitational classical field theory:

• Friedrich W. Hehl, Yuri N. Obukhov, Élie Cartan’s torsion in geometry and in field theory, an essay, Annales de la Fondation Louis de Broglie, 32 2-3 (2007) 157-194 [pdf, arXiv:0711.1535]

Discussion with an eye towards torsion constraints in supergravity:

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

  following:

  John Lott, Torsion constraints in supergeometry, Comm. Math. Phys. 133 (1990) 563-615 [doi:10.1007/BF02097010]

Cartan structural equations and Bianchi identities

On Cartan structural equations and their Bianchi identities for curvature and torsion of Cartan moving frames and (Cartan-)connections on tangent bundles (especially in first-order formulation of gravity):

The original account:

• 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 [doi:ASENS_1923_3_40__325_0]

Historical review:

• Erhard Scholz, §2 in: 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]

Further discussion:

• Shiing-Shen Chern, p. 748 of: A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics, Second Series, 45 4 (1944) 747-752 [doi:10.2307/1969302]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §I.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.2: pdf]

• Sigurdur Helgason, §I.8 in: Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]

• C. C. Briggs, A Sequence of Generalizations of Cartan’s Conservation of Torsion Theorem [arXiv:gr-qc/9908034]

• Loring Tu, §22 in: Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer (2017) [ISBN:978-3-319-55082-4]

• Thoan Do, Geoff Prince, An intrinsic and exterior form of the Bianchi identities, International Journal of Geometric Methods in Modern Physics 14 01 (2017) 1750001 [doi:10.1142/S0219887817500013, arXiv:1501.01123]

• Ivo Terek Couto, Cartan Formalism and some computations [pdf, pdf]

• Cadabra, Cartan structural equations and Bianchi identity

Generalization to supergeometry (motivated by supergravity):

• Julius Wess, Bruno Zumino, p. 362 of: Superspace formulation of supergravity, Phys. Lett. B 66 (1977) 361-364 [doi:10.1016/0370-2693(77)90015-6]

• Richard Grimm, Julius Wess, Bruno Zumino, §2 in: A complete solution of the Bianchi identities in superspace with supergravity constraints, Nuclear Phys. B 152 (1979) 255-265 [doi:10.1016/0550-3213(79)90102-0]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §III.3.2 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch III.3: pdf]