nLab Cartan structural equations



Differential geometry

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

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Cartan geometry (super, higher)

Riemannian geometry



In Riemann-Cartan differential geometry, what are called Cartan’s structural equations (équations de structure Cartan 1923, p. 368, see Scholz 2019, p. 53) are expressions for the torsion TT and the curvature RR of a Cartan moving frame ee with (Cartan-)connection ω\omega via the exterior derivative and wedge product of their differential form-representatives (shown as usual in components on any local chart with respect to a trivialized fiber bundles and using Einstein summation convention):

(1)T a = de a + ω a be b, R ab = dω ab + ω a cω cb. \begin{array}{ccccc} T^a &=& \mathrm{d}e^a &+& \omega^a{}_b \wedge e^b \,, \\ R^{ab} &=& \mathrm{d} \omega^{a b} &+& \omega^{a}{}_c \wedge \omega^{c b} \,. \end{array}

In the historically motivating case relating to the description of the field of gravity in what is now called first-order formulation, the representatives of the frame field and connection

(e a) a=0 d Ω dR 1(; 1+d), (ω ab=ω ba) a,b0 d Ω dR 1(;𝔰𝔬(1,d)) \begin{array}{rcl} \big(e^a\big)_{a = 0}^{d} &\in& \Omega^1_{dR}\big(-; \mathbb{R}^{1+d}\big) \,, \\ \big( \omega^{a b} \,=\, -\omega^{b a} \big)_{a,b \in 0}^d &\in& \Omega^1_{dR}\big(-; \mathfrak{so}(1,d)\big) \end{array}

are differential 1-forms which may jointly be understood as taking values in the Poincaré Lie algebra of a given dimension, and the two structural equations (1) jointly express the total curvature 2-form of this connection, broken up into its components.


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:

Generalization to supergeometry (motivated by supergravity):

Last revised on March 17, 2024 at 15:50:46. See the history of this page for a list of all contributions to it.