synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
In the wake of the movement of ideas which followed the general theory of relativity, I was led to introduce the notion of new geometries, more general than Riemannian geometry, and playing with respect to the different Klein geometries the same role as the Riemannian geometries play with respect to Euclidean space. The vast synthesis that I realized in this way depends of course on the ideas of Klein formulated in his celebrated Erlangen programme while at the same time going far beyond it since it includes Riemannian geometry, which had formed a completely isolated branch of geometry, within the compass of a very general scheme in which the notion of group still plays a fundamental role.
[Élie Cartan 1939, as quoted in Sharpe 1997, p. 171]
Cartan geometry is geometry of spaces that are locally (infinitesimally, tangentially) like coset spaces $G/H$, i.e. like Klein geometries. Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling without sliding’ the ‘model geometry’ $G/H$ along it. Hence Cartan geometry may be thought of as the globalization of the program of Klein geometry initiated in the Erlangen program.
Cartan geometry subsumes many types of geometry, such as notably Riemannian geometry, conformal geometry, parabolic geometry and many more. As a Cartan geometry is defined by principal connection data (hence by cocycles in nonabelian differential cohomology) this means that it serves to express all these kinds of geometries in connection data.
This is used notably in the first order formulation of gravity, which was the motivating example in the original text (Cartan 22). The physics literature tends to use the term “Cartan moving frame method” instead of “Cartan geometry”.
A Cartan geometry is a space equipped with a Cartan connection. See there for more.
(Riemann-Cartan geometry of $S^3$) We discuss the Riemannian geometry of the round 3-sphere $S^3$ as a $ISO(3)/O(3)$-Cartan geometry, hence via frame field and spin connection with vanishing torsion tensor (as one encounters it in the first-order formulation of gravity).
Among all $n$-spheres, this is particularly easy for the $S^3$, since its underlying smooth manifold is that of the Lie group $SU(2)$. This means that the frame field can be chosen to be globally defined
and to be given by left-invariant differential forms that satisfy the Maurer-Cartan equation
where $\epsilon^{i j k}$ is the Levi-Civita symbol in 3 dimensions, and we use the Einstein summation convention throughout.
In next defining the “spin connection” form
we shall use the convention where the torsion-tensor $\big(T^i\big)_{i = 1}^3$ and curvature-tensor $\big(R^{i j}\big)_{i,j=1}^3$ are formed with a relative minus sign as
This implies that the unique torsion-free spin connection is given by
since, by the above:
To find the curvature of this connection we compute
where we take the multi-Kronecker symbol is normalized as
and
to obtain:
hence
Notice that, therefore, with the conventions (1) the scalar curvature $\mathrm{R}$ of the $S^3$ comes out with a negative sign, since the Ricci tensor of $S^3$ is
so that
This may seem undesirable, but it is a common choice in practice (cf. e.g. Freund & Rubin 1980, below (4b)).
The original re-formulation of Riemannian geometry via coframe fields:
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, Comptes rendus hebdomadaires des séances de l’Académie des sciences, 174, 437-439, 593-595, 734-737, 857-860, 1104-1107 (January 1922).
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]
Élie Cartan, La géométrie des espaces de Riemann, Mémorial des sciences mathématiques 9 (1925) [numdam:MSM_1925__9__1_0]
and English translation of original lectures:
Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]
Élie Cartan (translated by Robert Hermann from Cartan’s lectures in 1951): Geometry of Riemannian Spaces, Lie Groups: History, Frontiers and Applications XIII, Math Sci Press (1983) [ark:/13960/s28rzmj9xrv]
Historical review:
Introductions:
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]
Benjamin McKay, An introduction to Cartan geometries, Proceedings of the 21st Winter School “Geometry and Physics”, Palermo (2002) [arXiv:2302.14457, eudml:220395]
Monographs:
Richard W. Sharpe, Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Graduate Texts in Mathematics 166, Springer (1997) [ISBN:9780387947327]
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Chapter I.3.7 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.3: pdf]
(these authors refer to Cartan geometries as “soft group manifolds”)
Andreas Čap, Jan Slovák, chapter 1 of: Parabolic Geometries I – Background and General Theory, AMS (2009) [ISBN:978-1-4704-1381-1]
For more see at Cartan connection – References.
Discussion in modal homotopy type theory is in
Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, 2017
Felix Wellen, Cartan Geometry in Modal Homotopy Type Theory (arXiv:1806.05966)
See also
wikipedia: Cartan connection
The blog discussion of Derek Wise, MacDowell-Mansouri gravity and Cartan geometry.
Last revised on May 18, 2024 at 11:40:22. See the history of this page for a list of all contributions to it.