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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
The Cartan geometry induced by parabolic subgroup inclusions is called parabolic geometry.
Parabolic geometries are Cartan geometries of type $(G, P)$, where $G$ is a semisimple Lie group and $P \subset G$ is a parabolic subgroup. The corresponding homogeneous spaces $G/P$ are the so–called generalized flag manifolds which are among the most important examples of homogeneous spaces. Under the conditions of regularity and normality, parabolic geometries always are equivalent to underlying structures. (Cap 05)
Parabolic geometries include conformal geometry, projective geometry, almost quaternionic structures, almost Grassmannian structures, hypersurface type CR structures, systems of 2nd order ODEs, and various bracket-generating distributions. An example of a Cartan geometry that is not parabolic is Riemannian geometry.
A key application of parabolic Cartan geometry is to the construction of curved generalizations of the BGG resolution.
All parabolic geometries admit a fundamental curvature quantity called harmonic curvature $\kappa_H$, which is a complete obstruction to flatness. The Weyl tensor is the specific instance of $\kappa_H$ in conformal geometry.
General discussion includes
Andreas Čap, H. Schichl, Parabolic Geometries and Canonical Cartan Connections Hokkaido Math. J. 29 no.3 (2000), 453-505
Andreas Čap, Two constructions with parabolic geometries, arXiv:0504389
Felipe Leitner, part 1, section 5 of Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry, 2007 (pdf)
Andreas Čap, Jan Slovák, Parabolic Geometries I – Background and General Theory, AMS 2009
Relation to twistor spaces is discussed in
Discussion of BGG sequences in parabolic geometry includes
for more see the references at BGG resolution – References – Curved generalization to parabolic Cartan geometries.
See also
A relation to Courant Lie 2-algebroids is discussed in
Stuart Armstrong, Rongmin Lu, Courant Algebroids in Parabolic Geometry (arXiv:1112.6425)
Xu Xiaomeng, Twisted Courant algebroids and coisotropic Cartan geometries, Journal of Geometry and Physics Volume 82, August 2014, Pages 124–131 (arXiv:1206.2282)
See also