nLab parabolic geometry

Idea

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.

Examples

• conformal geometry via conformal connections

• CR geometry

Related concepts

• tractor bundle

• Schubert calculus

• twistor space

References

General discussion:

• Andreas Čap, H. Schichl: Parabolic Geometries and Canonical Cartan Connections, Hokkaido Math. J. 29 3 (2000) 453-505 [pdf]

• Andreas Čap: Two constructions with parabolic geometries [arXiv:0504389rbrack;

• 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) [ams:surv-154]

Relation to twistor spaces:

• Andreas Čap: Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005) 143-172 [arXiv:math/0102097]

Discussion of BGG sequences in parabolic geometry includes

• David Calderbank, Tammo Diemer, Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J.Reine Angew.Math. 537 (2001) 67-103 (arXiv:math/0001158)

for more see the references at BGG resolution – References – Curved generalization to parabolic Cartan geometries.

See also

• Wikipedia, Parabolic geometry (differential geometry)

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

• Olivier Biquard, Rafe Mazzeo, Parabolic geometries as conformal infinities of Einstein metrics, Archivum Mathematicum, vol. 42 (2006), issue 5, pp. 85-104 (dml, pdf)