nLab
parabolic geometry

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

The Cartan geometry induced by parabolic subgroup inclusions is called parabolic geometry.

Parabolic geometries are Cartan geometries of type (G,P)(G, P), where GG is a semisimple Lie group and PGP \subset G is a parabolic subgroup. The corresponding homogeneous spaces G/PG/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 κ H\kappa_H, which is a complete obstruction to flatness. The Weyl tensor is the specific instance of κ H\kappa_H in conformal geometry.

Examples

References

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

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

Revised on July 21, 2016 03:32:13 by David Corfield (147.147.173.235)