# nLab parabolic geometry

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(ʃ \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$ʃ_{dR} \dashv \flat_{dR}$

• tangent cohesion

• differential cohomology diagram
• differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

$(\Re \dashv \Im \dashv \&)$

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

• 

\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{&#233;tale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& &#643; &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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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 $(G, P)$, 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.

## References

General discussion includes

• Andreas ?ap?, H. Schichl, Parabolic Geometries and Canonical Cartan Connections Hokkaido Math. J. 29 no.3 (2000), 453-505

• Andreas ?ap?, 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 ?ap?, Jan Slovák, Parabolic Geometries I – Background and General Theory, AMS 2009

Relation to twistor spaces is discussed in

• Andreas ?ap?, 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

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