NORMAL BGG SOLUTIONS AND POLYNOMIALS

@article{ap2012NORMALBS,
  title={NORMAL BGG SOLUTIONS AND POLYNOMIALS},
  author={A. {\vC}ap and A. Gover and M. Hammerl},
  journal={International Journal of Mathematics},
  year={2012},
  volume={23},
  pages={1250117}
}
First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the… Expand
Relative BGG sequences; II. BGG machinery and invariant operators
Abstract For a real or complex semisimple Lie group G and two nested parabolic subgroups Q ⊂ P ⊂ G , we study parabolic geometries of type ( G , Q ) . Associated to the group P , we introduce theExpand
Distinguished curves and integrability in Riemannian, conformal, and projective geometry
We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is a type of moving incidenceExpand
Projective geometry of Sasaki-Einstein structures and their compactification
We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projectiveExpand
Einstein metrics in projective geometry
It is well known that pseudo–Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certainExpand
The zero set of a twistor spinor in any metric signature
Using tractor methods, we exhibit the local structure of the zero set of a twistor spinor in any metric signature. It is given as the image under the exponential map of a distinguished totallyExpand
C-projective geometry
We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry andExpand
Detecting Einstein geodesics: Einstein metrics in projective and conformal geometry
Here we treat the problem: given a torsion-free connection do its geodesics, as unparametrised curves, coincide with the geodesics of an Einstein metric? We find projective invariants such that theExpand
Nearly Kähler Geometry and $(2,3,5)$-Distributions via Projective Holonomy
We show that any dimension 6 nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\mathrm{G}_2^{(*)}$ holonomy reduction. In the converse direction weExpand
Distinguished curves and first integrals on Poincaré-Einstein and other conformally singular geometries
We treat the problem of defining, and characterising in a practical way, an appropriate class of distinguished curves for Poincar\'e-Einstein manifolds, and other conformally singular geometries.Expand
Subriemannian Metrics and the Metrizability of Parabolic Geometries
We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R.Expand
...
1
2
...

References

SHOWING 1-10 OF 55 REFERENCES
Projective BGG equations, algebraic sets, and compactifications of Einstein geometries
TLDR
It is shown that a normal solution determines a canonical manifold stratification that reflects an orbit decomposition of the model, and questions concerning the zero locus of solutions are reduced to a study of the corresponding polynomial systems and algebraic sets. Expand
Infinitesimal automorphisms and deformations of parabolic geometries
We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de--Rham sequence associated to a certain linearExpand
Tractor calculi for parabolic geometries
Parabolic geometries may be considered as curved analogues of the homogeneous spaces G/P where G is a semisimple Lie group and P C G a parabolic subgroup. Conformal geometries and CR geometries areExpand
Holonomy reductions of Cartan geometries and curved orbit decompositions
We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initialExpand
On a new normalization for tractor covariant derivatives
A regular normal parabolic geometry of type G/P on a manifold M gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequencesExpand
Conformal geometry and 3-plane fields on 6-manifolds
The purpose of this note is to provide yet another example of the link between certain conformal geometries and ordinary differential equations, along the lines of the examples discussed by NurowskiExpand
Notes on Projective Differential Geometry
Projective differential geometry was initiated in the 1920s, especially by Elie Cartan and Tracey Thomas. Nowadays, the subject is not so well-known. These notes aim to remedy this deficit andExpand
Ricci-corrected derivatives and invariant differential operators
Abstract We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables usExpand
Coupling solutions of BGG-equations in conformal spin geometry
BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary andExpand
Differential equations and conformal structures
We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between theExpand
...
1
2
3
4
5
...