Normal Cartan Connection Research Articles

Conformal and projective tractors from 2-frame bundles by the dressing field method

This paper is devoted to the study of conformal and projective structures, and especially their connections, in the language of 2-frames, or [Formula: see text]-structures of second order. While their normal Cartan connections are well-known, we use the dressing field method to obtain their local, mostly gauge invariant, connections. Lastly, we recall how to obtain the tractor bundle from conformal structures, and then show that we can define a projective tractor bundle using the same kind of construction.

Homogeneous Integrable Legendrian Contact Structures in Dimension Five

We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using the techniques of parabolic differential geometry, we compute the associated regular, normal Cartan connection and give explicit formulas for the harmonic part of the curvature. The PDE system is trivializable by means of point transformations if and only if the harmonic curvature vanishes identically. In dimension five, the harmonic curvature takes the form of a binary quartic field, so there is a Petrov classification based on its root type. We give a complete local classification of all five-dimensional integrable Legendrian contact structures whose symmetry algebra is transitive on the manifold and has at least one-dimensional isotropy algebra at any point.

CR Embedded Submanifolds of CR Manifolds

We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Cartan (or tractor) connections via a CR Gauss formula. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more naive methods. We conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.

New relations between G2 geometries in dimensions 5 and 7

There are two well-known parabolic split [Formula: see text] geometries in dimension 5, [Formula: see text] distributions and [Formula: see text] contact structures. Here we link these two geometries with yet another [Formula: see text] related contact structure, which lives on a [Formula: see text]-manifold. More precisely, we present a natural geometric construction that associates to a [Formula: see text] distribution a [Formula: see text]-dimensional bundle endowed with a canonical Lie contact structure. We further study the relation between the canonical normal Cartan connections associated with the two structures and we show that the Cartan holonomy of the induced Lie contact structure reduces to [Formula: see text]. This motivates the study of the curved orbit decomposition associated with a [Formula: see text] reduced Lie contact structure on a [Formula: see text]-manifold. It is shown that, provided an additional curvature condition is satisfied, in a neighborhood of each point in the open curved orbit the structure descends to a [Formula: see text] distribution on a local leaf space. The closed orbit carries an induced [Formula: see text] contact structure.

3-dimensional left-invariant sub-Lorentzian contact structures

We provide a classification of ts-invariant sub-Lorentzian structures on 3 dimensional contact Lie groups. Our approach is based on invariants arising form the construction of a normal Cartan connection.

On conformal subriemannian fundamental graded Lie algebras and Cartan connections

Under some condition we classify the prolongations of conformal subriemannian fundamental graded Lie algebras. Also we show the existence of a normal Cartan connection associated with a bracket generating regular conformal subriemannian manifold.

Tanaka structures (non holonomic G-structures) and Cartan connections

Let h=h−k⊕⋯⊕hl (k>0, l≥0) be a finite dimensional graded Lie algebra, with a Euclidean metric 〈⋅,⋅〉 adapted to the gradation. The metric 〈⋅,⋅〉 is called admissible if the codifferentials ∂∗:Ck+1(h−,h)→Ck(h−,h) (k≥0) are Q-invariant (Lie(Q)=h0⊕h+). We find necessary and sufficient conditions for a Euclidean metric, adapted to the gradation, to be admissible, and we develop a theory of normal Cartan connections, when these conditions are satisfied. We show how the treatment from Cap and Slovak (2009), about normal Cartan connections of semisimple type, fits into our theory. We also consider in detail the case when h≔t∗(g) is the cotangent Lie algebra of a non-positively graded Lie algebra g.

Second order ODEs under area-preserving maps

We study the geometry of real analytic second order ODEs under the local real analytic diffeomorphism of \(\mathbb {R}^2\) which are area preserving, through the method of Cartan. We obtain a subdivision into three “parts”. The first one is the most symmetric case. It is characterized by the vanishing of an area-preserving relative invariant namely \(f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\). In this situation we associate a local affine normal Cartan connection on the first jet \(J^{1}(\mathbb {R},\mathbb {R})\) space whose curvature contains all the area-preserving relative differential invariants, to any second order ODE under study. The second case which includes all the Painleve transcendents is given by the ODEs for which \(f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\not \equiv 0\). In the latter case we give all necessary steps in order to obtain an \(e\)-structure on \(J^{1}(\mathbb {R},\mathbb {R})\) for a generic second order ODE equation of that type. Finally we give the method to reduce to an \(e\)-structure on \(J^{1}\) when \(f_{y^\prime y^\prime y^\prime y^\prime }\not \equiv 0\).

NORMAL BGG SOLUTIONS AND POLYNOMIALS

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 canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of local frames for any natural vector bundle associated to a parabolic geometry. We then prove that the coefficient functions of any normal solution of a first BGG operator with respect to such a frame are polynomials in the normal coordinates of the parabolic geometry. A bound on the degree of these polynomials in terms of representation theory data is derived. For geometries locally isomorphic to the homogeneous model of the geometry we explicitly compute the local frames mentioned above. Together with the fact that on such structures all solutions are normal, we obtain a complete description of all first BGG solutions in this case. Finally, we prove that in the general case the polynomial system coming from a normal solution is the pull-back of a polynomial system that solves the corresponding problem on the homogeneous model. Thus we can derive a complete list of potential normal solutions on curved geometries. Moreover, questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of polynomial systems and real algebraic sets.

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 sequences are constructed from the normal covariant derivative \nabla^\omega on the corresponding tractor bundle V, where \omega is the normal Cartan connection. The first operator D_0 in the sequence is overdetermined and it is well known that \nabla^\omega yields the prolongation of this operator in the homogeneous case M = G/P . Our first main result is the curved version of such a prolongation. This requires a new normalization of the tractor covariant derivative on V . Moreover, we obtain an analogue for higher operators D_i . In that case one needs to modify the exterior covariant derivative d^{\nabla^\omega} by differential terms. Finally we demonstrate these results on simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.

Subtleties concerning conformal tractor bundles

Abstract The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures.

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.

Prolongation on contact manifolds

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

Inclusions between Parabolic Geometries

Some of the well known Fefferman like constructions of parabolic geometries end up with a new structure on the same manifold. In this paper, we classify all such cases with the help of the classical Onishchik's lists [10] and we treat the only new series of inclusions in detail, providing the spinorial structures on the manifolds with generic free distributions. Our technique relies on the cohomological understanding of the canonical normal Cartan connections for parabolic geometries and the classical computations with exterior forms. Apart of the complete discussion of the distributions from the geometrical point of view and the new functorial construction of the inclusion into the spinorial geometry, we also discuss the normality problem of the resulting spinorial connections. In particular, there is a non-trivial subclass of distributions providing normal spinorial connections directly by the construction.

Tractor calculi for parabolic geometries

Parabolic geometries may be considered as curved analogues of the homogeneous spaces G / P G/P where G G is a semisimple Lie group and P ⊂ G P\subset G a parabolic subgroup. Conformal geometries and CR geometries are examples of such structures. We present a uniform description of a calculus, called tractor calculus, based on natural bundles with canonical linear connections for all parabolic geometries. It is shown that from these bundles and connections one can recover the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. We construct explicitly a family of fundamental first order differential operators, which are analogous to a covariant derivative, iterable and defined on all natural vector bundles on parabolic geometries. For an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus.

Twistor theory of manifolds with Grassmannian structures

AbstractAs a generalization of the conformal structure of type (2, 2), we study Grassmannian structures of type (n, m) forn, m≥ 2. We develop their twistor theory by considering the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.A Grassmannian structure of type (n, m) on a manifoldMis, by definition, an isomorphism from the tangent bundleTMofMto the tensor productV ⊗ Wof two vector bundlesVandWwith ranknandmoverMrespectively. Because of the tensor product structure, we have two null plane bundles with fibresPm-1(ℝ) andPn-1(ℝ) overM. The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka’s normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.Besides the integrability conditions corrsponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

The geometry of hyperbolic and elliptic CR-manifolds of codimension two

The general theory of parabolic manifolds is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two.

Third order ordinary differential equations and Legendre connections

We use N. Tanaka's theory of normal Cartan connections associated with simple graded Lie algebras to study Cartan's equivalence problem of single third order ordinary differential equations under contact transformations. As a result we obtain the complete structure equation with two differential inv riants, which is applied on general Legendre Grassmann bundles of three-dimensional contact manifolds.

Lie contact structures and normal Cartan connections

Lie contact structures and normal Cartan connections

A note on projective foliations

In [12], Nishikawa and Sato studied conformal and projective foliationsdefined as foliations whose second order transversal bundle is endowed witheither a conformal or a projective projectable structure. (See, for instance[7] for such structures on manifolds.) Namely, they proved the existence ofcorresponding projectable normal Cartan connections from which they deduce thatthe same strong Bott vanishing phenomenon like for Riemannian foliations holds.Then, Nishikawa studied characteristic classes of projective foliations in [13].Independently, I discussed conformal foliations in [16] using the classicaldefinition of conformal structures by means of Riemannian metrics, and Montesinos[11] proved the strong Bott vanishing theorem for conformal foliations, by usingthis classical approach.The aim of this Note is to present projective foliations by using the ,alternate approach to projective structures known as geometry of paths [6], andby constructing the normal connection with a vector bundle version of the originalCartan technique [4] . This approach will provide us not only with the Bott-Nishikawa-Sato vanishing theorem of [12, 13] but also with projectively invariantrepresentative forms of the real Pontrjagin classes of manifolds and of transversebundles of foliations. Furthermore, we shall obtain a cohomological obstructionto the existence of a transversal projective projectable (I am using the narrefoliate instead) structure.Beyond all this, since two approaches to projective structures on manifoldsare available, it seems natural to use them both in studying projective foliationsas well.109