Flag Structures Research Articles

On geometry of 2-nondegenerate CR structures of hypersurface type and flag structures on leaf spaces of Levi foliations

We construct canonical absolute parallelisms over real-analytic manifolds equipped with 2-nondegenerate, hypersurface-type CR structures of arbitrary odd dimension not less than 7 whose Levi kernel has constant rank belonging to a broad subclass of CR structures that we label as recoverable. For this we develop a new approach based on a reduction to a special flag structure, called the dynamical Legendrian contact structure, on the leaf space of the CR structure's associated Levi foliation. This extends the results of [23] from the case of regular CR symbols constituting a discrete set in the set of all CR symbols to the case of the arbitrary CR symbols for which the original CR structure can be uniquely recovered from its corresponding dynamical Legendrian contact structure. We find an explicit criterion for this recoverability. In particular, if the rank of the Levi kernel is 1 and the dimension of the CR manifold is not less than 7, then for each given signature of the reduced Levi form in the space of all CR symbols (which depend on continuous parameters) there are no more than 2 symbols for which the aforementioned recoverability fails, and while the present method is applicable for all but those 2 cases, they can be treated separately by the method of [23]. Our method clarifies the relationship between the bigraded Tanaka prolongation of regular symbols developed in [23] and their usual Tanaka prolongation, providing a geometric interpretation of conditions under which they are equal. Motivated by the search for homogeneous models with given nonregular symbols, we also describe a process of reduction from the original natural frame bundle, which is inevitable for structures with nonregular CR symbols. We demonstrate this reduction procedure for examples whose underlying manifolds have dimension 7 and 9.We show that, for any fixed rank r>1, in the set of all CR symbols associated with 2-nondegenerate, hypersurface-type CR manifolds of odd dimension greater than 4r+1 with rank r Levi kernel, the CR symbols not associated with any homogeneous model are generic, and, for r=1, the same result holds if the CR structure is pseudoconvex.

Geometric structures and configurations of flags in orbits of real forms

This is an introduction and a survey on geometric structures modelled on closed orbits of real forms acting on spaces of flags. We focus on 3-manifolds and the flag space of all pairs of a point and a line containing it in \({\mathbb{P}}({\mathbb{C}}^3)\). It includes a description of general flag structures which are not necessarily flat and a combinatorial description of flat structures through configurations of flags in closed orbits of real forms. We also review volume and Chern–Simons invariants for those structures.

Flag structures on real 3-manifolds

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space $$\mathbf{SL}(3,{{\mathbb {C}}})/B$$ , where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern–Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns–Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space.

Independent multistructures, flag structures, and sets of linearly independent polar elements

Let V be a finite dimensional complex vector space, and for each positive integer n let Sn(V) denote the vector space whose elements are the complex-valued symmetric n-linear functions on V×V×⋯×V (n-copies). If f∈S1(V), that is, f is a linear functional on V, then we define the symmetric n-linear function fn by fn(x1,x2,…,xn)=∏i=1nf(xi) for all x1,x2,…,xn∈V. It is well known that the elements fn, known as polar elements, span Sn(V). Our focus is the simple generation of sets of linearly independent polar elements of given order n. A basis for Sn(V) consisting entirely of polar elements is a polar basis. We introduce several new structures that are closely connected to the problem of generating independent sets of polar elements. In particular, we introduce the notion of independent multistructure of order n, n≥0, and depth p, p≥1, and show how such structures generate sets of linearly independent polar elements of cardinality (n+pp). When p=dim⁡(V) we obtain a basis for Sn(V) consisting entirely of polar elements. We also introduce the dual concept of flag structure of order n and depth p. Both of these concepts evolved from a consideration of subspace filters which were defined previously in [6].

Homological and Bloch invariants for $\mathbb Q$-rank one spaces and flag structures

We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for \mathbb Q -rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of ag structures can be recovered by a fundamental class construction.

Symmetries of trivial systems of ODEs of mixed order

We compute symmetry algebras of a system of two equations y(k)(x)=z(l)(x)=0, where 2⩽k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for (k,l)=(2,3) we get two non-isomorphic symmetry algebras of the same dimension. We explore how these symmetry algebras are related to both Sternberg prolongation of G-structures and Tanaka prolongation of graded nilpotent Lie algebras.Surprisingly, the case (k,l)=(2,3) provides an example of a linear subalgebra a in gl(5,R) such that the Sternberg prolongations of a and at are both of the same dimension, but are non-isomorphic.We also discuss the non-linear case and the link with flag structures on smooth manifolds.

PROLONGATION ON REGULAR INFINITESIMAL FLAG MANIFOLDS

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-) linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds M a conceptual method to rewrite these systems as systems of the form [Formula: see text], where [Formula: see text] is a linear connection on some vector bundle V over M and C : V → T* M ⊗ V is a (vector) bundle map. In particular, if the overdetermined system is linear, [Formula: see text] will be a linear connection on V and hence the dimension of its solution space is bounded by the rank of V. We will see that the rank of V can be easily computed using representation theory.

Flag enumerations of matroid base polytopes

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.

Flag structures on Seifert manifolds

We consider faithful projective actions of a cocompact lattice of SL(2,R) on the projective plane, with the following property: there is a common fixed point, which is a saddle fixed point for every element of infinite order of the the group. Typical examples of such an action are linear actions, ie, when the action arises from a morphism of the group into GL(2,R), viewed as the group of linear transformations of a copy of the affine plane in RP^{2}. We prove that in the general situation, such an action is always topologically linearisable, and that the linearisation is Lipschitz if and only if it is projective. This result is obtained through the study of a certain family of flag structures on Seifert manifolds. As a corollary, we deduce some dynamical properties of the transversely affine flows obtained by deformations of horocyclic flows. In particular, these flows are not minimal.

A note on generalized flag structures

Generalized flag structures occur naturally in modern geometry. By extending Stefan's well-known statement on generalized foliations we show that such structures admit distinguished charts. Several examples are included.

Characteristic classes of almost-flag structures

The aim of this paper is to present a basic theory of characteristic classes of almost-flag structures. Vanishing theorems for primary characteristic classes of almost-flag and almost-product structures are proved. Consequences of these theorems for secondary characteristic classes are drawn in the framework of Lehmann's theory. Two theorems on residues of flag structures complete the paper.