arXiv subjects

Zhangwen Guo

Publications and source records attributed to Zhangwen Guo.

4 recordsLinked to original sources

arXiv2026

Two Fefferman-type constructions involving almost Grassmann structures and path geometries

We introduce a Fefferman-type construction that associates an almost Grassmannian structure of type $(2,n+1)$ to every $(n+1)$-dimensional path geometry. We prove that the construction is normal and provide two equivalent characterizing conditions for all almost Grassmannian structures which locally arise from this construction: one in terms of certain parallel tractors and the other in terms of a Weyl connection of an almost Grassmann structure. We prove that the latter condition is independent of the choice of Weyl connection. We then introduce a related Fefferman-type construction associating an almost Grassmannian structure of type $(2,n+1)$ to every almost Grassmannian structure of type $(2,n)$. We prove that this construction is non-normal and characterize all almost Grassmannian structures which locally arise in this way in Cartan geometric terms.

math.DG↗

arXiv2026

Parallel (co-)tractors and the geometry of first BGG solutions on almost Grassmannian structures

We study the standard tractor bundle and the standard cotractor bundle of an almost Grassmann structure: We provide explicit formulae for their splitting operators, first BGG operators as well as prolongation connections. We characterize parallel tractors and cotractors as well as the solutions of the BGG operators in standard geometric terms. Moreover, we describe the geometry canonically endowed on the zero locus of a solution of the first BGG operators.

math.DG↗

arXiv2026

Weyl structures for path geometries

Path geometries provide a geometric encoding of systems of second order ODE, which serves as a model for the geometric theory of more general systems of ODE and for cone structures. They are an instance of the family of parabolic geometries, thus they are second order structures that are difficult to study using the usual tools of differential geometry. The general theory of parabolic geometries provides several efficient tools for the study of path geometries, but these use Cartan geometry methods and hence are not easily accessible. In this article, we build a bridge between these general methods and an elementary approach to path geometries. Motivated by the general theory of Weyl structures (but not using it), we first define a family of distinguished connections that is analogous to Webster-Tanaka connections in CR geometry. These are parametrized by (local) non-vanishing sections of a line bundle naturally associated to the geometry, and the dependence of this choice is described explicitly. We also discuss the Schouten tensor associated to such a choice and its dependence on the choice. We explain how these ingredients can be used to obtain an elementary approach to tractor calculus for path geometries and give examples of applications to the construction of invariant operators. A second major result that we prove is that in the case of path geometries, there is a smaller subclass of distinguished Weyl structures which does not seem to have an analog for any other type of parabolic geometries. This has interesting relations to the refinement of the de Rham complex induced by a path geometry via the machinery of BGG sequences. Again, all this is proved using elementary methods without reference to the general theory.

math.DG↗

arXiv2024

On Relative Tractor Bundles

This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated to the geometry in question. In many situations of interest, it is known that this sequence provides a resolution of a sheaf that can locally be realized as a pullback from a local leaf space of a foliation that is naturally available in this situation. An explicit description of the latter sheaf was only available under much more restrictive assumptions. For any geometry which admits relative tractor bundles, we construct a large family of such bundles for which we obtain a simple, explicit description of the resolved sheaves under weak assumptions on the torsion of the geometry. In particular, we discuss the cases of Legendrean contact structures and of generalized path geometries, which are among the most important examples for which the relative BGG machinery is available. In both cases, we show that essentially all relative tractor bundles are obtained by our construction and our description of the resolved sheaves applies whenever the BGG sequence is a resolution.

math.DG↗