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Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.


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Journal ArticleDOI
TL;DR: A local analysis of integrable GL(2)-structures of degree 4 is given in this article, where the main results are a structure theorem for integrably connected integrability, and an equivalence between local integrables and Hessian hydrodynamic hyperbolic PDEs in three variables.
Abstract: This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-structures are important because they naturally arise from a certain family of second-order hyperbolic PDEs in three variables that are integrable via hydrodynamic reduction. Familiar examples include the wave equation, the first flow of the dKP equation, and the Boyer--Finley equation. The main results are a structure theorem for integrable GL(2)-structures, a classification for connected integrable GL(2)-structures, and an equivalence between local integrable GL(2)-structures and Hessian hydrodynamic hyperbolic PDEs in three variables. This yields natural geometric characterizations of the wave equation, the first flow of the dKP hierarchy, and several others. It also provides an intrinsic, coordinate-free infrastructure to describe a large class of hydrodynamic integrable systems in three variables.

23 citations

Journal ArticleDOI
TL;DR: For every nonholonomic manifold, i.e., manifold with nonintegrable distribution the analog of the Riemann tensor is introduced in this paper, which is interpreted as modiÞcations of the Spencer cohomology.
Abstract: For every nonholonomic manifold, i.e., manifold with nonintegrable distribution the analog of the Riemann tensor is introduced. It is calculated here for the contact and Engel structures: for the contact structure it vanishes (another proof of Darboux’s canonical form); for the Engel distribution the target space of the tensor is of dimension 2. In particular, the Lie algebra preserving the Engel distribution is described. The tensors introduced are interpreted as modiÞcations of the Spencer cohomology and, as such, provide with a new way

23 citations

Journal ArticleDOI
TL;DR: In this article, a theory for predicting the strength of a bundle of sufficiently large number of elements, if the average breaking load and the breaking-elongation distribution of these elements are known, was developed.
Abstract: A theory is developed for predicting the strength of a bundle of sufficiently large number of elements, if the average breaking load and the breaking-elongation distribution of these elements are known. The usefulness of the theory is illustrated with data on cotton fibers.

23 citations

Proceedings ArticleDOI
18 Aug 2009
TL;DR: The main result of the paper is a characterization of trivializable oriented almost-Riemannian structures on compact oriented surfaces in terms of the topological invariants of the structure.
Abstract: Two dimensional almost-Riemannian geometries are metric structures on surfaces defined locally by a Lie bracket generating pair of vector fields. We study the relation between the topology of an almost-Riemannian structure on a compact oriented surface and the total curvature. In particular, we analyse the case in which there exist tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper is a characterization of trivializable oriented almost-Riemannian structures on compact oriented surfaces in terms of the topological invariants of the structure. Moreover, we present a Gauss- Bonnet formula for almost-Riemannian structures with tangency points.

23 citations

Book ChapterDOI
01 Jan 1988
TL;DR: In this article, the authors show connections between the notion of the value of a distribution at a point in the Łojasiewicz sense and the integrability of its Fourier transform.
Abstract: We show connections between the notion of the value of a distribution at a point in the Łojasiewicz sense and the integrability of its Fourier transform. We consider the one-dimensional case.

23 citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202159
202067
201953
201843
201733