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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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TL;DR: In this paper, the notion of optical geometry, defined to be a Lorentzian manifold equipped with a null line distribution, was studied from the perspective of intrinsic torsion.
Abstract: We study the notion of optical geometry, defined to be a Lorentzian manifold equipped with a null line distribution, from the perspective of intrinsic torsion. This is an instance of a non-integrable version of holonomy reduction in Lorentzian geometry. These generate congruences of null curves, which play an important r\^{o}le in general relativity. Conformal properties of these are investigated. We also extend this concept to generalised optical geometries as introduced by Robinson and Trautman.
6 citations
TL;DR: Two methods that flexibly extend the Mahalanobis distance on the extended Grassmann manifolds can be used to measure pattern (dis)similarity on the basis of the pattern structure.
Abstract: In pattern classification problems, pattern variations are often modeled as a linear manifold or a low-dimensional subspace. Conventional methods use such models and define a measure of similarity or dissimilarity. However, these similarity measures are deterministic and do not take into account the distribution of linear manifolds or low-dimensional subspaces. Therefore, if the distribution is not isotopic, the distance measurements are not reliable, as well as vector-based distance measurement in the Euclidean space. We previously systematized the representations of variational patterns using the Grassmann manifold and introduce the Mahalanobis distance to the Grassmann manifold as a natural extension of Euclidean case. In this paper, we present two methods that flexibly extend the Mahalanobis distance on the extended Grassmann manifolds. These methods can be used to measure pattern (dis)similarity on the basis of the pattern structure. Experimental evaluation of the performance of the proposed methods demonstrated that they exhibit a lower error classification rate.
6 citations
TL;DR: It is shown that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodeics from general relativity.
Abstract: In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.
6 citations
01 Dec 2019
TL;DR: In this paper, it was shown that a 3D non-cosymplectic quasi-Sasakian manifold admits Ricci almost soliton and the potential function is invariant in the orthogonal distribution of the Reeb vector field.
Abstract: In this paper it is shown that a three-dimensional non-cosymplectic quasi-Sasakian manifold admitting Ricci almost soliton is locally $$\phi $$-symmetric. It is proved that a Ricci almost soliton on a three-dimensional quasi-Sasakian manifold reduces to a Ricci soliton.
It is also proved that if a three-dimensional non-cosymplectic quasi-Sasakian manifold admits gradient Ricci soliton, then the potential function is invariant in the orthogonal distribution of the Reeb vector field $$\xi$$. We also improve some previous results regarding gradient Ricci soliton on three-dimensional quasi-Sasakian manifolds. An illustrative example is given to support the obtained results.
6 citations
Patent•
15 Oct 2011
TL;DR: In this paper, the pneumatic distributor has a horizontal feed pipe (4) for supplying the grit, and a manifold (5) is provided at an angle of 90[deg] with respect to the feed pipe.
Abstract: The pneumatic distributor has a horizontal feed pipe (4) for supplying the grit. A manifold (5) is provided at an angle of 90[deg] with respect to horizontal feed pipe. A corrugated pipe riser (6) is provided adjoint to the manifold. A header (7) with radial hose lines is arranged at upper end of manifold for supplying the grit. A guide element is provided in inner wall of manifold for guiding the flow of grit toward the center of manifold. The guide element is arranged to close the space between the element and inner wall of manifold.
6 citations