Topic
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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01 Jun 2014
TL;DR: The Locally Linear Embedding manifold learning method is modified to use an adaptive graph, and is used to extract low dimensional manifolds from hyperspectral image data collected during the SHARE 2012 campaign.
Abstract: Hyperspectral image data are traditionally analyzed using statistical models. However, as the spatial and spectral resolutions of the images improve as a result of advances in sensor technology, the data no longer maintain a Gaussian distribution; this is due to increased material diversity in the scene, i.e., clutter. This causes many statistical assumptions about the data — and subsequently, the algorithms based on those assumptions — to be flawed. In high dimensional data, manifold learning seeks to recover the embedded non-linear, lower-dimensional manifold upon which the data inherently lie. By recovering the lower-dimensional manifold, intrinsic structures and relationships within the data may be identified and exploited. Here, we consider the impacts of increasing material spectral clutter on the low dimensional manifolds recovered from high spatial resolution hyperspectral scenes for both single and multiple material classes. The Locally Linear Embedding manifold learning method is modified to use an adaptive graph, and is used to extract low dimensional manifolds from hyperspectral image data collected during the SHARE 2012 campaign.
4 citations
05 Sep 2020
TL;DR: In this article, the distributions in CR-light-like submanifolds of an indefinite Kaehler Statistical manifold have been characterized using second fundamental form and the necessary and sufficient conditions for integrability of the same have been obtained.
Abstract: In this paper, the distributions in CR-lightlike submanifolds of an indefinite Kaehler Statistical manifold have been characterized using second fundamental form and the necessary and sufficient conditions for integrability of the same have been obtained. Also, the conditions for the distributions to be totally geodesic with respect to the dual connections in the statistical manifold have been developed.
4 citations
Posted Content•
TL;DR: In this article, conformal flows of metrics restricted to the orthogonal distribution of a foliation were applied to study the question: which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive.
Abstract: We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case of integrable orthogonal distribution the flow velocity is proportional to the mixed scalar curvature. We observe that the mean curvature vector $H$ of $D$ satisfies along the leaves the forced Burgers equation, this reduces to the linear Schr\"{o}dinger equation, whose potential function is a certain "non-umbilicity" measure of $D$. On order to show convergence of the solution metrics $g_t$ as $t\to\infty$, we normalize the flow, and instead of a foliation consider a fiber bundle $\pi: M\to B$ of a Riemannian manifold $(M, g_0)$. In this case, if the "non-umbilicity" of $D$ is smaller in a sense then the "non-integrability", then the limit mixed scalar curvature function is positive. For integrable $D$, we give examples with foliated surfaces and twisted products.
4 citations
TL;DR: In this article, it was shown that the aggregate crystalline phase contribution to the fiber compliances is in terms of five independent compliances and that the intrinsic symmetry of the unit itself is unaltered.
Abstract: = ~~2:f ~ S,4’ = ~33~r .Si3’= ~28~~ and ~·6~ = 2B~11~
4 citations
Posted Content•
TL;DR: In this paper, the authors analyzed the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold and showed that certain holonomy reductions of the associated normal Tractor connections imply preferred connections with special properties, along with Riemannian structures on the manifold.
Abstract: This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, however, lie in the free 3-distributions. In these cases, there are normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$.
4 citations