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 Jan 2002
TL;DR: Several standard notions of the differential geometry, such as tangent and jet paces, Frolicher-Nijenhuis bracket, connections and curvature, can be generalized to the case of bundles whose fibres are distribution spaces as mentioned in this paper.
Abstract: Several standard notions of the differential geometry, such a tangent and jet paces, Frolicher-Nijenhuis bracket, connections and curvature, can be generalized to the case of bundles whose fibres are distribution spaces Moreover a classical connection on a finite-dimensional bundle yields a connection on an associated distributional bundle These idea can hed new light on field theories on a curved background
TL;DR: In this article, a geodesic distance is used to estimate the true distance distribution of a point cloud sampled from a manifold, which can be used to identify highly non-linear features with persistent homology.
Abstract: An increasingly popular approach to the analysis of neural data is to treat activity patterns as being constrained to and sampled from a manifold, which can be characterized by its topology. The persistent homology method identifies the type and number of holes in the manifold thereby yielding functional information about the coding and dynamic properties of the underlying neural network. In this work we give examples of highly non-linear manifolds in which the persistent homology algorithm fails when it uses the Euclidean distance which does not always yield a good approximation of the true distance distribution of a point cloud sampled from a manifold. To deal with this issue we propose a simple strategy for the estimation of the geodesic distance which is a better approximation of the true distance distribution and can be used to successfully identify highly non-linear features with persistent homology. To document the utility of our method we model a circular manifold, based on orthogonal sinusoidal basis functions and compare how the chosen metric determines the performance of the persistent homology algorithm. Furthermore we discuss the robustness of our method across different manifold properties and point out strategies for interpreting its results as well as some possible pitfalls of its application. Finally we apply this analysis to neural data coming from the Visual Coding - Neuropixels dataset recorded in mouse visual cortex after stimulation with drifting gratings at the Allen Institute. We find that different manifolds with a non-trivial topology can be seen across regions and stimulus properties. Finally, we discuss what these manifolds say about visual computation and how they depend on stimulus parameters.
01 Jan 2013
TL;DR: In this article, the integrability of GCR-light-like submanifolds of an indefinite Sasakian manifold was studied and necessary and sufficient conditions for various dis-tributions of GCL-lightlike sub-manifold of an infinite manifold were given.
Abstract: In this paper, we study GCR-lightlike submanifolds of indefinite Sasakian manifold. We give some necessary and sufficient conditions on integrability of various dis- tributions of GCR-lightlike submanifold of an indefinite Sasakian manifold. We also find the conditions for each leaf of holomorphic distribution and radical distribution is totally geodesic.
01 Apr 1997
TL;DR: In this paper, the degeneration locus I is defined as the set of points where the distribution has a Lie-bracket growth vector less than or equal I, a xed integer vector.
Abstract: Given a distribution of k-planes on a manifold, consider the degeneration locus I consisting of points where the distribution has Lie-bracket growth vector less than or equal I, a xed integer vector. We calculate the characteristic classes associated to the I for a generic two-plane distribution on a four-manifold. 1. Results and Background. 1.1. Generalities, Setting and Results. A distribution D of k-planes on an n-dimensional manifold Q can be thought of as either a subbundle DTQ of the tangent bundle or as a locally free sheaf of smooth vector elds. We use the same notation for both. Write D 2 = D +[ D ;D] and more generally D j+1 = D j +[ D ;D j ]. These are sheaves of modules of vector elds (over the ring of smooth functions). We are interested in distributions such that for r large enough we obtain all vector elds by this procedure: D r = T
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TL;DR: In this article, the Weyl structures on light-like hypersurfaces endowed with a conformal structure of certain type and specific screen distribution were investigated and necessary and sufficient conditions for a Weyl structure defined by the $1-$form of an almost contact structure given by an additional complex structure in case of an ambiant Kaehler manifold to be closed.
Abstract: We study Weyl structures on lightlikes hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambiant Lorentzian space $\mathbb{R}^{n+2}_{1}$ and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves. Finally, we establish necessary and sufficient conditions for a Weyl structure defined by the $1-$form of an almost contact structure given by an additional complex structure in case of an ambiant Kaehler manifold to be closed.