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.

The Circle as a Fermionic Distribution

Abstract: Let us consider the free loop space of a manifold. It is very well known that the equivariant cohomology of the free loop spce is related to the index theorem for a finite dimensional Dirac operator. The reader can see [Bis85], [Bis86], [GJP90], [JP90] for instance.

Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

Abstract: In this paper, we characterize the class of distributions on an homogeneous Lie group $\fN$ that can be extended via Poisson integration to a solvable one-dimensional extension $\fS$ of $\fN$ To do so, we introducte the $s'$-convolution on $\fN$ and show that the set of distributions that are $s'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions Moreover, we show that the $s'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behaviour Finally, we show that such distributions satisfy some global weak-$L^1$ estimates

Contact structures on Lie algebroids

Abstract: In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid structure on a vertical Liouville distribution over the big-tangent manifold of a Riemannain manifold.

Fluctuation geometry: a counterpart approach of inference geometry

Abstract: Starting from an axiomatic perspective, fluctuation geometry is developed as a counterpart approach of inference geometry. This approach is inspired on the existence of a notable analogy between the general theorems of inference theory and the general fluctuation theorems associated with a parametric family of distribution functions dp(I|?) = ?(I|?)dI, which describes the behavior of a set of continuous stochastic variables driven by a set of control parameters ?. In this approach, statistical properties are rephrased as purely geometric notions derived from the Riemannian structure on the manifold of stochastic variables I. Consequently, this theory arises as an alternative framework for applying the powerful methods of differential geometry for the statistical analysis. Fluctuation geometry has direct implications on statistics and physics. This geometric approach inspires a Riemannian reformulation of Einstein fluctuation theory as well as a geometric redefinition of the information entropy for a continuous distribution.

Manifold Curvature From Covariance Analysis

Abstract: Principal component analysis of cylindrical neighborhoods is proposed to study the local geometry of embedded Riemannian manifolds. At every generic point and scale, a high-dimensional cylinder orthogonal to the tangent space at the point cuts out a path-connected patch whose point-set distribution in ambient space encodes the intrinsic and extrinsic curvature. The covariance matrix of the points from that neighborhood has eigenvectors whose scale limit tends to the Frenet-Serret frame for curves, and to what we call the Ricci-Weingarten principal directions for submanifolds. More importantly, the limit of differences and products of eigenvalues can be used to recover curvature information at the point. The formula for hypersurfaces in terms of principal curvatures is particularly simple and plays a crucial role in the study of higher-codimension cases.