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.

Discretization of Parametrizable Signal Manifolds

Abstract: Transformation-invariant analysis of signals often requires the computation of the distance from a test pattern to a transformation manifold. In particular, the estimation of the distances between a transformed query signal and several transformation manifolds representing different classes provides essential information for the classification of the signal. In many applications, the computation of the exact distance to the manifold is costly, whereas an efficient practical solution is the approximation of the manifold distance with the aid of a manifold grid. In this paper, we consider a setting with transformation manifolds of known parameterization. We first present an algorithm for the selection of samples from a single manifold that permits to minimize the average error in the manifold distance estimation. Then we propose a method for the joint discretization of multiple manifolds that represent different signal classes, where we optimize the transformation-invariant classification accuracy yielded by the discrete manifold representation. Experimental results show that sampling each manifold individually by minimizing the manifold distance estimation error outperforms baseline sampling solutions with respect to registration and classification accuracy. Performing an additional joint optimization on all samples improves the classification performance further. Moreover, given a fixed total number of samples to be selected from all manifolds, an asymmetric distribution of samples to different manifolds depending on their geometric structures may also increase the classification accuracy in comparison with the equal distribution of samples.

On the Weyl tensor classification in all dimensions and its relation with integrability properties

Abstract: In this paper the Weyl tensor is used to define operators that act on the space of forms. These operators are shown to have interesting properties and are used to classify the Weyl tensor, the well-known Petrov classification emerging as a special case. Particularly, in the Euclidean signature this classification turns out to be really simple. Then it is shown that the integrability condition of maximally isotropic distributions can be described in terms of the invariance of certain subbundles under the action of these operators. Here it is also proved a new generalization of the Goldberg-Sachs theorem, valid in all even dimensions, stating that the existence of an integrable maximally isotropic distribution imposes restrictions on the optical matrix. Also the higher-dimensional versions of the self-dual manifolds are investigated. These topics can shed light on the integrability of Einstein's equation in higher dimensions.

On the distribution of resonances for some asymptotically hyperbolic manifolds

Abstract: We establish a sharp upper bound for the resonance counting function for a class of asymptotically hyperbolic manifolds in arbitrary dimension, including convex, cocompact hyperbolic manifolds in two dimensions. The proof is based on the construction of a suitable paramatrix for the absolute 5-matrix that is unitary for real values of the energy. This paramatrix is the ^-matrix for a model Laplacian corresponding to a separable metric near infinity. The proof of the upper bound on the resonance counting function requires estimates on the growth of the relative scattering phase, and singular values of a family of integral operators.

Differential invariants of generic parabolic Monge–Ampère equations

Abstract: Some new results on the geometry of classical parabolic Monge–Ampere equations (PMAs) are presented. PMAs are either integrable, or non-integrable according to the integrability of its characteristic distribution. All integrable PMAs are locally equivalent to the equation uxx = 0. We study non-integrable PMAs by associating with each of them a one-dimensional distribution on the corresponding first-order jet manifold, called the directing distribution. According to some property of this distribution, non-integrable PMAs are subdivided into three classes, one generic and two special. Generic PMAs are completely characterized by their directing distributions, and we study canonical models of the latter, projective curve bundles (PCB). A PCB is a one-dimensional sub-bundle of the projectivized cotangent bundle of a four-dimensional manifold. Differential invariants of projective curves composing such a bundle are used to construct a series of contact differential invariants for corresponding PMAs. These give a solution of the equivalence problem for generic PMAs with respect to contact transformations. The introduced invariants measure the nonlinearity of PMAs in an exact manner.