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.

A Symmetric Product for Vector Fields and its Geometric Meaning

Abstract: We introduce the notion of geodesic invariance for distributions on manifolds with a linear connection. This is a natural weakening of the concept of a totally geodesic foliation to allow distributions which are not necessarily integrable. To test a distribution for geodesic invariance, we introduce a symmetric, vector field valued product on the set of vector fields on a manifold with a linear connection. This product serves the same purpose for geodesically invariant distributions as the Lie bracket serves for integrable distributions. The relationship of this product with connections in the bundle of linear frames is also discussed. As an application, we investigate geodesically invariant distributions associated with a left-invariant affine connection on a Lie group.

Stochastic Development Regression on Non-Linear Manifolds

Abstract: We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded in the connection of the manifold. We propose an estimation procedure which applies the Laplace approximation of the likelihood function. A simulation study of the performance of the model is performed and the model is applied to a real dataset of Corpus Callosum shapes.

On projective and affine equivalence of sub-Riemannian metrics

Abstract: Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to $g$ is constantly proportional to $g$ (resp. conformal to $g$). In the Riemannian case the local classification of projectively and affinely equivalent metrics is classical (Levi-Civita, Eisenhart). In particular, a Riemannian metric which is not rigid satisfies the following two special properties: its geodesic flow possesses nontrivial integrals and the metric induces certain canonical product structure on the ambient manifold. These classification results were extended to contact and quasi-contact distributions by Zelenko. Our general goal is to extend these results to arbitrary sub-Riemannian manifolds, and we establish two types of results toward this goal: if a sub-Riemannian metric is not projectively conformally rigid, then, first, its flow of normal extremals has at least one nontrivial integral quadratic on the fibers of the cotangent bundle and, second, the nilpotent approximation of the underlying distribution at any point admits a product structure. As a consequence we obtain two types of genericity results: first, we show that a generic sub-Riemannian metric on a fixed pair $(M,D)$ is projectively conformally rigid. Second, we prove that, except for special pairs $(m,n)$, every sub-Riemannian metric on a rank $m$ generic distribution in an $n$-dimensional manifold is projectively conformally rigid. For the affine equivalence in both genericity results conformal rigidity can be replaced by usual rigidity.

On pseudo-Einstein real hypersurfaces

Abstract: Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c eq0$, $n\geq 3$. We show that the Ricci tensor $S$ of $M$ satisfies $S(X,Y)=ag(X,Y)$ for any vector fields $X$ and $Y$ on the holomorphic distribution, $a$ being a constant, if and only if $M$ is a pseudo-Einstein real hypersurface.

Parallelisability conditions for differentiable three-webs

Abstract: Our aim is to nd conditions under which a 3-web on a smooth 2n-dimensional manifold is locally equivalent with a web formed by three systems of parallel n-planes in R 2n. We will present here a new approach to this \classical" problem using projectors onto the distributions of tangent subspaces to the leaves of foliations forming the web. The parallelisability conditions for multicodimensional 3-webs were at rst formulated by S.S. Chern, 3]. Later on M. A. Akivis, 1], interpreted these conditions as vanishing of both torsion and curvature tensors of a certain connection intimately related to the web, so called canonical Chern connection of a 3-web (M. Kikkawa, 8]). We will nd parallelisability conditions formulated in terms of projectors of a web, and will verify that they are equivalent with those derived by Akivis. At the same time we will show that the problem of parallelisability of a 3-web is equivalent with integrability of a (P; B)-structure (a couple of polynomial structures deening a 3-web). All objects under consideration will be supposed of the class C 1 (smooth). A 3-web on a 2n-dimensional manifold is given by a triple of foliations (in general position) of codimension n which are usually deened by totally integrable systems of Pfaaan equations. For our purposes let us choose the following deenition. Deenition 1. Under a diierentiable 3-web W on a manifold M 2n of dimension 2n we will understand here a triple W = (D 1 ; D 2 ; D 3) of (smooth) n-dimensional integrable distributions which are pairwise complementary.