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.

Integrabilities of distribution D｛ξ｝on a nearly Sasakian manifold

Abstract: Some sufficient and necessary conditions were given for integrabilities of distribution D｛ξ｝ on a nearly Sasakian manifold,and generalized Bejancu's result was given also.

Constructing a Brownian Sheet with Values in a Compact Riemannian Manifold

Abstract: In this paper, we propose a new method of constructing a two-parameter random field W x (s, t ), x ∈ M , with values in a compact Riemannian manifold M possessing the property that the random processes W x ( · ,t )a ndW x M (s, · ) are Brownian motions on the manifold M with parameters t and s , respectively, issuing from the point x .( By aBrownian motion on a manifold M with parameter t we mean the diffusion process generated by the operator −(t/2)∆M , where ∆M is the Laplace operator on the manifold M.) For the case in which the manifold is a compact Lie group, the two-parameter random field constructed in the paper coincides with the Brownian sheet defined by Malliavin (1) in 1991. (Malliavin called this random field a Brownian motion with values in C((0, 1) ,M ) , which is the set of continuous functions defined on the closed interval (0, 1) and taking values in M.) Nevertheless, for the case in which the manifold is a compact Lie group, the method proposed in the present paper essentially differs from that used in Malliavin's paper. 1. FIRST STEP IN THE CONSTRUCTION OF THE RANDOM FIELD W x Suppose that M is a d-dimensional compact Riemannian manifold without boundary isomet- rically embedded in R m .B y aBrownian sheet with values in R m we mean the family of m independent standard Brownian sheets. Suppose that Wt,s is an n-dimensional Brownian sheet. Consider Wt,s as a process taking values in the space C((0, 1), R m ) . We denote this process by the symbol Wt . We introduce the following notation: if E is a locally convex space, then E t denotes C((0 ,t ) ,E ); if y ∈ C((0, 1), R m ) is a continuous function, then W y denotes the distribution of the process W y = y +Wt .I fψ ∈ C((0, 1), R m ) , then we define the process (W y )t = ψ(t )+ W y . Suppose that W y is the distribution of this process and E y,ψ is the expectation with respect to the measure W y . Further, Ue(M ) denotes the e -neighborhood of the manifold M. We consider W y for functions y and ψ satisfying the conditions: y(0) ∈ M , ψ(0) = 0 . The goal of this section is to prove the existence of a limit (given below) with respect to the family of bounded continu- ous cylindrical functions, where by a cylindrical function C((0, 1) × (0, 1), R m ) → R we mean a

On the Integrability of Orthogonal Distributions in Poisson Manifolds

Abstract: We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provided.

Lie Group Actions

Abstract: After an introduction which provides the reader with the basic notions and a number of examples, we show that a Lie group action gives rise to vector fields of a special type, called Killing vector fields We prove the Orbit Theorem, which states that the distribution spanned by the Killing vector fields is integrable and that its integral manifolds coincide with the connected components of the orbits of the action This way, every orbit gets endowed with the structure of an initial submanifold After this general part, we limit our attention to the important special class of proper Lie group actions Under this additional regularity assumption, one can prove the Tubular Neighbourhood Theorem (or Slice Theorem) which relates the action in a neighbourhood of an orbit to the isotropy representation at a point on that orbit and thus provides a normal form for the action near that orbit As an important special case, we discuss free proper actions and related bundle structures In the final two sections, we study invariant vector fields and make some elementary remarks on relative equilibria and relatively periodic integral curves This is relevant for the study of Hamiltonian systems with symmetries in Chap 10

Towards Shape Constellation Inference through Higher-Order MRF Optimization in Nonlinear Embeddings

Abstract: This paper introduces a novel approach for inferring articulated shape models from images. A low-dimensional manifold embedding is created from a training set of prior mesh models to establish the patterns of global shape variations. Local appearance is captured from neighborhoods in the manifold once the overall representation converges. Inference with respect to the manifold and shape parameters is performed using a Markov Random Field (MRF). Singleton and pairwise potentials measure the support from the data and shape coherence from neighboring models respectively, while higher-order cliques encode geometrical modes of variation in localized shape models. Optimization of model parameters is achieved using efficient linear programming and duality. The resulting model is geometrically intuitive, captures the statistical distribution of the underlying manifold and respects image support in the spatial domain. Experimental results on articulated bodies such as spinal column geometry estimation demonstrate the potentials of our approach.