An Introduction to Distributions and Foliations
نویسنده
چکیده
In smooth manifold theory, the notion of a tangent space makes it possible for differentiation to take place on an abstract manifold. In this paper, the notion of a distribution will be presented which makes it possible for integration to take place on an abstract manifold. The first section introduces terminology and builds intuition via an analogy to the concept of integral curves. The second section presents the Frobenius theorem–one of the foundational results in smooth manifold theory. The concluding section leaves the reader with foliations and a brief look at their connection to the Frobenius theorem.
منابع مشابه
Lightlike foliations of semi-Riemannian manifolds1
Using screen distributions and lightlike transversal vector bundles we develop a theory of degenerate foliations of semiRiemannian manifolds. We build lightlike foliations of a semiRiemannian manifold by suspension of a group homomorphism φ : π1(B, x0) → Isom(T ). We compute the basic cohomology groups of the flow determined by a lightlike Killing vector field on a complete semi-Riemannian mani...
متن کاملOn the k-nullity foliations in Finsler geometry
Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive...
متن کاملGeometric description of lightlike foliations by an observer in general relativity
We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension n, from a purely geometric point of view. Given an observer and a lightlike distribution Ω of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field U representing the observer and ...
متن کاملLectures on Partial Hyperbolicity and Stable Ergodicity
Weak integrability of the central foliation 56 6. Intermediate Foliations 58 6.1. Non-integrability of intermediate distributions 58 6.2. Invariant families of local manifolds 59 6.3. Insufficient smoothness of intermediate foliations 64 7. Absolute Continuity 69 7.1. The holonomy map 69 7.2. Absolute continuity of local manifolds 75
متن کاملTransversely Hessian foliations and information geometry
A family of probability distributions parametrized by an open domain Λ in Rn defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix is positive definite defining in this way a Riemannian metric on Λ. If we replace the "positive definite" assumption by "0-deformable" conditi...
متن کامل