Branched Manifold - Definition

Definition

Let K be a metrizable space, together with:

  1. a collection {Ui} of closed subsets of K;
  2. for each Ui, a finite collection {Dij} of closed subsets of Ui;
  3. for each i, a map πi: Ui → Din to a closed n-disk of class Ck in Rn.

These data must satisfy the following requirements:

  1. ∪j Dij = Ui and ∪i Int Ui = K;
  2. the restriction of πi to Dij is a homeomorphism onto its image πi(Dij) which is a closed class Ck n-disk relative to the boundary of Din;
  3. there is a cocycle of diffeomorphisms {αlm} of class Ck (k ≥ 1) such that πl = αlm · πm when defined. The domain of αlm is πm(Ul ∩ Um).

Then the space K is a branched n-manifold of class Ck.

The standard machinery of differential topology can be adapted to the case of branched manifolds. This leads to the definition of the tangent space TpK to a branched n-manifold K at a given point p, which is an n-dimensional real vector space; a natural notion of a Ck differentiable map f: K → L between branched manifolds, its differential df: TpK → Tf(p)L, the germ of f at p, jet spaces, and other related notions.

Read more about this topic:  Branched Manifold

Famous quotes containing the word definition:

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)