nLab
gluing function

Contents

Contents

Definition

Definition

(local chart and atlas and gluing function)

Given an nn-dimensional topological manifold XX (def. ), then

  1. an open subset UXU \subset X and a homeomorphism ϕ: nAAU\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U is also called a local coordinate chart of XX.

  2. an open cover of XX by local charts { nϕ iUX} iI\left\{ \mathbb{R}^n \overset{\phi_i}{\to} U \subset X \right\}_{i \in I} is called an atlas of the topological manifold.

  3. denoting for each i,jIi,j \in I the intersection of the iith chart with the jjth chart in such an atlas by

    U ijU iU j U_{i j} \coloneqq U_i \cap U_j

    then the induced homeomorphism

    nAAϕ i 1(U ij)Aϕ iAU ijAϕ j 1Aϕ j 1(U ij)AA n \mathbb{R}^n \supset \phantom{AA} \phi_i^{-1}(U_{i j}) \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow} U_{i j} \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow} \phi_j^{-1}(U_{i j}) \phantom{AA} \subset \mathbb{R}^n

    is called the gluing function or coordinate transformation from chart ii to chart jj.

graphics grabbed from Frankel

Last revised on June 28, 2017 at 11:11:32. See the history of this page for a list of all contributions to it.