Ivancevic_Applied-Diff-Geom

Introduction 7charts.The atlas containing all possible charts consistent with a given atlas iscalled the maximal atlas. Unlike an ordinary atlas, the maximal atlas of agiven atlas is unique.More generally, an atlas for a complicated space is constructed out ofthe following pieces of information:(i) A list of spaces that are considered simple.(ii) For each point in the complicated space, a neighborhood of thatpoint that is homeomorphic to a simple space, the homeomorphism beinga chart.(iii) We require the different charts to be compatible. At the minimum,we require that the composite of one chart with the inverse of another be ahomeomorphism (also known as a change of coordinates, or a transformationof coordinates, or a transition function, or a transition map) but weusually impose stronger requirements, such as C ∞ −smoothness. 6This definition of atlas is exactly analogous to the non–mathematicalmeaning of atlas. Each individual map in an atlas of the world gives aneighborhood of each point on the globe that is homeomorphic to the plane.While each individual map does not exactly line up with other maps that itoverlaps with (because of the Earth’s curvature), the overlap of two mapscan still be compared (by using latitude and longitude lines, for example).Different choices for simple spaces and compatibility conditions give differentobjects. For example, if we choose for our simple spaces the Euclideanspaces R n , we get topological manifolds. If we also require the coordinate6 Charts in an atlas may overlap and a single point of a manifold may be representedin several charts. If two charts overlap, parts of them represent the same region ofthe manifold, just as a map of Europe and a map of Asia may both contain Moscow.Given two overlapping charts, a transition function can be defined, which goes froman open Euclidean nD ball B n = {(x 1 , x 2 , ..., x n) ∈ R n |x 2 1 + x2 2 + ... + x2 n < 1} in R nto the manifold and then back to another (or perhaps the same) open nD ball in R n .The resultant map, like the map T in the circle example above, is called a change ofcoordinates, a coordinate transformation, a transition function, or a transition map.An atlas can also be used to define additional structure on the manifold. The structureis first defined on each chart separately. If all the transition maps are compatible withthis structure, the structure transfers to the manifold.This is the standard way differentiable manifolds are defined. If the transition functionsof an atlas for a topological manifold preserve the natural differential structureof R n (that is, if they are diffeomorphisms, i.e., invertible maps that are smooth inboth directions), the differential structure transfers to the manifold and turns it into adifferentiable, or smooth manifold.In general the structure on the manifold depends on the atlas, but sometimes differentatlases give rise to the same structure. Such atlases are called compatible.