Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 145exist between them), so their union is also an atlas. A manifold structureis a class of equivalent atlases.Finally, as charts ϕ i : M → R n were supposed to be 1-1 and onto maps,they can be either homeomorphisms, in which case we have a topological(C 0 ) manifold, or diffeomorphisms, in which case we have a smooth (C k )manifold.Slightly more precisely, a topological (respectively smooth) manifold isa separable space M which is locally homeomorphic (resp. diffeomorphic)to Euclidean space R n , having the following properties (reflected in Figure3.1):(1) M is a Hausdorff space: For every pair of points x 1 , x 2 ∈ M, there aredisjoint open subsets U 1 , U 2 ⊂ M such that x 1 ∈ U 1 and x 2 ∈ U 2 .(2) M is second–countable space: There exists a countable basis for thetopology of M.(3) M is locally Euclidean of dimension n: Every point of M has a neighborhoodthat is homeomorphic (resp. diffeomorphic) to an open subsetof R n .This implies that for any point x ∈ M there is a homeomorphism (resp.diffeomorphism) ϕ : U → ϕ(U) ⊆ R n , where U is an open neighborhoodof x in M and ϕ(U) is an open subset in R n . The pair (U, ϕ) is called acoordinate chart at a point x ∈ M, etc.3.3 Definition of a Differentiable ManifoldGiven a chart (U, ϕ), we call the set U a coordinate domain, or a coordinateneighborhood of each of its points. If in addition ϕ(U) is an open ballin R n , then U is called a coordinate ball. The map ϕ is called a (local)coordinate map, and the component functions (x 1 , ..., x n ) of ϕ, defined byϕ(m) = (x 1 (m), ..., x n (m)), are called local coordinates on U.Two charts (U 1 , ϕ 1 ) and (U 2 , ϕ 2 ) such that U 1 ∩ U 2 ≠ ∅ are calledcompatible if ϕ 1 (U 1 ∩U 2 ) and ϕ 2 (U 2 ∩U 1 ) are open subsets of R n . A family(U α , ϕ α ) α∈A of compatible charts on M such that the U α form a covering ofM is called an atlas. The maps ϕ αβ = ϕ β ◦ ϕ −1α : ϕ α (U αβ ) → ϕ β (U αβ ) arecalled the transition maps, for the atlas (U α , ϕ α ) α∈A , where U αβ = U α ∩U β ,