Ivancevic_Applied-Diff-Geom

144 Applied Differential Geometry: A Modern IntroductionFig. 3.1An intuitive geometrical picture behind the manifold concept (see text).On the other hand, to get some geometrical intuition behind the conceptof a manifold, consider a set M (see Figure 3.1) which is a candidate fora manifold. Any point x ∈ M 1 has its Euclidean chart, given by a 1–1and onto map ϕ i : M → R n , with its Euclidean image V i = ϕ i (U i ). Moreprecisely, a chart ϕ i is defined byϕ i : M ⊃ U i ∋ x ↦→ ϕ i (x) ∈ V i ⊂ R n ,where U i ⊂ M and V i ⊂ R n are open sets (see [Boothby (1986); Arnold(1978); De Rham (1984)]).Clearly, any point x ∈ M can have several different charts (see Figure3.1). Consider a case of two charts, ϕ i , ϕ j : M → R n , having in theirimages two open sets, V ij = ϕ i (U i ∩ U j ) and V ji = ϕ j (U i ∩ U j ). Then wehave transition functions ϕ ij between them,ϕ ij = ϕ j ◦ ϕ −1i : V ij → V ji , locally given by ϕ ij (x) = ϕ j (ϕ −1i (x)).If transition functions ϕ ij exist, then we say that two charts, ϕ i and ϕ j arecompatible. Transition functions represent a general (nonlinear) transformationsof coordinates, which are the core of classical tensor calculus.A set of compatible charts ϕ i : M → R n , such that each point x ∈ Mhas its Euclidean image in at least one chart, is called an atlas. Two atlasesare equivalent iff all their charts are compatible (i.e., transition functions1 Note that sometimes we will denote the point in a manifold M by m, and sometimesby x (thus implicitly assuming the existence of coordinates x = (x i )).