By Lee J.M.

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Now let Uα , xα be a chart on M containing x. By replacing U and Uα by U ∩ Uα we may assume that xα is defined on U = Uα . In this situation, if we let U ∗ := p(U ) then each restriction p|U : U → U ∗ is a homeomorphism. We define a chart map x∗α with domain Uα∗ by −1 x∗α := xα ◦ p|Uα : Uα∗ → Rn . ∗ Let xα and x∗β be two such chart maps with domains Uα∗ and Uβ∗ . If Uα∗ ∩ Uβ∗ = ∅ then we have to show that xβ∗ ◦ (x∗α )−1 is a C r diffeomorphism. Let x ¯ ∈ Uα∗ ∩ Uβ∗ ∗ ∗ ∗ ∗ ∗ and abbreviate Uαβ = Uα ∩ Uβ .

14 If we integrate the first order system of differential equations with initial conditions y=x y =x x(0) = ξ y(0) = θ we get solutions x (t; ξ, θ) = y (t; ξ, θ) = 1 2θ 1 2θ + 12 ξ et − + 12 ξ et + 1 2θ 1 2θ − 12 ξ e−t − 21 ξ e−t that depend on the initial conditions (ξ, θ). Now for any t the map Φt : (ξ, θ) → (x(t, ξ, θ), y(t, ξ, θ)) is a diffeomorphism R2 → R2 . This is a special case of a moderately hard theorem. 15 The map (x, y) → ( 1−z(x,y) x, 1−z(x,y) y) where z(x, y) = 1 − x2 − y 2 is a diffeomorphism from the open disk B(0, 1) = {(x, y) : x2 + y 2 < 1} onto the whole plane.

3. DIFFERENTIABLE MANIFOLDS AND DIFFERENTIABLE MAPS 21 not contained in the x, y plane. Every line p ∈ Uz intersects the plane z = 1 at exactly one point of the form (x( ), y( ), 1). We can define a bijection ψz : Uz → R2 by letting p → (x( ), y( )). This is a chart for P2 (R) and there are obviously two other analogous charts ψx , Ux and ψy , Uy which cover P2 (R). , un ) is on the line . 4 The graph of a smooth function f : Rn → R is the subset of the Cartesian product Rn × R given by Γf = {(x, f (x)) : x ∈ Rn }.