# Differential geometry, analysis, and physics by Lee J.M.

By Lee J.M.

Best differential geometry books

Geometric Phases in Classical and Quantum Mechanics

This paintings examines the attractive and significant actual idea referred to as the 'geometric phase,' bringing jointly assorted actual phenomena lower than a unified mathematical and actual scheme. a number of well-established geometric and topological equipment underscore the mathematical remedy of the topic, emphasizing a coherent point of view at a slightly refined point.

Lectures on Symplectic Geometry

Discusses differential geometry and hyperbolic geometry. For researchers and graduate scholars. Softcover.

Differential Geometry and Topology: With a View to Dynamical Systems

Obtainable, concise, and self-contained, this publication deals a superb advent to 3 similar topics: differential geometry, differential topology, and dynamical structures. issues of exact curiosity addressed within the ebook comprise Brouwer's mounted element theorem, Morse thought, and the geodesic stream.

Additional info for Differential geometry, analysis, and physics

Sample text

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 }.