By Ana Cannas da Silva

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

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Case M = Rn : For a compact k-dimensional submanifold X, take a neighborhood of the form U ε = {p ∈ M | distance (p, X) ≤ ε } . For ε sufficiently small so that any p ∈ U ε has a unique nearest point in X, define a projection π : U ε → X, p → point on X closest to p. If π (p) = q, then p = q + v for some v ∈ Nq X where Nq X = (Tq X)⊥ is the normal space at q; see Homework 5. Let h : U ε −→ Rn p −→ q + Lq v , where q = π (p) and v = p − π (p) ∈ Nq X. Then hX = idX and dh p = L p for p ∈ X. If X is not compact, replace ε by a continuous function ε : X → R+ which tends to zero fast enough as x tends to infinity.

If all of these sets were disjoint, then, since Area (Ui ) = Area (U) > 0 for all i, we would have Area A ≥ Area (U0 ∪ U1 ∪ U2 ∪ . ) = ∑ Area (Ui ) = ∞ . i To avoid this contradiction we must have ϕ (k) (U) ∩ ϕ (l) (U) = 0/ for some k > l, / ✷ which implies ϕ (k−l) (U) ∩ U = 0. Hence, eternal return applies to billiards... Remark. 3 clearly generalizes to volume-preserving diffeomorphisms in higher dimensions. 4. (Poincar´e’s Last Geometric Theorem) Suppose ϕ : A → A is an area-preserving diffeomorphism of the closed annulus A = R/Z×[−1, 1] which preserves the two components of the boundary, and twists them in opposite directions.

Yn ) be coordinate charts for X1 , X2 , with associated charts (T ∗ U1 , x1 , . . , xn , ξ1 , . . , ξn ), (T ∗ U2 , y1 , . . , yn , η1 , . . , ηn ) for M1 , M2 . 2 Method of Generating Functions 27 ϕ (x, ξ ) = (y, η ) ⇐⇒ ξ = dx f and η = −dy f . Therefore, given a point (x, ξ ) ∈ M1 , to find its image (y, η ) = ϕ (x, ξ ) we must solve the “Hamilton” equations ⎧ ∂f ⎪ ⎪ ξi = (x, y) ( ) ⎨ ∂ xi ∂f ⎪ ⎪ ⎩ ηi = − (x, y) . ( ) ∂ yi If there is a solution y = ϕ1 (x, ξ ) of ( ), we may feed it to ( ) thus obtaining η = ϕ2 (x, ξ ), so that ϕ (x, ξ ) = (ϕ1 (x, ξ ), ϕ2 (x, ξ )).