By Hatcher A.
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Extra resources for Algebraic topology. Errata (web draft, Nov. 2004)
15 A compact complex manifold M is Kobayashi hyperbolic iff every holomorphic map f : C → M is constant. Cf. [La, III, §2]. For example, if M is covered by a bounded domain in Cn , then it is Kobayashi hyperbolic. 16 Any holomorphic map between complex manifolds is distance non-increasing for the Kobayashi metric. 17 (Royden) The Teichm¨ uller metric on Teich(S) coincides with the Kobayashi metric. 18 Any holomorphic map f : Teich(S) → Teich(S) satisfies d(f X, f Y ) ≤ d(X, Y ) in the Teichm¨ uller metric.
This is the current [γ] associated to a closed loop. (ii) A positive weighted sum of closed curves Ci γi determines a current Ci [γi ]. For simplicity we suppressed the brackets in the future. Integral geometry. The integral geometric measure on G is deﬁned as follows. Fix an oriented geodesic γ ⊂ H. Then there is an injective map γ × S 1 → G sending (x, θ) to the unique unoriented geodesic δ(x, θ) through x making angle θ with γ. ) The measure µ = (1/2) sin θdθ dx can be shown to be independent of the choice of chart.
1. There is a lift of f to a map H → H that extends to the identity on R. 2. The map f is homotopic to the identity rel ideal boundary. 3. The map f is isotopic to the identity rel ideal boundary, through uniformly quasiconformal maps. See [EaM]. A Riemann surface marked by X is a pair (Y, g) where g : X → Y is a quasiconformal map. Two marked Riemann surfaces (Y, g) and (Z, h) are equivalent if there is a conformal isomorphism α : Y → Z such that f = h−1 ◦ α ◦ g : X → X is isotopic to the identity rel ideal boundary.
Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A.