/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ

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4 4.1 1 2 1 4 2 1

/ 2 4.1.1 n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ λ U λ (local chart, local coordinate) (M3) M A (M, A) (manifold) n (n-dimensional manifold) M M (M4) U λ U µ Φ λµ := ϕ µ ϕ 1 λ : ϕ λ(u λ U µ ) ϕ µ (U λ U µ ) µ, λ Λ C α M C α (C α -class manifold) 0 α α = ω C 0 (topological manifold) 4.1.2 (M1). (M1) M 3 3 M Hausdorff 2

/ 3 R 3 (M2) 18 4 (M2) 5 (a). 2 M M Λ Λ = {1, 2, 3,...} {ϕ λ } λ Λ M U λ M ϕ λ (U λ ) = U λ R 2 M 2 R 2 A ϕ λ : U λ U λ U λ ϕ λ M U λ M (a) M 6 (b). R n Λ λ Λ R n λ 4 (Pierre-Louis Moreau de Maupertuis, 1698 1759) 1738 5 6 U λ R 2

/ 4 4.1: Λ R n λ µ

/ 5 7 (b) λ R n λ ϕ λ U λ A 8 (M3). A M (M, A) M 9 M M A M = (M, A) (M4). A 1 M 1 M M 10 (M4) ϕ µ ϕ 1 λ : ϕ λ(u λ U µ ) ϕ µ (U λ U µ ) (b) (M4) 7 8 (atlas) 16 9 10

/ 6 4.1.3. 1. M 2. A M C C 1 3. M A, A A = A (M, A) (M, A ) 4.2 4.2.1 M = (M, A) M f : M R M f : M R M A M h : M R A {ϕ λ } λ Λ λ ϕ λ (U λ ) = U λ F λ = f ϕ 1 λ : U λ R F λ R n R

/ 7 M 4.2: M = (M, A) f : M R A ϕ λ : U λ U λ F λ = f ϕ 1 λ : U λ R x U λ U µ F λ (ϕ λ (x)) = F µ (ϕ µ (x)) f(x) 11 4.2.2 L.Bers Riemann Surfaces 1 Bers 11

/ 8 Bers M C ρ M C ρ C ρ C 1 C 1 (M4). M C 0 M f : M R C α C 1 C 1 C 1 Bers C 1 C 1 R n C 1 U R n F : U R C 1 x = (x 1,..., x n ) U x i x F x i (x) R C 1 C 1 M C 1 f : M R C 1 ϕ λ : U λ U λ F λ := f ϕ 1 λ : U λ R C 1 f f M 12 λ F λ C 1 f R n λ 4.2 C 1 M C 1 12

/ 9 C 1 C 1 f : M R F λ = f ϕ 1 λ : U λ R C 1 U λ U µ µ F µ = f ϕ 1 µ : ϕ µ (U λ U µ ) R C 1 M (M4) C 1 F µ = F λ (ϕ λ ϕ 1 µ ) M C 1 ϕ λ ϕ 1 µ C 1 F λ C 1 F µ = F λ (ϕ λ ϕ 1 µ ) C 1 F µ C 1 4.3 1 YES M C 1 ϕ λ ϕ 1 µ C1 F µ C 1 ϕ λ ϕ 1 µ F µ. M = (M, A) A = {ϕ λ : U λ R n } λ Λ f : M R M C 0 C 1 ϕ λ : U λ U λ F λ := f ϕ 1 λ C 1 : U λ R M R ϕ λ C 1 C 1 C 1 C α α [0, ] C α

/ 10. C 1 F λ f k f C 1 C 1 13. 1. C n 0 r n r C r 2. 4.3 1 M f : M R p f (2.1,1) 2.3 13

/ 11 M C 1 f C 1. A = {ϕ λ : U λ R n } λ Λ λ, µ Λ A = {ϕ : U ϕ R n } ϕ λ, ϕ µ A ϕ, ψ A 14 ϕ : U ϕ R n ϕ(u ϕ ) R n U ϕ ϕ : U ϕ U ϕ 4.3.1 1 M n = 8 15 M f : M R C 1 1 x, p M x p f(x) f(p) = A(x p) + o( x p ) x p M x p x p M p M ϕ : U ϕ U ϕ R 8 p U ϕ F ϕ = f ϕ 1 : U ϕ R ϕ(x) = x ϕ(p) = p U ϕ f(x) f(p) = F ϕ (x) F ϕ (p) p M f 14 ϕ λ, ϕ µ A λ µ ϕ, ψ A ϕ ψ 15

/ 12. f : M R C 1 ϕ F ϕ = f ϕ 1 : U ϕ R R 8 U ϕ C 1 1 x = (x 1,, x 8 ) U ϕ R 8 p = (p 1,, p 8 ) U ϕ R 8 p F ϕ 1 Taylor F ϕ (x) = F ϕ (p) + a 1 (x 1 p 1 ) + + a 8 (x 8 p 8 ) 1 + o( x p ) a 1,..., a 8 F ϕ p a i = F ϕ(x) x i 0.98 2 = F ϕ (p) x=p x i x p := (x 1 p 1 ) 2 + + (x 8 p 8 ) 2 Taylor a = (a 1,..., a 8 ) F ϕ a x 1 p 1 F ϕ (x) F ϕ (p) = (a 1 a 8 ). + o( x p ) x 8 p 8 F ϕ (x) F ϕ (p) = a (x p) + o( x p ) F ϕ F ϕ = F ϕ (x) F ϕ (p) x x = x p a x f(x) f(p) = a (x p) + o( x p ) f M x p A ϕ x, p U ϕ ϕ x p R 8 ϕ 16 x p 0 ϕ 16 x p U ϕ

/ 13 a R 8 ϕ p f ϕ x, p, a a ψ A p U ψ ψ : U ψ U ψ R 8 x U ψ ψ(x) = y = (y 1,, y 8 ) U ψ ψ(p) = q = (q 1,, q 8 ) U ψ q F ψ = f ψ 1 : U ψ R b = (b 1,..., b 8 ) y q 0 F ψ (y) F ψ (q) = b (y q) + o( y q ) x, p U ϕ U ψ f(x) f(p) = a (x p) + o( x p ) f(x) f(p) = b (y q) + o( y q ) cm x p y q 17. ϕ ψ Φ := ψ ϕ 1 ϕ(u ϕ U ψ ) U ϕ ψ(u ϕ U ψ ) U ψ C 1 M C 1 Φ : p q p 17

/ 14 4.3: Φ 2.2. 8 8 J = (J ij ) 1 i,j 8 x p y q = J(x p) + o( x p ) y 1 q 1 J 11 J 18 x 1 p 1. =...... y 8 q 8 J 81 J 88 x 8 p 8 i (1 i 8) + o( x p ) y i q i = J i1 (x 1 p 1 ) + + J i8 (x n p 8 ) + o( x p ) 1 J ij (1 j 8) J ij = y i(x) x j = y i (p) x=p x j Φ = ψ ϕ 1 p f J = (J ij ) Φ p DΦ Φ J f(x) f(p) x p o( x p ) o( y q ) f(x) f(y) a (x p) b (y q)

/ 15 y q J(x p) u v t u v a (x p) = a (J 1 J(x p)) = t aj 1 J(x p) ( t J 1 a) (y q) b (y q) b = t J 1 a 1 y q J(x p) a = t Jb ϕ ψ p q J = D(ψ ϕ 1 )(p). M C 1 C 1 f : M R p U ϕ U ψ x = ϕ(x) y = ψ(x) p = ϕ(p) q = ψ(p) f(x) f(p) = a (x p) + o( x p ) (x p) f(x) f(p) = b (y q) + o( y q ) (y q) a, b a = t Jb b = t J 1 a J = D(ψ ϕ 1 )(p) p = ϕ(p) F ϕ = f ϕ 1 C 1 a p C 1 J = D(ψ ϕ 1 )(p) p a = t Jb q = ψ(p) b q M C 1 F ϕ C 1 F ψ = f ψ 1 C 1

/ 16. x p y q = J(x p) + o( x p ) y q = O( x p ) y q = J(x p) 18 J = (J ij ) 1 i,j 8 K = max 1 i,j 8 J ij y i q i = J ij (x j a j ) 8K max x j a j 8K x a 1 j 8 1 j 8 y q = 1 i 8 y i q i 2 8 8M x a y q = O( x p ) f(x) f(p) o( y q ) = o( x p ) f(x) f(p) = F ψ (y) F ψ (q) = b (y q) + o( y q ) = b {J(x p) + o( x p )} + o( x p ) = b J(x p) + o( x p ) F ψ (y) F ψ (q) = t b J(x p) + o( x p ) = t ( t J b)(x p) + o( x p ) = ( t Jb) (x p) + o( x p ) F ϕ (x) F ϕ (p) = a (x p) + o( x p ) (a t Jb) (x p) = o( x p ) x p 0 a t Jb = 0 a = t Jb a t Jb c = (c 1,..., c 8 ) x = p + ϵe i e i i 1 0 ϵ 0 c (x p) = c i ϵ o( x p ) = o(ϵ) c i = o(1) a t Jb = c = (c 1,..., c 8 ) c i = 0 c = 0.. ϕ ψ U ϕ U ψ Φ = ψ ϕ 1 J = DΦ 18 x p o( x p ) x p /100

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