II 1 II 2012 II Gauss-Bonnet II

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1 II 1 II 212 II Gauss-Bonnet II Theorema egregium Gauss-Bonnet C 1.1. I R c : I R k k = 2 k = 3 c c (t) c : [a, b] R k b a c (t) dt c J I t = t(u) d : J R k d(u) = c(t(u)). (1) c(i) (2) d(u) = c(t(u)) c c(u) = c(t(u)) (3) ( 1.1) (4)

2 II 2 (5) c(t) dt du > c(u) ( 1.2) 1.2. c : I R k α I, c(α) = a R k s(t) = t α c (t) dt t s c(s) = c(t(s)) c (s) = 1 s a.. ds dt = c (t) > c(s) dc ds = dc dt dt ds = 1. c(t) c(s) t ċ, c s c, c c(t) dt du > c(u) 1.3. c(t) = (t, a cosh(t/a)) c : I R 2 e 1 = 1 ċ ċ 9 1 e 2 = 1 e 1 (e 1, e 2 ) c(t) Frenet. (1) ċ (2) 1.4. ė j = k ω k j e k ω 2 1 = ω1 2, ω1 1 = ω2 2 =. ω k j ωk j = e j e k e j e k = δ jk 1.5. c : I R 2 c c(t) κ(t) = ω2 1 ċ (t) = ė1(t) e 2 (t) ẋÿ ẍẏ = ċ (t) ( (ẋ)2 + (ẏ) 2) 3/2 ω 2 1 = κ(s) ( e 1 e 2) = ( e1 e 2 ) ( κ κ 1.6. c(t) κ(t) (1) φ : J I, t = φ(u) dt du > c(φ(u)) κ(u) κ(u) = κ(φ(u)) )

3 II 3 (2) F : R 2 R 2 F c(t) κ(t) κ(t) = κ(t) (3) F : R 2 R 2 F c(t) κ(t) κ(t) = κ(t). (1) Frenet {e 1, e 2 } e j (t) = e j (u) de j du = de j dt dt du, dc du = dc dt dt du (2)(3) F 2 A b F(x) = Ax + b A = 1 A = 1 F c(t) Frenet {Ae 1, A Ae 2 } (2)(3). R 1 R 1 R * c(t) = (a cos t, b sin t) Frenet 1.5. y = f (x) x 1.6. y = f (x) x α > x y = r r 2 x 2 r < 1/α r > 1/α 1.7. r = f (θ) θ 1.8. c(t) = (cosh t, sinh t) 1.9. c(s) c (s) = (cos θ(s), sin θ(s)) θ (s) = κ(s) 1.1. c(s) c (s) = κ(s) * c 1 (s), c 2 (s) κ 1 (s) = κ 2 (s) F : R 2 R 2 c 2 = c 1 F. c 1 () = c 2 (), c 1 () = c 2 () Frenet 2 F 1 (s), F 2 (s) ( ) κ(s) F j (s) = F j(s) κ(s) t F j F j = F j t F j = E d ds ( ) ( ) t F1 F 2 = (F1 ) t F 2 + F t κ 1 (F 2 ) = F 1 κ t( ) κ t F 2 + F 1 κ t F 2 = O *1 1 κ(t) c *2 1

4 II 4 F 1 t F 2 F 1 () = F 2 () F 1 t F 2 = E F 1 (s) = F 2 (s) 1 c 1 (s) = c 2 (s) c 1() = c 2 () c 1 (s) = c 2 (s) c : [a, b] R 2 c R b a [a, b) c (Mukhopadhyaya). c 4 κ (s) =. c : [, l] R 2 s = s = a (, a) κ (s) > (a, l) κ (s) < c() c(a) x x (, a) y > (a, l) y < c(s) = (x(s), y(s)) ( ) ( ) x (e 1 ) y = y = κe 2 = κ x < l κ (s)y(s)ds = l κ(s)y (s)ds = l x (s)ds = [x (s)] l 3 κ 1 c(), c(a) c m c = 1 2π l κ(s)ds, l = c c m c = 1 2π b a κ(t) ċ dt = 1 2π b a ẋÿ ẍẏ (ẋ) 2 + (ẏ) 2 dt. c (s) = (cos θ(s), sin θ(s)) θ c θ() [, l] m θ(l) = θ() + 2πm l κ(s)ds = l θ (s)ds = θ(l) θ() = 2πm c (s) c, c 1 C 2 C : [, 1] I R 2, C(, t) = c (t), C(1, t) = c 1 (t)

5 II 5 c u (t) = C(u, t) c c 1 c u (t) u c u (t) c c (Whitney). c, c 1. m c c 1 1 c () = c 1 () = (, ), c () = c 1 () = (1, ) c j (s) = (cos θ j(s), sin θ j (s)) θ j () = m θ j (1) = 2πm Θ(u, s) = (1 u)θ (s) + uθ 1 (s) D(u, s) = s (cos Θ(u, t), sin Θ(u, t))dt Θ(, s) = θ (s), Θ(1, s) = θ 1 (s) D(, s) = c (s), D(1, s) = c 1 (s) C(u, s) = D(u, s) sd(u, 1) C(u, s + 1) C(u, s) = D(u, s + 1) D(u, s) D(u, 1) = s+1 s 1 (cos Θ(u, t), sin Θ(u, t)) t 1 C(u, s + 1) = C(u, s) c u (s) = C(u, s) 1 D(, 1) = D(1, 1) = (, ) C(, s) = c (s), C(1, s) = c 1 (s) c u (s) c u = C s = D D(u, 1) = (cos Θ(u, s), sin Θ(u, s)) s 1 (cos Θ(u, t), sin Θ(u, t))dt [, 1] 1 1 m c u m = Θ(u, t) u θ 1 (s) θ (s) c 1 c : [a, b] R 2 a t 1 < t 2 < b c(t 1 ) = c(t 2 ) 2 *3 P Q sgn P Q +1 Q sgn P Q = 1 Q *4 *3 *4 -

6 II (Whitney ). P 1 ±1 c c µ = l κ(s) ds, l = c µ 2π. (1) c (s) = (x (s), y (s)) y (s) l = y(l) y() = l y (s)ds y (s) x c (s) = (x (s), y (s)) (2) c () = c (a) = c (l) = (1, ) c (s) = (cos θ(s), sin θ(s)) a l a µ = κ(s) ds + κ(s) ds l θ (s)ds + θ (s)ds = θ(a) θ() + θ(l) θ(a) 2π a (3) ±1 ±1 κ(s) (2) (, a) (a, l) θ (s) θ(s) ±π c (s) (1) c(t) = (a cos t, b sin t) < a < b c(t) = (a cos t, b sin t) c F F c m F c = m c r 2 = 2a 2 cos 2θ a

7 II f (z) z 1 z = 1 f (z) (1) c(t) = f (e it ) (2) c(t) (x + iy)(u iv) = (xu + yv) i(xv yu) κ(t) = Re( f f e it ) + f f ( f f ) 3/2 (3) c z < 1 f (z) 1.4 c(s) c (s) κ(s) = c (s) c(s) c (s) e 2 e 1 (s) = c (s), e 2 (s) = 1 κ(s) e 1 (s), e 3(s) = e 1 (s) e 2 (s) Frenet Frenet ( e 1 (s) e 2 (s) e 3 (s)) = ( e 1 (s) e 2 (s) e 3 (s) ) κ(s) κ(s) τ(s) τ(s) e 2 e 3 τ(s) c(s) c(s) c(s) κ(s) = κ(s) τ(s) = τ(s) κ(s) > τ(s) c(s) κ(s) τ(s). 3 κ df ds = F κ τ = FK τ F() = E t K = K d ds (Ft (F)) = F t F + F t F = FK t F + F t (FK) = FK t F FK t F = O F t F = F() t F() = E F(s) F(s) (helix) c(t) = (a cos t, a sin t, bt) Frenet

8 II c Frenet e 1 = ċ ċ, e ċ ( c ċ) 2 = ċ ( c ċ), e 3 = ċ c ċ c κ(t) = ċ c, τ(t) = det(ċ, c, c(3) ) c (3) = d3 c ċ 3 ċ c 2 dt 3 a (b c) = (a c)b (a b)c, (a b) c = det(a, b, c) c(t) = (a cos t, b sin t, ct) 1.2. c(t) c V n ρ : V V R V {e 1, e 2,..., e n }. A = ρ(e i, e j ). V ρ(x, x) ρ t PAP g g ρ g t PAP = P 1 AP g G ρ A g t PGP t PAP t PGP = E g ρ t PAP λe = t PAP λ t PGP = P 2 A λg = t PAP p (A λg)pp = p Pp ρ g 1 A

9 II 9 A M m m x 2 t xax M x 2 x m, M 2.1. g ρ M m mg(x, x) ρ(x, x) Mg(x, x) x g ρ. g R 3 S uv D p : D R 3, p(u, v) = (x(u, v), y(u, v), z(u, v)) p u p v. (1) S = p(d) R 3 (u, v) D S p(u, v) D u = (u, v) S (2) S p 2.3. u S p u (u) p v (u) 2 S u T u S = { ξ p u (u) + ζ p v (u) ξ, ζ R }. (ξ, ζ) (u(s, t), v(s, t)) u s v s u t v t p(s, t) = p(u(s, t), v(s, t)) S p S T p S ( ) us u (p s, p t ) = (p u, p v ) t v s v t 2.4. T u S (ξ 1 p u + ζ 1 p v ) (ξ 2 p u + ζ 2 p v ) = ( ) ( E F ξ 1 ζ 1 F G ) ( ξ2 ζ 2 ), E = p u p u, F = p u p v, G = p v p v S 1 E, F, G 1 1 d p = p u du + p v dv I = d p d p = Edudu + 2Fdudv + Gdvdv

10 II 1 R 3 E, F, G 1 R 3 u = ξ p u (u) + ζ p v (u) T u S u = E(ξ) 2 + 2Fξζ + G(ζ 2 ) 2 = I(u, u) u, w T u S θ cos θ = I(u, w) u w S c(t) = p(u(t), v(t)), a t b ċ(t) = u(t)p u + v(t)p v L(c) = b a ċ(t) dt = b U D p(u) S EG F2 dudv U a E( u)2 + 2F u v + G( v) 2 dt dµ = EG F 2 dudv S ( ). ( ). ( ). x 2 a + y2 2 b + z2 = 1, p(u, v) = (a cos u cos v, b cos u sin v, c sin u) 2 c2 x 2 a + y2 2 b z2 = 1, p(u, v) = (a cosh u cos v, b cosh u sin v, c sinh u) 2 c2 x 2 a + y2 2 b z2 = 1, p(u, v) = (a sinh u cos v, b sinh u sin v, c cosh u) 2 c2 ( ). z = x2 a 2 + y2 b 2, p(u, v) = (u, v, u2 /a 2 + v 2 /b 2 ), p(u, v) = (au cos v, bu sin v, u 2 ) ( ). z = x2 a 2 y2 b 2, p(u, v) = (u, v, u2 /a 2 v 2 /b 2 ), p(u, v) = (a(u + v), b(u v), 4uv) ( ). xz (x R) 2 + z 2 = r 2, R > r > z, p(u, v) = ((R + r cos u) cos v, (R + r cos u) sin v, r sin u) ( ). xz x > x = f (u) >, z = g(u) z p(u, v) = ( f (u) cos v, f (u) sin v, g(u)) ( ). xy x = f (u), y = g(u) z p(u, v) = ( f (u), g(u), v). C 2 z = f (x, y) p(u, v) = (u, v, f (u, v)) 2.1. u = u(s, t), v = v(s, t) (s, t) 1 Ẽ, F, G (Ẽ ) ( ) ( ) ( ) F us v = s E F us u t F G u t v t F G v s v t

11 II u = u(s, t), v = v(s, t) Ẽ G F 2 dsdt = EG F 2 dudv u 2.6. p(u, v) = ( f (u), g(u), v), a u b, c v d ( f (u), g(u)) d c 2.7. z = f (x, y) p(u, v) = (u, v, f (u, v)) * 5 EG F2 = 1 + ( f u ) 2 + ( f v ) S γ(s) = p(u(s), v(s)) γ(s) γ (s) S γ (s) = κ g + κ n κ g T γ(s) S κ n S e = p u p v p u p v * 6 κ n = κ n e = γ (s) e = d ds (p uu + p v v ) e = ( p uu (u ) 2 + 2p uv u v + p vv (v ) 2) e γ γ u = γ T u S u u e S 2.5. T u S 2 II = Ldudu + 2Mdudv + Ndvdv, L = p uu e, M = p uv e, N = p vv e 2 L, M, N 2 II(u, u) u 2.6. I II 2.7. ( ) ( ) L M E F λ = (EG F M N F G 2 )λ 2 (EN + GL 2FM)λ + (LN F 2 ) = *5 1 *6 1

12 II 12 κ (( ) ( )) L M E F κ x = M N F G (Euler ). u S κ 1, κ 2 κ 1 κ 2 κ 1 u φ u II(u, u) = κ 1 cos 2 φ + κ 2 sin 2 φ κ 1 = κ 2. u (,, 1) z = f (x, y) f x (, ) = f y (, ) = p(u, v) = (u, v, f (u, v)) p u = (1,, ), p v = (, 1, ) 1 2 z κ 1 x κ 2 y 2 II II = κ 1 dudu + κ 2 dvdv u = (cos φ, sin φ, ) κ 1 cos 2 φ + κ 2 sin 2 φ κ 1 = κ 2 φ II = κ 1 dudu + κ 1 dvdv κ K Gauss H K = κ 1 κ 2 = LN M2 EG F, H = κ 1 + κ 2 EN + GL 2FM = 2 2 2(EG F 2 ). S. 2.1 p(u, v) = (a cos u cos v, b cos u sin v, c sin u) p u = ( a sin u cos v, b sin u sin v, c cos u) p v = ( a cos u sin v, b cos u cos v, ) p uu = ( a cos u cos v, b cos u sin v, c sin u) p uv = (a sin u sin v, b sin u cos v, ) p vv = ( a cos u cos v, b cos u sin v, ) e = 1 ( bc cos u cos v, ca cos u sin v, ab sin u)

13 II 13 π/2 u π/2 cos u 2 = b 2 c 2 cos 2 u cos 2 v + c 2 a 2 cos 2 u sin 2 v + a 2 b 2 sin 2 u = a 2 b 2 c 2 ( x 2 /a 4 + y 2 /b 4 + z 2 /c 4) e 1 E = a 2 sin 2 u cos 2 v + b 2 sin 2 u sin 2 v + c 2 cos 2 u F = (a 2 b 2 ) cos u sin u cos v sin v G = a 2 cos 2 u sin 2 v + b 2 cos 2 u cos 2 v L = abc, M =, N = abc cos2 u EG F 2 = 2 cos 2 u, LN M 2 = a 2 b 2 c 2 cos 2 u/ 2 K = a2 b 2 c 2 4 = 1 a 2 b 2 c 2 ( x 2 /a 4 + y 2 /b 4 + z 2 /c 4) 2 H = abc(a2 sin 2 u cos 2 v + b 2 sin 2 u sin 2 v + c 2 cos 2 u + a 2 sin 2 v + b 2 cos 2 v) 2 3 a = p(α, β) S p n f (u, v) = p(u, v) n *7 f u (α, β) = f v (α, β) =, f uu (α, β) = L, f uv (α, β) = M, f vv (α, β) = N a S K > S a a K < K > K < K =. a = (,, ), n = (,, 1) S 2 z = f (x, y) f x (, ) = f y (, ) = LN M 2 = f xx (, ) f yy (, ) ( f xy (, )) 2 p f (, ) xy p f xy 2.8. u = u(s, t), v = v(s, t) 2 (u, v) 2 L, M, N (s, t) 2 L, M, Ñ ( ) ( ) ( ) ( ) L M us v = s L M us u t M Ñ u t v t M N v s v t 2.9. p(u, v) K, H c ( ) cp(u, v) K/c 2, H/c 2.1. ( 2.8) 1 2 ( f (u)) 2 + (g (u)) 2 = ( 2.7) ( f (u)) 2 + (g (u)) 2 = *7 a n

14 II S a S H(a) a. p : U S p(, ) = a a S λ(u, v) C λ(u, v) 1, λ(, ) = 1 (, ) V V U U V λ p s (u, v) = p(u, v) + sλ(u, v)h(u, v)e(u, v) p s p a p s E s, F s, G s E s = E + sλhl + O(s 2 ), F s = F + sλhm + O(s 2 ), G s = G + sλhn + O(s 2 ) E s G s (F s ) 2 = EG F 2 + sλh(gl + EN 2FM) + O(s 2 ) = (EG F 2 )(1 + 2sλH 2 + O(s 2 )) Es G s (F s ) 2 = EG F 2 (1 + sλh 2 + O(s 2 )) p s A(s) A () = S λh 2 dµ A() e S γ(t) = p(u(t), v(t)) e e = 1 d (e(γ(t)) e(γ(t)) = dt d dt (e(γ(t)) = e uu + e v v T γ(t) S A : T p S T p S A(ξ p u + ζ p v ) = e u ξ e v ζ A g : S S 2 g(p) = e 1 g ) (1) II(u, u) = I (A(u), u) (2) A

15 II 15. (1) u = ξ 1 p u + ζ 1 p v, u = ξ 2 p u + ζ 2 p v I (A(u), u) = ( e u ξ 1 e v ζ 1 ) ξ 2 p u + ζ 2 p v = ξ 1 ξ 2 e u p u ξ 1 ζ 2 e u p v ζ 1 ξ 2 e v p u ζ 1 ζ 2 e v p v = ξ 1 ξ 2 L + ξ 1 ζ 2 M + ζ 1 ξ 2 M + ζ 1 ζ 2 N = II(u, u) (2) {p u, p v } II A κ 1 = κ 2 A p p II = κi. 1 A = λe II(u, u) = I (A(u), u) = λi (u, u) S S ( ). S p(u, v) Au = λu A(ξ p u + ζ p v ) = e u ξ e v ζ e u = λp u, e v = λp v e uv = e vu λ v p u + λp uv = λ u p v + λp vu λ u = λ v = λ λ = 2 ( ) λ p 1 λ e a a 1/ λ ( ) K (1) (2) (3) S κ 1 cos 2 φ + κ 2 sin 2 φ = φ (1)(3) κ 1 =, κ 2 κ 1 φ = κ 1 = κ 2 = 2.8 S z = f (x, y), f (, ) = f x (, ) = f ( y, y) = xy f (x, y) = f xx f yy ( f xy ) 2 < (, ) 2 γ(t), γ() = (,, ) xy γ xy (, ) γ ()

16 II p(u, v) = ( f (u) cos v, f (u) sin v, g(u)), ( f (u)) 2 + (g (u)) 2 = 1 a f (u) = a f (u) ( 2.6) p(u, v) = ( f (u) cos v, f (u) sin v, g(u)), ( f (u)) 2 + (g (u)) 2 = x = 3u + 3uv 2 u 3, y = v 3 3v 3u 2 v, z = 3(u 2 v 2 ) Enneper ( 2.1) Theorema egregium 2 R 3 Gauss 1 Gauss (Theorema egregium) (u, v) (u 1, u 2 ) *8 1 φ u j = φ / j 2 φ u 1 u 2 = φ /21 g 11 = p /1 p /1 = E g 12 = g 21 = p /1 p /2 = F g 22 = p /2 p /2 = G ( ) ( ) g = g i j du i du j g11 g 12 E F = g 21 g 22 F G i, j 2 h 11 = p /11 e = p /1 e /1 = L, h 12 = h 21 = p /12 e = p /1 e /2 = M, h 22 = p /22 e = p /2 e /2 = N, ( ) ( ) h = h i j du i du j h11 h 12 L M = h 21 h 22 M N i, j *8

17 II 17 (g i j ) (g i j ) ( ) g 11 g 12 g 21 g 22 = 1 EG F Γ k i j Christoffel ( ) G F g F E ik g k j = δ i j Γ k i j = 1 g kh (g ih/ j + g jh/i g i j/h ) 2 h. (1) 1 Γ k i j = Γk ji Christoffel 6 (2) Christoffel u R 3 u = ( g 11 u p /1 + g 12 u p /2 ) p/1 + ( g 21 u p /1 + g 22 u p /2 ) p/2 + (u e)e k. u {p /1, p /2, e} u p / j 3.3. (1) p /i j = Γ k i j p /k + h i j e (Gauss ) k (2) e /i = g jk h ki p / j (Weingareten ) k, j. g i j/k = (p /i p / j ) /k = p /ik p / j + p /i p / jk 2p /i j p /k = g ik/ j + g jk/i g i j/k 2 h i j = p /i j e (1) (2) 3.4. R i jkl = Γi jl/k Γi jk/l + Γ m l j Γi mk Γ m k j Γi ml R i jkl = 3.5. (1) R i jkl = h ik h jl h il h jk (Gauss ) (2) h i j/k + h l j Γ l ki = h ik/ j + h lk Γ l ji (Codazzi-Mainardi ) l l. Gauss u k p /i jk = Γ l i j/k p /l + Γ l i j p /lk + h i j/k e + h i j e /k l m l m g im R m jkl Gauss Weingarten {p /1, p /2, e} p /i jk = Γl i j/k + Γ m i j Γl mk h i j g ml h mk p /l + h i j/k + Γ m i j h mk e l m m p /i jk p /ik j = p /l Gauss e Codazzi-Mainardi. (1) 1 R i jkl, R i jkl k, l (2) Gauss R i jkl i, j R i jkl i, j, k, l R i jkl i, j, k, l 1 2 R 1212 = det h R i jkl ± det h m

18 II 18 Gauss Theorema egregium Gauss 1 2 det h det g 3.6 (Gauss). Gauss K = R 1212 det g K 1 u 1 u 2 D 2 g = g i j du i du j h = h i j du i du j g p : U R 3 1 g 2 h 2 p 3.7 ( Bonnet ). g, h Gauss Codazzi-Mainardi p : U R 3 1 g 2 h p R g = g i j du i du j = du 1 du 1 + h 2 du 2 du 2 Christoffel Γ i jk K = h /11 h g = g i j du i du j = e 2σ (du 1 du 1 + du 2 du 2 ) Christoffel Γ i jk K = σ σ = u /11 + u /22 e 2σ 3.2 γ(t) = p(u 1 (t), u 2 (t)) S X(t) γ X(t) T γ(t) S X(t) = ξ 1 (t)p /1 (γ(t)) + ξ 2 (t)p /2 (γ(t)) = ξ j (t)p / j (γ(t)) γ (t) γ X(t) {p /1, p /2, e} d dt X(t) = (ξ j ) p / j + ξ j p / jk (u k ) = (ξh ) + ξ j Γ h jk (uk ) p /h + ξ j h jk (u k ) e j j,k h 1 S 2 S 2 2 h(x(t), γ (t))e j,k j j,k d 3.8. dt X(t) D dt X(t) = (ξh ) + ξ j Γ h jk (uk ) p /h h j,k X(t) γ. 1 1 R 3 g(u, w)

19 II S γ γ X, Y d ( D ) dt g (X(t), Y(t)) = g dt X(t), Y(t) + g (X(t), D ) dt Y(t). R 3 (X(t) Y(t)) = X (t) Y(t) + X(t) Y (t) X(t) Y(t) e 3.1. S γ γ X D dt X γ γ S γ(t) γ (t) T γ(t) S. γ (t) D dt γ γ(t) = p(u 1 (t), u 2 (t)) X(t) (ξ h ) + Γ h jk (uk ) ξ j = h = 1, 2 1 X() γ(t) = p(u 1 (t), u 2 (t)) (u h ) + Γ h jk (u j ) (u k ) = h = 1, 2 j,k j,k 2 γ(), γ (). γ (t) = (u 1 ) p /1 + (u 2 ) p /2 γ(t), a t b S C : ( ε, ε) [a, b] S R 3 C(s, a) = γ(a), C(s, b) = γ(b), C(, t) = γ(t) C(s, t) = γ s (t) γ s γ = γ C γ γ s L(s) s = C t L(s) = b a C t = γ (t) γ (t) = c C t dt L () = 1 c b a 2 C s t C t dt Y(t) = C s (, t) γ Y(a) = Y(b) = γ C L () = 1 c b a d ( Y(t) γ (t) ) Y(t) γ (t)dt = 1 dt c b a Y(t) D dt γ (t)dt

20 II 2 D dt γ (α) α (a, b) Y(t) = λ(t) D dt γ (t), λ(t) 1, λ(a) = λ(b) =, λ(α) = 1 L () < γ S a a = (,, ) T a S xy γ() = (,, ), γ () = (u 1, u 2, ) γ γ(u 1, u 2 ; t) t (u 1 ) 2 + (u 2 ) 2 = 1 ε > t < ε = γ(u 1, u 2 ; t) γ(tu 1, tu 2 ; 1) = γ(u 1, u 2 ; t) p(u 1, u 2 ) = γ(u 1, u 2 ; 1) (u 1 ) 2 + (u 2 ) 2 ε 2 p p(t, ) p(, ) γ(t, ; 1) (,, ) γ(1, ; t) (,, ) p /1 (, ) = lim = lim lim = (1,, ) t t t t t t p /2 (, ) = (, 1, ) (, ) p(u 1, u 2 ) {p /1, p /2 } a (u 1 ) 2 + (u 2 ) 2 c 2 a c u 1 = r cos θ, u 2 = r sin θ p(r, θ) a θ γ(r) = p(r, θ) = p(r cos θ, r sin θ; 1) = p(cos θ, sin θ; r) 1 p r p r = 1 p r = cos θp /1 + sin θp /2, p θ = r sin θp /1 + r cos θp / p(r, θ) lim r + h r (r, θ) = 1 g = drdr + h 2 (r, θ)dθdθ. E = p r p r = 1 θ = 2p rθ p r = 2(p θ p r ) r 2p θ p rr

21 II 21 p(r, θ) r p rr 2 (p θ p r ) r = p θ p r r p θ (, θ) = F = p θ p r = G = p θ p θ r 2 G r = 2p rθ p θ, G rr = 2p rrθ p θ + 2p rθ p rθ p rθ = sin θp /1 + cos θp /2 + ru lim G r =. r lim G rr = 2 r G lim r r = 1 2 h(r, θ) = G lim r + h(r, θ)/r = 1 lim r + h r (r, θ) = a r a r L(r) A(r) L(r) = 2πr π 3 K(a)r3 + o(r 3 ), A(r) = πr 2 π 12 K(a)r4 + o(r 4 ). p(r, θ) a r > p(r, θ). θ 2π r L(r) = 2π h(r, θ)dθ, A(r) = ds 2π h(s, θ)dθ = r L(s)ds L(r) K = h rr h lim r + h rr /h = K(a) lim r + h rr = h rr lim r + r lim r + h rrr = K(a) Taylor h rr h = lim r + h r = K(a) h(r, θ) = r 1 3! K(a)r3 + o(r 3 ) L(r) K(a) = lim r + 6πr 3L(r) πr γ X(t), Y(t) g (X(t), Y(t)) 3.4. γ g(γ, γ )

22 II p(u 1, u 2 ) = ( f (u 1 ) cos u 2, f (u 1 ) sin u 2, g(u 1 )), ( f ) 2 + (g ) 2 = 1 γ(t) = ( f (t) cos α, f (t) sin α, g(t)) δ(t) = ( f (a) cos t, f (a) sin t, g(a)) 3.7. R 1/R Gauss-Bonnet ABC A p(r, θ) γ(t) = p(r(t), θ(t)) I = drdr + h(r, θ) 2 dθdθ, p θ p θ = h 2 r hh r (θ ) 2 =, θ + 2 h r h r θ + h θ h (θ ) 2 = γ (t) p r (γ(t)) φ(t) φ (t) = θ (t)h r. 3.1 Γ 1 11 = Γ1 12 = Γ2 11 =, Γ1 22 = hh r, Γ 1 12 = h 1 h r, Γ 2 22 = h 1 h θ u 1 = r, u 2 = θ r + Γ 1 i j (u j ) (u k ) = r hh r (θ ) 2 = i j θ + Γ 2 i j (u j ) (u k ) = θ + 2h 1 h r r θ + h 1 h θ (θ ) 2 = i j γ(t) = (r(t), θ(t) 1 γ(t) { p r (γ(t)), γ (t) = r p r (γ(t)) + θ p θ (γ(t)) = cos φ(t)p r (γ(t)) + sin φ(t) 1 h p θ(γ(t)) } 1 h p θ(γ(t)) r = cos φ(t), θ = h 1 sin φ(t) r = (sin φ)φ = hh r (θ ) 2 = h r (sin φ)θ sin φ φ = h r θ sin φ = γ A ±p r γ(t) p(±t + c, θ) φ π θ φ = h r θ =

23 II (Gauss). KdA = A + B + C π ABC K da. A p(r, θ) α θ β, r m(θ) B p(r, α) C p(r, β) γ(θ) = p(m(θ), θ) BC γ(θ) BC β KdA = dθ ABC α m(θ) h rr h hdr = β α h r (, θ) h r (m(θ), θ)dθ 1 h r (, θ) = 1 β α A 2 h r = dφ dθ β α h r (m(θ), θ)dθ = β α dφ dθ = φ(β) φ(α) dθ φ γ A φ(α) = π B, φ(β) = C.. R ( π)r 2 S χ(s ) = + * (Gauss-Bonnet ). S KdA = 2πχ(S ). S T j, 1 j N S KdA = N j=1 S T j KdA = N (T j π) T j L 2πL KdA = 2πL πn *9 4 I S j=1

24 II 24 M 3N = 2M N = 2M 2N KdA = 2πL πn = 2π(L M + N) = 2πχ(S ) S R 2 D 1 g = g 11 du 1 du 1 + 2g 12 du 1 du 2 + g 22 du 2 du 2 R H = {(x, y) y > } g = dx2 + dy 2 H = {z C Imz > } H 1 R S L(2, R) H ( ) a b A =, φ c d A (z) = az + b cz + d A ad bc = 1 Im(φ A (z)) = Imz cz + d 2 φ A H H φ AB = φ A φ B E φ E φ A H H 3.2. (1) φ A H (2) 2 z 1, z 2 H φ A (z 1 ) = z 2 A S L(2, R) (3) z H φ A (z ) = z A S L(2, R) F z F z S O(2) (4) φ A (z) = z A = ±E. (1) z(t) H H φ A (z) = φ A (z) (t) H = z (cz + d) 2 1 Im(φ A (z)) u H = 1 Imz u φ A (z) = 1 Im(φ A (z)) cz + d 2 z (t) = 1 Imz y 2 z (t) = z (t) H φ A 2 φ A

25 II 25 (2) a, b z 2 = az 1 + b z j a = Im(z 2 )/Im(z 1 ) > b = z 2 az 1 A = ( ) a b/ a 1/ S L(2, R) a φ A (z) = az + b φ A (z 1 ) = z 2 a b (3) z = i φ A (i) = i A = c d a = d b = c ad bc = 1 F i = S O(2) z φ P (i) = z P S L(2, R) A F z φ P 1 AP(i) = (φ P ) 1 φ A φ P (i) = i P 1 AP S O(2) F z = P (S O(2)) P 1 F z S O(2) (4) φ A (z) = z z 2 b = c = a = d ad bc = a 2 = 1 A = ±E x 2x y y =, y + (x ) 2 y (y ) 2 y x y /y y /y = log y = log y + A y = By y = Ce Bt H (1) (2). Az z + λz + λ z + D =, λ 2 AD > (1) (2) Cauchy-Riemann φ A γ(t) 1 A S L(2, R) φ A (γ()) = i P S O(2) φ PA (γ) () φ PA (γ(t)) i γ(t) φ (PA) 1 w H v T w H w v =

26 II 26 L P P L (a, b, c, d) = a c b d b c a d H 2 z 1, z 2 (z 1, z 1, z 2, z 2 ) = (i, i, ai, ai) z 1 z 2 z 1 z 2 = z 1 z 2 2 z 1 z 2 z 1 z 2 z 1 z 2 < 1 2 (a 1)2 a > 1 (a + 1) 2 a 1 a + 1 = z 1 z 2 z 1 z 2, a = z 1 z 2 + z 1 z 2 z 1 z 2 z 1 z 2 z 1, z 1, z 2, z 2 i, i, ai, ai S L(2, R) z 1, z 2 i, ai z 1 z 2 i ai {iy 1 y a} H 2 z 1, z 2 a 1 1 dy = log a y d(z 1, z 2 ) = log z 1 z 2 + z 1 z 2 z 1 z 2 z 1 z 2

( ) ( )

( ) ( ) 20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))