24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x

24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t), x 2 (t),, x n (t) C (ii) x 1(t), x 2(t),, x n(t) C = γ([a, b]) (γ ). (1) a 0 γ : (, ) R 3 ; t (a cos t, a sin t, bt) (2) ( (ii) ) γ : [ 1, 1] R 2 ; t (t 2, t 3 ) x 1(t) = 2t, x 2(t) = 3t 2, x 1(0) { = x 2(0) = 0 x = t 2, y = t 3 x 3/2 (t 0), 0 x 1, y = x 3/2 (t < 0) C 1.2. γ : [a, b] R n γ : [ã, b] R n γ ( ), C h: [ã, b] [a, b] h > 0, h(ã) = a, h( b) = b (h < 0, h(ã) = b, h( b) = a), γ( t) = γ(h( t)) (ã t b) 1.3. (R n ) R n v = (v 1, v 2,, v n ) R n p R n v p p R n, T p R n def = {v p v R n } p R n, T p R n p R n T p R n, v p + w p = (v + w) p, λ(v p ) = (λv) p, v p, w p = v, w, 1.4. γ : [a, b] R n ; t (x 1 (t), x 2 (t),, x n (t)) γ (t) = (x 1(t), x 2(t), x n(t)) γ(t) T γ(t) R n 1

(0, 0,, 0) 1.1 (ii) γ (t) = x 1(t) 2 + x 2(t) 2 + + x n(t) 2 ( ) γ (t) = lim t 0 = lim t 0 γ(t + t) γ(t) t ( x1 (t + t) x 1 (t) t ), T γ(t) C def = {λγ (t) λ R} T γ(t) R n 1 T γ(t) C γ(t) C, T γ(t) C γ(t) C 1.2. 1.5. γ : [a, b] R n L(γ) L(γ) =. γ : [0, 2π] R 3 ; t (a cos t, a sin t, bt) γ (t) = ( a sin t, a cos t, b), γ (t) = a 2 + b 2 =: l L(γ) = 2π 0 l dt = 2πl = 2π a 2 + b 2 b a γ (t) dt γ : [0, 1] R 3 ; t (a cos 2πt, a sin 2πt, 2πbt) 1.6. γ : [a, b] R n, γ : [ã, b] R n γ, L(γ) = L( γ), γ, h 1.2 γ ( t) = γ (h( t))h ( t), γ ( t) = γ (h( t)) h ( t) t = h( t), L(γ) = b a γ (t) dt = eb ea γ (h( t)) h ( t) d t = eb ea γ ( t) d t = L( γ). γ : [a, b] R n s γ. ( ) t s = s(t) = γ (u) du = lt, t = s 0 ( l γ(t) = γ(s/l) = a cos s l, a sin s ) l, bs =: γ(s) (0 s 2πl) l γ : [a, b] R n s(t) = t a γ (u) du s(a) = 0, s(b) = L(γ), s (t) = γ (t) > 0... s(t) t 2

t s(t) h: [0, L(γ)] [a, b] γ : [0, L(γ)] R 3 ; s γ(h(s)) γ γ γ (s) = γ (h(s))h (s) = γ (h(s))s (h(s)) 1 = γ (t(s)) γ (t(s)) 1... γ (s) = 1, 1 1.7. γ : [a, b] R n t def γ (t) = 1 (a t b) 1.8. γ : [a, b] R n, γ,. 1.7 t = s(t) + a., b = L(c) + a.. (1) y = x2 2 (0 x 1) γ : [0, 1] R2 ; t (t, t 2 /2), s(t) : s(t) = t 0 1 + t2 dt = 1 2 1 + u2 du = 1 { t 1 + t 2 2 + log(t + 1 + t 2 ) { t 1 + t 2 + log (t + )} 1 + t 2 (2) x2 a + y2 = 1, 0 < b < a s(t) 2 b2 t s(t) = a 1 ε2 cos 2 u du 0 ε } 1.3. 1.9. γ : [a, b] R n Y, t [a, b] T γ(t) R n Y(t), Y(t) = (y 1 (t), y 2 (t),, y n (t)) γ(t), y 1 (t), y 2 (t),, y n (t) C Y(t) T γ(t) C ( t), Y, Y(t) T γ(t) C ( t), 1.10. Y, Z γ : [a, b] R n, Y(t), Z(t) = Y (t), Z(t) + Y(t), Z (t), Y, Y (t), Y(t) = 0, t Y (t) Y(t) 3

Y(t) = (y 1 (t),, y n (t)) γ(t), Z(t) = (z 1 (t),, z n (t)) γ(t), Y (t) = (y 1(t),, y n(t)) γ(t), Y(t), Z(t) = (y 1 (t)z 1 (t) + + y n (t)z n (t)) = (y 1(t)z 1 (t) + y 1 (t)z 1(t)) + + (y n(t)z n (t) + y n (t)z n(t)) = Y (t), Z(t) + Y(t), Z (t)., γ : [a, b] R 3, γ 1.11. γ (s) 0 ( s) e(s) = γ (s), e(s) = 1, e γ e γ ( ) 1.10, e (s) (= γ (s) 0) e(s), n(s) = e (s) = γ (s) e (s) γ (s), n(s) = 1, n γ n γ b(s) = e(s) n(s), b(s) = 1, b γ b γ s {e(s), n(s), b(s)} T γ(s) R 3 e, n, b γ 1.12. ( ) κ(s) = e (s) (= γ (s) ) κ(s) 0, γ (s) 0 ( s), κ(s) > 0, e (s) = κ(s)n(s), 1.13, n (s) = τ(s)b(s) κ(s)e(s) τ(s) 1.13. n (s) = τ(s)b(s) κ(s)e(s) n (s) = ϕ(s)e(s) + ψ(s)n(s) + τ(s)b(s), 1.10, 0 = n (s), n(s) = ψ(s), 0 = n(s), e(s) = n (s), e(s) + n(s), e (s) = ϕ(s) + κ(s), ϕ(s) = κ(s). τ(s) = n (s), b(s) 1.14. ( ) γ : [a, b] R 3, γ, γ (s) 0 ( s), 3 : e (s) = κ(s)n(s) n (s) = τ(s)b(s) κ(s)e(s) b (s) = τ(s)n(s), e(s) e (s) 0 κ(s) 0 e(s) d n(s) = n (s) = κ(s) 0 τ(s) n(s) ds b(s) b (s) 0 τ(s) 0 b(s) 4

b (s) = ϕ(s)e(s) + ψ(s)n(s) 0 = b(s), e(s) = b (s), e(s) + b(s), e (s) = ϕ(s) 0 = b(s), n(s) = b (s), n(s) + b(s), n (s) = ψ(s) + τ(s), ϕ(s) = 0, ψ(s) = τ(s) 1.4. 1.15. γ : [a, b] R 3 t, γ (t) γ (t) γ : [0, L(c)] R 3 γ : s(t) = t a γ (u) du t [a, b] s(t) [0, L(c)] s [0, L(c)] h(s) [a, b] γ(s) = γ(h(s)) γ(s(t)) = γ(t) C (γ ) γ(t),,,, C γ(s(t)), e(t) def = ẽ(s(t)), n(t) def = ñ(s(t)), b(t) def = b(s(t)), κ(t) def = κ(s(t)), τ(t) def = τ(s(t)) (,.) 1.16. γ : [a, b] R 3 t, γ (t) γ (t) C = γ([a, b]) C γ(t),,, e = γ γ n = γ, γ γ γ, γ γ γ, γ γ γ, γ γ = (γ γ ) γ (γ γ ) γ b = γ γ γ γ κ = γ γ γ 3 τ = γ γ, γ = det(γ, γ, γ ) γ γ 2 γ γ 2. n = b e 5

γ (s(t))s (t) = γ (t)... e(t) = ẽ(s(t)) = γ (t) ( ) γ (t) γ ẽ (s(t))s (t) (t) =... γ (t) ẽ (s(t)) = 1 γ (t)... n(t) = ñ(s(t)) = ẽ (s(t)) ẽ (s(t)) = ( ) γ (t) γ (t) = γ (t) 2 γ (t) γ (t), γ (t) γ (t) γ (t) 4 = (γ (t) γ (t)) γ (t) γ (t) 4 ( c, a b c, b a = (a b) c) κ(t) = κ(s(t)) = ẽ (s(t)) = γ (t) γ (t) γ (t) 3 (γ (t) γ (t)) γ (t) (γ (t) γ (t)) γ (t) b(t) = b(s(t)) = ẽ(s(t)) ñ(s(t)) = e(t) n(t) = γ (t) γ (t) γ (t) 2 γ (t) γ (t), γ (t) γ (t) γ (t) γ (t) γ (t) = γ (t) 2 γ (t) γ (t) γ (t) 2 γ (t) γ (t) = γ (t) γ (t) γ (t) γ (t) τ(t) = τ(s(t)) = ñ (s(t)), b(s(t)) = 1 γ (t) n (t), b(t) = = 1 γ (t) γ (t) γ (t) n (t), γ (t) γ (t) 1 γ (t) γ (t) γ (t) γ (t) 2 (γ (t) γ (t)) γ (t) γ (t), γ (t) γ (t) = det(γ (t), γ (t), γ (t)) γ (t) γ (t) 2 n (t) = (ñ(s(t))) = ñ (s(t))s (t)... ñ (s(t)) = n (t) γ (t), n(t) = a(t)γ (t) + b(t)γ (t), n (t) = a(t)γ (t) + [γ (t) γ (t) ] (a(t) n(t) ) 1.5. γ : [a, b] R 3, γ, γ (s) 0 ( s) C = γ([a, b]). 6

s 0 [a, b] γ 3 s 0 = 0 (a, b) 1.17. γ(s) = γ(0)+ (s 16 ) ( 1 s3 κ(0) 2 e(0)+ 2 s2 κ(0) + 1 ) 6 s3 κ (0) n(0)+ 1 6 s3 κ(0)τ(0)b(0)+o(s 3 ). f(s) = o(s 3 ) def lim s 0 f(s)/s 3 = 0 0 x(s) = x(0) + x (0)s + 1 2 x (0)s 2 + 1 6 x (0)s 3 + o(s 3 ) γ(s) = γ(0) + sγ (0) + 1 2 s2 γ (0) + 1 6 s3 γ (0) + o(s 3 ) = γ(0) + se(0) + 1 2 s2 κ(0)n(0) + 1 6 s3 γ (0) + o(s 3 ) (n = κe + τb) γ = (κn) = κ n + κn = κ 2 e + κ n + κτb 1.18. ˆγ(s) = γ(0) + se(0) + 1 2 s2 κ(0)n(0) + 1 6 s3 κ(0)τ(0)b(0), γ s = 0 2 C γ(0) s γ(0) + se(0) γ γ(0) (1 ) 3 s γ(0) + se(0) + 1 2 s2 κ(0)n(0) γ γ(0) 2. (1), γ(0) b(0) C γ(0) (osculating plane) (2), xy y = κ(0) 2 x2, C γ(0) τ(0), ˆγ, γ(0) C. γ : (, ) R 3 ; γ(s) = (cos s/ 2, sin s/ 2, s/ 2) 1.19. γ : [a, b] R 3, γ C, C, κ(s) = e (s) = γ (s), κ 0 γ 0 7

1.6. R 3, 1.20. R 3 ϕ: R 3 R 3 def ϕ(x) ϕ(y) = x y (x, y R 3 ), ϕ 2 ϕ(x) = Ux + q (x R 3 ) U (, t UU = U t U = I), q R 3, ( det U = ±1 ) det U = 1, ϕ 1.21. ϕ(s), ψ(s) [a, b] C, ϕ(s) > 0 ( s) γ : [a, b] R 3 s.t. κ(s) = ϕ(s), τ(s) = ψ(s), γ, 1, cf. 6, 7 ( 5 ). 2. R 3 2.1.,, 2.1. R 3 ( ), R 3 S : p S, S p U, R 2 D, f : D R 3 ; (u, v) f(u, v) = (x(u, v), y(u, v), z(u, v)) : (i) f(d) = U, f : D U (ii) x(u, v), y(u, v), z(u, v) C (iii) (u, v) D (, ) f x y z (u, v) = (u, v), (u, v), (u, v) u ( u u u ) f x y z (u, v) = (u, v), (u, v), (u, v) v v v v f : D U S ( U ), (u, v) S ( U ) (U = S, S, ), f u, x u, f u, x u,. 8

(1) ( ) z = ϕ(x, y) ((x, y) D) C S = {(x, y, ϕ(x, y)) (x, y) D} f : D R 3 ; (x, y) (x, y, ϕ(x, y)) (1-1) z = ax 2 + by 2 (a, b > 0) z = ax 2 by 2 (a, b > 0) (1-2) z = r 2 x 2 y 2 (r > 0, x 2 + y 2 < r 2 ), r ( ) ( ) (2) ( ) D = {(θ, ϕ) 0 < θ < π, 0 < ϕ < 2π} f : D R 3 ; (θ, ϕ) (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) U = f(d) (3) ( ) R > r > 0 D = {(θ, ϕ) 0 < θ < 2π, 0 < ϕ < 2π} f : D R 3 ; (θ, ϕ) ((R + r cos θ) cos ϕ, (R + r cos θ) sin ϕ, r sin θ) U = f(d) f u (u, v), f v (u, v) 2.2. f : D U (U S ) S, p = f(u, v) S f u (u, v) p, f v (u, v) p ( ) T p R 3 2 T p S : T p S = {af u (u, v) p + bf v (u, v) p a, b R}. T p S p S, p S 2.3. S, γ : [a, b] R 3 γ([a, b]) S, γ : [a, b] S 2.4. γ : [a, b] S S, t 0 (a, b), f : D U, γ(t 0 ) S U, γ((t 0 ε, t 0 +ε)) U, D γ 0 : (t 0 ε, t 0 +ε) D, f(γ 0 (t)) = γ(t) ( t (t 0 ε, t 0 +ε)) γ 0 2.5. S p S T p S : T p S = {γ (0) γ : ( ε, ε) S γ(0) = p S }., v T p S, S γ : ( ε, ε) S, γ(0) = p, γ (0) = v, T p S p 9

γ : ( ε, ε) S S, f : D U, γ(0) S U ε, γ(( ε, ε)) U γ 0 : ( ε, ε) D 2.4, γ 0 (t) = (u(t), v(t)), γ (0) = f u (u(0), v(0))u (0) + f v (u(0), v(0))v (0), T p S v T p S v = af u (u, v) + bf v (u, v) (cf. 2.2)., γ 0 : ( ε, ε) D γ 0 (t) = (u + at, v + bt), γ = f γ 0, γ (0) = f u (u, v)a + f v (u, v)b = v. 2.6. (1) S C, S ϕ, S f : D U, ϕ f D C (2) S Y, p S Y p T p R 3, Y S C p S Y(p) T p S, p S Y(p) T p S 2.7. S, S N, p S N(p) = 1 S, S. (1) S, N S f : D U S, N(p) = ± f u(u, v) f v (u, v) (p = f(u, v)) f u (u, v) f v (u, v) (2), (1) +, D = {(v, u) (u, v) D}, ι: D D; (v, u) (u, v) f = f ι: D U; (v, u) f(v, u) = f(u, v) f v (v, u) f u (v, u) = f v (u, v) f u (u, v) = f u (u, v) f v (u, v) f u (u, v) f v (u, v) f u (u, v) f v (u, v) = f u(u, v) f v (u, v) f u (u, v)) f v (u, v). (1),,,, n,, (2), 2.2. V, n = dim V 2.8. Φ: V V R, v 1 V v 2 V Φ(v 1, v 2 ) R, v 2 V v 1 V Φ(v 1, v 2 ) R v 1, v 2 Φ(v 1, v 2 ) = Φ(v 2, v 1 ) Φ 10

, (V,, )., : V V R; (v 1, v 2 ) v 1, v 2, T : V V, Φ: V V R; (v 1, v 2 ) T v 1, v 2 2.9. T : V V, Φ, v 1, v 2 V T v 1, v 2 = v 1, T v 2. Φ: V V R, T : V V Φ(, ) = T ( ),, Φ T (, ) 2.10. T : V V, V T, V T, V, T {e 1,, e n } V, T A = (a ij ) T (e j ) = n a ije i T (e j ), e k = e j, T (e k ), T (e j ), e k = A n a ij e i, e k = a kj, e j, T (e k ) = e j, n a ik e i = a jk A, n U = (u ij ) U 1 AU = t UAU V {f 1,, f n } f j = n u ije i, {f 1,, f n } T t UAU ( ). 2.3. S, N S 2.11. v T p S, S γ : ( ε, ε) S, γ(0) = p, γ (0) = v ( 2.5). (1) S C ϕ v ϕ = D v ϕ = d dt ϕ(γ(t)) t=0 (2) S Y D v Y = d dt Y(γ(t)) t=0 T p S 11

2.12. (1) v ϕ, D v Y well-defined (2) v ϕ, D v Y v (3) v Y, Z = D v Y, Z + Y, D v Z. (4) D v N T p S. (1) v = af u (u, v) + bf v (u, v), γ(t) = f(u(t), v(t)) (cf. 2.4), γ (0) = f u (u, v) u (0) + f v (u, v) v (0) = v, u (0) = a, v (0) = b. d dt Y(γ(t)) t=0 = d dt Y(f(u(t), v(t))) t=0 = (Y f) u (u(0), v(0)) u (0) + (Y f) v (u(0), v(0)) v (0) D v Y well-defined. = a(y f) u (f 1 (p)) + b(y f) v (f 1 (p)) (2) v = af u (u, v)+bf v (u, v), w = cf u (u, v)+df v (u, v) T p M, λ, µ R λv+µw = (λa + µc)f u (u, v) + (λb + µd)f v (u, v), (1) (3) D λv+µw Y = (λa + µc)(y f) u + (λb + µd)(y f) v = λ(a(y f) u + b(y f) v ) + µ(c(y f) u + d(y f) v ) = λd v Y + µd w Y. v Y, Z = d dt Y(ϕ(γ(t))), Z(ϕ(γ(t))) t=0 d = dt Y(ϕ(γ(t))) t=0, Z(p) + Y (p), d dt Z(ϕ(γ(t))) t=0 = D v Y, Z + Y, D v Z. (4) N, N = 1 (3), D v N, N = 0., D v N T p S. 2.13. v T p S, Σ(v) = D v N,, Σ(v) T p S, Σ: T p S T p S Σ (shape operator) 2.14. T p S I : T p S T p S R, II : T p S T p S R I(v, w) = v, w, II(v, w) = I(Σ(v), w) (v, w T p S), S 1 (1st fundamental form), 2 (2nd fundamental form) g ij (u 1, u 2 ) = I(f ui (u 1, u 2 ), f uj (u 1, u 2 )), h ij (u 1, u 2 ) = II(f ui (u 1, u 2 ), f uj (u 1, u 2 )) 12

. (1), (u, v) (u 1, u 2 ) (2) II 2.15 2.15. (1) h ij = N(f(u 1, u 2 )), f ui u j (u 1, u 2 ) h ij = h ji (2) II, II(v, w) = II(w, v) (v, w T p S), Σ (1) γ(t) = f(u 1 +t, u 2 ), γ(0) = f(u 1, u 2 ), γ (0) = f u1 (u 1, u 2 ), Σ(f u1 (u 1, u 2 )) = D fu1 (u 1,u 2 )N = d dt N(f(u 1 + t, u 2 )) t=0 = u 1 N(f(u 1, u 2 )), Σ(f u2 (u 1, u 2 )) = N(f(u 1, u 2 )), u 2 h ij = N(f(u 1, u 2 )), f uj (u 1, u 2 ) u i = u i N(f(u1, u 2 )), f uj (u 1, u 2 ) + N(u 1, u 2 ), f ui u j (u 1, u 2 ) = N(u 1, u 2 ), f ui u j (u 1, u 2 ). (2) v = af ui + bf uj, w = cf ui + df uj, (1) II(v, w) = acii(f ui, f ui ) + adii(f ui, f uj ) + bcii(f uj, f ui ) + bdii(f uj, f uj ) = ach ii + adh ij + bch ji + bdh jj = ach ii + (ad + bc)h ij + bdh jj = = II(w, v) 2.16. S Σ: T p S T p S det Σ, trσ 1/2, p S,, K, H. (1), T : V V (V ), det T, trt,, (2) Σ: T p S T p S κ 1, κ 2 p S K = κ 1 κ 2, H = 1 2 (κ 1 + κ 2 ) 2.17. S K, H K = h 11h 22 h 2 12, H = g 22h 11 2g 12 h 12 + g 11 h 22 g 11 g 22 g12 2 2(g 11 g 22 g12) 2 13

Σ(f uj (u 1, u 2 )) T p S (p = f(u 1, u 2 )), ( ) Σ(f uj (u 1, u 2 )) = Σ = ( 2 σ ij (u 1, u 2 )f ui (u 1, u 2 ) (j = 1, 2) σ 11 (u 1, u 2 ) σ 12 (u 1, u 2 ) σ 21 (u 1, u 2 ) σ 22 (u 1, u 2 ), T p S {f u1 (u 1, u 2 ), f u2 (u 1, u 2 )} Σ σ ij, g ij, h ij ( ( ) g 11 g 12 2.18. Î = ), g 21 g ÎI = h 11 h 12, 22 h 21 h 22 (1) Î (2) Σ = Î 1 ÎI (1) 0 A 2 = ( f u1 f u2 sin θ) 2 = f u1 2 f u2 2 (1 cos 2 θ) = f u1 2 f u2 2 ( f u1 f u2 cos θ) 2 = f u1 2 f u2 2 f u1, f u2 2 = g 11 g 22 g 2 12 = det Î, det Î 0., Î (2) ( ) f uk (u 1, u 2 ) 2 Σ(f uj ), f uk = σ ij f ui, f uk = h jk, 2 h jk = σ ij g ik h kj = ) 2 σ ij g ik. 2 g ki σ ij ÎI = Î Σ.. g 11, g 12 = g 21, g 22 E, F, G h 11, h 12 = h 21, h 22 L, M, N 2.17 ( ) ( E F Î =, ÎI = F G L M ) M N K = det Σ 2 LN M = (det Î) 1 det ÎI = ( ) ( EG F 2 ) 1 G F L M Σ = = EG F 2 F E M N... H = 1 2 tr Σ GL 2F M + EN = 2(EG F 2 ) ( 1 EG F 2 GL F M F L + EM GM F N F M + EN ) 14

2.4. S R 3, p, T p S xy, N(p) e 3 = (0, 0, 1) S p, xy U z = ϕ(x, y),, ϕ(0, 0) = 0, N(ϕ(x, y)) = ( ϕx, ϕy,1),, N(p) = (0, 0, 1), ϕ ϕ 2 x(0, 0) = x +ϕ 2 y +1 ϕ y (0, 0) = 0, (0, 0) ϕ 24 (1), p K(p) = ϕ xx (0, 0)ϕ yy (0, 0) ϕ xy (0, 0) 2 ϕ (0, 0) K(p) > 0 (ϕ xx ϕ yy ϕ 2 xy)(0, 0) > 0 { ϕ xx (0, 0) > 0 ϕ (0, 0) ϕ xx (0, 0) < 0 ϕ (0, 0) K(p) < 0 (ϕ xx ϕ yy ϕ 2 xy)(0, 0) < 0 (0, 0) ϕ ( ),, S f : D S f(u 1 + u 1, u 2 ) f(u 1, u 2 ) f u1 (u 1, u 2 ) u 1 f(u 1, u 2 + u 2 ) f(u 1, u 2 ) f u2 (u 1, u 2 ) u 2 = : g 11 g 22 g 2 12 u 1 u 2 2.19. f : D S S A(S) A(S) = g 11 g 22 g12 2 du 1 du 2 da =. g 11 g 22 g 2 12 = f u1 f u2 D g 11 g 22 g 2 12 du 1 du 2, S. (1) S = {(x, y, ϕ(x, y))} z = ϕ(x, y) f(x, y) = (x, y, ϕ(x, y)) f x = (1, 0, ϕ x ), f y = (0, 1, ϕ y ), f x f y = ( ϕ x, ϕ y, 1), f x f y = ϕ 2 x + ϕ 2 y + 1 15

ϕ(x, y) = r 2 x 2 y 2 x ϕ x = r2 x 2 y, ϕ y 2 y = r2 x 2 y, f r 2 x f y = r = dxdy = = x 2 +y 2 <r 2 r2 x 2 2πr2 y2 r 2 x 2 y 2 (2) f(θ, ϕ) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) f θ = (r cos θ cos ϕ, r cos θ sin ϕ, r sin θ) f ϕ = ( r sin θ sin ϕ, r sin θ cos ϕ, 0) f θ f ϕ = (r 2 sin 2 θ cos ϕ, r 2 sin 2 θ sin ϕ, r 2 cos θ sin θ), f θ f ϕ = r 2 sin θ = r 2 sin θ = 4πr 2 0<θ<π,0<ϕ<2π, S, N S : N = f u 1 f u2 f u1 f u2 D D, 0 t, f t : D R 3 f t (u 1, u 2 ) = f(u 1, u 2 ) + tn(f(u 1, u 2 )) ((u 1, u 2 ) D ) S = f(d ), S t = f t (D ), S t f t : D S t S t 1 (g t) ij (i, j = 1, 2) : (g t ) ij = (f t ) ui, (f t ) uj A(S t ) = (gt ) 11 (g t ) 22 (g t ) 12 du 1 du 2 D 2.20. d dt A(S t) t=0 = 2 H(f(u 1, u 2 )) g 11 g 22 g12 2 du 1 du 2 = 2 H da. D S 2.20, : D H > 0 A(S t) t D H < 0 A(S t) t D H = 0 A(S t) 2.20 (f t ) ui = f ui + t(n f) ui (g t ) ij = (f t ) ui, (f t ) uj = f ui + t(n f) ui, f uj + t(n f) uj = f ui, f uj + t f ui, (N f) uj + t (N f) ui, f uj + t 2 (N f) ui, (N f) uj... t (g t) ij t=0 = f ui, (N f) uj + (N f) ui, f uj = I(f ui, Σ(f uj )) I(Σ(f ui ), f uj ) = II(f uj, f ui ) II(f ui, f uj ) = 2h ij 16

...... t ((g t) 11 (g t ) 22 (g t ) 12 2 ) t=0 = 2h 11 g 22 + g 11 ( 2h 22 ) 2g 12 ( 2h 12 ) = 2(h 11 g 22 + g 11 h 22 2g 12 h 12 ) (g t ) 11 (g t ) 22 (g t ) 2 12 t=0 = 1 t 2 (g 11g 22 g 2 12 ) 1/2 ( 2)(h 11 g 22 + g 11 h 22 2g 12 h 12 ) d (g t ) 11 (g t ) 22 (g t ) 2 12 du 1 du 2 t=0 = dt D = g 22h 11 + g 11 h 22 2g 12 h 12 g11 g 22 g 2 12 = 2H g 11 g 22 g 12 2 D t = 2 H g 11 g 22 g 122 du 1 du 2 D (g t ) 11 (g t ) 22 (g t ) 12 2 t=0 du 1 du 2 S,,, S d S t, dt A(S t) t=0 = 0, 2.5., S, 1 γ : [a, b] S S, C = γ([a, b]) U f : D U, D γ 0 : [a, b] D γ = f γ 0 (cf. 2.4). γ 0 (t) = (u 1 (t), u 2 (t)) γ(t) = f(u 1 (t), u 2 (t)) L(γ) = b a γ (t) dt γ (t) = f u1 (u 1 (t), u 2 (t))u 1 (t) + f u2 (u 1 (t), u 2 (t))u 2 (t) γ 2 = f u1 u 1 + f u2 u 2, f u1 u 1 + f u2 u 2 = g 11 u 1 2 + 2g 12 u 1 u 2 + g 22 u 2 2, 2.21. L(γ) = b a ( g 11 (u 1 (t), u 2 (t))u 1 (t) 2 + 2g 12 (u 1 (t), u 2 (t))u 1 (t)u 2 (t) +g 22 (u 1 (t), u 2 (t))u 2 (t) 2) 1/2 dt : S, 1 (u 0, v 0 ) D, a, b D γ 0 : [0, T ] D; t (u 0 + at, v 0 + bt), γ = f γ 0 L(γ) = T 0 (g 11 (u 0 + at, v 0 + bt)a 2 + ) 1/2 dt 17

d dt L(γ) T =0 = g 11 (u 0, v 0 )a 2 + 2g 12 (u 0, v 0 )ab + g 22 (u 0, v 0 )b 2 ) 1/2 a = 1, b = 0, = g 11 (u 0, v 0 ) 1/2. S 2,, S., 1 2.22 ( ). S K 1 (i) K (ii) K S 2 ( 1 ), K, S R 3 (, R 3 \ S ),, 2.22 2.23. S X v T p S v X = Π(D v X) = D v X D v X, N(p) N(p) T p S, Π: T p R 3 T p S v X X v 2.24. Y S, X S, ϕ S C, v T p S : (i) D v (ϕy) = (v ϕ)y(p) + f(p)d v Y. (ii) v (ϕx) = (v ϕ)x(p) + f(p) v X., S f : D U, U, S Y f Y f Y (, U = D, Y U D ) 2.25. fui f uj = af u1 + bf u2, a, b 1 E, F, G, i = j = 1 fu1 f u1 = af u1 + bf u2, f u1, f u2, ae + bf = f u1 u 1, f u1 = 1 2 f u 1, f u1 u1 = 1 2 E u 1, 18

af + bg = f u1 u 1, f u2 = f u1, f u2 u1 f u1, f u2 u 1 = f u1, f u2 u1 1 2 f u 1, f u1 u2 = F u1 1 2 E u 2. a, b 2.26 ( ). (1) fu1 ( fu2 X) fu2 ( fu1 X) = I(X, Σ(f u2 ))Σ(f u1 ) I(X, Σ(f u1 ))Σ(f u2 ). fui X = X ui X ui, N N = X ui + X, N ui N, fu1 ( fu2 X) = ( fu2 X) u1 + fu2 X, N u1 N = [X u2 X u2, N N] u1 + [X u2 X u2, N N], N u1 N = X u2 u 1 X u2 u 1, N N X u2, N u1 N X u2, N N u1 + X u2, N u1 N = X u2 u 1 X u2 u 1, N N + X, N u2 N u1., fu1 ( fu2 X) fu2 ( fu1 X) = I(X, Σ(f u2 ))Σ(f u1 ) I(X, Σ(f u1 ))Σ(f u2 ). X = f u1, (1) f u2 I( fu1 ( fu2 f u1 ) fu2 ( fu1 f u1 ), f u2 ) (2) = I(f u1, Σ(f u2 ))I(Σ(f u1 ), f u2 ) I(f u1, Σ(f u1 ))I(Σ(f u2 ), f u2 ) = II(f u1, f u2 )I(f u1, f u2 ) II(f u1, f u1 )II(f u2, f u2 ) = h 2 12 h 11 h 22. 2.27. (2), 1 E, F, G 2 fu2 f u1 = af u1 + bf u2, 2.25, a, b 1 E, F, G fu1 ( fu2 f u1 ) = fu1 (af u1 + bf u2 ) = (a f) u1 f u1 + a fu1 f u1 + (b f) u1 f u2 + b fu1 f u2. 2.25 19

3. 3.1. (, ). S R 3, f : D U S γ : [a, b] S S U, γ s, D γ 0 : [a, b] D γ = f γ 0 (cf. 2.4).., s γ 0 3.1. t(s) = γ (s), n(s) = N(γ(s)), n g (s) = n(s) t(s) ({t(s), n g (s), n(s)} T γ(s) R 3 ), t (s) n g (s) n(s), n g (s) κ g (s) γ γ(s) (geodesic curvature) : κ g = t (s), n g (s) = γ (s), n g (s).. {t(s), n g (s)} T γ(s) S, n g (s) T γ(s) S t(s) 90 3.2. γ (total geodesic curvature), b κ g ds = κ g (s) ds γ 3.3 ( ). γ 0 : [a, b] R 2 R 2, R 0, P, Q (R 0 γ 0 ) C, ( b ) (P u + Qv ) dt = (P (u(t), v(t))u (t) + Q(u(t), v(t))v (t)) dt = (Q u P v ) dudv γ 0 a R 0 ( ) 3.2. ( ). a 3.4. γ U, R, K da = 2π κ g ds R γ U 1 e 1 ( e 1 = f u / E). e 2 = N e 1, U p, {e 1 (p), e 2 (p)} T p S e 1 1, fu e 1 e 1, 1 1 2.12 (3), fu e 1 = P e 2, fv e 1 = Qe 2 0 = v e 1, e 1 = 2 D v e 1, e 1 = 2 v e 1, e 1. 20

P, Q, (P u + Qv ) ds = (u fu e 1, e 2 + v fv e 1, e 2 ) ds γ 0 γ 0 = u fu e 1 + v fv e 1, e 2 ds γ 0 = e 1, e 2 ds γ 0 = (κ g θ ) ds γ 0 = κ g ds 2π γ 0, e 1 = D γ e 1, γ = f u u + f v v, 2 2 ( ) κ g = e 1, e 2 + θ t = γ, t = cos θe 1 + sin θe 2 t = sin θθ e 1 + cos θe 1 + cos θθ e 2 + sin θe 2, t, e 2 = cos θ e 1, e 2 + cos θθ, t, e 2 = κ g n g, e 2 = κ g n t, e 2 = κ g cos θe 2 sin θe 1, e 2 = κ g cos θ, 3, fv ( fu e 1 ) = fv (P e 2 ) = P v e 2 + P fv e 2 = P v e 2 + P ( Qe 1 ) 4 u, v, fv ( fu e 1 ) fu ( fv e 1 ) = (P v Q u )e 2, ( 2.26) fu ( fv e 1 ) fv ( fu e 1 ) = I(e 1, Σ(f v ))Σ(f u ) I(e 1, Σ(f u ))Σ(f v ) 3 2 (a b) c = a, c b b, c a γ 0 θ ds = 2π γ 0 3 cos θ = 0 ẽ 1 = cos αe 1 + sin αe 2, cos(θ α)( ẽ 1, ẽ 2 + θ κ g ) = 0 ẽ 1, ẽ 2 = e 1, e 2 4 fv e 1 = Qe 2 fv e 2 = Qe 1, 3 0 = f v e 1, e 2 = fv e 1, e 2 + e 1, fv e 2 21

(Σ(f u ) Σ(f v )) e 1, Σ(f u ) Σ(f v ) = λn, λn e 1 = λe 2 λ, N = f u f v / f u f v = f u f v / g 11 g 22 g 12 2, 4, λ g 11 g 22 g 122 = Σ(f u ) Σ(f v ), f u f v a b, c d = a, c b, d a, d b, c 5 I(Σ(f u ), f u )I(Σ(f v ), f v ) I(Σ(f u ), f v )I(Σ(f v ), f u ) = h 11 h 22 h 12 2, λ = (h 11 h 22 h 12 2 )/ g 11 g 22 g 122 = K g 11 g 22 g 12 2 Q u P v = K g 11 g 22 g 12 2, (Q u P v ) dudv = K g 11 g 22 g 122 dudv = K da R 0 R 0 R 0 γ (γ = γ 1 γ 2 γ 3 ), θ, δ i, θ ds = 2π, γ 0 : 3 3 3 θ ds = θ ds = 2π δ i = α i π. γ 0 γ 0i, α i = π δ i, 3.5. U γ α i, 3 K da = α i π κ g ds 5 R γ = det(a, b, c d) = det(c d, a, b) = (c d) a, b = c, a d d, a c, b = = (a b) c, d = det(a b, c, d) = det(c, d, a b) = c d, a b = 22

. (1), 0 = 3 α i π, π (2) ( ), K = 1, κ g = 0, A(ABC) = A + B + C π κ g = 0 : γ(s) = (cos s, sin s, 0) ( ), t = γ = ( sin s, cos s, 0), t = ( cos s, sin s, 0) = n. 1. R 0 γ 0, R 0. γ 0 (t) = (cos t, sin t) (0 t 2π), P (u, v) = v, Q(u, v) = u, 2π P du + Q dv = (sin t (cos t) cos t (sin t) ) dt = 2π, γ 0 0 ( 2π ) (Q u P v ) dudv = ( 2) dudv = 2π. R 0 R 0 2. γ θ ds = 2π 3.3. ( ). 3.6. S, S 3.7. S, v, e, f,,, χ(s) = v e + f S. χ(s) S. (1) χ(s 2 ) = 2. ( ) (2) χ(t 2 ) = 0. (18 ) (3) Σ n (n 2) n, χ(σ n ) = 2 2n. ( 2,, ) 3.8. S, K da = 2πχ(S) S 23

S ( ) T (1),, T (f), f K da def = K da S T (j), 3.4 K da = T (j) 3 (α (j) ) i π κ g ds γ (j) j=1 S,, T (j), T (k) c (jk), γ (j) i, γ (k) i γ (j) i γ (k) i, κ g ds + κ g ds = 0 γ (j) i γ (k) i f, κ g ds = 0 γ (j) v, e, f,,, f 3 K da = (α (j) ) i πf = 2πv πf S, 2π j=1,,, 3f = 2e, f = 2f 2e K da = 2πv + π(2f 2e) = 2π(v e + f) = 2πχ(S) S 3.9. S, S,, n ( 2), j=1 24