2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

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1 I R c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t, t ) c : I R 2 κ(t) κ(t) := det(c (t), c (t))/ c (t) 3.

2 2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) = r > 0 (1) c(t) = (r cos t, r sin t) κ(t) = 1/r (2) a 0 c(t) = (r cos(at), r sin(at)) a > 0 κ(t) = 1/r a < 0 κ(t) = 1/r r > 0 y = r 2 x 2 c(t) = (t, r 2 t 2 ) t ( r, r) κ(t) x 2 /a 2 + y 2 /b 2 = 1 a > b > 0

3 c 1, c 2 : I R 2 (1) c 1 c 2 g SO(2) v R 2 t I c 2 (t) = gc 1 (t) + v (2) c 1 c 2 g O(2) v R 2 t I c 2 (t) = gc 1 (t) + v g SO(2) v R 2 g O(2) c 1, c 2 : I R 2 c 1 c 2. κ c1 = κ c

4 , f : R m R n C R m (x 1,..., x m ) f = (f 1,..., f n ) f p R m (Jf) p := ( ) fi (p) : R m R n. x j f : R m R n g : R n R l C p R m (J(g f)) p = (Jg) f(p) (Jf) p 1.5 I I R t : I I (i) (ii) C (iii) s I t (s) > 0 (iii) s I t (s) < c : I R 2 t : I I (1) c t : I R 2 (2) s I κ c (t(s)) = κ c t (s) 1 κ c κ c t c c t

5 c : I R 2 c t I c (t) = c : I R 2 t = t(s) c t c(t) = (x(t), y(t)) n(t) := ( y (t), x (t)) R c (t) = κ c (t)n(t) ( y (t) x (t) ) = ( ) ( x (t) y (t) c : I R 2 t I c = κ c c ).

6 c(t) n(t) = ( y (t), x (t)) n(t) c : I R 2 e(t) := c (t) {e(t), n(t)} Frenet n (t) = κ c (t)e(t) c : I R 2 t I Frenet c : I R 2 Frenet {e(t), n(t)} ( e n ) = ( e n ) ( 0 κ κ 0 Frenet C - κ : I R κ c : I R 2 ).

7 D R φ : D R 3 (i) φ C (ii) (u, v) D rank (Jφ) (u,v) = 2 (Jφ) (u,v) := (φ u, φ v ) (u,v) Jacobi φ u φ v rank (Jφ) (u,v) = 2 {φ u, φ v } (1) xy φ : R 2 R 3 : (u, v) (u, v, 0) (2) c : I R 2 : t (x(t), y(t)) φ : I R R 3 : (u, v) (x(u), y(u), v) (1) ( ) φ : R 2 R 3 : (u, v) (0, 0, 0) (2) ( ) φ : R 2 R 3 : (u, v) (x(u), y(u), z(u)) (3) φ : R 2 R 3 : (u, v) (u 3, u 2, v) (ii) (Jφ) (u,v)

8 c : I R 2 : t c(t) = (x(t), z(t)) t I x(t) > 0 φ : R I R 3 : (u, v) cos u sin u 0 sin u cos u x(v) 0 z(v) φ c c xz z (1) ( ) φ : R 2 R 3 : (u, v) (cos u, sin u, v). (2) ( ) φ : R ( π/2, π/2) R 3 : (u, v) (cos u cos v, sin u cos v, sin v). (3) ( ) φ : R 2 R 3 : (u, v) (cos u(2 + cos v), sin u(2 + cos v), sin v). R 2.

9 a = t (a 1, a 2, a 3 ) b = t (b 1, b 2, b 3 ) R 3 ( ) ( ) ( )) a b := (det t a2 b 2 a3 b, det 3 a1 b, det 1. a 3 b 3 a 1 b 1 a 2 b 2 a b a b a b = 0 a b φ : D R 3 φ n(u, v) := φ u (u, v) φ v (u, v)/ φ u (u, v) φ v (u, v). φ u φ v φ : D R 3 (1) E := φ u, φ u F := φ u, φ v G := φ v, φ v (2) L := φ uu, n M := φ uv, n N := φ vv, n (3) ( E F Î := F G (4) A := Î 1 ÎI ) ( L M, ÎI p := M N (5) K := det(a) H := (1/2)tr(A) A Î ).

10 K H (1) K = H = 0 (2) r > 0 φ(u, v) = (r cos u, r sin u, v) A = ( r ) 1 ( r ), K = 0, H = 1/(2r). (3) r > 0 φ(u, v) = (r cos u cos v, r sin u cos v, r sin v) K = /r 2 H = 1/r r > c z φ c φ c : I R 2 : t (x(t), y(t)) κ φ : I R R 3 : (u, v) (x(u), y(u), v) φ K = 0 H = κ/2 Frenet ( ) ( ) x y = κ. y x

11 E F G φ : D R 3, R p D I p : R 2 R 2 R I p (X, Y ) := (Jφ) p X, (Jφ) p Y. I p (Jφ) p p D I p R 2. R 3 X, Y = t XY p D X, Y R 2 I p (X, Y ) = t X Î 1 p Y. Îp I p p D Îp. I p A p D (1) A p I p X, Y R 2 I p (AX, Y ) = I p (X, AY ) (2) A p A φ λ 1 λ 2

12 φ 1, φ 2 : D R 3 (1) φ 1 φ 2 g SO(3) w R 3 p D φ 2 (p) = gφ 1 (p) + w. (2) φ 1 φ 2 g O(3) w R 3 p D φ 2 (p) = gφ 1 (p) + w φ 1, φ 2 : D R 3 E i F i φ 1 φ 2 φ 2 = gφ + w g O(3) w R 3 (1) E 2 = E 1 F 2 = F 1 G 2 = G 1 (2) n 2 = (det g)gn 1 (3) L 2 = (det g)l 1 M 2 = (det g)m 1 N 2 = (det g)n 1 (4) A 2 = (det g)a 1 (5) K 2 = K 1 H 2 = (det g)h 1

13 D D R ξ : D D (i) ξ (ii) ξ C (iii) q D det(jξ) q > 0 (iii) φ : D R 3 ξ : D D φ ξ : D R φ : D R 3 (1) Î = t (Jφ)(Jφ) (2) L = φ u, n u M = φ u, n v = φ v, n u N = φ v, n v (3) ÎI = t (Jφ)(Jn) φ : D R 3 φ := φ ξ : D R 3 A A (1) Î = t (Jξ) Î (Jξ) (2) n = n ξ (3) ÎI = t (Jξ) ÎI (Jξ) (4) A = (Jξ) 1 A (Jξ) (5) K = K H = H

14 φ : D R 3 A n = n(u, v) D R m F : D R n C F p D (df ) p : R m R n 1 : X lim (F (p + tx) F (p)). t 0 t (df ) p (JF ) p φ : D R 3 n : D R 3 (dn) (u,v) n, n = X R 2 (dn) (u,v) (X), n (u,v) = 0 (dn) (u,v) : R 2 span{φ u (u, v), φ v (u, v)} A (dn) (u,v) (e 1, e 2 ) = (n u, n v ) = (φ u, φ v )A A n

15 φ : D R p D II p : R 2 R 2 R II p (X, Y ) := I p (A p X, Y ). II p A p I p A p I p II p p D X, Y R 2 II p (X, Y ) = t XÎI py. ÎI p L M N A

16 (V 1,, 1 ) (V 2,, 2 ) f : (V 1,, 1 ) (V 2,, 2 ) X, Y V 1 X, Y 1 = f(x), f(y ) 2 φ : D R 3 p D I p φ 1 : D 1 R 3 φ 2 : D 2 R 3 (1) ξ : D 1 D 2 ξ C ξ 1 C (2) ξ : D 1 D 2 ξ p D 1 (dξ) p : (R 2, (I 1 ) p ) (R 2, (I 2 ) ξ(p) φ 1 (u, v) := (u, v, 0) φ 2 (u, v) := (r cos u, r sin u, v) r > 0 ξ φ 1 φ 2 ξ : R 2 R 2 : (x 1, x 2 ) ((1/r)x 1, x 2 ).

17 D R 2 p = (u, v) D u v X : D R 3 C u X : D R 3 : p (X u )(p), v X : D R 3 : p (X v )(p). φ : D R 3 u φ v φ φ : D R 3 X : D R 3 X φ (i) X : D R 3 C (ii) p D X(p) Span{φ u (p), φ v (p)} φ uu φ uv φ vv φ : D R 3 X : D R 3 C X : D R 3 X X := X X, n n n φ p D Span{φ u (p), φ v (p)} n(p) φ : D R 3 X : D R 3 C X : D R 3 φ φ uu φ : D R 3 φ uu φ uv φ vv E F G φ uu = Γ 1 11φ u + Γ 2 11φ v φ u φ v

18 u v u v φ : D R 3 X : D R 3 φ u X v X : D R 3 φ u X := ( u X), v X := ( v X) φ : D R 3 X Y : D R 3 φ α β : D R C (1) u (X + Y ) = u X + u Y v (X + Y ) = v X + v Y (2) u (αx) = α u X + α u X v (αx) = α v X + α v X φ : D R 3 X : D R 3 φ u v E F G 2.13 K LN M φ : D R 3 X Y : D R 3 φ u v X, Y = X vu, Y X v, n n u, Y φ : D R 3 K E F G LM N 2 = v u φ u u v φ u, φ v.

19 19 3 M C 3.1 M p M T p M X F (X) := {ξ : X R} F (X) ξ, η F (X) a R x X (ξ + η)(x) := ξ(x) + η(x), (aξ)(x) := aξ(x), (ξη)(x) := ξ(x)η(x). (3.1.1) F (X)

20 20 3 M C C F (M) C (M) := {ξ : M R : C }. (3.1.2) C (M) p M v : C (M) R M p (i) v : C (M) R (ii) ξ, η C (M) v(ξη) = v(ξ)η(p) + ξ(p)v(η) p M M p T p M := {v : C (M) R : p }. (3.1.3) T p M F (C (M)) T p M C (M) ε > 0 c : ( ε, ε) M C c (0) c 0 c (0) : C (M) R : ξ d dt (ξ c)(0). (3.1.4) C c : ( ε, ε) M M c (0) c (0) T p M c : ( ε, ε) M C p := c(0) (U, φ) M p U M m φ = (x 1,..., x m ) x i : U R

21 ( ) p ( ) p : C (M) R : ξ (ξ φ 1 )(φ(p)). (3.1.5) i ( ) p F (C (M)) ( ) p (1) a 1,..., a m R a i ( ) p (2) {( x 1 ) p,..., ( x m ) p } (1) ( ) p p span{( x 1 ) p,..., ( x m ) p } T p M T p M M

22 X : D R 3 p D X p M C (M) X : C (M) C (M) (i) (ii) X ξ, η C (M) X(ξη) = (Xξ)η + ξ(xη) M X(M) X(M) C (M) X, Y X(M) f C (M) X + Y, fx X(M) ξ C (M) (X + Y )ξ := Xξ + Y ξ, (fx)ξ := f(xξ). (3.2.1) X X(M) p M X p p M X p : C (M) R : ξ (Xξ)(p). (3.2.2) X X : M T M(:= T p M) : p X p. (3.2.3) p M p M X p T p M

23 (U, φ) M φ = (x 1,..., x m ) f 1,..., f m C (U) f i fi : U T U : q f i (q)( ) q. (3.2.4) X X(M) X : M T M X U : U T U f 1,..., f m C (U) X U = f i. (3.2.5) X f i

24 V n V V := {f : V R : } V F (V ) V F (V ) V {e 1,..., e n } V {ω 1,..., ω n } V ω i V, ω i (e j ) = δ ij S 2 (V ) := {Ω : V V R :, } S 2 (V ) F (V V ) ω 1, ω 2 V ω 1 ω 2 S 2 (V ) ω 1, ω 2 ω 1 ω 2 : V V R : (X, Y ) (1/2)(ω 1 (X)ω 2 (Y ) + ω 2 (X)ω 1 (Y )) {ω 1,..., ω n } V {ω i ω j 1 i j n} S 2 (V ), V, S 2 (V )

25 T p M Tp M Tp M F : M N C p M (df ) p : T p M T F (p) N : v (df ) p v F p (df ) p v : C (N) R : ξ v(ξ F ). T p M c (0) T p M F : M N C p M c (0) T p M (df ) p (c (0)) = (F c) (0). (df ) p (c (0)) c F : M N C - p M p M (U, φ = (x 1,..., x m )) F (p) N (V, ψ = (y 1,..., y n )) (df ) p ( a i ( ) p ) = ( b j ( y j ) F (p) ). t (b 1,..., b n ) = J(ψ F φ 1 ) φ(p) t(a 1,..., a m ) p M (U, φ = (x 1,..., x m )) i x i C (U) x i (dx i ) p T p M p M (U, φ = (x 1,..., x m )) x i {((dx 1 ) p,..., (dx m ) p } {( x 1 ) p,..., ( x m ) p }

26 M (U, φ = (x 1,..., x m )) T M T M := Tp M. p M f i C (U) ω U 1 ω := m i=1 f idx i : U T U : p m i=1 f i(p)(dx i ) p. X p M X p T p M 1 ω p M ω p T p M ω : M T M M 1 (i) (ii) p M ω p T p M (U, φ) ω U U 1 1 ω 1 ω 2 ω 1 ω 2 ω 1 ω 2 : M S 2 (T M) := S 2 (Tp M). p M

27 M, g p M g p : T p M T p M R g : M S 2 (T M) M g U C g ij C (U) s.t. g U = g ij dx i dx j. C C C M := R n g = dx dx 2 n g g p ( a i ( ) p, b i ( ) p ) := a i b i. M := {(x, y) R 2 y > 0} g = (1/y 2 )(dx 2 + dy 2 ) M (M, g).

28 (M, g) (M, g) M X : C (M) C (M) X(M) X, Y X(M) [X, Y ] X Y [X, Y ] : C (M) C (M) : f X(Y f) Y (Xf) (1) [, ] : X(M) X(M) X(M) (2) [X, Y ] = [Y, X] (3) [X, [Y, Z] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 (4) [X, fy ] = (Xf)Y + f[x, Y ] f C (M) (U, φ = (x 1,..., x m )) X = ξ i X = ξ i Y = η j [X, Y ] = i,j x j ( ξ i η j [, x j ] = 0 X(U) x j ) ξ η i j x j.

29 Levi-Civita (M, g) : X(M) X(M) X(M) : (X, Y ) X Y Levi-Civita 2g( X Y, Z) = Xg(Y, Z) + Y g(z, X) Zg(X, Y ) + g([x, Y ], Z) + g([z, X], Y ) + g(x, [Z, Y ]). Koszul Xg(Y, Z) X g(y, Z) C (M) g(y, Z) : M R : p g p (Y p, Z p ) : X(M) X(M) X(M) Levi-Civita (i) X Y Y X = [X, Y ] (ii) Xg(Y, Z) = g( X Y, Z) + g(y, X Z) (i) (ii) Levi-Civita Levi-Civita : X(M) X(M) X(M) (1) (2) fx Y = f X Y (3) X (fy ) = (Xf)Y + f X Y (U, φ = (x 1,..., x m )) x j Levi-Civita φ : D R 3 D g u X v X X X (D, g) u v Levi-Civita

30 (M, g) R (M, g) R(X, Y )Z := [X,Y ] Z X Y Z + Y X Z. R : X(M) X(M) X(M) X(M) : (X, Y, Z) R(X, Y )Z [ X, Y ] := X Y Y X R(X, Y ) := [X,Y ] [ X, Y ] X, Y, Z, W X(M) (1) R (2) R(X, Y ) = R(Y, X) (3) g(r(x, Y )Z, W ) = g(r(x, Y )W, Z) (4) fr(x, Y )Z = R(fX, Y )Z = R(X, fy )Z = R(X, Y )(fz) f C (M) (1) (2). (3), (ii). (4), R (R(X, Y )Z) p X p Y p Z p X p = X p Y p = Y p Z p = Z p (R(X, Y )Z) p = (R(X, Y )Z ) p g [, ] Levi-Civita R n R 0

31 σ T p M 2 X, Y X(M) {X p, Y p } σ σ K σ = g(r(x, Y )X, Y ) p σ X, Y X(M) R K σ σ {X p, Y p } {X p, Y p} σ T p M g(r(x, Y )X, Y ) p = g(r(x, Y )X, Y ) p RH 2 1 X := y x Y := y y [X, Y ] = X X X = Y Y X = Y Y = 0

1 1.1 [ ]., D R m, f : D R n C -. f p D (df) p : (df) p : R m R n f(p + vt) f(p) : v lim. t 0 t, (df) p., R m {x 1,..., x m }, (df) p (x i ) =

1 1.1 [ ]., D R m, f : D R n C -. f p D (df) p : (df) p : R m R n f(p + vt) f(p) : v lim. t 0 t, (df) p., R m {x 1,..., x m }, (df) p (x i ) = 2004 / D : 0,.,., :,.,.,,.,,,.,.,,.. :,,,,,,,., web page.,,. C-613 e-mail tamaru math.sci.hiroshima-u.ac.jp url http://www.math.sci.hiroshima-u.ac.jp/ tamaru/index-j.html 2004 D - 1 - 1 1.1 [ ].,. 1.1.1

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