R C Gunning, Lectures on Riemann Surfaces, Princeton Math Notes, Princeton Univ Press 1966,, (4),,, Gunning, Schwarz Schwarz Schwarz, {z; x}, [z; x] =

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1 Schwarz 1, x z = z(x) {z; x} {z; x} = z z 1 2 z z, = d/dx (1) a 0, b {az; x} = {z; x}, {z + b; x} = {z; x} {1/z; x} = {z; x} (2) ad bc 0 a, b, c, d 2 { az + b cz + d ; x } = {z; x} (3) z(x) = (ax + b)/(cx + d) (a, b, c, d ad bc 0 {z; x} = 0, {x; x} = 0 (2) (4) {z; x} = 0, z(x) = (ax + b)/(cx + d) (a, b, c, d ad bc 0 ), {z; x} Schwarz, z x Schwarz Schwarz Hermann Amadeus Schwarz ( ), Schwarz, Schwarz,, Schwarz, schwarz,,, Schwarz,,,, 4 1

2 R C Gunning, Lectures on Riemann Surfaces, Princeton Math Notes, Princeton Univ Press 1966,, (4),,, Gunning, Schwarz Schwarz Schwarz, {z; x}, [z; x] = z z, x ax + b a, b C, a 0 Aff(1, C); x ax + b cx + d a, b, c, d C, ad bc 0 PGL(2, C) C, ( ),,, {z; x} = 0 z(x) PGL(2, C) [z; x] = 0 z(x) Aff(1, C) Schwarz, z x, x y [z; y] = [z; x] dx + [x; y] dy ( dx )2 {z; y} = {z; x} + {x; y} dy 2

3 2 Schwarz Schwarz, Schwarz Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrisce Reihe eine algebraische Funktion ihres vierten Elementes darstellt, Journal für reine und angewandte Mathematik 75(1872), d 2 z γ (α + β + 1)x + dx2 x(1 x) dz dx αβ x(1 x) z = 0 Gauss (α, β, γ), Schwarz,, d 2 z dx + pdz 2 dx + qz = 0, z 1 z 2, Wronski z 2 dz 1 /dx z 1 dz 2 /dx exp( pdx) Gauss x γ (1 x) γ α β 1,, γ, α + β, z 1 z 2, z 2 /z 1,, a, b, c, d s = (az 1 + bz 2 )/(cz 1 + dz 2 ), s {s; x} = 2q 1 2 p2 dp dx 2q Schwarz, z 2 = sz 1 z 2 = sz 1+s z 1 z 2 = sz 1 +2s z 1+s z 1 z 2 z 2 + pz 2 + qz 2 = 0, z 1 z 1 + pz 1 + qz 1 = 0, 2s z 1 + (s + ps )z 1 = 0,, p + s = 2 z 1 s z 1 3

4 , p + s s = 2 z 1z 1 (z 1) 2 = 2q + 2 z1 2 p z 1 z 1 + = 2 z 1( pz 1 qz 1 ) (z 1) 2 2 z 1 z 1 z 1/z 1 p s, Gauss (1) {s; x} = 1 λ2 + 1 µ2 2x 2 2(1 x) ν2 λ 2 µ 2 2 2x(1 x),, λ = 1 γ, µ = γ α β, ν = α β, {s; x} x s(x) = z 2 (x)/z 1 (x), x = 0, s = x a u(x), u(x) x = 0 u(0) 0, Schwarz {s; x} = a2 1 4x [x = 0 x ],, (1) Gauss x = 0, 1, Schwarz,,,,, (1), Schwarz z

5 , p = 0, {s; x} = 2q, x q, s {s; x} = 2q(x), z + qz = 0 z 1, z 2, s = z 2 /z 1 s x ( ) ( ) P 1 (, C { } ) x s = [z 1, z 2 ] P 1, {s; x} P 1, s 1 s 2,, ( ) {s 1 ; x} = {s 2 ; x},,, n, m P m ( (Grassmann) ), Schwarz, ( ), n, m (1) n = 1, m 2 ( ), (2) n = m ( ), (3) n = m 1 ( ) ( n = k 2 (k, 2k) ) 5

6 4 1 P n,, P n PGL(n + 1),, P n Euclid ( ) Frenet-Serret ( ),,, Halphen, Laguerre, Forsyth, Y Se-ashi, A geometric construction of Laguerre-Forsyth s canonical forms of linear ordinary differential equations, Advanced Studies in Pure Mathematics 22(1993), , n = 2 P 2 z(x) = [z 1 (x), z 2 (x), z 3 (x)] z z z z z z 1 z 1 z 1 z 1 z 2 z 2 z 2 z 2 z 3 z 3 z 3 z 3, z 1, z 2, z 3 = 0 z + p 1 z + p 2 z + p 3 z = 0,,,,, ρ, ρz,, 6

7 ,, ρ, p 1 = 0, z + P 2 z + P 3 z = 0, f (2) {f; x} = 1 4 P 2, x y = f(x), z w = f z, w y d 3 w dy 3 + Rw = 0 R = (P 3 P 2/2)/(f ) 3,,, (2),,,,,, R R, Rdy 3,, Laguerre-Forsyth, R, p 1, p 2, p 3, z, Schwarz 5 Schwarz x = (x 1,, x n ) z(x) = [z 0 (x),, z n (x)] P n P n PGL(n + 1),, 7

8 ( ),,, J Liouville, P Pepin, L Fuchs, F Klein, F Brioschi, C Jordan, E Goursat, 1870, Schwarz P Painlevé, Compte Rendus E Goursat Compte Rendus A Boulanger, Contribution à l étude des équations différentielles linéaires homogènes intégrable algébriquement, Journal de L École Polytechnique, 4(1898), (x, y) (x, y) (u, v), (u, v) (U, V ) = au + bv + c a u + b v + c a u + b v + c, a u + b v + c, u, v, U, V (x, y), a, b, c, a, I(u, v) = u xxv x v xx u x, J(u, v) = v yyu y u yy v y, u x v y v x u y u x v y v x u y M(u, v) = u xxv y v xx u y + 2(u xy v x v xy u x ), 3(u x v y v x u y ) N(u, v) = v yyu x u yy v x + 2(v xy u x u xy v x ) 3(u x v y v x u y ) 8

9 , I(u, v) = I(U, V ), J(u, v) = J(U, V ), M(u, v) = M(U, V ), N(u, v) = N(U, V ),, Schwarz,,, (x, y) (u, v), z 0, z 1, z 2 z 0 = ρ = (u x v y u y v x ) 1/3, z 1 = uρ, z 2 = uρ, (x, y) P 2 [1, u, v] = [z 0, z 1, z 2 ], z 0, z 1, z 2 z, z xx z x z y z z 0 xx z 0 x z 0 y z 0 z 1 xx z 1 x z 1 y z 1 z 2 xx z 2 x z 2 y z 2 = 0, z yy z x z y z z 0 yy z 0 x z 0 y z 0 z 1 yy z 1 x z 1 y z 1 z 2 yy z 2 x z 2 y z 2 z xy z x z y z z 0 xy z 0 x z 0 y z 0 z 1 xy z 1 x z 1 y z 1 z 2 xy z 2 x z 2 y z 2 = 0 = 0, z xx, z xy, z yy z 0 x z 0 y z 0 z 1 x z 1 y z 1 z 2 x z 2 y z 2 = ρ x ρ y ρ u x ρ u y ρ 0 v x ρ v y ρ 0 = (ρ) 3 (u x v y u y v x ) = 1, z xx z xy z yy = Mz x Iz y + Az, = Nz x Mz y + Bz, = Jz x + Nz y + Cz 9

10 , A = 2(M 2 +IN) M x +I y, B = IJ MN+M y +N x, C = 2(N 2 +JM) N y +J x (x, y) P 2 (x, y) z = [z 0, z 1, z 2 ] P 2, z xx = p 1 z x + q 1 z y + r 1 z, z xy = p 2 z x + q 2 z y + r 2 z, z yy = p 3 z x + q 3 z y + r 3 z z ρz ρ, p i, q i, r i, ρ, p 1 + q 2 = 0, q 3 + p 2 = 0,,,, n, (x 1,, x n ) (z 1,, z n ), j(z, x) = (ji k ) ; ji k = z k / x i Jacobi, Ji k (z, x) = x k / z i,, det j(z, x) 0 σ(z, x) = 1 log det j(z, x), n + 1 σ γ k ij(z, x) = l 2 z l x i x j J k l (z, x) i(z, x) = σ x i,, z(x) Schwarz S k ij(z; x) = γ k ij(z, x) δ k i σ j (z, x) δ k j σ i (z, x) 10

11 n = 1, n 2 n = 2, Schwarz, S k ij(z; x) = S k ji(z; x), k S k ik = 0 Schwarz A PGL(n + 1), Az z S k ij(az; x) = S k ij(z; x) S k ij(z; x) = 0 z = Ax, x y S k ij(z; y) p,q,r S r pq(z; x)j p i (x; y)j q j (x; y)j k r (x, y) = S k ij(x; y) : z 1 z 2 S k ij(z 1 ; x) = S k ij(z 2 ; x) i, j, k ρ = (det j(z, x)) 1/(n+1), z 0 = ρ, ρz 1,, ρz n (3) z ij = Sijz k k + 1 S l k n 1 ikslj k l,k k x k Sk ij 6 Schwarz,, moduli M moduli, 11

12 DAllcock, J Carlson, and D Toledo, A complex hyperbolic structure for moduli of cubic surfaces, Compte Rendus Acad Sci 326(1998), 49 54, P 3 moduli, B 4 = {[z 0,, z 4 ] P 4 z z 4 2 < z 0 2 } (M B 4 M f : M x z B 4 M K3 Gauss 3,, 2, 3, 4,,, P 2 blowup, P 2,, 3 6, [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1], x 1 x x 3 x 4 D(x) := x 1 x 2 x 3 x 4 (x 1 1)(x 2 1)(x 3 1)(x 4 1)(x 1 x 4 x 2 x 3 ) 12

13 (x 1 x 2 )(x 1 x 3 )(x 2 x 4 )(x 3 x 4 ) {(x 1 1)(x 4 1) (x 2 1)(x 3 1)} {x 1 (x 2 1)(x 3 1)x 4 (x 1 1)x 2 x 3 (x 4 1)}, moduli M {x = (x 1,, x 4 ) C 4 D(x) 0} f : M P 4 x = (x 1, x 2, x 3, x 4 ) Schwarz Sij k, z ij = k S k ijz k + S ij z S ij (3), Sij k, Schwarz PGL(4), Sij k x, D(x) = 0,, Sij k,, M,, M, M E 6 Weyl,,, Sij k, : n 2 z = (z 1,, z n ) x 1 = 0, z 1 (x) = (x 1 ) α v 1, z 2 (x) = v 2,, z n (x) = v n, z x = (x1 ) α 1 u,, v j (1 j n) u x 1, S k ij{z; x}, 1 α 1 n+1 S1j{z; k x} + δj k S1j{z; 1 x}, x 1 S11{z; k x}, S11{z; 1 x} n 1 n+1 2 i, j, k n, x 1, (x 1 ) 1 S 1 ij{z; x},, Sij k x 13 α 1 x 1

14 , 9 Appell-Lauricella, TTerasoma and K Matsumoto: Theta constants associated to cubic threefolds, mathag/ T Sasaki and M Yoshida, The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces II, mathag/ , 1, 7 Schwarz,, Teichmüller, Nehari x < 1 z {z; x} 6(1 z 2 ) 2, {z; x} 2(1 z 2 ) 2, z, g C 3g 3 Bers Schwarz, O Kobayashi and M Wada, Circular geometry and the Schwarzian, to appear in Far East Jounral of Mathematical Sciences, Riemann (M, g M ), (N, g N ) Schwarz, H Sato, Schwarzian derivatives of contact diffeomorphisms, Lobachevskii Jounral of Math 4(1999), y = f(x, y, y ) (x, y) y = 0, I, J, M, N, y = f(x, y, y, y ) (x, y, y ) 14

15 y = 0, Schwarz 1 2 / / 15

変位変位とは物体中のある点が変形後に別の点に異動したときの位置の変化でありベクトル量である変位には物体の変形の他に剛体運動剛体変位が含まれている剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,

変位変位とは物体中のある点が変形後に別の点に異動したときの位置の変化でありベクトル量である変位には物体の変形の他に剛体運動剛体変位が含まれている剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, 変位変位とは物体中のある点が変形後に別の点に異動したときの位置の変化でありベクトル量である変位には物体の変形の他に剛体運動剛体変位が含まれている剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +