83 ( Intrinsic ( (1 V v i V {e 1,, e n } V v V v = v 1 e 1 + + v n e n = v i e i V V V V w i V {f 1,, f n } V w 1
V w = w 1 f 1 + + w n f n = w i f i V V V {e 1,, e n } V {e 1,, e n } e 1 (e 1 e n e n = E E n A = (a i j, B = (bi j ( A i j a i j w i = a i j ej, v i = b j i e j V {w 1,, w n }, V {v 1,, v n } w 1 (v 1 v n w n 2
e 1 = A = AB e n (e 1 e n B {w 1,, w n } {v 1,, v n } AB = E B A E ( n, E m ( n m ( En A En B En A + B = 0 E m 0 E m 0 E m ( En A 0 E m ( En A 0 E m ω X, Y dω(x, Y = X(ω(Y Y (ω(x ω([x, Y ] Intrinsic 3
n R n U r (Pfaff ω p = 0 (1 p r U Pfaff ω p = a p i dxi a p i C (U Pfaff U R n ω p = 0 (1 p r U r Pfaff R m N U f : N( R m U f ω p 0, (1 p r m f(n Pfaff ω p = 0 (1 p r U P P n r Pfaff ω p = 0 (1 p r 4
R n r Pfaff i, j 1 i, j n p, q 1 p, q r α, β r + 1 α, β n Σ R n U r Pfaff U r Pfaff ω p (p = 1,, r rank( r (U, x i ω p = a p i dxi ω 1 a 1 1 a 1 n = ω r dx 1 a1 r a r n dx n (a p i rank r ωr+1,, ω n {ω 1,, ω n } (1 U Pfaff (a i j (bi j (1 X 1, X n X i = b j i x j 5
E = ω 1 ω r (X 1 X n a 1 1 a 1 n dx 1 = ( x 1 a n 1 a n n dx n a 1 1 a 1 n b 1 1 b 1 n = x 1 b 1 1 b 1 n b n 1 b n n a n 1 a n n b n 1 b n n U r Pfaff ω 1,, ω r U P D n r (P = {X T P (R n ω 1 (X = 0,, ω r (X = 0} {ω 1,, ω n } {X 1,, X n } D n r = Span{X r+1,, X n } dim D n r (P = n r U r Pfaff U n r D n r U r Pfaff ω 1,, ω r Pfaff n r D n r 6
ω p ω r+1,, ω n {X 1,, X n } D n r R n (N, f Pfaff ω p = 0 (1 p r N X 0 = (f ω p (X = ω p (f (X (N, f D n r ω p = 0 (1 p r D n r U R n U r Pfaff ω 1,, ω r U P n r D n r (P = {X T P (R n : ω 1 (X = 0, ω r (X = 0} Pfaff ω p = 0 (1 p r D n r 7
Pfaff ω p = 0 (1 p r Pfaff r Pfaff ω p (p = 1, r ω p = ω p i dxi ω p i Rn ω p i = ωp i (x1,, x n (ω p i rank r xi r (ω p q (a p q {θ p } θ p = a p qω q = dx p + b p αdx α b p α = a p qω q α θ 1 a 1 1 a 1 r ω 1 = θ r a r 1 a r r ω r 8
a 1 1 a 1 r ω1 1 ωr 1 ωr+1 1 ω 1 n = a r 1 a r r ω1 r ωr r ωr+1 r ωn r 1 0 0 b 1 r+1 b 1 n = 0 0 1 b r r+1 b r n dx r+1,, dx n {θ 1,, θ r, dx r+1,, dx n } dx 1 dx r dx r+1 dx n dx 1 dx r dx r+1 dx n R n 1 0 0 b 1 r+1 b 1 n dx 1 0 0 1 b r r+1 b r n dx r 0 0 0 1 0 0 dx r+1 0 0 1 dx n {θ 1,, θ r, dx r+1,, dx n } X 1, X n 9
θ 1 (X 1 X n = E θ n ( En A 1 = ( En A 0 E m 0 E m 1 0 0 b 1 r+1 b 1 n ( x 1 0 0 1 b r r+1 b r n x n 0 0 0 1 0 0 0 0 1 r Pfaff ω p (p = 1, r 1 r Pfaff n-r D n r r+1 n R 5 2 x = x(u, v y = y(u, v z = z(u, v x x y y x x = a, = b, = c, = d, = e, u v u v u v = f Pfaff dx = x x du + dv = adu + bdv u v 10
dy = y y du + dv = cdu + ddv u v dz = z z du + dv = edu + fdv u v θ 1 = dx adu vdv = 0 θ 2 = dy cdu ddv = 0 θ 3 = dz edu fdv = 0 θ 4 = du θ 5 = dv θ 1, θ 2, θ 3, θ 4, θ 5 1 0 0 a b dx 0 1 0 c d dy 0 0 1 e f dz 0 0 0 1 0 du 0 0 0 0 1 dv X 1, X 2 X 1 = u + x u x + y u y + z u z = u + a x + c y + e z X 2 = v + b x + d y + f z X 1, X 2 a b ( x y z u v c d e f 1 0 0 1 11
Pfaff Pfaff U R n U Pfaff ω 1 = 0,, ω r = 0 p(1 p r dω p = θq p ω q r 2 θq p ( 29 11 U R n U Pfaff ω 1 = 0,, ω r = 0 (x 1,, x n ω p = ω p i dxi (1 p r pfaff η p r r (a p q η p = a p qω q {ω p } {η p } 12
( (a p q (b p q ω p = b p qη q dω p = θ p q ω q dη p dη p = d(a p qω q = da p q ω q + a p qdω q = da p q ω q + a p qθ q r ω r = (da p q + a p rθ r q ω q = (da p q + a p rθ r q b q sη s = (da p s + a p rθ r s b s qη q θ p q = (da p s + a p rθ r sb s q dη p = θ p q η q {η p } {ω p } ω p a p qω q = dx p + b p αdx α {ω 1,, ω r } n r Pfaff {ω 1,, ω n }, {X 1,, X n } X r+1, X n X r+1, X n 13
{ω i ω j 1 i < j n} 2 dω p = c p ij ωi ω j i<j {ω i }, {X i } ω p (X α = 0 (1 p r, r + 1 α n c p αβ = dωp (X α, X β = X α (ω p (X β X β (ω p (X α ω p ([X α, X β ] = ω p ([X α, X β ] ω 1 = 0,, ω r = 0, dω p = θq p ω q (ω α ω β c p αβ = 0 ωp ([X α, X β ] = 0 (1 q r [X α, X β ] {X r+1,, X n } {X r+1,, X n } {X r+1,, X n } c p αβ = 0 dω p ω α ω β Pfaff Pfaff Intrinsic [X, Y ] = XY Y X 14
R n (R n (1d R (2 f df d(df = 0 (3ω k (R n, η (R n d(ω η = dω η + ( 1 k ω dη d : (R n (R n Pfaff d 1 R 3 ω 1 = P dx + Qdy + Rdz ω 2, ω 3 {ω 1, ω 2, ω 3 } {X 1, X 2, X 3 } dω 1 = aω 1 ω 2 + bω 1 ω 3 + cω 2 ω 3 ω 1 c = 0 15
dω 1 ω 1 = cω 2 ω 3 ω 1 = cω 1 ω 2 ω 3 ω 1 dω 1 ω 1 = 0 ω = P dx + Qdy + Rdz P x = P x dω ω = (P y dy dz + P z dz dx + Q x dx dy + Q z dz dx + R x dx dz + R y dy dz (P dx + Qdy + Rdz = (P (R y Q z + Q(P z R x + R(Q x P y dx dy dz = 0 P (R y Q z + Q(P z R x + R(Q x P y = 0 2 16
X 1 = x 1 + (x3 + 3(x 1 2 x 4 X 2 = x 2 + x2 x 4 X 3 = x 3 + x1 x 4 x 4 (0, 0, 0 = 0 ( X 1 X 2 X 3 x 4 = ( x 1 x 2 x 3 x 4 1 0 0 0 0 1 0 0 0 0 1 0 x 3 + 3(x 1 2 x 2 x 1 1 1 0 0 0 dx 1 0 1 0 0 dx 2 0 0 1 0 dx x 3 3(x 1 2 x 2 x 1 1 dx 4 {X 1, X 2, X 3 } Pfaff (x 3 + 3(x 1 2 dx 1 x 2 dx 2 x 1 dx 3 + dx 4 = 0 dx 4 = (x 3 + 3(x 1 2 dx 1 + x 2 dx 2 + x 1 dx 3 ( 17