a145a143a89 a77a193a88(a24)
2001a3511a2192a134
1 a77a193a123a41a37
a152a21a183a162a89a24a154a77a193a88,a135a232a73a243a77a193a222a97a126a123a207a196a243a114a199a106a166a176a62a52a105
a7a155a123,a89a52a79a143a243a77a193a88a165,a77a193a123a89a74a117a159a128a128a106a223a116a217a198a205a18a123a60
a111,a13a89a74a205a18a123a60a111a35a154a116a77a193a123a30a127a15a139a105a245a117a159,a243a77a193a123a60a111a46a178
a21a9a234,a0a143,a77a193a139a199a128a154,a210a52a183a162a34a189a199a52a2433a145a56a45a176a62,a196a49a28a23
a122a68,a28a44a49a152a67,a44a49a243a23a122a68a199a222a62a123a77a34a176a123a117a159a249a217a198a31a143a123,a24
a116a176a145a49a128,a210a23a88a68a234,a183a123a89a27a203a86a88a24a248,a196a49a183a46a76a27a178a89a1a16a97
a238a133a92a24a5,a89a199a97a238a76a150a97a148a243a202a47a135a73a1a91a86a222a73a88a116a123,a199a139a199a128
a154,a210a52a79a122a105a13a37a48a27a117a122a59a92,a0a143a31a143a52a77a193a17? a77a193a210a52a67a222a24
a199a59a77a123a192a220,a183a178a199a123a154a123a43a41a44a143a220a199a35a106a123a60a106,a89a220a199a35a106a183a119
a41u,v,u,va24a196a119a41a107a106(parameter),a34a38u,va35a154a123a30a127,a89a74a154a123a27a236
a210a196a234a24a1992a145a123a77a193x(u,v) = (x1(u,v),x2(u,v),x3(u,v)),a149a52a79a143a139u,v,
a196a49a28a44a49a38a122a24a199a107a106a123a138a52a158a106,a108a128a38a122a12a24a199a107a106a35a154,a247v a52
a158a106,a13u a35a154,a196a49a210a139a24a30a77a34,a199a123a107a106a52u,a51a248,a28a44a49a38a122ua88
a35,a13va35a154,a79a36a243a77a193a176a62,a139a220a28a77a34,a199a123a107a106a24a28a52v a127a149a158a106,a24
a28ua52a127a149a158a106,a233a149a89a220a28a77a34,a154a24a30a77a34a243a154a24a154a209a139a24a30a35a34,a196a49
a243a24a199a154x,a183a162a210a139a220a30a134a34,a183a34a189a89a220a30a134a34a88a173a92,a166a233a155a128,a137a219
a143a89,xu,xv a88a52a51a24a199a48a53,a88a178a134(a34a117)a35a7,a217a165xu,xva210a52xa123a37a222a73
a47a233u,va70a160a135a73,a221a86a183a128,a199a123a37a222a232xu ×xv negationslash= 0,a34a38a89a220a199a48a53a88a173
a92,a196a49a199a162a210a60a196a24a199a178a193,a89a199a178a193a183a119a41a77a193a243a89a24a154a123a35a193,a89a199
a35a193a243xa89a24a154a139a199a18a134a123a48a53,a89a199a48a53a123a134a34a24a196a119a41a27a34,a215a248a27a34
a123a48a53a139a24a199a92a160a37a222,a79a36a15a119a41a27a37a222,a89a199a27a37a222a139a220a199a160a10,a199a44
a49a53a222a18,a15a44a49a53a5a18,a139a220a199a48a53a166a80a35a39a123a160a10,a183a160a10a199a38a122xu,xv
a203a27a37a222a52a24a199a141a67a123a43a41a41a37,a52a24a199a141a67a248,a166a233a155a128,a183a119a89a199a27a37
1
a222e3,a63a34a247e3a52a199a92a160a37a222,a149a52e3a210a119a22a30a71
(e3,e3) = 1 (a92a160a37a222),e3 = xu ×xv|x
u ×xv|2
,(4.1)
a0a143a243a89a248a123a160a10a131a5,(xu,xv,e3)a210a52a141a67a248,a89a30,a113a239a42det(xu,xv,e3)a52
a116a123,a253det(xu,xv,e3) > 0,a196a49a52a141a67a248,a25a243,a183a123a89a199a48a27a203a24a196a123a48
a27a52a88a51a123,a24a196a123a86a222a128a128a126u,v a107a106a21a48a114a199a123a77a193a123a135a73a1a91,a79a36
a210a7a118a159a234,a198a162a89a176a139a24a199a98a154,a79a143xu a203xv a88a24a189a18a134,a0a143a183a162a123a108
a85a52a243a149Euclid a1a91a139a24a199a221a222,a196a49a198a162a126a123a52a58a18a134a123a43a41a248,a13a1a91
a52a24a248a123,a98a52a73a219a48a193a123a218a42a210a7a118a133a236a234,a13a183a82e1,e2a52a92a160a37a222,a199
a162a52a144a35a18a134a123,a196a49a25a243e1,e2,e3a174a199a37a222a209a52a92a160a37a222,a13a37a144a35a18a134,
a63a37a79a143a7a199a52a24a199a141a67a248,a196a49a199a123a113a239a42a97a148a52a116a123,a98a52a79a143a89a174a199
a37a222a209a52a144a35a18a134a123a92a160a37a222,a196a49a113a239a42a127a149±1,a199a123a178a48a127a1491,a196a49
a183a25a243a52a119a89a199a113a239a42a127a1491 a123,a79a36a89a52a24a199a116a98a123a43a41a248,a199a123a113a239a42
a127a1491,a149a52
e1,e2,e3 (a141a116a98a41a37),(ei,ej) = δij,(e1,e2,e3) = 1,1 ≤ i,j,k ≤ 3,(4.2)
2 a77a193a123a135a73a42a249a217a1a91
a0a143a135a73a1a91a14a143a248a17? a89a30a210a88a144a52a139a24a199a43a41a248,a13a52a139a24a24(family)a43
a41a248,a164a139a1a199a35a106(a35a123a107a106),a0a143a33a7a155a123a24a199a25a54a210a52a24a199a43a41a248a203
a199a248a163a123a43a41a248a52a14a143a24a199a7a248,a7a234a137a89a199a7a248a52a135a73a1a91a33a204a7a123a175
a11,a196a49a183a25a243a139a24a24a43a41a248,a7a48a128a52a139a220a199a35a106u,v,a4a150a44a49a139a245a199a35
a106,a0a143a7a73a89a199a248a163a43a41a248a203a199a123a7a248,a183a210a178a199a135a73a234,a25a243a183a89a199xa52
a37a222,eia15a209a52a37a222,a196a49a183a210a70a70a23dx,a23da203dei,a89a52a24a199a37a222a123a135a73.a98
a52a79a143e1,e2,e3 a52a24a199a41a37,a52a34a117a195a7a123,a13a183a162a52a2433a145a56a45a165a255a88a123,
a196a49a127a91a24a199a37a222a23a108a52e1,e2,e3 a123a34a117a28a92,a196a49a183a44a49a178deia85a196
dei = ωijej,(4.3)
2
a89a176a183a126a123a52Einsteina123a110a82,a152a42a139a24a199a141a106a173a133a123a155a210a35a28,a79a143a183a162
a123a56a45a523 a145a123,i,j,ka209a52a441a1163,a0a143ωijej a210a52a233j a35a28,a196a49a89a523 a49,
a253j = 1,2,3,a1393a49,a89a52Einstein a243a135a73a1a91a90a159a123a110a82,Einsteina164a41a234
a24a71a47a60,a7a48a128,a44a3a28a7a139a199a106a248,a199a7a139a199a141a106,a253xi,ia89a199a141a106a209a85
a196a5a41,Einsteina128a88a85a5a41,a198a210a85a234a222a41,a79a36a128a117a123a135a73a1a91a123a86a176
a62,a222a41a58a158a245,a43a41a123a89a199a141a41a209a85a196a222a41,a217a34,a222a5a150a139a7a248,a196a49,
a183a126a5a41a85a196a218a42(4.3),a218a42(4.3)a52a228a253a123a218a42,a199a44a43a220a199a248a163a43a41a123
a7a248,a24a199dei a203a198a117a123a43a41ei a123a7a248,ωija52a31a143a17? a199a52a24a39a135a73a42,a79
a143dei a52a24a39a135a73a42,a199a123a138a52a37a222a123a24a39a135a73a42,a196a49a28a152a42a85a196a199a123a73
a222a123a155,a210a1393a199a73a222,a13a37a154a24a199a209a52deia123a73a222,a0a143ωija52a24a39a135a73a42,a79
a143i,ja209a52a441 a1163,a196a49a24a225a1399a199,a98a52a89a74ωij a88a52a127a63a123,a89a52a1a91a222,
a197a166a222a33a228a253a123a24a199a218a42,a79a143a183a162a24a35a209a52a116a98a123a248a58,a196a49ωija233a149a5
a41i,ja52a39a233a193a123,a253
ωij + ωji = 0,(4.4)
a143a31a143a17? a79a143ei,ej a119a22(ei,ej) = δij,a28a178a199a135a73,a145a79a143 deltaij a52a158a106,
a196a49a135a73a199a131a128a520a199a149a52a210a21a122a116a5a193a218a42:
(dei,ej) + (ei,dej) = 0,(4.5)
a132a149dei = ωikek,a63a37ei,eka52a144a35a116a98a123,a196a49a132a222a42a210a122a116ωij+ωji = 0,a253
a218a42(4.4),a233a149a196a139a123i,ja441a1163,ωija52a39a233a193a123,a79a36(ωij)a23a196a24a1993×3a123
a48a106,a0a143a89a199a48a106a52a39a233a193a123,a199a88a52a1399a199a195a163,a34a19a222,a144a1393a199,a63a37
a79a143a39a233a193,a196a49a204(a233a110)a34a222a123a195a163a209a520,a217a198a123a233a149a204a34a52a39a233a193a123.
a15a210a52a139
(ωij) =
0 ω12 ω13
ω12 0 ω23
ω13?ω23 0
,(4.6)
a79a36a144a16a53a199a195a163,a253a144a139ω12,ω13,ω23,a893a199a209a52a24a39a135a73a42,a13a893a199
a24a39a135a73a42a233a77a193a1a91a117a159a52a58a158a7a155a123,a253a199a162a209a44a49a132a135a73a42a117a44a43.
a202a47a207a196a60a106a88,a180a73a219a123a30a127,a209a52a89a60a106,a255a88a60a106a221a86a52a178a60a106a135a73
3
a234,a178a199a123a135a219a42a143a248a106,a76a27a4a58a88a243a14a149a24a39a135a73a42,a217a34,a24a39a135a73
a42a52a178a89a199a175a11a65a122a59a92a234,a183a13a62a123a30a127a17a178a203a28a162a144a235a44a27a41a123a76a106
a166a27Darbouxa123a86,a199a139a143a76a253a119a41a21a77a193a88a22,a89a171a86a105a138a122a23,a88a44a44
a234a52a27a169a123,a198a88a126a135a73a42,a198a126a123a52a160a135a73,a196a49a139a142a245a218a42a85a229a117a159a24
a74,a126a135a73a42a123a155,a24a39a135a73a42a85a229a117a210a59a92a245a234,a196a49a183a89a176a126a234a24a39a135
a73a42,a126a116a98a41a37,a38a77a193a88a58a158a59a92,a89a74a28a162a243a202a47a135a73a1a91a86a165a105a232
a21a73a116,a98a52a89a171a48a27a105a139a72,a79a143a24a35a192a220a209a59a92a234,a183a34a189a77a193a52a189a53
a123,a253a243a221a123a30a127a139a24a199a39a30a168a48a53,a79a143a189a234a53a131a128,a243a24a199a154,a199a139a220a199
a35a37a222a123a155,a199a123a27a37a222a210a113a92a104a189a234,a79a36,a77a193a189a53a131a128,a154a24a154a24a189a139
a24a199a251a189a123a92a160a27a37a222,a88a52a199a123a139a123a37a222,a0a143,a77a193a222a139a24a199a105a7a155a123
a1a91a136a232,a210a52a24a199a154a28a24a199a92a160a35a37a222,a253xa203e1,a89a52a25a83a196a162a14a145a45a123
a33a59a92a123a60a89,a15a52a33a7a155a123a60a89,a0a143a233a149x + e1,a199a245a234a24a199a145a106,a79a143
a251a189a234xa131a128,e1a89a199a92a160a35a37a222a164a44a49a221a87,a196a49xa123a27a236a522a145a123,a0a143a154
a24a199xa123a35a37a222a164a122a281a145,a89a52a79a143a199a44a49a52a24a199a203,a221a24a199a203a177,a199a52a92
a160a123,a196a49a89a199a56a45a523a145a123,a13a89a1993a145a56a45a183a119a41E:
E = {xe1|x ∈ M}(a203a45); dimE = 3,(4.7)
a139a24a1993a145a123a56a45a221a86a128a5a241a24a1993a145a56a45,a89a199a10a39a7a155a244a234,a25a243a142a245a106
a166,a212a174a209a137a7a89a199a10a39,a24a96a28a126a198a117a123a22a111a117a195a85a1a91a88a234,a128a128a137a7
a222a193a139a24a199a203a87,a89a199a183a162a119a41a14a145a45,a89a30a203a177a52a24a199a14a145,a79a36a89a199a14
a145a45a119a41a203a45,a79a143a14a145a52a203,a196a49Ea52a132a22a111M(a77a193a183a119a199a143M,a52a24a199
a22a111)a5a241a117a123a24a199a203a45,a0a143a243a24a154a7a139e1,a210a139e2a234,a89a52a79a143e2 a52a203a199
a18a134a123a24a199a92a160a35a37a222,a51a30a44a49a132a198a117a123a189a53a104a189a5a117,a79a36e1,e2a210a189
a5a117a234,e3a52a27a37a222a104a108a15a189a234,a196a49xe1a89a199a92a160a35a37a222a203a41a37a52a24a195a47.
a139a234a92a160a35a37a222a15a44a49a232a5a24a199a41a37,a104a108a139a234a41a37,a28a210a44a49a82a145a24a199a35
a37a222a143e1,a196a49a893a145a56a45a210a52a183a162a123a77a193a196a139a89a74a41a37e1,e2,e3 a196a196a123a56
a45,a149a52a183a162a210a139a222a193a123a218a42(4.7),a183a128Ea523a145a123,Ea139a24a199a110a243,a110a116a198
a117a123a77a193M,a79a143a28a139a89a199a35a37a222,a199a139a24a199a198a154x,a132xe1a178a199a110a143x,a89a52
a24a199a44E a116Ma123a110a243,a7a207a196a77a193a123a135a73a1a91,a92a44a77a193a88a234,a24a189a7a126E.
4
Ea203a198a117a123a77a193a139a183a35a123a7a248,a166a98a183a89a234a89a171a1a91a123a7a248,a126Ea123a80a255a243
a149,E a123a56a45a523a145a123a56a45,a199a222a62a139a24a39a135a73a42,a89a74a135a73a42a243Ea222a62a209a52
a104a189a123,a0a143a248a234a183a89a123e1,e2,e3a89a1a199a92a160a37a222,a104a108a77a193a123a154xa15a52a24a199
a37a222,a199a123a160a152a37a222a15a52a24a199a37a222,x a52u,va123a60a106,dx a104a108a52u,v a123a24a199
a24a39a135a73a42,a199a15a44a49a44a143e1,e2,e3 a123a24a199a34a117a28a92,a98a52a34a19a222,a199a24a189
a52e1,e2 a123a28a92,a89a52a79a143e3 a52a27a37a222,a196a49a199a24a189a52a243a35a193a222a62,a13e1,e2a209
a52a35a37a222,a196a49dxa24a189a52e1,e2a123a34a117a28a92,a199a123a248a106a183a119a41ω1,ω2,a253a122a116a183
a162a145a24a199a218a42:
dx = ω1e1 + ω2e2,(4.8)
a196a49a183a25a243a1395a199a24a39a135a73a42,ω1,ω2,ω13,ω23,ω12,a89a28a135a73a42a58a158a7a155,a199
a162a209a139a59a92a123a1a91a63a66,a89a199a210a128a210a135a73a42a255a88a1a91a117a159a38a122a175a11a59a92a13
a37a142a52,a25a243a183a162a210a139a218a42( 4.8),a183a162a82a218a42(4.3),(4.4)a42a143a183a162a123a145a19a199
a218a42,a183a243a3a193a89a44,a135a73a42a139a199a33a76a123a128a154,a210a52a135a73a220a39a128a520,a253a233
a149dx,d(dx) = 0,a233a149a127a91a60a106a123a105a135a73a220a39a24a189a127a1490,a89a210a35a104a149a243a56
a45a127a63a82a24a199a75a173,a242a82a199a123a30a142,a13a30a142a88a242a139a30a142,a82a30a142a82a220a39a24a189
a52a127a1490,a79a143a89a199a117a159,a89a171a106a166a136a232a210a139a196a162a123a51a174(homology)a117a159,a25
a243a28a7a180a31a143a192a220,a209a52a51a174a117a159,a58a158a7a155,a88a44a183a162a25a243a178a218a42(4.8)a38
a30dx a242da24a5a6a210a127a1490,a13a178a141a30a123a48a13a210a122d( omega1e1) + d(ω2e2),a253
d(ω1e1) + d(ω2e2) = 0,(4.9)
a213a63a104a105a135a73a3a193a139a24a199a24a39a123a155,a135a73a145a19a199a79a6a7a149a82,a15a13a211a131,a28
a210a178a199a162a135a73a234,a135a73a131a128,a210a21a25a196a122a116a123a42a6a52e1,e2,e3a123a34a117a28a92,a0
a143a199a123a248a106a52a19a39a135a73a42,a13a233a149a89a199a248a106a52a19a39a135a73a42a123a37a222a7a127a1490a123
a155,a196a139a123a248a106a209a7a127a1490,a149a52a28a152a42a13e1,e2a123a248a106a1430 a123a155,a210a122a116a183a5
a193a123a145a174a199a218a42:
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(4.10)
a183a88a45a251a178a199a41a234,a89a199a121a210a105a59a92,a44a193a253a44a122,a89a52a7a155a244a234a123a24a199a218
a42,a89a74a218a42a28a162a15a142a250a122a99,a79a143a243a202a47a86a222a23a88a68,a89a52a132a149a202a47a88
5
a245a161a126a135a73a42,a142a245a124a15a88a204a126a135a73a42,a217a34a89a105a59a92,a196a49a183a210a122a116a218
a42(4.10),a108a128a13e3a123a248a106a1430,a210a122a116a145a143a199a218a42:
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(4.11)
a89a52a58a158a58a158a7a155a123a218a42,a132a89a74a210a122a116a196a162a123Levi-Civitaa178a113a117,a25a243,
a89a199a15a119a201a100(connection),a233a149a201a100,a202a47a73a24a253a86,a44a49a89a222a105a200a105a200,
a217a34a105a59a92,a183a128a243a89a199a60a111a131a5,Levi-C ivitaa201a100a210a52ω12a89a199a24a39a135a73
a42,a213a63ω12 a52Ea176a62a123a24a199a24a39a135a73a42,a199a210a189a1a91a123a117a159,a38a122a183a44a49a178
a89a199a37a222a215a248a24a30a77a34a178a113a123a35a196,a89a52a31a143a63a135a17? a210a52ω12a89a199a24a39a135
a73a42a132a48a199(4.10)a113a92a104a189,a79a143a152a42a139a24a199ωprime12,a38a122a55a92a51a248a123a48a199a42,
a253
dω1 =?ω2 ∧ωprime12; dω2 = ω1 ∧ωprime12,(4.12)
a183a137a7a121a210ω12 = ωprime12,a253a144a139a24a199a44a21a117,a144a139a24a199ω12 a55a92a48a199(4.10),a15
a210a52a128,a34a38a139a12a105a24a199ωprime12a55a92a48a199(4.10),a0a143a183a162a178a220a199a48a199a35a62,a210
a122a116
ω1 ∧(ωprime12?ω12) = 0; ω2 ∧(ωprime12?ω12) = 0,(4.13)
a13a220a199a24a39a135a73a42a152a42a35a198a127a1490a123a155,a13a37a152a42a217a165a24a199a88a520,a0a143a217
a199a0a199a23a108a52a199a123a245a106,a89a52a105a83a106a33a59a92a123a192a220,a28a162a174a24a174a210a241a117
a234,ω1,ω2a52a88a1430 a123,a13a37a88a98a88a127a1490,a63a37a34a117a195a7,a79a36,ωprime12? ω12a17
a127a149ω1a123a245a106,a145a127a149ω2a123a245a106,a98a52ω1,ω2 a34a117a195a7,a196a49a199a7a127a1490.
a79a36a152a42a242a139a24a199ωprime12a55a92a51a248a123a48a199,a199a24a189a127a149?mega12,a89a52a113a92
a104a189a123,a0a143a183a126a89a199a90a159a196a162a123Levi-Civitaa178a113a117,a183a162a25a243a139de1 =
ω12e2 +?mega13e3,a183a162a178e3a89a24a49a82a62,a82a62a52a31a143a63a135a17? a34a38a139a24a199
a37a222,a28a178a199a123e3a82a62a131a128,a210a52a215a248e3a123a48a53a82a89a199a116a98a61a107,a196a49a28a178a199
a82a62a234a123a63a135a210a52a82a199a123a116a98a61a107,a28a178a199a82a62a123a155,a183a25a243a88a242a52a202a47a123
a135a73a234,a52a99a123a135a73a234,a183a126Da44a43a89a199a99a123a135a73,a149a52a210a122a116a89a28a218a42:
De1 = ω12e2,De2 =?ω12e1,(4.14)
6
a89a52a76a183a166a98a121a210a123a52a132a1a91a113a92a104a189a123,a89a52a79a143ω12a52a113a92a104a189a123,a34
a38De1 = 0a123a155,a183a210a128a89a199a37a222a52Levi-Civitaa63a66a5a178a113,a253a183a128a243a89a171a60
a111a5a199a210a52a178a113a123,a199a52a178a113a123,a89a52a139a99a123a63a66,a243Euclid a1a91a123a30a127,a89
a210a52a202a47a178a113,a25a243a52a202a47a178a113a123a77a18,a15a210a52De1 = 0a30a52Levi-Civitaa178
a113,a196a49a243a77a193a222,a28a126a24a39a135a73a42,a166a11a89a199a178a113a7a118a10a195,a13De1 = 0a123
a155,a233a149a37a222a52a199a135a73a48a199,a89a199a135a73a48a199a88a24a189a139a137,a98a52a34a38a28a139a24a30
a77a34,a28a210a44a49a215a248a89a30a77a34a70a137,a233a149a77a34a117a128,a196a139a123a89a74a60a106a209a52ta123
a60a106,a215a248a89a30a77a34a70a137,a44a49a70a116a137,a150a149a89a199a30a71,a183a128a89a199a37a222a210
a52Levi-Civita a178a113,a196a49a44a89a199a189a66a44a49a234a137a89a199a178a113a117a203a77a34a123a160a10a139
a7,a34a38a139a24a199a154,a12a105a164a139a24a154,a0a143a28a201a89a220a154a139a220a30a88a51a123a77a34,a28a178
a51a24a199a37a222a215a62a24a30a77a34a178a113,a63a215a145a19a30a77a34a178a113,a23a116a123a24a196a88a52a51a24
a199a37a222,a150a149a89a199a178a113a117a166a77a34a123a160a10a139a7a248,a52a202a47a123a178a113a117a123a77a18,a202
a47a123a30a127,a178a113a117a243Euclid a178a193a176a222a52a253a233a123a90a189a234a123,a25a243a89a199a30a127a52a203
a77a34a123a160a10a139a7a248,a89a210a52a201a100,a183a166a98a178x a89a199a37a222da220a39a127a1490,a13a51a248
a123,a183a44a49a233eia89a199a37a222a15d a220a39,a253d(dei) = 0,a79a36a174a241a117d(dei) = 0,a210
a139:
dωij = ωik ∧ωkj,(4.15)
a34a19a222,a105a59a92a143,a183a233dei = ωijej a70a135a73,a108a128a79a143ωija52a24a39a135a73a42,a196a49
a97a148a139a199a139a82,a253a139a123ωij ∧dej,a149a52a139
d(dei) = dωijej?ωij ∧dej = 0,(4.16)
a98a52dej = ωjkek,a196a49a62a24a199dωijeja149a143dωikek a150a139a7a248,a89a52a79a143a28a233
a149j,ka52a70a90,a44a85a196j,a15a44a85a196k,a209a52a441a1163,a195a196a162a123,a79a36a210a122a116
a218a42(4.15),a89a52a24a199a228a253a123a218a42,a23a248a102a38,a217a34a105a59a92,a131a217a52a2433a145
a123a60a111,a105a59a92,a34a19a222,a141a30a89a229a117a523a49a131a90,a98a52a893a49a52a233a149j a70
a90,a79a143ω a52a39a233a193a123,a196a49a183a162a117a23ωik,i negationslash= k,a221a86a210a117a23a23ω13,a243a5
a193a0a199a218a42a23d omega13 a97a148a52ω1jωj3,a98a52a34a19a222,ja144a21a127a1492,a89a52a79
a143j = 1a30ω11 = 0,j = 3 a30ω33 = 0,a196a49a141a30a23a248a52a89a143a248a131a90,a217a34a210a144
7
a139a24a49,a196a49a28a210a122a116
dω13 = ω12 ∧ω23,(4.17)
a51a248a44a49a122a116
dω23 =?ω12 ∧ω13,(4.18)
a89a74a24a196a210a52a196a162a123Weingartena218a42,a58a158a59a92,a196a49a89a74a218a42a209a52a44a59a92
a123a105a135a73a193a253a122a116a123,Weingartena218a42a105a139a63a135,a28a7a199a203a198a117a123ω1,ω2a123
a218a42a7a118a123a155,a113a92a52a24a199a111a231,a253a213a63a116a89a74a218a42a166a218a42(4.10)a35a145.a89a199
a111a231a52a139a1a91a63a66a123,a25a243a199a126a117a189a196a162a123Betti Transform ation,a89a199a183a88
a251a89a234,Weingartena218a42a203a198a117dωia123a218a42a123a35a145a117a52a139a253a47a123a1a91a63a66a123.
3 a77a193a123a228a253a88a35a42
a222a193a183a255a88a234De1a123Levi-Civita a178a113a117,a152a42a82a89a199a37a222a243e3a89a199a48a53a123
a155,a183a210a122a116ω1 ∧ω13 +ω2 ∧ω23 = 0,a253a218a42(4.1 1),a79a36,ω13,ω23a209a52ω1,ω2
a123a34a117a28a92,a183a44a49a85a196
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(4.19)
a132a89a199a183a122a116a89a199a77a193a123a7a155a123a88a35a42,a24a199a77a193a166a12a24a199a77a193a139a31a143a73
a47? a89a199a73a47a243a149a77a71,a77a71a52a77a193a222a123a60a106,a7a48a128,a24a196a123a155,a183a162a126a145
a24a228a253a42a203a145a19a228a253a42a117a44a43:
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (4.20)
a145a19a228a253a42 II = (?dx,de3) = aω2 + 2bω1ω2 + cω22,(4.21)
a233a149a145a24a228a253a42,a183a119a199(I).a199a210a52a77a193a123ds2 = (dx,dx),a132a149dx =
ω1e1 + omega2e2,a98a52e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49a210a122a116ω21 + ω22.
a79a36ωa895a199a24a39a135a73a42a209a139a173a7a123a1a91a63a66,ω1,ω2 a123a178a48a90a210a52a77a193a123
a221a222,a2433a145a56a45a176a62,a77a193a104a108a139a24a199a221a222,a0a143a199a123a221a222a52a31a143a17? a210
a52a199a123Riemann a221a222,a52a24a1992a39a123a135a73a42,a89a1992a39a135a73a42a59a92a244a234,a210
8
a52ω21 + ω22,a0a143,a183a15a44a49a233e3a89a199a37a222a82?(de3,dx),a76a183a166a98a196a85a123,a199
a210a127a149aω2 + 2bω1ω2 + cω22,a89a15a52a24a1992a39a135a73a42,a89a199a24a196a119a41a145a19a228a253
a42,a0a143a139a24a199a145a24a228a253a42,a164a139a199a145a19a228a253a42,a28a210a82a199a123a1a110a138,a1a110
a138a123a90a210a52a + c,a1a110a138a123a232a210a52ac? b2,a24a196a143,H = a+c2 a119a41a77a193a123a165
a77a71,K = ac? b2 a119a41a77a193a123Gaussa77a71,a89a176a7a155a244a234,Gaussa77a71a139a142
a245a139a85a123a117a159,a79a36a89a52a14a143a248a445a199a24a39a135a73a42a132a199a162a123a34a117a28a92a123a7a248
a210a122a116a77a193a123a88a35a42,a165a77a71a203Gaussa77a71a89a220a199a88a35a42a195a85a77a193a123a1a91
a117a159,a7a48a128,a44a49a121a210Gauss a77a71a52a116a123a155,a77a193a52a243a123,Gaussa77a71a52a139
a123a155,a77a193a210a139a145a154,a210a54a106a145a154,a142a245a1a91a117a159a209a44a49a126a89a220a199a77a71a117a195
a85,a196a49a52a77a193a176a62a220a199a33a204a7a123a88a35a42,a89a199a88a35a42a52a60a106,a49a128a183a162
a122a116a123a52a24a39a135a73a42,a13a24a39a135a73a42a88a76a142a52a46a54a196a190a52a31a143a63a135,a98a132
a199a123a228a174a44a49a122a241a60a106a117,a89a60a106a104a108a52a234a137a122a7a118a56a249,a34a122a116a165a77a71
a203Gauss a77a71,a165a77a71a127a1490,a24a196a119a41a244a66a77a193,a196a162a123a244a66a77a193,a222a190a210
a52a28a178a24a30a85a20a123a77a34a56a243a61a6a121a176a62a196a196a123a77a193a210a52a193a232a33a66a123a77a193,
a199a123a165a77a71a1430,a196a49a89a139a59a92a123a1a91a63a66,a33a163,a4a150a25a243a209a139a105a245a7a149a244
a66a77a193a123a207a196,Gaussa77a71a205a7a155,Gaussa77a71a52a127a149ω13 ∧ ω23,a243a77a193a222a15
a24a248a139Gaussa110a243,a77a193a154a154a139a24a199a92a160a27a37a222,a178a89a199a92a160a27a37a222a23a143a24
a199a204a187a1431 a123a69a193a123a154,a210a178a77a193a110a243a116a69a193a222a86a234,a154a199a154a139a24a199a92a160a27
a37a222,a28a2430a154a152a24a199a92a160a27a37a222a203a199a178a113,a199a123a224a154a210a243a92a160a69a193a222,a0a143
a233a149a196a139a123a154a209a41a89a199a232a5a123a155,a210a243a92a160a69a193a222a122a116a24a199a75a173,a243a89a199a110
a243a5,a220a199a193a232a195a163a123a7,a253a199a123a52(imag e)a123a193a232a195a163a203a198a117a193a232a195a163a123
a7a210a52Gauss a77a71,a89a248a24a5a6a210a23a241a117a234,a196a49a89a199Gaussa77a71a139a105a59a92
a123a1a91a63a66,a89a30,a77a193a123a255a88a52a24a199a77a18,a77a193a123a30a127a139a77a34,a77a34a15a139a24
a199Gaussa110a243,a0a199Gaussa110a243a52a82a92a160a35a37a222,a217a34a15a44a49a82a92a160a27a37a222.
a0a143a77a34a123a30a127,Gaussa110a243a178a35a34a110a243a116a92a160a203a222a62,a178a92a160a203a123a221a222a250
a198a117a89a199a221a222a248a210a52a77a71,a25a243a210a178a89a199a10a39a77a18a116a176a145,a77a18a1162a145,a253a77
a18a116a77a193,a196a49a183a132a89a199a77a193Gaussa110a243a116a24a199a92a160a69a193a222a62,a89a220a199a193a232
a123a7a210a52Gaussa77a71,a196a49a89a52a24a199a58a158a7a108a123a1a91a221a222,a25a243a139a199a105a7a155
a123a7a248,a222a62a139dωik = ωij ∧ ωjk,a25a243a183a178a89a199a218a42a126a116ω12,i,k = 1,2,a149
9
a52dω12 = ω13 ∧ ω32,a98a52ωa52a39a233a193a123,a196a49ω32 =?ω23,a89a176ω13 ∧ ω23 a210
a52ac?b2,a15a210a52Gaussa77a71,a196a49a183a210a122a116a218a42
dω12 =?Kω1 ∧ω2,(4.22)
a217a165,Ka52Gaussa77a71,a89a52a24a199a13Gauss a58a158a175a199a123a218a42,a196a49Gaussa178a89
a199a189a174a119a41Theorem Egregious,Egregiousa52a110a182a9a241,a63a135a141a89a52a24a199a234
a88a122a123a189a174,a143a31a143a17? a199a121a210a234Gaussa77a71a144a203a77a193a123Riemann a221a222a139
a7,a203a89a199a77a193a243a56a45a123a160a152a195a7,a79a143a243a89a199a218a42a176a62,ω12a46a178a121a210
a234a144a203a77a193a123Riemanna221a222a139a7,a108a128ω1 ∧ ω2a210a52a243a89a199a221a222a5a123a193a232a195
a163,a104a108a144a203ds2a139a7,a253a144a203Riemanna221a222a139a7,a196a49a175a108Gaussa77a71a52a24
a199a77a193a243a56a45a176a62a123a24a199a88a35a42,a98a52a199a144a203a77a193a123Riemanna221a222a139a7.
a166a233a155a128,a28a178a77a193a35a166(deform)a24a5a6,a38a122Riemanna221a222a88a35,Gaussa77
a71a210a88a35,a196a49Gaussa77a71a139a89a143a173a7a123a117a159,a183a46Gau ssa41a234a142a245a7a155
a123a90a164a224a123a136a42,a89a199a189a174a23a108a52a198a1a47a97a220a123a24a199a136a42,a52a105a1a47a123
a24a199a136a42,a34a19a222,a183a162a210a178a89a74a48a199a126a105a135a73a135a73a24a5a6,a122a116a228a253a218
a42,a108a128a210a122a116a89a74a218a42,a5a39,a183a7a89a132a89a199a218a42a121a210Gauss-Bonneta218
a42,Gauss-Bonneta218a42a52a77a18a174a110a111a174a110a131a90a127a149180o,a82a89a199a218a42a77a18a116
a77a193a123a60a111,Gauss-Bonneta218a42a123a97a126a58a158a131a18,a121a210a123a30a127,a183a41a170a188
a126a203a45,a89a203a45a227a135a126a122a133a236a24a74,a34a189a89a199a45a52a24a199a133a123a34a45(complex
line),a0a143a89a199a133a34a45a123a14a145a52a30a133a34,a243a133a34a176a62,a253a233a138a127a1491a123a133a106
a52a24a199a203a177,a210a52a203a45,a89a199a10a39a243a212a174a222a52a228a253a123,a25a243a76a27a180a122a105a245
a123a52a98a1a91( symplectic geometry),a92a139a98a136a232(symplectic structure)a88a212
a139a126,a28a7a178a98a1a91a126a116a222a6a197a166a123a155,a137a7a28a222a24a199a133a34a45,a210a52a183a162
a25a243a196a89a123a192a220a123a24a199a77a18,a183a162a25a243a892a145,a24a116a212a174a123a155,a56a45a166a30a45
a28a243a24a229a524a145,a196a49a28a228a253a123a56a45a524a145,a2434 a145a56a45a104a165a139a24a199a133a34a45.
a79a36a210a122a116a89a199Gauss a77a71,a25a243a89a199a77a71a522a39a135a73a42,a199da24a5a6a127a1490,
a210a122a116Maxwella48a199,a196a49a135a232a73a243a1a91a222a97a126a52a142a245a212a174a123a228a250,a33a163
a183a243a21a41a166a22a236a147a222a85a234a24a159a169a57,a119a41a21Gauss-Bonneta218a42a166Maxwella48
a199a22,a28a162a152a42a21a73a116,a44a49a86a23a24a23,a183a46a178a89a159a169a57a252a117a201a76a27,a98a52
10
a25a243a79a122a105,a150a73a116,a5a39a183a46a89Gauss-Bonneta218a42a123a24a199a121a210,a89a199a203a183
a162a123a211a42a139a58a158a183a35a123a7a248,a2435 0a35a3,a243a102a210a220a8a201a76,a183a115a135a73a1a91,
a210a122a116Gauss-Bonneta218a42a123a89a199a121a210,a202a47a86a222a133a236a245a234,Gauss-Bonneta218
a42a204a7a63a135a52a128,a28a139a127a91a24a199a77a193,a77a193a222a62a139a24a199a245a30a111,a178a89a199a245
a30a111a123Eulera1a110a106a44a43a171a171a77a71a123a28a92,a89a52Gauss-Bonneta218a42,a183a122a116
a89a199a121a210,a183a46a89a52Gauss-Bonneta218a42a33a59a92a90a33a7a108a123a24a199a121a210,a210a52
a178dω12a24a135a73a210a122a116Gauss-Bonneta218a42a232a73a123a42a6,a108a128a183a243a24a202a143a1a35
a123a30a127,a116a234Princeton a41a176a145a123,a104a108a7a108a13a108a143,a183a46a27a6a178a89a199a77a18
a116a176a145,a89a199a52a44a49a41a116a123,a49a128,a183a164a41a234a74a47a123a211a42,a196a49a44a49a128a183a89
a199Gauss-Bonneta218a42a123a121a210a243
a183a196a41a123a211a42a165a52a183a33a245a161a123a24a199,a183a46a25a243a44a49a136a101a234.
11
2001a3511a2192a134
1 a77a193a123a41a37
a152a21a183a162a89a24a154a77a193a88,a135a232a73a243a77a193a222a97a126a123a207a196a243a114a199a106a166a176a62a52a105
a7a155a123,a89a52a79a143a243a77a193a88a165,a77a193a123a89a74a117a159a128a128a106a223a116a217a198a205a18a123a60
a111,a13a89a74a205a18a123a60a111a35a154a116a77a193a123a30a127a15a139a105a245a117a159,a243a77a193a123a60a111a46a178
a21a9a234,a0a143,a77a193a139a199a128a154,a210a52a183a162a34a189a199a52a2433a145a56a45a176a62,a196a49a28a23
a122a68,a28a44a49a152a67,a44a49a243a23a122a68a199a222a62a123a77a34a176a123a117a159a249a217a198a31a143a123,a24
a116a176a145a49a128,a210a23a88a68a234,a183a123a89a27a203a86a88a24a248,a196a49a183a46a76a27a178a89a1a16a97
a238a133a92a24a5,a89a199a97a238a76a150a97a148a243a202a47a135a73a1a91a86a222a73a88a116a123,a199a139a199a128
a154,a210a52a79a122a105a13a37a48a27a117a122a59a92,a0a143a31a143a52a77a193a17? a77a193a210a52a67a222a24
a199a59a77a123a192a220,a183a178a199a123a154a123a43a41a44a143a220a199a35a106a123a60a106,a89a220a199a35a106a183a119
a41u,v,u,va24a196a119a41a107a106(parameter),a34a38u,va35a154a123a30a127,a89a74a154a123a27a236
a210a196a234a24a1992a145a123a77a193x(u,v) = (x1(u,v),x2(u,v),x3(u,v)),a149a52a79a143a139u,v,
a196a49a28a44a49a38a122a24a199a107a106a123a138a52a158a106,a108a128a38a122a12a24a199a107a106a35a154,a247v a52
a158a106,a13u a35a154,a196a49a210a139a24a30a77a34,a199a123a107a106a52u,a51a248,a28a44a49a38a122ua88
a35,a13va35a154,a79a36a243a77a193a176a62,a139a220a28a77a34,a199a123a107a106a24a28a52v a127a149a158a106,a24
a28ua52a127a149a158a106,a233a149a89a220a28a77a34,a154a24a30a77a34a243a154a24a154a209a139a24a30a35a34,a196a49
a243a24a199a154x,a183a162a210a139a220a30a134a34,a183a34a189a89a220a30a134a34a88a173a92,a166a233a155a128,a137a219
a143a89,xu,xv a88a52a51a24a199a48a53,a88a178a134(a34a117)a35a7,a217a165xu,xva210a52xa123a37a222a73
a47a233u,va70a160a135a73,a221a86a183a128,a199a123a37a222a232xu ×xv negationslash= 0,a34a38a89a220a199a48a53a88a173
a92,a196a49a199a162a210a60a196a24a199a178a193,a89a199a178a193a183a119a41a77a193a243a89a24a154a123a35a193,a89a199
a35a193a243xa89a24a154a139a199a18a134a123a48a53,a89a199a48a53a123a134a34a24a196a119a41a27a34,a215a248a27a34
a123a48a53a139a24a199a92a160a37a222,a79a36a15a119a41a27a37a222,a89a199a27a37a222a139a220a199a160a10,a199a44
a49a53a222a18,a15a44a49a53a5a18,a139a220a199a48a53a166a80a35a39a123a160a10,a183a160a10a199a38a122xu,xv
a203a27a37a222a52a24a199a141a67a123a43a41a41a37,a52a24a199a141a67a248,a166a233a155a128,a183a119a89a199a27a37
1
a222e3,a63a34a247e3a52a199a92a160a37a222,a149a52e3a210a119a22a30a71
(e3,e3) = 1 (a92a160a37a222),e3 = xu ×xv|x
u ×xv|2
,(4.1)
a0a143a243a89a248a123a160a10a131a5,(xu,xv,e3)a210a52a141a67a248,a89a30,a113a239a42det(xu,xv,e3)a52
a116a123,a253det(xu,xv,e3) > 0,a196a49a52a141a67a248,a25a243,a183a123a89a199a48a27a203a24a196a123a48
a27a52a88a51a123,a24a196a123a86a222a128a128a126u,v a107a106a21a48a114a199a123a77a193a123a135a73a1a91,a79a36
a210a7a118a159a234,a198a162a89a176a139a24a199a98a154,a79a143xu a203xv a88a24a189a18a134,a0a143a183a162a123a108
a85a52a243a149Euclid a1a91a139a24a199a221a222,a196a49a198a162a126a123a52a58a18a134a123a43a41a248,a13a1a91
a52a24a248a123,a98a52a73a219a48a193a123a218a42a210a7a118a133a236a234,a13a183a82e1,e2a52a92a160a37a222,a199
a162a52a144a35a18a134a123,a196a49a25a243e1,e2,e3a174a199a37a222a209a52a92a160a37a222,a13a37a144a35a18a134,
a63a37a79a143a7a199a52a24a199a141a67a248,a196a49a199a123a113a239a42a97a148a52a116a123,a98a52a79a143a89a174a199
a37a222a209a52a144a35a18a134a123a92a160a37a222,a196a49a113a239a42a127a149±1,a199a123a178a48a127a1491,a196a49
a183a25a243a52a119a89a199a113a239a42a127a1491 a123,a79a36a89a52a24a199a116a98a123a43a41a248,a199a123a113a239a42
a127a1491,a149a52
e1,e2,e3 (a141a116a98a41a37),(ei,ej) = δij,(e1,e2,e3) = 1,1 ≤ i,j,k ≤ 3,(4.2)
2 a77a193a123a135a73a42a249a217a1a91
a0a143a135a73a1a91a14a143a248a17? a89a30a210a88a144a52a139a24a199a43a41a248,a13a52a139a24a24(family)a43
a41a248,a164a139a1a199a35a106(a35a123a107a106),a0a143a33a7a155a123a24a199a25a54a210a52a24a199a43a41a248a203
a199a248a163a123a43a41a248a52a14a143a24a199a7a248,a7a234a137a89a199a7a248a52a135a73a1a91a33a204a7a123a175
a11,a196a49a183a25a243a139a24a24a43a41a248,a7a48a128a52a139a220a199a35a106u,v,a4a150a44a49a139a245a199a35
a106,a0a143a7a73a89a199a248a163a43a41a248a203a199a123a7a248,a183a210a178a199a135a73a234,a25a243a183a89a199xa52
a37a222,eia15a209a52a37a222,a196a49a183a210a70a70a23dx,a23da203dei,a89a52a24a199a37a222a123a135a73.a98
a52a79a143e1,e2,e3 a52a24a199a41a37,a52a34a117a195a7a123,a13a183a162a52a2433a145a56a45a165a255a88a123,
a196a49a127a91a24a199a37a222a23a108a52e1,e2,e3 a123a34a117a28a92,a196a49a183a44a49a178deia85a196
dei = ωijej,(4.3)
2
a89a176a183a126a123a52Einsteina123a110a82,a152a42a139a24a199a141a106a173a133a123a155a210a35a28,a79a143a183a162
a123a56a45a523 a145a123,i,j,ka209a52a441a1163,a0a143ωijej a210a52a233j a35a28,a196a49a89a523 a49,
a253j = 1,2,3,a1393a49,a89a52Einstein a243a135a73a1a91a90a159a123a110a82,Einsteina164a41a234
a24a71a47a60,a7a48a128,a44a3a28a7a139a199a106a248,a199a7a139a199a141a106,a253xi,ia89a199a141a106a209a85
a196a5a41,Einsteina128a88a85a5a41,a198a210a85a234a222a41,a79a36a128a117a123a135a73a1a91a123a86a176
a62,a222a41a58a158a245,a43a41a123a89a199a141a41a209a85a196a222a41,a217a34,a222a5a150a139a7a248,a196a49,
a183a126a5a41a85a196a218a42(4.3),a218a42(4.3)a52a228a253a123a218a42,a199a44a43a220a199a248a163a43a41a123
a7a248,a24a199dei a203a198a117a123a43a41ei a123a7a248,ωija52a31a143a17? a199a52a24a39a135a73a42,a79
a143dei a52a24a39a135a73a42,a199a123a138a52a37a222a123a24a39a135a73a42,a196a49a28a152a42a85a196a199a123a73
a222a123a155,a210a1393a199a73a222,a13a37a154a24a199a209a52deia123a73a222,a0a143ωija52a24a39a135a73a42,a79
a143i,ja209a52a441 a1163,a196a49a24a225a1399a199,a98a52a89a74ωij a88a52a127a63a123,a89a52a1a91a222,
a197a166a222a33a228a253a123a24a199a218a42,a79a143a183a162a24a35a209a52a116a98a123a248a58,a196a49ωija233a149a5
a41i,ja52a39a233a193a123,a253
ωij + ωji = 0,(4.4)
a143a31a143a17? a79a143ei,ej a119a22(ei,ej) = δij,a28a178a199a135a73,a145a79a143 deltaij a52a158a106,
a196a49a135a73a199a131a128a520a199a149a52a210a21a122a116a5a193a218a42:
(dei,ej) + (ei,dej) = 0,(4.5)
a132a149dei = ωikek,a63a37ei,eka52a144a35a116a98a123,a196a49a132a222a42a210a122a116ωij+ωji = 0,a253
a218a42(4.4),a233a149a196a139a123i,ja441a1163,ωija52a39a233a193a123,a79a36(ωij)a23a196a24a1993×3a123
a48a106,a0a143a89a199a48a106a52a39a233a193a123,a199a88a52a1399a199a195a163,a34a19a222,a144a1393a199,a63a37
a79a143a39a233a193,a196a49a204(a233a110)a34a222a123a195a163a209a520,a217a198a123a233a149a204a34a52a39a233a193a123.
a15a210a52a139
(ωij) =
0 ω12 ω13
ω12 0 ω23
ω13?ω23 0
,(4.6)
a79a36a144a16a53a199a195a163,a253a144a139ω12,ω13,ω23,a893a199a209a52a24a39a135a73a42,a13a893a199
a24a39a135a73a42a233a77a193a1a91a117a159a52a58a158a7a155a123,a253a199a162a209a44a49a132a135a73a42a117a44a43.
a202a47a207a196a60a106a88,a180a73a219a123a30a127,a209a52a89a60a106,a255a88a60a106a221a86a52a178a60a106a135a73
3
a234,a178a199a123a135a219a42a143a248a106,a76a27a4a58a88a243a14a149a24a39a135a73a42,a217a34,a24a39a135a73
a42a52a178a89a199a175a11a65a122a59a92a234,a183a13a62a123a30a127a17a178a203a28a162a144a235a44a27a41a123a76a106
a166a27Darbouxa123a86,a199a139a143a76a253a119a41a21a77a193a88a22,a89a171a86a105a138a122a23,a88a44a44
a234a52a27a169a123,a198a88a126a135a73a42,a198a126a123a52a160a135a73,a196a49a139a142a245a218a42a85a229a117a159a24
a74,a126a135a73a42a123a155,a24a39a135a73a42a85a229a117a210a59a92a245a234,a196a49a183a89a176a126a234a24a39a135
a73a42,a126a116a98a41a37,a38a77a193a88a58a158a59a92,a89a74a28a162a243a202a47a135a73a1a91a86a165a105a232
a21a73a116,a98a52a89a171a48a27a105a139a72,a79a143a24a35a192a220a209a59a92a234,a183a34a189a77a193a52a189a53
a123,a253a243a221a123a30a127a139a24a199a39a30a168a48a53,a79a143a189a234a53a131a128,a243a24a199a154,a199a139a220a199
a35a37a222a123a155,a199a123a27a37a222a210a113a92a104a189a234,a79a36,a77a193a189a53a131a128,a154a24a154a24a189a139
a24a199a251a189a123a92a160a27a37a222,a88a52a199a123a139a123a37a222,a0a143,a77a193a222a139a24a199a105a7a155a123
a1a91a136a232,a210a52a24a199a154a28a24a199a92a160a35a37a222,a253xa203e1,a89a52a25a83a196a162a14a145a45a123
a33a59a92a123a60a89,a15a52a33a7a155a123a60a89,a0a143a233a149x + e1,a199a245a234a24a199a145a106,a79a143
a251a189a234xa131a128,e1a89a199a92a160a35a37a222a164a44a49a221a87,a196a49xa123a27a236a522a145a123,a0a143a154
a24a199xa123a35a37a222a164a122a281a145,a89a52a79a143a199a44a49a52a24a199a203,a221a24a199a203a177,a199a52a92
a160a123,a196a49a89a199a56a45a523a145a123,a13a89a1993a145a56a45a183a119a41E:
E = {xe1|x ∈ M}(a203a45); dimE = 3,(4.7)
a139a24a1993a145a123a56a45a221a86a128a5a241a24a1993a145a56a45,a89a199a10a39a7a155a244a234,a25a243a142a245a106
a166,a212a174a209a137a7a89a199a10a39,a24a96a28a126a198a117a123a22a111a117a195a85a1a91a88a234,a128a128a137a7
a222a193a139a24a199a203a87,a89a199a183a162a119a41a14a145a45,a89a30a203a177a52a24a199a14a145,a79a36a89a199a14
a145a45a119a41a203a45,a79a143a14a145a52a203,a196a49Ea52a132a22a111M(a77a193a183a119a199a143M,a52a24a199
a22a111)a5a241a117a123a24a199a203a45,a0a143a243a24a154a7a139e1,a210a139e2a234,a89a52a79a143e2 a52a203a199
a18a134a123a24a199a92a160a35a37a222,a51a30a44a49a132a198a117a123a189a53a104a189a5a117,a79a36e1,e2a210a189
a5a117a234,e3a52a27a37a222a104a108a15a189a234,a196a49xe1a89a199a92a160a35a37a222a203a41a37a52a24a195a47.
a139a234a92a160a35a37a222a15a44a49a232a5a24a199a41a37,a104a108a139a234a41a37,a28a210a44a49a82a145a24a199a35
a37a222a143e1,a196a49a893a145a56a45a210a52a183a162a123a77a193a196a139a89a74a41a37e1,e2,e3 a196a196a123a56
a45,a149a52a183a162a210a139a222a193a123a218a42(4.7),a183a128Ea523a145a123,Ea139a24a199a110a243,a110a116a198
a117a123a77a193M,a79a143a28a139a89a199a35a37a222,a199a139a24a199a198a154x,a132xe1a178a199a110a143x,a89a52
a24a199a44E a116Ma123a110a243,a7a207a196a77a193a123a135a73a1a91,a92a44a77a193a88a234,a24a189a7a126E.
4
Ea203a198a117a123a77a193a139a183a35a123a7a248,a166a98a183a89a234a89a171a1a91a123a7a248,a126Ea123a80a255a243
a149,E a123a56a45a523a145a123a56a45,a199a222a62a139a24a39a135a73a42,a89a74a135a73a42a243Ea222a62a209a52
a104a189a123,a0a143a248a234a183a89a123e1,e2,e3a89a1a199a92a160a37a222,a104a108a77a193a123a154xa15a52a24a199
a37a222,a199a123a160a152a37a222a15a52a24a199a37a222,x a52u,va123a60a106,dx a104a108a52u,v a123a24a199
a24a39a135a73a42,a199a15a44a49a44a143e1,e2,e3 a123a24a199a34a117a28a92,a98a52a34a19a222,a199a24a189
a52e1,e2 a123a28a92,a89a52a79a143e3 a52a27a37a222,a196a49a199a24a189a52a243a35a193a222a62,a13e1,e2a209
a52a35a37a222,a196a49dxa24a189a52e1,e2a123a34a117a28a92,a199a123a248a106a183a119a41ω1,ω2,a253a122a116a183
a162a145a24a199a218a42:
dx = ω1e1 + ω2e2,(4.8)
a196a49a183a25a243a1395a199a24a39a135a73a42,ω1,ω2,ω13,ω23,ω12,a89a28a135a73a42a58a158a7a155,a199
a162a209a139a59a92a123a1a91a63a66,a89a199a210a128a210a135a73a42a255a88a1a91a117a159a38a122a175a11a59a92a13
a37a142a52,a25a243a183a162a210a139a218a42( 4.8),a183a162a82a218a42(4.3),(4.4)a42a143a183a162a123a145a19a199
a218a42,a183a243a3a193a89a44,a135a73a42a139a199a33a76a123a128a154,a210a52a135a73a220a39a128a520,a253a233
a149dx,d(dx) = 0,a233a149a127a91a60a106a123a105a135a73a220a39a24a189a127a1490,a89a210a35a104a149a243a56
a45a127a63a82a24a199a75a173,a242a82a199a123a30a142,a13a30a142a88a242a139a30a142,a82a30a142a82a220a39a24a189
a52a127a1490,a79a143a89a199a117a159,a89a171a106a166a136a232a210a139a196a162a123a51a174(homology)a117a159,a25
a243a28a7a180a31a143a192a220,a209a52a51a174a117a159,a58a158a7a155,a88a44a183a162a25a243a178a218a42(4.8)a38
a30dx a242da24a5a6a210a127a1490,a13a178a141a30a123a48a13a210a122d( omega1e1) + d(ω2e2),a253
d(ω1e1) + d(ω2e2) = 0,(4.9)
a213a63a104a105a135a73a3a193a139a24a199a24a39a123a155,a135a73a145a19a199a79a6a7a149a82,a15a13a211a131,a28
a210a178a199a162a135a73a234,a135a73a131a128,a210a21a25a196a122a116a123a42a6a52e1,e2,e3a123a34a117a28a92,a0
a143a199a123a248a106a52a19a39a135a73a42,a13a233a149a89a199a248a106a52a19a39a135a73a42a123a37a222a7a127a1490a123
a155,a196a139a123a248a106a209a7a127a1490,a149a52a28a152a42a13e1,e2a123a248a106a1430 a123a155,a210a122a116a183a5
a193a123a145a174a199a218a42:
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(4.10)
a183a88a45a251a178a199a41a234,a89a199a121a210a105a59a92,a44a193a253a44a122,a89a52a7a155a244a234a123a24a199a218
a42,a89a74a218a42a28a162a15a142a250a122a99,a79a143a243a202a47a86a222a23a88a68,a89a52a132a149a202a47a88
5
a245a161a126a135a73a42,a142a245a124a15a88a204a126a135a73a42,a217a34a89a105a59a92,a196a49a183a210a122a116a218
a42(4.10),a108a128a13e3a123a248a106a1430,a210a122a116a145a143a199a218a42:
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(4.11)
a89a52a58a158a58a158a7a155a123a218a42,a132a89a74a210a122a116a196a162a123Levi-Civitaa178a113a117,a25a243,
a89a199a15a119a201a100(connection),a233a149a201a100,a202a47a73a24a253a86,a44a49a89a222a105a200a105a200,
a217a34a105a59a92,a183a128a243a89a199a60a111a131a5,Levi-C ivitaa201a100a210a52ω12a89a199a24a39a135a73
a42,a213a63ω12 a52Ea176a62a123a24a199a24a39a135a73a42,a199a210a189a1a91a123a117a159,a38a122a183a44a49a178
a89a199a37a222a215a248a24a30a77a34a178a113a123a35a196,a89a52a31a143a63a135a17? a210a52ω12a89a199a24a39a135
a73a42a132a48a199(4.10)a113a92a104a189,a79a143a152a42a139a24a199ωprime12,a38a122a55a92a51a248a123a48a199a42,
a253
dω1 =?ω2 ∧ωprime12; dω2 = ω1 ∧ωprime12,(4.12)
a183a137a7a121a210ω12 = ωprime12,a253a144a139a24a199a44a21a117,a144a139a24a199ω12 a55a92a48a199(4.10),a15
a210a52a128,a34a38a139a12a105a24a199ωprime12a55a92a48a199(4.10),a0a143a183a162a178a220a199a48a199a35a62,a210
a122a116
ω1 ∧(ωprime12?ω12) = 0; ω2 ∧(ωprime12?ω12) = 0,(4.13)
a13a220a199a24a39a135a73a42a152a42a35a198a127a1490a123a155,a13a37a152a42a217a165a24a199a88a520,a0a143a217
a199a0a199a23a108a52a199a123a245a106,a89a52a105a83a106a33a59a92a123a192a220,a28a162a174a24a174a210a241a117
a234,ω1,ω2a52a88a1430 a123,a13a37a88a98a88a127a1490,a63a37a34a117a195a7,a79a36,ωprime12? ω12a17
a127a149ω1a123a245a106,a145a127a149ω2a123a245a106,a98a52ω1,ω2 a34a117a195a7,a196a49a199a7a127a1490.
a79a36a152a42a242a139a24a199ωprime12a55a92a51a248a123a48a199,a199a24a189a127a149?mega12,a89a52a113a92
a104a189a123,a0a143a183a126a89a199a90a159a196a162a123Levi-Civitaa178a113a117,a183a162a25a243a139de1 =
ω12e2 +?mega13e3,a183a162a178e3a89a24a49a82a62,a82a62a52a31a143a63a135a17? a34a38a139a24a199
a37a222,a28a178a199a123e3a82a62a131a128,a210a52a215a248e3a123a48a53a82a89a199a116a98a61a107,a196a49a28a178a199
a82a62a234a123a63a135a210a52a82a199a123a116a98a61a107,a28a178a199a82a62a123a155,a183a25a243a88a242a52a202a47a123
a135a73a234,a52a99a123a135a73a234,a183a126Da44a43a89a199a99a123a135a73,a149a52a210a122a116a89a28a218a42:
De1 = ω12e2,De2 =?ω12e1,(4.14)
6
a89a52a76a183a166a98a121a210a123a52a132a1a91a113a92a104a189a123,a89a52a79a143ω12a52a113a92a104a189a123,a34
a38De1 = 0a123a155,a183a210a128a89a199a37a222a52Levi-Civitaa63a66a5a178a113,a253a183a128a243a89a171a60
a111a5a199a210a52a178a113a123,a199a52a178a113a123,a89a52a139a99a123a63a66,a243Euclid a1a91a123a30a127,a89
a210a52a202a47a178a113,a25a243a52a202a47a178a113a123a77a18,a15a210a52De1 = 0a30a52Levi-Civitaa178
a113,a196a49a243a77a193a222,a28a126a24a39a135a73a42,a166a11a89a199a178a113a7a118a10a195,a13De1 = 0a123
a155,a233a149a37a222a52a199a135a73a48a199,a89a199a135a73a48a199a88a24a189a139a137,a98a52a34a38a28a139a24a30
a77a34,a28a210a44a49a215a248a89a30a77a34a70a137,a233a149a77a34a117a128,a196a139a123a89a74a60a106a209a52ta123
a60a106,a215a248a89a30a77a34a70a137,a44a49a70a116a137,a150a149a89a199a30a71,a183a128a89a199a37a222a210
a52Levi-Civita a178a113,a196a49a44a89a199a189a66a44a49a234a137a89a199a178a113a117a203a77a34a123a160a10a139
a7,a34a38a139a24a199a154,a12a105a164a139a24a154,a0a143a28a201a89a220a154a139a220a30a88a51a123a77a34,a28a178
a51a24a199a37a222a215a62a24a30a77a34a178a113,a63a215a145a19a30a77a34a178a113,a23a116a123a24a196a88a52a51a24
a199a37a222,a150a149a89a199a178a113a117a166a77a34a123a160a10a139a7a248,a52a202a47a123a178a113a117a123a77a18,a202
a47a123a30a127,a178a113a117a243Euclid a178a193a176a222a52a253a233a123a90a189a234a123,a25a243a89a199a30a127a52a203
a77a34a123a160a10a139a7a248,a89a210a52a201a100,a183a166a98a178x a89a199a37a222da220a39a127a1490,a13a51a248
a123,a183a44a49a233eia89a199a37a222a15d a220a39,a253d(dei) = 0,a79a36a174a241a117d(dei) = 0,a210
a139:
dωij = ωik ∧ωkj,(4.15)
a34a19a222,a105a59a92a143,a183a233dei = ωijej a70a135a73,a108a128a79a143ωija52a24a39a135a73a42,a196a49
a97a148a139a199a139a82,a253a139a123ωij ∧dej,a149a52a139
d(dei) = dωijej?ωij ∧dej = 0,(4.16)
a98a52dej = ωjkek,a196a49a62a24a199dωijeja149a143dωikek a150a139a7a248,a89a52a79a143a28a233
a149j,ka52a70a90,a44a85a196j,a15a44a85a196k,a209a52a441a1163,a195a196a162a123,a79a36a210a122a116
a218a42(4.15),a89a52a24a199a228a253a123a218a42,a23a248a102a38,a217a34a105a59a92,a131a217a52a2433a145
a123a60a111,a105a59a92,a34a19a222,a141a30a89a229a117a523a49a131a90,a98a52a893a49a52a233a149j a70
a90,a79a143ω a52a39a233a193a123,a196a49a183a162a117a23ωik,i negationslash= k,a221a86a210a117a23a23ω13,a243a5
a193a0a199a218a42a23d omega13 a97a148a52ω1jωj3,a98a52a34a19a222,ja144a21a127a1492,a89a52a79
a143j = 1a30ω11 = 0,j = 3 a30ω33 = 0,a196a49a141a30a23a248a52a89a143a248a131a90,a217a34a210a144
7
a139a24a49,a196a49a28a210a122a116
dω13 = ω12 ∧ω23,(4.17)
a51a248a44a49a122a116
dω23 =?ω12 ∧ω13,(4.18)
a89a74a24a196a210a52a196a162a123Weingartena218a42,a58a158a59a92,a196a49a89a74a218a42a209a52a44a59a92
a123a105a135a73a193a253a122a116a123,Weingartena218a42a105a139a63a135,a28a7a199a203a198a117a123ω1,ω2a123
a218a42a7a118a123a155,a113a92a52a24a199a111a231,a253a213a63a116a89a74a218a42a166a218a42(4.10)a35a145.a89a199
a111a231a52a139a1a91a63a66a123,a25a243a199a126a117a189a196a162a123Betti Transform ation,a89a199a183a88
a251a89a234,Weingartena218a42a203a198a117dωia123a218a42a123a35a145a117a52a139a253a47a123a1a91a63a66a123.
3 a77a193a123a228a253a88a35a42
a222a193a183a255a88a234De1a123Levi-Civita a178a113a117,a152a42a82a89a199a37a222a243e3a89a199a48a53a123
a155,a183a210a122a116ω1 ∧ω13 +ω2 ∧ω23 = 0,a253a218a42(4.1 1),a79a36,ω13,ω23a209a52ω1,ω2
a123a34a117a28a92,a183a44a49a85a196
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(4.19)
a132a89a199a183a122a116a89a199a77a193a123a7a155a123a88a35a42,a24a199a77a193a166a12a24a199a77a193a139a31a143a73
a47? a89a199a73a47a243a149a77a71,a77a71a52a77a193a222a123a60a106,a7a48a128,a24a196a123a155,a183a162a126a145
a24a228a253a42a203a145a19a228a253a42a117a44a43:
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (4.20)
a145a19a228a253a42 II = (?dx,de3) = aω2 + 2bω1ω2 + cω22,(4.21)
a233a149a145a24a228a253a42,a183a119a199(I).a199a210a52a77a193a123ds2 = (dx,dx),a132a149dx =
ω1e1 + omega2e2,a98a52e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49a210a122a116ω21 + ω22.
a79a36ωa895a199a24a39a135a73a42a209a139a173a7a123a1a91a63a66,ω1,ω2 a123a178a48a90a210a52a77a193a123
a221a222,a2433a145a56a45a176a62,a77a193a104a108a139a24a199a221a222,a0a143a199a123a221a222a52a31a143a17? a210
a52a199a123Riemann a221a222,a52a24a1992a39a123a135a73a42,a89a1992a39a135a73a42a59a92a244a234,a210
8
a52ω21 + ω22,a0a143,a183a15a44a49a233e3a89a199a37a222a82?(de3,dx),a76a183a166a98a196a85a123,a199
a210a127a149aω2 + 2bω1ω2 + cω22,a89a15a52a24a1992a39a135a73a42,a89a199a24a196a119a41a145a19a228a253
a42,a0a143a139a24a199a145a24a228a253a42,a164a139a199a145a19a228a253a42,a28a210a82a199a123a1a110a138,a1a110
a138a123a90a210a52a + c,a1a110a138a123a232a210a52ac? b2,a24a196a143,H = a+c2 a119a41a77a193a123a165
a77a71,K = ac? b2 a119a41a77a193a123Gaussa77a71,a89a176a7a155a244a234,Gaussa77a71a139a142
a245a139a85a123a117a159,a79a36a89a52a14a143a248a445a199a24a39a135a73a42a132a199a162a123a34a117a28a92a123a7a248
a210a122a116a77a193a123a88a35a42,a165a77a71a203Gaussa77a71a89a220a199a88a35a42a195a85a77a193a123a1a91
a117a159,a7a48a128,a44a49a121a210Gauss a77a71a52a116a123a155,a77a193a52a243a123,Gaussa77a71a52a139
a123a155,a77a193a210a139a145a154,a210a54a106a145a154,a142a245a1a91a117a159a209a44a49a126a89a220a199a77a71a117a195
a85,a196a49a52a77a193a176a62a220a199a33a204a7a123a88a35a42,a89a199a88a35a42a52a60a106,a49a128a183a162
a122a116a123a52a24a39a135a73a42,a13a24a39a135a73a42a88a76a142a52a46a54a196a190a52a31a143a63a135,a98a132
a199a123a228a174a44a49a122a241a60a106a117,a89a60a106a104a108a52a234a137a122a7a118a56a249,a34a122a116a165a77a71
a203Gauss a77a71,a165a77a71a127a1490,a24a196a119a41a244a66a77a193,a196a162a123a244a66a77a193,a222a190a210
a52a28a178a24a30a85a20a123a77a34a56a243a61a6a121a176a62a196a196a123a77a193a210a52a193a232a33a66a123a77a193,
a199a123a165a77a71a1430,a196a49a89a139a59a92a123a1a91a63a66,a33a163,a4a150a25a243a209a139a105a245a7a149a244
a66a77a193a123a207a196,Gaussa77a71a205a7a155,Gaussa77a71a52a127a149ω13 ∧ ω23,a243a77a193a222a15
a24a248a139Gaussa110a243,a77a193a154a154a139a24a199a92a160a27a37a222,a178a89a199a92a160a27a37a222a23a143a24
a199a204a187a1431 a123a69a193a123a154,a210a178a77a193a110a243a116a69a193a222a86a234,a154a199a154a139a24a199a92a160a27
a37a222,a28a2430a154a152a24a199a92a160a27a37a222a203a199a178a113,a199a123a224a154a210a243a92a160a69a193a222,a0a143
a233a149a196a139a123a154a209a41a89a199a232a5a123a155,a210a243a92a160a69a193a222a122a116a24a199a75a173,a243a89a199a110
a243a5,a220a199a193a232a195a163a123a7,a253a199a123a52(imag e)a123a193a232a195a163a203a198a117a193a232a195a163a123
a7a210a52Gauss a77a71,a89a248a24a5a6a210a23a241a117a234,a196a49a89a199Gaussa77a71a139a105a59a92
a123a1a91a63a66,a89a30,a77a193a123a255a88a52a24a199a77a18,a77a193a123a30a127a139a77a34,a77a34a15a139a24
a199Gaussa110a243,a0a199Gaussa110a243a52a82a92a160a35a37a222,a217a34a15a44a49a82a92a160a27a37a222.
a0a143a77a34a123a30a127,Gaussa110a243a178a35a34a110a243a116a92a160a203a222a62,a178a92a160a203a123a221a222a250
a198a117a89a199a221a222a248a210a52a77a71,a25a243a210a178a89a199a10a39a77a18a116a176a145,a77a18a1162a145,a253a77
a18a116a77a193,a196a49a183a132a89a199a77a193Gaussa110a243a116a24a199a92a160a69a193a222a62,a89a220a199a193a232
a123a7a210a52Gaussa77a71,a196a49a89a52a24a199a58a158a7a108a123a1a91a221a222,a25a243a139a199a105a7a155
a123a7a248,a222a62a139dωik = ωij ∧ ωjk,a25a243a183a178a89a199a218a42a126a116ω12,i,k = 1,2,a149
9
a52dω12 = ω13 ∧ ω32,a98a52ωa52a39a233a193a123,a196a49ω32 =?ω23,a89a176ω13 ∧ ω23 a210
a52ac?b2,a15a210a52Gaussa77a71,a196a49a183a210a122a116a218a42
dω12 =?Kω1 ∧ω2,(4.22)
a217a165,Ka52Gaussa77a71,a89a52a24a199a13Gauss a58a158a175a199a123a218a42,a196a49Gaussa178a89
a199a189a174a119a41Theorem Egregious,Egregiousa52a110a182a9a241,a63a135a141a89a52a24a199a234
a88a122a123a189a174,a143a31a143a17? a199a121a210a234Gaussa77a71a144a203a77a193a123Riemann a221a222a139
a7,a203a89a199a77a193a243a56a45a123a160a152a195a7,a79a143a243a89a199a218a42a176a62,ω12a46a178a121a210
a234a144a203a77a193a123Riemanna221a222a139a7,a108a128ω1 ∧ ω2a210a52a243a89a199a221a222a5a123a193a232a195
a163,a104a108a144a203ds2a139a7,a253a144a203Riemanna221a222a139a7,a196a49a175a108Gaussa77a71a52a24
a199a77a193a243a56a45a176a62a123a24a199a88a35a42,a98a52a199a144a203a77a193a123Riemanna221a222a139a7.
a166a233a155a128,a28a178a77a193a35a166(deform)a24a5a6,a38a122Riemanna221a222a88a35,Gaussa77
a71a210a88a35,a196a49Gaussa77a71a139a89a143a173a7a123a117a159,a183a46Gau ssa41a234a142a245a7a155
a123a90a164a224a123a136a42,a89a199a189a174a23a108a52a198a1a47a97a220a123a24a199a136a42,a52a105a1a47a123
a24a199a136a42,a34a19a222,a183a162a210a178a89a74a48a199a126a105a135a73a135a73a24a5a6,a122a116a228a253a218
a42,a108a128a210a122a116a89a74a218a42,a5a39,a183a7a89a132a89a199a218a42a121a210Gauss-Bonneta218
a42,Gauss-Bonneta218a42a52a77a18a174a110a111a174a110a131a90a127a149180o,a82a89a199a218a42a77a18a116
a77a193a123a60a111,Gauss-Bonneta218a42a123a97a126a58a158a131a18,a121a210a123a30a127,a183a41a170a188
a126a203a45,a89a203a45a227a135a126a122a133a236a24a74,a34a189a89a199a45a52a24a199a133a123a34a45(complex
line),a0a143a89a199a133a34a45a123a14a145a52a30a133a34,a243a133a34a176a62,a253a233a138a127a1491a123a133a106
a52a24a199a203a177,a210a52a203a45,a89a199a10a39a243a212a174a222a52a228a253a123,a25a243a76a27a180a122a105a245
a123a52a98a1a91( symplectic geometry),a92a139a98a136a232(symplectic structure)a88a212
a139a126,a28a7a178a98a1a91a126a116a222a6a197a166a123a155,a137a7a28a222a24a199a133a34a45,a210a52a183a162
a25a243a196a89a123a192a220a123a24a199a77a18,a183a162a25a243a892a145,a24a116a212a174a123a155,a56a45a166a30a45
a28a243a24a229a524a145,a196a49a28a228a253a123a56a45a524a145,a2434 a145a56a45a104a165a139a24a199a133a34a45.
a79a36a210a122a116a89a199Gauss a77a71,a25a243a89a199a77a71a522a39a135a73a42,a199da24a5a6a127a1490,
a210a122a116Maxwella48a199,a196a49a135a232a73a243a1a91a222a97a126a52a142a245a212a174a123a228a250,a33a163
a183a243a21a41a166a22a236a147a222a85a234a24a159a169a57,a119a41a21Gauss-Bonneta218a42a166Maxwella48
a199a22,a28a162a152a42a21a73a116,a44a49a86a23a24a23,a183a46a178a89a159a169a57a252a117a201a76a27,a98a52
10
a25a243a79a122a105,a150a73a116,a5a39a183a46a89Gauss-Bonneta218a42a123a24a199a121a210,a89a199a203a183
a162a123a211a42a139a58a158a183a35a123a7a248,a2435 0a35a3,a243a102a210a220a8a201a76,a183a115a135a73a1a91,
a210a122a116Gauss-Bonneta218a42a123a89a199a121a210,a202a47a86a222a133a236a245a234,Gauss-Bonneta218
a42a204a7a63a135a52a128,a28a139a127a91a24a199a77a193,a77a193a222a62a139a24a199a245a30a111,a178a89a199a245
a30a111a123Eulera1a110a106a44a43a171a171a77a71a123a28a92,a89a52Gauss-Bonneta218a42,a183a122a116
a89a199a121a210,a183a46a89a52Gauss-Bonneta218a42a33a59a92a90a33a7a108a123a24a199a121a210,a210a52
a178dω12a24a135a73a210a122a116Gauss-Bonneta218a42a232a73a123a42a6,a108a128a183a243a24a202a143a1a35
a123a30a127,a116a234Princeton a41a176a145a123,a104a108a7a108a13a108a143,a183a46a27a6a178a89a199a77a18
a116a176a145,a89a199a52a44a49a41a116a123,a49a128,a183a164a41a234a74a47a123a211a42,a196a49a44a49a128a183a89
a199Gauss-Bonneta218a42a123a121a210a243
a183a196a41a123a211a42a165a52a183a33a245a161a123a24a199,a183a46a25a243a44a49a136a101a234.
11
