a145a202a89 a77a193a88(a19)
a9Gauss-Bonneta218a42
2001a3511a21923a134
1 a77a193a88a21a48a123a59a144
a105a176a108a145a166a76a27a68a193a234,a183a243a26a211a176a212a234a1a21,a28a162a44a49a23a241a117a183a164a150
a139a113a92a80,a88a44a183a250a122a183a164a52a203a76a27a89a89a89a74a192a220,a0a143,a183a152a21a7a89a123
a52Gauss-Bonneta218a42,a89a199a218a42a139a35a104a123a63a66,a15a139a35a104a123a187a36,a131a217a203a183
a199a124a123a211a42a15a139a7a248,a196a49a183a7a10a24a10a183a203a89a199a175a11a52a14a143a248a123a7a248,a183
a162a222a39a89a116a77a193a88,a77a193a88a52a135a73a1a91a176a62a33a173a7a123a24a92a73,a79a143a142a245a135
a73a1a91a123a25a54a243a2433a145a56a45a176a1232a145a77a193a123a60a89a46a178a151a9a234,a51a30,a79a143a199
a52a2433a145a56a45a176a62,a89a199a1a91a123a60a89a52a44a49a23a68a123,a88a52a113a92a126a83a106a117a44a43.
a77a193a88a139a105a159a123a187a36,a33a0a123a104a108a52Monge,Monge a52a27a41a123a76a106a166a27,a198
a144a11a9a233a117a163a139a74a217a196,a196a49a198a248a234a41a76a166a115a71a131a105,a198a233a27a41a123a115a184a139
a105a245a107a47,a198a52a252a187a85a142a5a123a24a199a252a202,a208a207a252a187a85a41a47,a198a52a252a187a85a117a127
a123a48a1a92a159,a4a150a164a203a248a252a187a85a86a132a249a75a67,a79a143a198a123a107a47,a27a41a123a176a211a166
a68(Ploytechnique)a210a79a193a229a117a234,a105a159a24a227a30a207,a27a41a33a80a123a166a9a209a243a176a211
a166a68,a183a46a176a211a166a68a15a142a54a25a243a123a56a149,a139a142a245a80a123a166a9,a190a152a128,a27a41a24a199
a33a76a123a106a166a27Poincare a210a52Ploytechniquea123a166a9,Mongea52a145a24a199a85a7a149
a135a73a1a91a86a123a124,a198a123a86a210a119a41a21a135a232a73a243a1a91a222a123a97a126a22,a89a15a52a183a7a89
a123a11a248,a79a36,a27a41a123a115a184a243a135a73a1a91a139a24a199a35a104a123a4a58,a248a234Mongea253a124
a131a105,a132a149a198a243a176a211a123a107a47,a198a139a105a245a166a9,a209a52a243a135a73a1a91a139a35a104a224a26a123
a124,a108a128a243a135a73a1a91a197a76a21a48a131a165,a7a118a116a24a74a123a52a27a41a12a105a24a199a76a106a166
a27Darboux,Darbo uxa52a27a41a41a166a211a123a179a86a159,a196a49a243a198a123a30a207,a198a52a243a41
a5a142a139a105a245a107a47a123a24a199a124,a198a88a157a52a179a86a159,a15a52a174a169a76a166a174a166a211a123a211a159.
a198a123a33a76a123a211a42a52a143a253a21a77a193a88a22,a183a46,a89a143a76a253a52a106a166a169a26a176a62a123a207a123
a24a199a169a26,a25a243a44a234a132a149a199a52a27a169a123,a105a245a124a88a23a89a199a86,a183a46a89a74a124a233a135
1
a73a1a91a98a232a24a154a234a137,a198a162a97a148a23a89a199a86,Darbouxa123a86a89a122a58a158a80,a221a105a234
a105a245a97a238,a2431941 a35,a183a243a220a8a201a76a115a86,a115a21a135a73a1a91a22,a15a89a116a77a193a88.
a89a116a77a193a88a30,a104a108a210a23Darbouxa123a86,a210a46a116Darbouxa123a86a176a62,a24a199a204a7
a123a48a27a52a126a217a196a41a37,a15a210a52a102a126a217a196a41a37a27,a198a126a122a58a158a180a142,a41a122a58a158
a131a164a224,Darbouxa227a135a88a126a123a24a154a52a198a88a126a105a135a73,a183a46,a183a123a49a52a89a135a232
a73,a13a135a232a73a28a7a89a116a245a195,a245a35a106a123a30a127,a89a199a105a135a73a88a21a27a189,a89a52a79a143
a243a245a35a106a123a30a127,a33a139a72a123a211a228a52a105a135a73,a105a135a73a44a49a28,a62,a44a49a198,a44a49
a135a73,a196a49a139a105a245a83a106a123a228a174a44a49a126a116a105a135a73,a51a30,a24a199a105a135a73a15a52a24a199
a42a6,a89a199a42a6a201a163a105a245a106a166a175a11,a88a11a52a199a123a1a91,a164a52a199a123a73a219,a209a201a28
a105a245a97a238,a79a36a52a58a158a139a126a123,Darbouxa123a98a154a52a198a150a139a126a105a135a73,a198a126a217
a196a41a37a27,a98a52a150a139a105a135a73,a79a36,a139a105a245a211
a42,a88a126a105a135a73,a14a143a205a17? a198a15a52a164a7a126a135a232a73a123,a88a126a105a135a73,a198
a126a160a135a73,a126a160a135a73a7a105a135a73a139a122a245a234,a79a143a28a123a77a193a522a145a123a56a45,a196a49
a233a149a220a199a35a106,a253a77a193a123a107a106u,v,a28a7a233ua70a160a135a73,a233va70a160a135a73,a89a176a62
a139a105a245a160a135a73,a13a126a105a135a73a210a59a92a245a234,a98a52a198a88a102a126a105a135a73,a89a52a105a219
a5a123a47a60,Darbouxa52a21a25a24a39a105a135a73a42a123a145a24a124,a24a199a52Darboux,a24a199
a52Frobenius,a198a162a220a199a124a33a0a21a25a89a199a192a220a123,a98a52a127a116a97a126a116a77a193a207a196a123
a30a127,a88a127a119a143a31a143,a198a150a139a126,a15a142a52a132a149a4a58a123a7a248,a198a85a234a105a135a73a131
a128,a120a209a88a195a234,a196a49a198a88a126a105a135a73a234,a183a166a98a89a234,a2431941a35,a183a166a28a243a102
a210a115a89a199a49,a183a105a7a108a143a46,a143a31a143a88a126a105a135a73a17?a196a49a183a210a126a105a135a73a46a27
a6a41Darbouxa196a41a123a211a42,a221a86a128a150a232a41a77a193a88a90a24a74a1a91a123a255a88,a183a102a126
a105a135a73,a79a36a183a122a116a24a199a105a80a123a234a137.
2 a77a193a88a228a253a19a142a123a195a250
a31a143a119a105a135a73a17?a210a52a28a21a25a7a207a196a77a193a123a155,a77a193a52a24a1992a145a123a22a111,a199a243
a202a47a56a45a176a62a522a145a123,a196a49a199a222a62a127a63a24a199a154a52a220a199a35a106,a47a158a210a119a41a107
a106,a98a52a25a243a17,a183a162a210a119a199a219a92a43a41u,v,a79a36a199a123a43a41a522a199a35a106a123a60a106,
a196a49a52a89a199a30a71a38a199a243a154a24a154a139a24a199a35a178a193,a89a199a35a178a193a104a108a105a7a155,a79a143
2
a183a162a150a139a27a6a207a196a133a236a123a67a111,a183a162a144a21a207a196a33a59a92a123a152a134a34,a178a193a89a74a192
a220,a35a178a193a203a77a193a139a33a183a35a123a7a248,a0a143,a92a128a183a35a123a7a248a88a234,a24a189a7a137a219
a143a21a234a137a251a7a118a205a253a47a123a24a74a175a11,a139a24a199a35a178a193,a243a89a199a35a178a193a222a522 a145
a123,a149a52a154a24a154a210a139a142a245a37a222,a15a210a52a35a37a222,a35a37a222a210a52a203a89a199a77a193a35a35
a123a37a222,a79a143a89a199a77a193a52a243Euclid a56a45a176a62,a196a49a183a44a49a89a89a199a37a222a123a159a221.
a143a59a92a229a68,a183a33a149a255a88a159a221a127a1491 a123a37a222,a253a92a160a37a222,a196a49a139a24a87a92a160
a35a37a222,a203a89a74a92a160a35a37a222a18a134a123a139a12a105a24a199a37a222,a183a162a34a189a199a52a82a196a92a160
a123,a0a143a89a199a37a222a183a162a119a41a92a160a27a37a222,a7a213a63a123a52a89a176a210a139a24a199a1a91a25a54
a21a9a234,a79a143a34a38a89a199a77a193a65a178a234a123a155,a92a160a27a37a222a44a49a53a222a18,a15a44a49a53
a5a18,a166a233a155a128,a89a199a77a193a248a234a52a24a1992a145a123a22a111a131a105,a199a164a139a24a199a189a53,a243
a77a193a222a28a52a126a48a53a221,a164a52a203a31a48a53a221,a89a199a221a196a52a105a88a24a248a123,a196a49,a28a7
a189a14a143a248a6a221a196a52a126a48a53,a89a210a52a7a201a77a193a24a199a189a53,a189a53a139a234a131a128,a199a123
a92a160a27a37a222a210a189a234,a92a160a27a37a222a243a89a199a48a53a44a49a53a222a18,a15a44a49a53a5a18,a189
a234a24a199a131a128,a89a199a77a193a15a210a189a53a234,a89a52a105a173a7a123a24a199a10a39,a175a108a35a139a123a144
a52a24a199a110a82,a98a52a89a52a24a199a105a173a7a123a10a39,Mobiusa52a121a41a149a76a123a1a91a166a27.
a79a143a28a7a189a53,Mobiusa21a25a139a74a77a193a88a21a189a53,a89a104a108a52a105a139a63a135a123a24a71a47
a60,a28a162a76a27a209a127a119a123a89a199a67a111,a210a52a252a24a60a146,a28a178a199a221a24a87a203a229a117a123a155,
a210a122a116a196a162a123Mobius a77a193,a199a150a27a6a189a53,a89a52a1a91a222a105a139a63a135a123a24a71a47
a60,a0a143a183a162a34a189a77a193a46a178a104a189a234a24a199a48a53,a139a234a89a248a189a53a123a77a193a131a128,a1
a91a60a89a52a14a143a248a123a17?a132a92a160a27a37a222e3,a217a165e3 a52a2433a145a56a45,a28a21a25a139a24a71
a47a34,a210a52a128a28a92a213a255a88a77a193a88a234,a28a24a189a7a188a126a77a193a222a123a92a160a35a37a222,a183
a119a89a199a92a160a35a37a222a143e1,a89a248a183a210a139a234a24a199a41a37,a199a139a234a145a24a199a92a160a37a222a90
a145a174a199a92a160a37a222,a152a42a56a45a52a189a53a123,a145a19a199a92a160a37a222e2 = e3 ×e1a15a210a113a92
a104a189a234,a196a49a183a210a139a199a92a160a41a37,a92a160a41a37a210a52a174a199a92a160a37a222a149a76a24a189a123
a39a147,a52a144a35a18a134a123,a143a31a143a92a160a41a37a243a1a91a123a207a196a131a165a52a89a143a173a7? a210a52
a79a143a1a91a52a202a226a228a196a107a207a196a56a45a243a228a196a131a5a88a35a123a1a91a117a159,a13a89a228a196a107
a210a52a41a37a196a196a123a56a45,a79a143a52a139a24a199a63a37a144a139a24a199a228a196a178a24a199a41a37a35a143a217
a199a123a41a37,a150a149a92a14a123a92a160a41a37a203a89a199a228a196a107a123a195a163a196a24a24a233a97,a88a98a52a24
a24a233a97,a13a37a233a97a226a209a95a192a90a24a35a123a117a159,a196a49a228a196a107a105a7a155,a79a143a56a89a123
3
a228a196a88a127a119a243a137a219a123a60a89a131a5a152a91a44a49a255a174,a13a234a41a37a131a128,a210a44a49a255a174a234.
a41a37a210a52a37a222a234,a13a37a222a24a196a52a1393a199a73a222a123a37a222,a13a154a24a199a73a222a52a60a106,a210
a44a49a178a199a135a73,a28,a62a31a143a123,a37a222a139a28,a62a123a228a174,a15a139a135a73a123a228a174,a243a236
a171a63a66a5,a164a44a49a139a232a73a123a228a174,a196a49a183a25a243a210a44a49a135a73,a183a207a196a77a193a123
a30a127,a88a144a24a199a41a37,a0a143a243a77a193a123a154a154,a89a248a123a41a37a139a245a232a17? a34a38a28a65
a122e1a123a155,a51a30a89a199a77a193a52a189a53a123,a89a199a41a37a210a113a92a189a234,e1a52a31a143a17? e1a52
a89a199a77a193a243a89a24a154a123a92a160a35a37a222,a0a143a89a199a77a193a139a245a232a92a160a35a37a222a17? a154a154
a139a24a87a243a35a178a193a222a62a127a149a92a160a37a222,a13a77a193a522a145a123,a196a49a199a162a196a196a123a56a45
a523a145a22a111,a89a52a79a143a89a199a154a52a243a77a193a222a35a196,a522a145a123,a25a243a243a154a189a234a131a128,
a92a160a35a37a222a44a49a119a248a199a221a24a87,a196a24a199a203a177,a196a49a199a52a145a28a24a145,a523a145,a89
a1993a145a56a45a58a158a7a155,a183a46a25a243a34a19a222,a28a162a7a234a137a135a232a73a221a86a234a137a203a135a232
a73a5a86a123a106a166a221a86a243a106a166a165a123a97a126,a89a199a60a89a52a33a59a92a123,a51a30a52a33a139a126a123.
a196a49a183a139a24a1993a145a56a45,a132a149a154a24a154a139a199a203a177,a25a243a139a199a214a9a119a41a203a45,a221a86
a203a177a45,a45a52bundle,a196a49a28a7a207a196a77a193a123a1a91a117a159,a126a89a199a137a219a123a48a27,a24
a189a7a255a88a199a123a203a45,a255a88a203a45a234a131a128,a24a35a209a59a92a234,a79a143a24a35a209a52a37a222,
a13a52a37a222a123a155,a199a139a73a222,a210a44a49a135a73,a210a44a49a126a83a106a221a86a135a73a123a228a174,a183
a162a52a243a255a88a135a232a73,a183a162a34a189a141a116a31a143a60a106a209a44a49a135a73,a183a119a243a89a199a77a193a222
a123a154a143x,a0a143dx a52a24a199a37a222,a210a52a44a198a154a203a248a89a199a154a123a37a222,x a52u,v a123a60
a106,a13u,va52a77a193a222a123a219a92a43a41,a196a49a28a44a49a85a241dx,a34a38x a33a155a243a77a193a222,
a0a143dxa24a189a52e1 a166e2a123a34a117a28a92,a196a49a243a89a199a143a48,a183a210a223a73a188a126a105a135a73a123
a10a39,a34a19a222dxa52a24a199a37a222a138a123a24a39a135a73a42,a196a49a199a52e1 a166e2 a123a34a117a28a92,
a199a123a28a92a248a106a52a24a39a135a73a42,a196a49dxa44a49a85a143
dx = ω1e1 + ω2e2,(5.1)
(dx,dx) a210a52a183a162a77a193a123a169a121a221a222,a79a143e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49
ω21 + ω22 = ds2 (5.2)
a210a52a169a121a221a222,a152a42a89a199a56a249a234,a89a233a149a202a47a89a135a73a1a91a59a92a245a234,a79
a143a202a47a135a73a1a91,a169a121a221a222a7a85a196gijdxidxj,a89a52a79a143a243a35a56a45a176a196a188a126
4
a123a43a41a52a127a63a123Ca rtesiana43a41,a199a88a24a189a18a134,a15a88a24a189a52a92a160,dx a127
a149ω1e1 + ω2e2,a98a52a183a162a105a135a73a139a199a228a253a123a117a159,a210a52a242a126a24a39a123a155,a199a127
a1490,a89a210a52a202a47a128a123a160a135a73a44a49a52a98a166a123a30a71,a24a248a123,a15a210a52a122a116a123a160a135
a73a166a135a73a123a39a147a195a7,a196a49a28a210a178da126a116dxa222a62,a24a189a127a1490,a28a178a141a30a48a13
a123a155,a210a122a116d(ω1e1) + d(ω2e2),a213a63a104a105a135a73a3a193a139a24a199a24a39a79a42a123a155,a135
a73a145a19a199a79a6a7a149a82,a15a13a211a131,a44a49a122a116
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(5.3)
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(5.4)
a183a204a243a5a193a201a241ω12,ω13,ω23,a17a108a126a135a232a73a234,a196a49a44a49a178(xe1e2,e3)a123a135
a73a44a143e1,e2,e3a123a34a117a28a92,a89a199a34a117a28a92a178deia85a196ωijej,a25a243a183a126a135a73a1
a91a202a47a123a110a82,a34a38a139a24a199a141a106a7a173a133a123a155,a210a44a43a35a28,i,j,ka441a1163,a28
a178dei a85a196ωijej,ωija123a1a91a63a66a105a210a23,a28a25a243a139a24a28a41a37,a89a28a41a37a203a24
a28a107a106a139a7a248,a13a233a149a89a24a28a41a37,a210a139a24a199a249a163a41a37,a89a199a249a163a41a37a203a198
a117a41a37a123a7a248a210a52ωij,a89a7a248a52a132a24a39a135a73a42a117a44a43a123,a79a36a210a139
dei = ωijej,(5.5)
a89a28a48a199a42a105a7a155,a199a210a44a43a220a199a249a163a41a37a144a35a123a7a248,a243a89a199a60a89a131a5,
a135a73a1a91a203a197a166a88a76a24a248,a197a166a128a128a35a106a52a30a45,a196a49a24a199a41a37a203a248a30a45a243
a35a196,a79a36a28a114a199a41a37a144a139a24a199a35a106,a209a52a30a45ta123a60a106,a25a243a183a162a52a24a199a77
a193,a154a154a139a142a245a41a37,a196a49a183a89a41a37a123a107a106a523,a89a52a79a143a139a35a193a123a219a92a43
a41,a145a139a35a37a222a243a178a193a176a62a35a166a123a43a41,a196a49a183a25a243a89a199a7a35a106a523,a164a79
a143E a89a56a45a523a145a123,a7a35a106a176a234,a196a49a89a52a139a198a79a38a122a105a135a73a139a72,a183a162a46
a82dei a85a196a48a199(5.5),ωija233a149i,ja52a39a233a193a123,a89a52a79a143a183a123a41a37a52a92a160a41a37,
a253a79a143(ei,ej) = δij,a196a49a199a52a39a233a193a123,a79a36ωija34a19a222a105a59a92,a28a178(ωij)a85
a241a117,a199a52a24a199a48a106,a89a199a48a106a52a39a233a193a123,a196a49a243a233a110a34a123ωa127a1490,a217a155
a123a233a248a233a110a34a52a39a233a193a123,a79a36a34a19a222a144a1393a199a24a39a135a73a42,ω12,ω13,ω23.
a183a46a183a222a39a121a210a234ω12a132dω1,dω2a123a48a199(5.3)a113a92a104a189,a89a52a24a199a173a7a123a189
5
a174,a89a52a38a122Levi-Civitaa241a214a123a173a7a189a174,a183a25a243a178a48a199(5.5)a70a105a135a73,a79
a143d(dei) = 0,a196a49a141a30a123a155,a183a210a122a70dωij,a136a42a122a116a123a52
dωij = ωik ∧ωkj,(5.6)
a79a36a89a74ωa131a45a139a105a59a92a123a7a248,a59a92a122a88a122a234,a79a143a31a143a17?a79a143a233
a149dωij = ωik ∧ωkj,i,ja52a88a35a127a123,a152a42a35a127a234a123a155,ωii a520,a89a52a79a143ωa52
a39a233a193a123,a196a49a28a82i negationslash= j,a152a42ka127a149i,a11ωii = 0; ka7a127a149j,ωjj = 0,a196a49k
a88a127a149i,a88a127a149j,a79a143a183a162a52a2433a145a56a45,ka144a139a24a199a44a21a117,a79a36a89a199a23a248
a105a0a219a123a48a199a42,a199a123a141a30a144a139a24a49,a183a222a39a178a199a85a5a117a234,a210a52
dω12 = ω13 ∧ω32,(5.7)
dω13 = ω12 ∧ω23,(5.8)
dω23 = ω21 ∧ω13,(5.9)
a131a217a52a122a116dω12 = ω13 ∧ω32a89a199a218a42,a98a52ω a52a39a233a193a123,a196a49a210a122a116
dω12 =?ω13 ∧ω23,(5.10)
a13ω13,ω23a209a52ω1,ω2a123a34a117a28a92:
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(5.11)
a89a166a28a210a122a116a5a193a89a199a218a42:
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(5.12)
Ka52Gaussa77a71,Gaussa77a71a210a52K = ac?b2,a89a199a218a42a88a122a234,a104a35Gaussa88
a52a89a248a122a116a89a199a218a42,a52a126a107a123a48a27a122a116a218a42,a199a119a41Theorem Egregium,
a126a165a169a89,a199a52a24a199a219a201a123a189a174,a201a123a189a174,a28a251a251a23a23a199,a199a52a105a201a123,a79
a143a183a162a123Ea52a92a160a37a222a45,a199a89a199a203a177a45a233a149a77a193M a139a24a199a61a107,a233a149a203a177
a45,a139a89a199a92a160a37a222,a183a82a199a123a198a154a210a52a199a123a61a107,a79a36a183a162a25a243a123a1a91a7
a44a3a10a39a222a7a118a133a236a234,a210a52a128,a88a144a52a139a24a199a77a193a221a86a139a24a199a77a34,a25a243
6
a139a220a199a56a45,a139Ea89a199a56a45a90a77a193M,a47a34a222,a139a234a77a193M,a108a128a139a132a196a139a199
a123a92a160a37a222a196a196a123a56a45a52E,a199a52a77a193a123a203a177a45,a25a243a47a158a119a89a199a77a193a52a142
a56a45,a199a52a243a142a222a123a24a199a56a45,a196a49a210a139a24a1993a145a123a203a177a45,a199a123a142a56a45a52a183
a162a123a77a193,a243a89a199a60a89a131a5,a183a162a139a1a199a24a39a135a73a42,a243a56a45a176a62a139a24a39a135
a73a42ω1,ω2,a108a128a139ω12,ω13,ω23,a24a2255a199a24a39a135a73a42,a217a165ω13,ω23a52ω1,ω2a123
a34a117a28a92(a218a42(5.11)),a199a44a43a77a193a123a1a91a117a159,a183a162a139
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (5.13)
a145a19a228a253a42 II = (?dx,de3) = ω1 ∧ω13 + ω2 ∧ω23 = aω2 + 2bω1ω2 + cω22.
(5.14)
a15a210a52ω1∧ω13 a28a222ω2∧ω23a52a77a193a196a162a123a145a19a228a253a42,a145a24a228a253a42a52ω21+ω22,
a52a220a199a35a106ω1,ω2a123a178a48a90.a145a19a228a253a42a233a149a145a24a228a253a42a139a1a110a138,a1a110a138a123
a83a106a90a24a196a119a41a165a77a71H = a+c2,a1a110a138a123a232a210a52Gaussa77a71,a196a49Gaussa77
a71a52ac?b2,a13a183a162a25a243a139a24a199a219a201a123a218a42a210a52dω12 =?Kω1 ∧ω2,a243a89
a199a218a42a141a30,ω1 ∧ω2 a210a52a77a193a123a193a232a195a163,a199a104a108a144a203a77a193a123a221a222a139a7,a166
a199a123a160a152a150a139a7a248,a144a203a77a193a123a145a24a228a253a42a139a7,a243a199a123a38a30,ω12 a52a183a162
a123Levi-Civitaa201a100,a132a149a141a30a132a145a24a228a253a42a113a92a104a189,a196a49a38a30a15a52a144a166a145
a24a228a253a42a139a7,a79a36a183a162a127a119Gaussa77a71a144a203a145a24a228a253a42a139a7,a89a52Gaussa58
a158a122a63a123a136a42.
3 Gauss-Bonneta218a42
a183a162a44a49a132dω12 =?Kω1 ∧ ω2 a122a241a117Gaussa139a214a123a89a199a189a174,a183a25a243a128,
Gauss-Bonnet a218a42a15a193a47a210a132a89a199a218a42a122a241a117,a0a143,a104a35a183a15a89a143a46,a79
a143a34a38a28a139a24a199a85a20a123a77a193,a139a234a89a199Gauss a77a71a198a49a193a232a195a163,a28a104a108a105
a7a108a143a175a199a123a232a73a52a31a143,a139a24a199a193a232a195a163a198a49a24a199a60a106(Gaussa77a71),a178
a199a243a77a193a222a232a24a40a122a241a123a106a52a88a52a203a89a199a77a193a123a1a91a139a7?a0a143,a89a199a210
a52Gauss-Bonneta218a42,Ma52a85a20a123a189a53a77a193,a11
integraldisplay integraldisplay
KdA = 12piχ(M),(5.15)
7
a243a183a162a89a199a218a42a165,a183a162a178ω1 ∧ ω2 a119a193a232a195a163,a183a162a15a44a49a119a199a41dA,
KdAa210a52a89a199a232a73,a28a193a47a210a46,a183a162a243a85a20a123a77a193,a196a49a152a42a139a24a199a232
a73,a199a44a49a85a196da31a143a123,a0a143a199a123a232a73a243a97a148a52a243a77a193a123a30a142a222,a44a49a35
a143a24a199a30a142a232a73,a89a52a183a13a41a89a123Stokesa189a174a204a7a123a92a73,a196a49a204a88a204a132
a89a199a122a116Gaussa77a71a123a232a73a520,a89a23a108a52a70a216a123,Gaussa77a71a232a73a88a204a127
a1490,a89a52a79a143a28a82a24a1992a145a123a69a193,a199a44a49a82a24a196a123a221a222a38a199a123Gaussa77a71
a127a1491,a0a143a243a77a193a222a123a232a73a104a108a88a204a127a1490,a196a49a89a199a119a174a52a70a216a123,a104
a35a79a36a183a46a70a243a31a143a143a48,a70a243a183a162a89a199a218a42dω12 =?KdAa88a52a243a77a193a222,
a52a243a203a177a45a222a62,a89a199a77a193a52a243a203a177a45a222,a196a49a183a137a7a178a89a199a77a193a10a176a116
a203a177a45a222a62,a10a176a52a31a143a63a135a17? a210a52a243a154a24a154a28a82a89a199a154a42a143a198a154a123a92
a160a35a37a222,a196a49a166a233a155a128,a183a162a24a196a119a89a199a143a37a222a156,a79a36a28a243a24a1992a145a22
a111a222a193a7a82a24a199a37a222a156,a89a199a37a222a52a92a160a37a222a203a77a193a35a35,a253a143a92a160a37a222
a156,a89a52a88a52a44a21? a177a34a127a201a28a24a199a77a193,a28a21a88a21a154a154a141a189a24a199a92a160a35
a37a222,a221a86a35a37a222,a38a122a199a52a203a153a123,a4a150a52a44a49a135a73a123,a89a248a123a37a222a156a88
a24a189a139,a89a52a24a199a95a192a123a189a174,a114a199a123a89a24a227a123a21a48a52a77a193a88,a221a86a52a114
a14a77a193a88a203a77a193a123a221a222a89a171a171a123a7a248a243a139a88a245100a35a3a52a204a7a123a175a11,a0
a199a30a127a33a204a7a123a236a147a52a21Mathematische Annalena22,a92a45a142a106a166a123a165a101a52
a243a121a41,a21Mathematische Annalena22a222a62a7a149a89a199a175a11a139a105a245a169a57,a0a30
a127,Einsteina52a21Mathematische Annalena22a123a24a199a31a246,a89a199a175a11a113a92a180a56
a249a234a137a7a24a74a30a45,a0a143a217a165a139a24a199a79a163a52a34a38a28a139a24a199a85a20a123a77a193,a28
a82a154a154a123a63a37a189a53a123a92a160a27a37a222,a253a82a154a154a104a189a123a92a160a27a37a222,a178a199a110a243
a116a92a160a69a222a86,a89a199a47a158a119a41Gauss a110a243,a196a49Gaussa110a243a178a28a123a77a193a110a243
a116a92a160a69a222a86,a89a248a210a178a24a1992a145a77a193a110a243a116a12a24a1992a145a77a193,a220a199a209a52a189a53
a123,a149a52a89a176a210a139a24a199a95a192a88a35a42,a119a41Mapping Degree,a253a110a243a123a39a106,a243
a178a193a123a60a111,a28a139a24a30a77a34,a196a49a92a160a69a210a52a92a160a203,a0a143a28a178a89a30a85a20a123
a77a34a110a243a116a92a160a203a222a62,a196a190a199a221a234a245a232a87a17?a89a210a52Mapping Degree(a110a243
a123a39a106),a89a199a189a66a243a176a145,a4a150a522a145a210a88a89a143a59a92a234,a23a108,a183a162a178a77a193
a110a243a116a92a160a69a222a62,a108a128a82a92a160a69a222a62a123a193a232a195a163,a89a210a52Gauss a77a71,a196
a49a89a199a110a243a210a127a149Gaussa77a71a123a232a73,a178a89a199a233a193a56a45a123a193a232a195a163a126a110a243a110
8
a195a117a131a128a90a198a117a77a193a123a193a232a195a163a24a248,a253a143a70a199a123a114a199a123a232a73,a210a122a116a110
a243a123Degree,a44a49a121a210a89a199Degree a243a44a49a189a53a123a77a193a123a60a89a5,a199a52a77a193
a123Eulera106a123 12pi,a105a44a234a123,a183a76a150a150a139a30a45a178a89a199a113a92a89a113,a183a46a183a5a39
a242a45a251a89a89,a183a25a243a76a151a89a89a89a199a121a210a52a14a143a248a123,a23a108a183a162a46a139a228a253a218
a42dω12 =?KdA,a183a162a164a7a178a77a193a56a116Ea222a86,a196a49a7a189a24a199a92a160a37a222a156,
a89a88a24a189a44a21,a89a248a123a92a160a37a222a156a7a139a73a154,a15a210a52a128a152a42a28a227a142a114a14a92a160
a37a222a156a139a73a154a123a155,a199a0a52a44a21a123,a89a15a137a7a121a210,a139a234a73a154a131a128,a210a23a89a199
a73a154a123a117a159,a7a152a128,a28a23a23a89a74a92a160a37a222a156,a152a89a1a199a67a195a67a81a196,a89a52a37
a222a156a123a73a154a123a44a21a117,a44a21a37a222a156a139a142a245a88a51a123a48a53,a98a52a243a73a154a139a30a127a89
a199a37a222a53a105a18,a15a139a30a127a37a222a53a176a18,a15a139a30a127a210a54a145a174a199a67,a199a52a203a24a24
a118a77a34a35a35,a196a49a255a88a37a222a156a123a73a154a123a117a159a52a58a158a139a63a135a123a175a11,a0a143a183
a243a127a91a77a193a24a189a139a24a199,a63a37a37a88a144a24a199,a13a52a139a142a245a139a73a154a123a92a160a37a222a156.
a243a89a199a60a111a131a5,a28a7a178a77a193a35a196a66a76,a35a196a24a199a196a162complex(a133a111),a28a11
a189a199a243a186a154,a79a143a196a162a123complex,a210a52a128,a139a74a186a154,a139a74a30,a139a74a193,a243a186
a154,a189a24a199a127a91a123a37a222,a0a143a145a19a90a178a89a199a37a222a156a210a116a30,a82a24a199a30a123a155,a152
a42a37a222a156a46a178a243a186a154a104a189a234,a0a143a105a142a52a23a241a117a44a49a178a199a210a116a114a199a123a30,a25
a243a52a193a234,a34a38a52a193a123a155,a7a48a128a139a199a174a110a111,a25a243a37a222a156a46a178a243a30a222a209a189
a234,a52a88a52a21a234a210a159a116a199a123a19a92,a89a210a88a210a23a234,a104a108a89a52a24a189a44a49a123,a34
a38a28a227a142a199a139a73a154,a7a48a128a52a24a199a174a110a111,a0a143a28a210a243a174a110a111a165a45a82a24a154,
a108a128a44a89a154a203a116a30a222,a0a143a89a199a37a222a156a210a178a199a210a116a19a92,a98a52a89a248a189a123a37a222
a156a243a165a45a123a0a154a24a189a52a73a154,a79a143a165a45a123a0a154a150a27a6a189a24a199a44a49a203a153a123a37
a222a156,a38a122a154a154a144a139a24a199a37a222,a243a89a24a154a210a150a27a6a189,a196a49a166a98a89a199a174a132a44
a49a121a210,a34a38a28a227a142a89a37a222a156a139a73a154,a89a248a123a37a222a156a52a63a243a123,a202a47a24a189a139
a73a154,a243a143a69a222,a7a253a90a123a155,a89a90a203a143a69a35a35a123a30a127,a199a123a48a53a52a199a37a222
a156,a24a189a139a24a154a150a139a90,a150a232a139a24a154,a13a89a24a154a210a105a133a236a234,a0a143a89a171a248a6a154
a123a207a196a52a105a139a63a135,a15a105a7a155a123a175a11,a75a47a37a222a156a243a73a154a123a117a159,a105a59a92a123
a24a199a48a27a52a183a162a44a49a243a73a154a189a24a199a114a106,a89a24a196a210a119a41a199a123a141a41a213
I(s) = lim 12pi
integraldisplay
ω1,(5.16)
9
a210a52a128,a165a45a139a73a154,a73a154a142a163a123a37a222a156a52a113a92a104a189a123,a196a49a28a243a89a154a152a199
a66a203a177,a0a143a203a177a123a30a142a37a222a156a46a178a139a234,a196a49a28a243a89a24a154a178a199a110a243a116a24a199
a203a177,a79a143a219a92a123a30a127,a28a34a189a52Euclida1a91,a34a189a126a178a113a34,a28a210a178a199a110a116
a203a177,a0a143a119a203a177a52a245a232a39a17? a89a210a52a37a222a156a123a141a41,a5a39a183a121a210a34a38a28a139
a234a89a74,a28a210a82a92a160a37a222a123a37a222a156,a63a37a178a89a199a77a193a10a176a116Ea176a62,a253a10a176
a116a203a177a45a176a62,a199a210a35a89a199a77a193a143a139a43a154a123,a17a139a43a154,a145a139a73a154,a89a199a73a154
a65a222a86a210a52a43a154,a0a143a89a74a43a154a123a141a41a28a243a24a229a210a52Euler a158a106,a210a127a149a89
a199a232a73,a253a89a199a232a73a139a63a66,a199a52Eulera158a106,a183a46a135a232a73a139a24a199a173a7a123a97a126
a52a243a133a35a60a106a88,a28a7a89a106,a33a139a63a135a123a106a210a52a133a106,a105a110a99,a183a23a165a41a106a166
a36,a165a41a124a212a34a19a234,a165a41a124a150a139a133a106,a133a106a7a155a122a88a122a234,a183a7a89a24a154a133
a35a60a106,a183a7a121a83a106a228a253a189a174,a133a35a60a106a131a128,a127a91a83a106a48a199a42a209a139a137,a89a52
a88a122a234a123a24a199a136a42,a89a199a136a42,a104a35Eulera88a204a121,a105a245a124a209a121a88a241a117,Gauss
a21a121a210,a198a52a163a83a33a149a76a123a106a166a27.
10
a9Gauss-Bonneta218a42
2001a3511a21923a134
1 a77a193a88a21a48a123a59a144
a105a176a108a145a166a76a27a68a193a234,a183a243a26a211a176a212a234a1a21,a28a162a44a49a23a241a117a183a164a150
a139a113a92a80,a88a44a183a250a122a183a164a52a203a76a27a89a89a89a74a192a220,a0a143,a183a152a21a7a89a123
a52Gauss-Bonneta218a42,a89a199a218a42a139a35a104a123a63a66,a15a139a35a104a123a187a36,a131a217a203a183
a199a124a123a211a42a15a139a7a248,a196a49a183a7a10a24a10a183a203a89a199a175a11a52a14a143a248a123a7a248,a183
a162a222a39a89a116a77a193a88,a77a193a88a52a135a73a1a91a176a62a33a173a7a123a24a92a73,a79a143a142a245a135
a73a1a91a123a25a54a243a2433a145a56a45a176a1232a145a77a193a123a60a89a46a178a151a9a234,a51a30,a79a143a199
a52a2433a145a56a45a176a62,a89a199a1a91a123a60a89a52a44a49a23a68a123,a88a52a113a92a126a83a106a117a44a43.
a77a193a88a139a105a159a123a187a36,a33a0a123a104a108a52Monge,Monge a52a27a41a123a76a106a166a27,a198
a144a11a9a233a117a163a139a74a217a196,a196a49a198a248a234a41a76a166a115a71a131a105,a198a233a27a41a123a115a184a139
a105a245a107a47,a198a52a252a187a85a142a5a123a24a199a252a202,a208a207a252a187a85a41a47,a198a52a252a187a85a117a127
a123a48a1a92a159,a4a150a164a203a248a252a187a85a86a132a249a75a67,a79a143a198a123a107a47,a27a41a123a176a211a166
a68(Ploytechnique)a210a79a193a229a117a234,a105a159a24a227a30a207,a27a41a33a80a123a166a9a209a243a176a211
a166a68,a183a46a176a211a166a68a15a142a54a25a243a123a56a149,a139a142a245a80a123a166a9,a190a152a128,a27a41a24a199
a33a76a123a106a166a27Poincare a210a52Ploytechniquea123a166a9,Mongea52a145a24a199a85a7a149
a135a73a1a91a86a123a124,a198a123a86a210a119a41a21a135a232a73a243a1a91a222a123a97a126a22,a89a15a52a183a7a89
a123a11a248,a79a36,a27a41a123a115a184a243a135a73a1a91a139a24a199a35a104a123a4a58,a248a234Mongea253a124
a131a105,a132a149a198a243a176a211a123a107a47,a198a139a105a245a166a9,a209a52a243a135a73a1a91a139a35a104a224a26a123
a124,a108a128a243a135a73a1a91a197a76a21a48a131a165,a7a118a116a24a74a123a52a27a41a12a105a24a199a76a106a166
a27Darboux,Darbo uxa52a27a41a41a166a211a123a179a86a159,a196a49a243a198a123a30a207,a198a52a243a41
a5a142a139a105a245a107a47a123a24a199a124,a198a88a157a52a179a86a159,a15a52a174a169a76a166a174a166a211a123a211a159.
a198a123a33a76a123a211a42a52a143a253a21a77a193a88a22,a183a46,a89a143a76a253a52a106a166a169a26a176a62a123a207a123
a24a199a169a26,a25a243a44a234a132a149a199a52a27a169a123,a105a245a124a88a23a89a199a86,a183a46a89a74a124a233a135
1
a73a1a91a98a232a24a154a234a137,a198a162a97a148a23a89a199a86,Darbouxa123a86a89a122a58a158a80,a221a105a234
a105a245a97a238,a2431941 a35,a183a243a220a8a201a76a115a86,a115a21a135a73a1a91a22,a15a89a116a77a193a88.
a89a116a77a193a88a30,a104a108a210a23Darbouxa123a86,a210a46a116Darbouxa123a86a176a62,a24a199a204a7
a123a48a27a52a126a217a196a41a37,a15a210a52a102a126a217a196a41a37a27,a198a126a122a58a158a180a142,a41a122a58a158
a131a164a224,Darbouxa227a135a88a126a123a24a154a52a198a88a126a105a135a73,a183a46,a183a123a49a52a89a135a232
a73,a13a135a232a73a28a7a89a116a245a195,a245a35a106a123a30a127,a89a199a105a135a73a88a21a27a189,a89a52a79a143
a243a245a35a106a123a30a127,a33a139a72a123a211a228a52a105a135a73,a105a135a73a44a49a28,a62,a44a49a198,a44a49
a135a73,a196a49a139a105a245a83a106a123a228a174a44a49a126a116a105a135a73,a51a30,a24a199a105a135a73a15a52a24a199
a42a6,a89a199a42a6a201a163a105a245a106a166a175a11,a88a11a52a199a123a1a91,a164a52a199a123a73a219,a209a201a28
a105a245a97a238,a79a36a52a58a158a139a126a123,Darbouxa123a98a154a52a198a150a139a126a105a135a73,a198a126a217
a196a41a37a27,a98a52a150a139a105a135a73,a79a36,a139a105a245a211
a42,a88a126a105a135a73,a14a143a205a17? a198a15a52a164a7a126a135a232a73a123,a88a126a105a135a73,a198
a126a160a135a73,a126a160a135a73a7a105a135a73a139a122a245a234,a79a143a28a123a77a193a522a145a123a56a45,a196a49
a233a149a220a199a35a106,a253a77a193a123a107a106u,v,a28a7a233ua70a160a135a73,a233va70a160a135a73,a89a176a62
a139a105a245a160a135a73,a13a126a105a135a73a210a59a92a245a234,a98a52a198a88a102a126a105a135a73,a89a52a105a219
a5a123a47a60,Darbouxa52a21a25a24a39a105a135a73a42a123a145a24a124,a24a199a52Darboux,a24a199
a52Frobenius,a198a162a220a199a124a33a0a21a25a89a199a192a220a123,a98a52a127a116a97a126a116a77a193a207a196a123
a30a127,a88a127a119a143a31a143,a198a150a139a126,a15a142a52a132a149a4a58a123a7a248,a198a85a234a105a135a73a131
a128,a120a209a88a195a234,a196a49a198a88a126a105a135a73a234,a183a166a98a89a234,a2431941a35,a183a166a28a243a102
a210a115a89a199a49,a183a105a7a108a143a46,a143a31a143a88a126a105a135a73a17?a196a49a183a210a126a105a135a73a46a27
a6a41Darbouxa196a41a123a211a42,a221a86a128a150a232a41a77a193a88a90a24a74a1a91a123a255a88,a183a102a126
a105a135a73,a79a36a183a122a116a24a199a105a80a123a234a137.
2 a77a193a88a228a253a19a142a123a195a250
a31a143a119a105a135a73a17?a210a52a28a21a25a7a207a196a77a193a123a155,a77a193a52a24a1992a145a123a22a111,a199a243
a202a47a56a45a176a62a522a145a123,a196a49a199a222a62a127a63a24a199a154a52a220a199a35a106,a47a158a210a119a41a107
a106,a98a52a25a243a17,a183a162a210a119a199a219a92a43a41u,v,a79a36a199a123a43a41a522a199a35a106a123a60a106,
a196a49a52a89a199a30a71a38a199a243a154a24a154a139a24a199a35a178a193,a89a199a35a178a193a104a108a105a7a155,a79a143
2
a183a162a150a139a27a6a207a196a133a236a123a67a111,a183a162a144a21a207a196a33a59a92a123a152a134a34,a178a193a89a74a192
a220,a35a178a193a203a77a193a139a33a183a35a123a7a248,a0a143,a92a128a183a35a123a7a248a88a234,a24a189a7a137a219
a143a21a234a137a251a7a118a205a253a47a123a24a74a175a11,a139a24a199a35a178a193,a243a89a199a35a178a193a222a522 a145
a123,a149a52a154a24a154a210a139a142a245a37a222,a15a210a52a35a37a222,a35a37a222a210a52a203a89a199a77a193a35a35
a123a37a222,a79a143a89a199a77a193a52a243Euclid a56a45a176a62,a196a49a183a44a49a89a89a199a37a222a123a159a221.
a143a59a92a229a68,a183a33a149a255a88a159a221a127a1491 a123a37a222,a253a92a160a37a222,a196a49a139a24a87a92a160
a35a37a222,a203a89a74a92a160a35a37a222a18a134a123a139a12a105a24a199a37a222,a183a162a34a189a199a52a82a196a92a160
a123,a0a143a89a199a37a222a183a162a119a41a92a160a27a37a222,a7a213a63a123a52a89a176a210a139a24a199a1a91a25a54
a21a9a234,a79a143a34a38a89a199a77a193a65a178a234a123a155,a92a160a27a37a222a44a49a53a222a18,a15a44a49a53
a5a18,a166a233a155a128,a89a199a77a193a248a234a52a24a1992a145a123a22a111a131a105,a199a164a139a24a199a189a53,a243
a77a193a222a28a52a126a48a53a221,a164a52a203a31a48a53a221,a89a199a221a196a52a105a88a24a248a123,a196a49,a28a7
a189a14a143a248a6a221a196a52a126a48a53,a89a210a52a7a201a77a193a24a199a189a53,a189a53a139a234a131a128,a199a123
a92a160a27a37a222a210a189a234,a92a160a27a37a222a243a89a199a48a53a44a49a53a222a18,a15a44a49a53a5a18,a189
a234a24a199a131a128,a89a199a77a193a15a210a189a53a234,a89a52a105a173a7a123a24a199a10a39,a175a108a35a139a123a144
a52a24a199a110a82,a98a52a89a52a24a199a105a173a7a123a10a39,Mobiusa52a121a41a149a76a123a1a91a166a27.
a79a143a28a7a189a53,Mobiusa21a25a139a74a77a193a88a21a189a53,a89a104a108a52a105a139a63a135a123a24a71a47
a60,a28a162a76a27a209a127a119a123a89a199a67a111,a210a52a252a24a60a146,a28a178a199a221a24a87a203a229a117a123a155,
a210a122a116a196a162a123Mobius a77a193,a199a150a27a6a189a53,a89a52a1a91a222a105a139a63a135a123a24a71a47
a60,a0a143a183a162a34a189a77a193a46a178a104a189a234a24a199a48a53,a139a234a89a248a189a53a123a77a193a131a128,a1
a91a60a89a52a14a143a248a123a17?a132a92a160a27a37a222e3,a217a165e3 a52a2433a145a56a45,a28a21a25a139a24a71
a47a34,a210a52a128a28a92a213a255a88a77a193a88a234,a28a24a189a7a188a126a77a193a222a123a92a160a35a37a222,a183
a119a89a199a92a160a35a37a222a143e1,a89a248a183a210a139a234a24a199a41a37,a199a139a234a145a24a199a92a160a37a222a90
a145a174a199a92a160a37a222,a152a42a56a45a52a189a53a123,a145a19a199a92a160a37a222e2 = e3 ×e1a15a210a113a92
a104a189a234,a196a49a183a210a139a199a92a160a41a37,a92a160a41a37a210a52a174a199a92a160a37a222a149a76a24a189a123
a39a147,a52a144a35a18a134a123,a143a31a143a92a160a41a37a243a1a91a123a207a196a131a165a52a89a143a173a7? a210a52
a79a143a1a91a52a202a226a228a196a107a207a196a56a45a243a228a196a131a5a88a35a123a1a91a117a159,a13a89a228a196a107
a210a52a41a37a196a196a123a56a45,a79a143a52a139a24a199a63a37a144a139a24a199a228a196a178a24a199a41a37a35a143a217
a199a123a41a37,a150a149a92a14a123a92a160a41a37a203a89a199a228a196a107a123a195a163a196a24a24a233a97,a88a98a52a24
a24a233a97,a13a37a233a97a226a209a95a192a90a24a35a123a117a159,a196a49a228a196a107a105a7a155,a79a143a56a89a123
3
a228a196a88a127a119a243a137a219a123a60a89a131a5a152a91a44a49a255a174,a13a234a41a37a131a128,a210a44a49a255a174a234.
a41a37a210a52a37a222a234,a13a37a222a24a196a52a1393a199a73a222a123a37a222,a13a154a24a199a73a222a52a60a106,a210
a44a49a178a199a135a73,a28,a62a31a143a123,a37a222a139a28,a62a123a228a174,a15a139a135a73a123a228a174,a243a236
a171a63a66a5,a164a44a49a139a232a73a123a228a174,a196a49a183a25a243a210a44a49a135a73,a183a207a196a77a193a123
a30a127,a88a144a24a199a41a37,a0a143a243a77a193a123a154a154,a89a248a123a41a37a139a245a232a17? a34a38a28a65
a122e1a123a155,a51a30a89a199a77a193a52a189a53a123,a89a199a41a37a210a113a92a189a234,e1a52a31a143a17? e1a52
a89a199a77a193a243a89a24a154a123a92a160a35a37a222,a0a143a89a199a77a193a139a245a232a92a160a35a37a222a17? a154a154
a139a24a87a243a35a178a193a222a62a127a149a92a160a37a222,a13a77a193a522a145a123,a196a49a199a162a196a196a123a56a45
a523a145a22a111,a89a52a79a143a89a199a154a52a243a77a193a222a35a196,a522a145a123,a25a243a243a154a189a234a131a128,
a92a160a35a37a222a44a49a119a248a199a221a24a87,a196a24a199a203a177,a196a49a199a52a145a28a24a145,a523a145,a89
a1993a145a56a45a58a158a7a155,a183a46a25a243a34a19a222,a28a162a7a234a137a135a232a73a221a86a234a137a203a135a232
a73a5a86a123a106a166a221a86a243a106a166a165a123a97a126,a89a199a60a89a52a33a59a92a123,a51a30a52a33a139a126a123.
a196a49a183a139a24a1993a145a56a45,a132a149a154a24a154a139a199a203a177,a25a243a139a199a214a9a119a41a203a45,a221a86
a203a177a45,a45a52bundle,a196a49a28a7a207a196a77a193a123a1a91a117a159,a126a89a199a137a219a123a48a27,a24
a189a7a255a88a199a123a203a45,a255a88a203a45a234a131a128,a24a35a209a59a92a234,a79a143a24a35a209a52a37a222,
a13a52a37a222a123a155,a199a139a73a222,a210a44a49a135a73,a210a44a49a126a83a106a221a86a135a73a123a228a174,a183
a162a52a243a255a88a135a232a73,a183a162a34a189a141a116a31a143a60a106a209a44a49a135a73,a183a119a243a89a199a77a193a222
a123a154a143x,a0a143dx a52a24a199a37a222,a210a52a44a198a154a203a248a89a199a154a123a37a222,x a52u,v a123a60
a106,a13u,va52a77a193a222a123a219a92a43a41,a196a49a28a44a49a85a241dx,a34a38x a33a155a243a77a193a222,
a0a143dxa24a189a52e1 a166e2a123a34a117a28a92,a196a49a243a89a199a143a48,a183a210a223a73a188a126a105a135a73a123
a10a39,a34a19a222dxa52a24a199a37a222a138a123a24a39a135a73a42,a196a49a199a52e1 a166e2 a123a34a117a28a92,
a199a123a28a92a248a106a52a24a39a135a73a42,a196a49dxa44a49a85a143
dx = ω1e1 + ω2e2,(5.1)
(dx,dx) a210a52a183a162a77a193a123a169a121a221a222,a79a143e1,e2a52a144a35a18a134a123a92a160a37a222,a196a49
ω21 + ω22 = ds2 (5.2)
a210a52a169a121a221a222,a152a42a89a199a56a249a234,a89a233a149a202a47a89a135a73a1a91a59a92a245a234,a79
a143a202a47a135a73a1a91,a169a121a221a222a7a85a196gijdxidxj,a89a52a79a143a243a35a56a45a176a196a188a126
4
a123a43a41a52a127a63a123Ca rtesiana43a41,a199a88a24a189a18a134,a15a88a24a189a52a92a160,dx a127
a149ω1e1 + ω2e2,a98a52a183a162a105a135a73a139a199a228a253a123a117a159,a210a52a242a126a24a39a123a155,a199a127
a1490,a89a210a52a202a47a128a123a160a135a73a44a49a52a98a166a123a30a71,a24a248a123,a15a210a52a122a116a123a160a135
a73a166a135a73a123a39a147a195a7,a196a49a28a210a178da126a116dxa222a62,a24a189a127a1490,a28a178a141a30a48a13
a123a155,a210a122a116d(ω1e1) + d(ω2e2),a213a63a104a105a135a73a3a193a139a24a199a24a39a79a42a123a155,a135
a73a145a19a199a79a6a7a149a82,a15a13a211a131,a44a49a122a116
dω1 =?ω2 ∧ω12;dω2 = ω1 ∧ω12,(5.3)
0 = dω3 = ω1 ∧ω13 + ω2 ∧ω23,(5.4)
a183a204a243a5a193a201a241ω12,ω13,ω23,a17a108a126a135a232a73a234,a196a49a44a49a178(xe1e2,e3)a123a135
a73a44a143e1,e2,e3a123a34a117a28a92,a89a199a34a117a28a92a178deia85a196ωijej,a25a243a183a126a135a73a1
a91a202a47a123a110a82,a34a38a139a24a199a141a106a7a173a133a123a155,a210a44a43a35a28,i,j,ka441a1163,a28
a178dei a85a196ωijej,ωija123a1a91a63a66a105a210a23,a28a25a243a139a24a28a41a37,a89a28a41a37a203a24
a28a107a106a139a7a248,a13a233a149a89a24a28a41a37,a210a139a24a199a249a163a41a37,a89a199a249a163a41a37a203a198
a117a41a37a123a7a248a210a52ωij,a89a7a248a52a132a24a39a135a73a42a117a44a43a123,a79a36a210a139
dei = ωijej,(5.5)
a89a28a48a199a42a105a7a155,a199a210a44a43a220a199a249a163a41a37a144a35a123a7a248,a243a89a199a60a89a131a5,
a135a73a1a91a203a197a166a88a76a24a248,a197a166a128a128a35a106a52a30a45,a196a49a24a199a41a37a203a248a30a45a243
a35a196,a79a36a28a114a199a41a37a144a139a24a199a35a106,a209a52a30a45ta123a60a106,a25a243a183a162a52a24a199a77
a193,a154a154a139a142a245a41a37,a196a49a183a89a41a37a123a107a106a523,a89a52a79a143a139a35a193a123a219a92a43
a41,a145a139a35a37a222a243a178a193a176a62a35a166a123a43a41,a196a49a183a25a243a89a199a7a35a106a523,a164a79
a143E a89a56a45a523a145a123,a7a35a106a176a234,a196a49a89a52a139a198a79a38a122a105a135a73a139a72,a183a162a46
a82dei a85a196a48a199(5.5),ωija233a149i,ja52a39a233a193a123,a89a52a79a143a183a123a41a37a52a92a160a41a37,
a253a79a143(ei,ej) = δij,a196a49a199a52a39a233a193a123,a79a36ωija34a19a222a105a59a92,a28a178(ωij)a85
a241a117,a199a52a24a199a48a106,a89a199a48a106a52a39a233a193a123,a196a49a243a233a110a34a123ωa127a1490,a217a155
a123a233a248a233a110a34a52a39a233a193a123,a79a36a34a19a222a144a1393a199a24a39a135a73a42,ω12,ω13,ω23.
a183a46a183a222a39a121a210a234ω12a132dω1,dω2a123a48a199(5.3)a113a92a104a189,a89a52a24a199a173a7a123a189
5
a174,a89a52a38a122Levi-Civitaa241a214a123a173a7a189a174,a183a25a243a178a48a199(5.5)a70a105a135a73,a79
a143d(dei) = 0,a196a49a141a30a123a155,a183a210a122a70dωij,a136a42a122a116a123a52
dωij = ωik ∧ωkj,(5.6)
a79a36a89a74ωa131a45a139a105a59a92a123a7a248,a59a92a122a88a122a234,a79a143a31a143a17?a79a143a233
a149dωij = ωik ∧ωkj,i,ja52a88a35a127a123,a152a42a35a127a234a123a155,ωii a520,a89a52a79a143ωa52
a39a233a193a123,a196a49a28a82i negationslash= j,a152a42ka127a149i,a11ωii = 0; ka7a127a149j,ωjj = 0,a196a49k
a88a127a149i,a88a127a149j,a79a143a183a162a52a2433a145a56a45,ka144a139a24a199a44a21a117,a79a36a89a199a23a248
a105a0a219a123a48a199a42,a199a123a141a30a144a139a24a49,a183a222a39a178a199a85a5a117a234,a210a52
dω12 = ω13 ∧ω32,(5.7)
dω13 = ω12 ∧ω23,(5.8)
dω23 = ω21 ∧ω13,(5.9)
a131a217a52a122a116dω12 = ω13 ∧ω32a89a199a218a42,a98a52ω a52a39a233a193a123,a196a49a210a122a116
dω12 =?ω13 ∧ω23,(5.10)
a13ω13,ω23a209a52ω1,ω2a123a34a117a28a92:
ω13 = aω1 + bω2,ω23 = bω1 + cω2,(5.11)
a89a166a28a210a122a116a5a193a89a199a218a42:
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(5.12)
Ka52Gaussa77a71,Gaussa77a71a210a52K = ac?b2,a89a199a218a42a88a122a234,a104a35Gaussa88
a52a89a248a122a116a89a199a218a42,a52a126a107a123a48a27a122a116a218a42,a199a119a41Theorem Egregium,
a126a165a169a89,a199a52a24a199a219a201a123a189a174,a201a123a189a174,a28a251a251a23a23a199,a199a52a105a201a123,a79
a143a183a162a123Ea52a92a160a37a222a45,a199a89a199a203a177a45a233a149a77a193M a139a24a199a61a107,a233a149a203a177
a45,a139a89a199a92a160a37a222,a183a82a199a123a198a154a210a52a199a123a61a107,a79a36a183a162a25a243a123a1a91a7
a44a3a10a39a222a7a118a133a236a234,a210a52a128,a88a144a52a139a24a199a77a193a221a86a139a24a199a77a34,a25a243
6
a139a220a199a56a45,a139Ea89a199a56a45a90a77a193M,a47a34a222,a139a234a77a193M,a108a128a139a132a196a139a199
a123a92a160a37a222a196a196a123a56a45a52E,a199a52a77a193a123a203a177a45,a25a243a47a158a119a89a199a77a193a52a142
a56a45,a199a52a243a142a222a123a24a199a56a45,a196a49a210a139a24a1993a145a123a203a177a45,a199a123a142a56a45a52a183
a162a123a77a193,a243a89a199a60a89a131a5,a183a162a139a1a199a24a39a135a73a42,a243a56a45a176a62a139a24a39a135
a73a42ω1,ω2,a108a128a139ω12,ω13,ω23,a24a2255a199a24a39a135a73a42,a217a165ω13,ω23a52ω1,ω2a123
a34a117a28a92(a218a42(5.11)),a199a44a43a77a193a123a1a91a117a159,a183a162a139
a145a24a228a253a42 I = ds2 = (dx,dx) = ω21 + ω22; (5.13)
a145a19a228a253a42 II = (?dx,de3) = ω1 ∧ω13 + ω2 ∧ω23 = aω2 + 2bω1ω2 + cω22.
(5.14)
a15a210a52ω1∧ω13 a28a222ω2∧ω23a52a77a193a196a162a123a145a19a228a253a42,a145a24a228a253a42a52ω21+ω22,
a52a220a199a35a106ω1,ω2a123a178a48a90.a145a19a228a253a42a233a149a145a24a228a253a42a139a1a110a138,a1a110a138a123
a83a106a90a24a196a119a41a165a77a71H = a+c2,a1a110a138a123a232a210a52Gaussa77a71,a196a49Gaussa77
a71a52ac?b2,a13a183a162a25a243a139a24a199a219a201a123a218a42a210a52dω12 =?Kω1 ∧ω2,a243a89
a199a218a42a141a30,ω1 ∧ω2 a210a52a77a193a123a193a232a195a163,a199a104a108a144a203a77a193a123a221a222a139a7,a166
a199a123a160a152a150a139a7a248,a144a203a77a193a123a145a24a228a253a42a139a7,a243a199a123a38a30,ω12 a52a183a162
a123Levi-Civitaa201a100,a132a149a141a30a132a145a24a228a253a42a113a92a104a189,a196a49a38a30a15a52a144a166a145
a24a228a253a42a139a7,a79a36a183a162a127a119Gaussa77a71a144a203a145a24a228a253a42a139a7,a89a52Gaussa58
a158a122a63a123a136a42.
3 Gauss-Bonneta218a42
a183a162a44a49a132dω12 =?Kω1 ∧ ω2 a122a241a117Gaussa139a214a123a89a199a189a174,a183a25a243a128,
Gauss-Bonnet a218a42a15a193a47a210a132a89a199a218a42a122a241a117,a0a143,a104a35a183a15a89a143a46,a79
a143a34a38a28a139a24a199a85a20a123a77a193,a139a234a89a199Gauss a77a71a198a49a193a232a195a163,a28a104a108a105
a7a108a143a175a199a123a232a73a52a31a143,a139a24a199a193a232a195a163a198a49a24a199a60a106(Gaussa77a71),a178
a199a243a77a193a222a232a24a40a122a241a123a106a52a88a52a203a89a199a77a193a123a1a91a139a7?a0a143,a89a199a210
a52Gauss-Bonneta218a42,Ma52a85a20a123a189a53a77a193,a11
integraldisplay integraldisplay
KdA = 12piχ(M),(5.15)
7
a243a183a162a89a199a218a42a165,a183a162a178ω1 ∧ ω2 a119a193a232a195a163,a183a162a15a44a49a119a199a41dA,
KdAa210a52a89a199a232a73,a28a193a47a210a46,a183a162a243a85a20a123a77a193,a196a49a152a42a139a24a199a232
a73,a199a44a49a85a196da31a143a123,a0a143a199a123a232a73a243a97a148a52a243a77a193a123a30a142a222,a44a49a35
a143a24a199a30a142a232a73,a89a52a183a13a41a89a123Stokesa189a174a204a7a123a92a73,a196a49a204a88a204a132
a89a199a122a116Gaussa77a71a123a232a73a520,a89a23a108a52a70a216a123,Gaussa77a71a232a73a88a204a127
a1490,a89a52a79a143a28a82a24a1992a145a123a69a193,a199a44a49a82a24a196a123a221a222a38a199a123Gaussa77a71
a127a1491,a0a143a243a77a193a222a123a232a73a104a108a88a204a127a1490,a196a49a89a199a119a174a52a70a216a123,a104
a35a79a36a183a46a70a243a31a143a143a48,a70a243a183a162a89a199a218a42dω12 =?KdAa88a52a243a77a193a222,
a52a243a203a177a45a222a62,a89a199a77a193a52a243a203a177a45a222,a196a49a183a137a7a178a89a199a77a193a10a176a116
a203a177a45a222a62,a10a176a52a31a143a63a135a17? a210a52a243a154a24a154a28a82a89a199a154a42a143a198a154a123a92
a160a35a37a222,a196a49a166a233a155a128,a183a162a24a196a119a89a199a143a37a222a156,a79a36a28a243a24a1992a145a22
a111a222a193a7a82a24a199a37a222a156,a89a199a37a222a52a92a160a37a222a203a77a193a35a35,a253a143a92a160a37a222
a156,a89a52a88a52a44a21? a177a34a127a201a28a24a199a77a193,a28a21a88a21a154a154a141a189a24a199a92a160a35
a37a222,a221a86a35a37a222,a38a122a199a52a203a153a123,a4a150a52a44a49a135a73a123,a89a248a123a37a222a156a88
a24a189a139,a89a52a24a199a95a192a123a189a174,a114a199a123a89a24a227a123a21a48a52a77a193a88,a221a86a52a114
a14a77a193a88a203a77a193a123a221a222a89a171a171a123a7a248a243a139a88a245100a35a3a52a204a7a123a175a11,a0
a199a30a127a33a204a7a123a236a147a52a21Mathematische Annalena22,a92a45a142a106a166a123a165a101a52
a243a121a41,a21Mathematische Annalena22a222a62a7a149a89a199a175a11a139a105a245a169a57,a0a30
a127,Einsteina52a21Mathematische Annalena22a123a24a199a31a246,a89a199a175a11a113a92a180a56
a249a234a137a7a24a74a30a45,a0a143a217a165a139a24a199a79a163a52a34a38a28a139a24a199a85a20a123a77a193,a28
a82a154a154a123a63a37a189a53a123a92a160a27a37a222,a253a82a154a154a104a189a123a92a160a27a37a222,a178a199a110a243
a116a92a160a69a222a86,a89a199a47a158a119a41Gauss a110a243,a196a49Gaussa110a243a178a28a123a77a193a110a243
a116a92a160a69a222a86,a89a248a210a178a24a1992a145a77a193a110a243a116a12a24a1992a145a77a193,a220a199a209a52a189a53
a123,a149a52a89a176a210a139a24a199a95a192a88a35a42,a119a41Mapping Degree,a253a110a243a123a39a106,a243
a178a193a123a60a111,a28a139a24a30a77a34,a196a49a92a160a69a210a52a92a160a203,a0a143a28a178a89a30a85a20a123
a77a34a110a243a116a92a160a203a222a62,a196a190a199a221a234a245a232a87a17?a89a210a52Mapping Degree(a110a243
a123a39a106),a89a199a189a66a243a176a145,a4a150a522a145a210a88a89a143a59a92a234,a23a108,a183a162a178a77a193
a110a243a116a92a160a69a222a62,a108a128a82a92a160a69a222a62a123a193a232a195a163,a89a210a52Gauss a77a71,a196
a49a89a199a110a243a210a127a149Gaussa77a71a123a232a73,a178a89a199a233a193a56a45a123a193a232a195a163a126a110a243a110
8
a195a117a131a128a90a198a117a77a193a123a193a232a195a163a24a248,a253a143a70a199a123a114a199a123a232a73,a210a122a116a110
a243a123Degree,a44a49a121a210a89a199Degree a243a44a49a189a53a123a77a193a123a60a89a5,a199a52a77a193
a123Eulera106a123 12pi,a105a44a234a123,a183a76a150a150a139a30a45a178a89a199a113a92a89a113,a183a46a183a5a39
a242a45a251a89a89,a183a25a243a76a151a89a89a89a199a121a210a52a14a143a248a123,a23a108a183a162a46a139a228a253a218
a42dω12 =?KdA,a183a162a164a7a178a77a193a56a116Ea222a86,a196a49a7a189a24a199a92a160a37a222a156,
a89a88a24a189a44a21,a89a248a123a92a160a37a222a156a7a139a73a154,a15a210a52a128a152a42a28a227a142a114a14a92a160
a37a222a156a139a73a154a123a155,a199a0a52a44a21a123,a89a15a137a7a121a210,a139a234a73a154a131a128,a210a23a89a199
a73a154a123a117a159,a7a152a128,a28a23a23a89a74a92a160a37a222a156,a152a89a1a199a67a195a67a81a196,a89a52a37
a222a156a123a73a154a123a44a21a117,a44a21a37a222a156a139a142a245a88a51a123a48a53,a98a52a243a73a154a139a30a127a89
a199a37a222a53a105a18,a15a139a30a127a37a222a53a176a18,a15a139a30a127a210a54a145a174a199a67,a199a52a203a24a24
a118a77a34a35a35,a196a49a255a88a37a222a156a123a73a154a123a117a159a52a58a158a139a63a135a123a175a11,a0a143a183
a243a127a91a77a193a24a189a139a24a199,a63a37a37a88a144a24a199,a13a52a139a142a245a139a73a154a123a92a160a37a222a156.
a243a89a199a60a111a131a5,a28a7a178a77a193a35a196a66a76,a35a196a24a199a196a162complex(a133a111),a28a11
a189a199a243a186a154,a79a143a196a162a123complex,a210a52a128,a139a74a186a154,a139a74a30,a139a74a193,a243a186
a154,a189a24a199a127a91a123a37a222,a0a143a145a19a90a178a89a199a37a222a156a210a116a30,a82a24a199a30a123a155,a152
a42a37a222a156a46a178a243a186a154a104a189a234,a0a143a105a142a52a23a241a117a44a49a178a199a210a116a114a199a123a30,a25
a243a52a193a234,a34a38a52a193a123a155,a7a48a128a139a199a174a110a111,a25a243a37a222a156a46a178a243a30a222a209a189
a234,a52a88a52a21a234a210a159a116a199a123a19a92,a89a210a88a210a23a234,a104a108a89a52a24a189a44a49a123,a34
a38a28a227a142a199a139a73a154,a7a48a128a52a24a199a174a110a111,a0a143a28a210a243a174a110a111a165a45a82a24a154,
a108a128a44a89a154a203a116a30a222,a0a143a89a199a37a222a156a210a178a199a210a116a19a92,a98a52a89a248a189a123a37a222
a156a243a165a45a123a0a154a24a189a52a73a154,a79a143a165a45a123a0a154a150a27a6a189a24a199a44a49a203a153a123a37
a222a156,a38a122a154a154a144a139a24a199a37a222,a243a89a24a154a210a150a27a6a189,a196a49a166a98a89a199a174a132a44
a49a121a210,a34a38a28a227a142a89a37a222a156a139a73a154,a89a248a123a37a222a156a52a63a243a123,a202a47a24a189a139
a73a154,a243a143a69a222,a7a253a90a123a155,a89a90a203a143a69a35a35a123a30a127,a199a123a48a53a52a199a37a222
a156,a24a189a139a24a154a150a139a90,a150a232a139a24a154,a13a89a24a154a210a105a133a236a234,a0a143a89a171a248a6a154
a123a207a196a52a105a139a63a135,a15a105a7a155a123a175a11,a75a47a37a222a156a243a73a154a123a117a159,a105a59a92a123
a24a199a48a27a52a183a162a44a49a243a73a154a189a24a199a114a106,a89a24a196a210a119a41a199a123a141a41a213
I(s) = lim 12pi
integraldisplay
ω1,(5.16)
9
a210a52a128,a165a45a139a73a154,a73a154a142a163a123a37a222a156a52a113a92a104a189a123,a196a49a28a243a89a154a152a199
a66a203a177,a0a143a203a177a123a30a142a37a222a156a46a178a139a234,a196a49a28a243a89a24a154a178a199a110a243a116a24a199
a203a177,a79a143a219a92a123a30a127,a28a34a189a52Euclida1a91,a34a189a126a178a113a34,a28a210a178a199a110a116
a203a177,a0a143a119a203a177a52a245a232a39a17? a89a210a52a37a222a156a123a141a41,a5a39a183a121a210a34a38a28a139
a234a89a74,a28a210a82a92a160a37a222a123a37a222a156,a63a37a178a89a199a77a193a10a176a116Ea176a62,a253a10a176
a116a203a177a45a176a62,a199a210a35a89a199a77a193a143a139a43a154a123,a17a139a43a154,a145a139a73a154,a89a199a73a154
a65a222a86a210a52a43a154,a0a143a89a74a43a154a123a141a41a28a243a24a229a210a52Euler a158a106,a210a127a149a89
a199a232a73,a253a89a199a232a73a139a63a66,a199a52Eulera158a106,a183a46a135a232a73a139a24a199a173a7a123a97a126
a52a243a133a35a60a106a88,a28a7a89a106,a33a139a63a135a123a106a210a52a133a106,a105a110a99,a183a23a165a41a106a166
a36,a165a41a124a212a34a19a234,a165a41a124a150a139a133a106,a133a106a7a155a122a88a122a234,a183a7a89a24a154a133
a35a60a106,a183a7a121a83a106a228a253a189a174,a133a35a60a106a131a128,a127a91a83a106a48a199a42a209a139a137,a89a52
a88a122a234a123a24a199a136a42,a89a199a136a42,a104a35Eulera88a204a121,a105a245a124a209a121a88a241a117,Gauss
a21a121a210,a198a52a163a83a33a149a76a123a106a166a27.
10
