a145a24a89 a77a193a88(a174)
a9Gauss-Bonneta218a42(a153)
2001a3511a21930a134
1 a135a232a73a243a133a35a60a106a88a165a97a126a59a144
a183a164a97a148a242a89a220a39,a89a220a39a183a139a199a15a153,a189a247a89a24a154a133a35a60a106a88,a79a143a243a106
a166a165,a105a7a155a123a24a71a47a34,a51a30a243a106a166a36a222a15a52a58a158a7a155a123a24a71a47a60,a210a52
a139a133a106,a89a199a133a106a38a122a106a166a59a92,a133a60a106a139a142a245a164a224,a139a63a135a123a117a159,a79a36,
a89a38a122a89a74a60a106a243a97a126a222a1a47a139a126a255,a196a49,a183a189a247a89a24a89,a7a152a128,a133a35a60
a106a139a24a199a105a173a7a123a117a159,a127a63a123a83a106a48a199a243a133a35a60a106a131a165a24a189a139a137,a89a52a24
a199a88a122a234a123a47a60,a79a143a88a11a28a14a143a248a85a24a199a48a199,a28a7a52a227a142a137a52a133a106a123a155,
a199a24a189a139a137,a190a152,x2 + 1 = 0,a0a143a199a139a199a137a210a52√?1,a196a49√?1a210a89a143a248
a6a139a126a255,a88a98a152a36,a133a106a203a34a106a24a248,a44a49a28a62,a139a51a248a123a117a159,a196a49,a199
a44a49a228a174,a51a30a199a221a57a234a142a245a97a238a52a34a106a88a21a221a57a123,a183a46a183a123a49a243a44a199a165
a24a189a204a139a199a56a105,a243a56a105a123a30a127,a183a46a73a220a39a89a133a35a60a106,a183a189a247a89,a24a199a52
a183a166a98a89a123a83a106a123a228a253a189a174,a210a52a128a127a63a123a83a106a123a48a199a243a133a106a173a165a24a189a139
a137,a89a199a52a105a10a121a210a123,a137a7a106a166a222a99a123a10a39,a7a48a128,a149a76a106a166a27a152Euler,
a198a46a27a6a121a210,a98a150a139a21a196a213,a183a46Gauss a52a183a162a163a83a33a149a76a123a106a166a27,a198
a105a35a52a123a30a127a210a139a24a199a121a210,a15a210a52a133a106a137a7a24a74a1a91a123a117a159,a88a113a92a52a83
a106a123a175a11,a183a189a247a5a39a89a133a106a123a30a127a121a210a89a199a189a174;a51a30,a133a35a60a106a33a204a7
a123a24a199a189a174a52Picarda189a174,a210a52a128,a34a38a233a149a24a199a133a35a60a106,a82a199a123a60a106a138a243
a133a178a193a176a62a196a82a123a160a152,a199a178a114a199a133a178a193a209a152a212a234,a217a165a15a142a44a49a86a171a24a154,
a220a154,a89a52a88a122a234a123,a210a52a128,a60a106a152a42a52a24a199a92a24a60a106a123a155,a199a73a89a122a58a158
a131a254a225,a44a49a128a139a88a245a178a178a193a209a152a212a234,a139a63a135a123a24a71a47a60a52a89a199a189a174a52a133
a35a60a106a176a88a123a189a174,a44a49a188a126a183a162a25a243a7a89a123Gauss-Bonneta218a42a117a121a210,a89
a128a210a23a229a117a150a139a7a248a123a24a74a48a27a203a10a39,a136a42a52a139a7a248a123,a89a52a106a166a222a58a158
a7a155,a139a63a135a123a175a11.
1
2 a7a149a166a243a123a7a196a117
a89a199a49a79a136a101a234,a28a162a243a89a199a49a85a199a231a183,a33a80a52a7a196,a28a21a234a7a3a73a116a24
a199a175a11,a89a52a205a7a155a123,a183a46a28a162a209a52a76a166a9,a76a166a9a73a176a127a115a184a33a128a123a24
a227,a49a128a116a246a204a222a86,a253a38a243a166a68,a243a166a98a92a160a176a62,a33a7a155a123a24a189a7a7a196.
a88a7a52a127a144a19a119a28a41a31a143,a28a242a41a31a143,a89a199a33a160,a7a7a196,a7a7a3a21a73a175
a11,a7a7a3a21a234a73a133a7a3a73a123a175a11,a0a143,a104a108a28a73a123a175a11a88a24a189a92a55,a28a246
a30a15a88a24a189a21a234a122a116a73a153,a88a44,a28a165a45a178a44a24a74a107a52,a178a44a24a74a70a216,a44a49
a38a122a28a123a166a175a93a116a159a90,a13a38a122a28a93a116a159a90a123a210a52a7a178a44a89a248a123a67a153,a196a49
a183a243a185a76a27a7a7a196,a245a24a154a143a89a229a117,a28a162a4a150a7a21a234a28a132a24a199a76a14,a144a35
a231a183a73a175a11,a221a86a62a68a19a68a105a144a19,a51a166a117a41a231a183,a89a52a105a139a80a255a123,a7a3a7
a178a106a166a46a24a46,a221a86a233a127a63a123a166a175,a28a7a3a139a199a135a46,a250a122a139a199a31a143a248a123a217
a196,a233a149a28,a233a149a89a199a166a175a123a127a35a44a49a15a28,a51a30a28a233a166a175a123a21a197a15a44a49a15
a28,a196a49a89a52a105a138a122a213a63a123a24a71a47a60,a230a130a28a162a31a68a24a5a89a199a44a21a117.
3 Gauss-Bonneta218a42a123a121a210
a222a39,Gauss-Bonneta218a42a183a150a139a121a210a92,a196a49a183a11a178a121a210a128a92a234,a183a222a39a89
a123Gauss-Bo nneta218a42a210a52,a34a38a243a56a45a176a62a139a24a199a77a193,a199a52a24a199a114a199a123
a77a193,a63a37a34a38a89a199a77a193a52a189a53a123,a253a199a123a27a34a139a24a189a123a48a53,a149a52a89a248a6,
Gaussa77a71Ka210a52a77a193a222a123a24a199a60a106,a183a44a49a178a89a199a60a106a233a149a77a193a222a123a193a232
a221a222a70a232a73,a89a199a232a73a52a24a1992a173a232a73,a70a199a232a73a131a128,a136a42a89a199a232a73a127a149a24
a199a158a106(2pi)a198a49a77a193a123Euler a43a117a106,a253
integraldisplay integraldisplay
KdA = 2piχ(M) (6.1)
Eulera43a117a106a210a52a178a77a193a35a196a66a76a131a128,a55a149a24a154a7a108a123a30a71,a178a199a35a113a131
a128,a217a186a154a199a106?a30a123a199a106+a193a123a199a106,a89a2483a199a106a123a116a139a123a90a210a119a41a89a199a77
a193a123Eulera43a117a106,a104a77a193a52a69a193a123a155,a199a123Eulera43a117a106= 2,a152a42a199a52a199a162
a193,a199a123Eulera43a117a106a520,a28a162a44a49a65a24a65,a210a21a122a116a89a199.a152a42a77a193a52a199a189
2
a53a77a193,a89a52a199a123a141a24a123a95a192a88a35a42,a24a196a89a229a117,a34a38a69a222a28a1a199a162,a162a123
a199a106a210a203Eulera43a117a106a139a199a7a248,a89a162a123a199a106a202a47a119a77a193a123a90a194(genus),a89
a52a77a193a33a173a7a123a95a192a88a35a42,a139a63a135a123a52,a89a77a193a123a117a159,a77a193a222a62a60a106a123
a117a159a203a90a194a139a183a35a123a7a248,a196a49a90a194a52a95a192a88a35a42,a199a107a47a116a77a193a123a1a91a117
a159a90a137a219a117a159,a139a58a158a131a173a7a123a107a47,a196a49a114a199a89a74a7a248a52a105a253a163a123,a35a104
a253a163a123,a79a36,a15a52a58a158a7a155,a58a158a139a63a135a123,a183a222a39a121a210Gauss-Bonneta218
a42,a33a7a155a123a218a42a210a52
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(6.2)
a183a25a243a173a133a24a40,a7a207a196a77a193a88a123a155,a24a189a7a207a196a77a193a222a123a41a37,a34a38a82a89
a199a41a37,a38a41a37a1233a199a92a160a37a222a144a35a18a134,a63a37a183a162a34a189a199a52a199a141a67a248,a253
a243a220a199a131a45a160a10a24a199a141a67a221a86a38a67,a183a162a34a38a52a141a67a248,a0a143,a233a149a89a248a6
a41a37,a34a38a28a127a119a145a24a199a37a222a131a128,a217a199a220a199a37a222a210a104a189a234,a79a143a183a162a34a189
a145a174a199a52a77a193a123a92a160a27a37a222,a0a143a145a24a199,a145a174a199a189a234a123a155,a145a19a199a15a210a189
a234,a47a34a222,a183a89a52a24a199a92a160a41a37,a51a30a52a141a67a248(a141a67a41a37),a89a210a113a92a189a234.
a196a49a233a149a243a24a199a154a123a196a139a89a171a248a6a123a41a37,a24a225a89a171a41a37a139a92a107a106a248(one
parameter family),a52a202a226a234a24a199a35a106,a77a193a522a145a123,a242a28a222a89a154a123a41a37a139
a24a199a107a106,a196a49a77a193a196a139a41a37a52a24a1993a145a123a56a45,3a145a56a45a139xa89a199a186a154,a189a199
a243a77a193a123a160a152,a199a86a171a220a199a145,a108a128a242a82a24a199a35a34a48a53,a145a139a24a199a145,a79a143a35
a34a243a35a193a176a62a44a49a221,a196a49a145a245a234a24a145,a89a248a6a210a122a116a196a139a41a37a248a196a196a56a45
a1233a145a123a117a159,a139a234a41a37a248,a139a31a143a80a255a17? a79a143a139a234a37a222,a28a44a49a126a218a42a117
a44a43a241a117,a37a222a139a73a222,a89a73a222a210a139a106,a183a162a180a106a166a33a7a155a123a7a139a106,a28a7
a139a106a123a155,a195a85a52a239a104a123,a63a37a97a126a123a30a127a28a44a49a10a137a116a123a209a52a106,a243a236a171
a63a66a5a143a31a143a135a232a73a7a155? a183a46a106a166a204a7a123a248a123a52a207a196a60a106,a207a196a220a199a248a58
a123a7a248,a25a243a89a7a248a17,a60a106a88a80a180a234,a196a49a135a232a73a52a178a89a199a7a248a34a117a154,a79
a36a44a49a126a83a106,a37a222a44a49a28,a252a199a106a248a117a198,a196a49a135a232a73a204a7a123a196a210a52a178a56
a45a123a174a88,a178a60a106a123a174a88a34a117a154,a83a106a154,a139a234a83a106a49a128,a28a210a44a49a174,a196a49
a210a139a126,a79a36a15a173a7,a0a143a139a234a41a37a196a122a116a123a137a219a123a47a34a52a31a143a17? a183a178a89
3
a199a41a37a119a41xe1e2e3,
E = {xe1e2e3|Ma189a53 e3a52a27a37a222,e1 mbox4#%} (6.3)
e3a52a92a160a123a27a37a222,x,e1,e2,e3 a209a52a37a222,a196a49a199a162a123a135a73a15a52a24a199a37a222,a135
a73a131a128a52a24a39a135a73a42a37a222a138,a79a36,a199a162a44a49a44a143e1,e2,e3a123a24a199a34a117a28a92.
a183a178dxa44a143a34a117a28a92,a122a116a123a248a106a183a119a41ω1,ω2,
dx = ω1e1 + ω2e2,(6.4)
ω1 ∧ ω2a210a52a77a193a123a193a232a221a222,a52a24a1992a39a135a73a42,a199a104a108a44a49a126a117a41a199a173a232
a73a123a232a73a60a106(integral),a196a49a178a199a232a73a123a155,a210a122a116a89a199a77a193a123a193a232,a183a89
a123a7a149a77a193a123a174a88a123a89a74a136a42,a28a243a135a73a1a91a86a222a73a88a116,a152a42a28a88a21a113a92
a122a73,a88a21a113a92a195a123a155,a150a139a7a248,a79a143a89a74a19a142a76a150a52a202a47a135a73a1a91a44a49
a89a24a199a219,a183a89a24a220a39a210a178a199a89a113a234,a89a15a121a210a89a199a48a27a123a128a154,a199a123a128a154
a204a7a52a183a243a207a1963a145a56a45a123Euc lid a1a91,Euclida1a91a33a80a52a126a116a98a41a37,a79
a143a116a98a117a243Euclida1a91a88a35,a52a139a63a66a123,a196a49a33a80a126a116a98a41a37,a0a143,a24a196
a123a135a73a1a91a123a86a127a126a116a77a193a88a123a30a127,a199a88a126a116a98a41a37,a28a7a46a47a123a27a6.
a7a152a128,a178a193a137a219a1a91,a28a88a126a116a98a41a37,a28a123a220a199a43a41a88a18a134,a4a150a149a199a18
a123a48a53a88a52a92a160a123a48a53,a28a86a65a65a23a210a127a119a10a245a234,a102a38a245a234,a88a52a88a44a21
a41,a44a49a41a116,a210a52a102a38a245a234,a139a63a135a123a24a71a47a60,a104a108a183a162a209a127a119,a43a41a248
a58a52a27a41a123a83a166a27,a106a166a27a136a11a16a21a25a123,a198a62a24a39a126a43a41a123a30a127a88a52a116a98
a41a37,a209a52a127a63a123a41a37,a198a126a127a63a123a41a37a252a117a255a174a89a171a1a91a123a175a11,a88a127a119
a52a253a160a11a9a56a234a199a116a98a41a37,a49a128a28a243a86a222a23a116a123a209a52a116a98a41a37,a196a49,a183
a123a41a37a52xe1e2e3,a894a199a209a52a37a222,a199a162a123a135a73a15a122a116a37a222a138a123a24a39a135a73,a196
a49a154a24a199a44a49a44a143e1,e2,e3a123a34a117a28a92,a132a149a183a162a52a243a24a1993a145a123a56a45,a0a143
a89a210a52a222a193a85a123a89a199a218a42(6.4)a90a5a193a123a218a42:
dei = ωijej,(6.5)
a89a30a127,a79a143a52a24a39a135a73a42,a196a49a89a171a34a117a28a92a52e1,e2,e3a123a34a117a28a92,a199a123
a248a106a52a24a39a135a73a42,a88a242a52a60a106a234,a49a3a152a42a52a60a106a123a155,a199a34a117a28a92a123
4
a248a106a52a60a106,a25a243,a248a106a52a24a39a135a73a42,a89a74a24a39a135a73a42a173a7a122a105,a79a143a199
a195a85a24a199a41a37a203a199a248a163a41a37a123a7a248,a199a248a163a41a37a196a24a154a154,a203a198a117a35a139a245
a232? a35a139a52a24a199a135a73,a210a52a183a162a123ωia203ωij,a89a1a199a135a73a42a139a59a92a123a7a248,a33
a7a155a123a52ωij,a28a23a199a105a102a38,i,ja441a1163,a98a52a79a143a41a37a52a116a98a123a92a160a37a222,
a196a49ωija233a149i,ja52a39a233a193a123,a79a36,a28a178ωija85a196a24a1993 × 3 a123a48a106a123a155,a89a199
a48a106a52a39a233a193a123,a199a123a233a110a34a123a195a163a209a520,a63a37a233a149a233a110a34a199a52a39a233a193a123,
a196a49a144a1393a199a93a116a7a255a174a123a24a39a135a73a42,a28a7a126a41a37a117a207a196a1a91a123a89a171a60
a89,a243a197a166a105a7a108,a197a166a89a24a199a212a14a243a0a14a35a196,a0a143a199a123a160a152a210a52a30a45a123
a60a106,a79a36,a89a41a37a52a30a45a123a60a106,a89a171a60a106a243a197a166a222a52a24a199a35a106a123a60a106.
a79a143a243a197a166a222,a243a196a197a166a222,a93a116a123a35a106a52a30a45,a144a139a24a199,a98a52a7a207a196a1
a91a123a155,a60a89a117a122a133a236,a44a21a89a199a41a37a52a203a245a149a24a199a35a106a139a7a248,a44a49a52a245
a35a106a123a60a106,a79a36a89a131a45a210a139a74a7a248,a89a7a248a210a52a28a70d(dei),a183a89a44,a28a126
a222da123a155,da126a220a39a520,a196a49a28a178a89a199a7a248a85a241a117a123a155,a210a122a116dωij a52a24a199a42
a6,a44a49a126a217a198a123ωa117a44a43,a89a42a6a52
dωij = ωik ∧ωkj,(6.6)
a28a122a116a89a248a6a24a28a48a199,a89a52a139a63a66a123,a79a143ωija52a24a39a135a73a42,a28a178a199a135
a73a123a155a522a39a135a73a42,a13a243a141a30a52a220a199a24a39a135a73a42a35a198,a196a49a15a522a39a135a73
a42,a79a36a89a28a48a199a88a175a216,a89a28a48a199a58a158a7a155,a199a162a83a44a228a196a107a114a199a123a117
a159,a89a28a48a199a23a248a133a236,a217a34a58a158a59a92,a79a143a89a74ωij a52a39a233a193a123,a196a49a152
a42i negationslash= j a123a155,a190a152,a152a42i = 1,j = 2,a0a143k = 3,a89a52a79a143k a7a52a127a1491,a149
a52a139ω11 = 0,a13a7a52ka127a1492,a0a143a139ω22 = 0,a196a49a89a28a48a199a42a23a248a133a236,a141
a30a144a139a24a49,a105a59a92a143,a28a164a44a49a122a116a24a199a1a47a60a111,a210a122a116
dω12 = ω13 ∧ω32 =?ω13 ∧ω23,(6.7)
a89a199a218a42a7a155a244a234,a183a162a243a89a199a60a111a210a141a116a24a199a99a123a60a89,a51a248a28a162a39a135a232
a73a123a30a127,a24a196a144a139a24a199a56a45,a76a150a24a196a88a52a178a193a210a523a145a56a45,a44a52a183a162a25
a243a139a220a199a56a45,a24a199a52a41a37a158a106a196a123a56a45,a523a145a123;a12a24a199a52a183a1622a145a123a77
a193,a196a49a183a139a24a1992a145a77a193a164a139a24a1993a145a123a56a45,a893a145a56a45a52a199a41a37,a79a36a152
5
a42a24a199a41a37,a28a82a199a198a154a123a155,a183a162a128a199a210a61a107a116a77a193a222a86a234,a89a248a6a210a139
a199a61a107,a25a243a199a139a199a214a35a119a41a14a145a45,a25a243a52a203a177a45a234,a14a145a52a203a177,a139a24
a178a203a177,a13a114a199a123a203a177a196a196a123a56a45a210a52a183a198a117a123a77a193,a183a162a119a198a117a123a77a193a143
a142a56a45,a252a51a24a199a198a154a123a196a139a92a160a35a37a222a210a196a14a145,a149a52a232a196a14a145a45,a199a210
a54a183a162a31a113a145a123,a139a24a30a24a30a123a34,a33a59a92a123a14a145a45a52a199a123a14a145a52a134a34,a0
a143a199a52a134a34a45,a183a65a248a178a199a7a48a196a24a178a77a6,a28a139a80a245a77a6,a154a24a202a77a6a52
a30a134a34,a0a143a139a80a245a77a6,a114a199a77a6a196a24a199a56a45,a89a210a52a183a162a123a14a145a45,a89a52
a134a34a123a60a89,a183a162a25a243a41a123a60a89a52a203a177a45,a89a199a10a39a52a135a232a73a176a62a24a199a99a123
a10a39,a210a52a128,a28a88a52a255a88a24a199a56a45,a13a52a28a243a255a88a220a199a56a45,a63a37a89a220a199a56
a45a131a45a139a183a35a123a7a248,a24a199a52a203a177a196a196a123a56a45,a24a199a52a183a162a123a142a56a45,a15a210
a52a198a117a123a77a193,a89a220a199a56a45a131a45a139a183a196a128a123a89a199a7a248,a89a199a7a248a139a63a135a244a234,
a173a7a244a234,a79a143a139a5a193a123a7a248
dω12 =?Kω1 ∧ω2,(6.8)
a141a30a52a77a193a222a123a42a6,a89a52a79a143Ka52Gaussa77a71,ω1 ∧ ω2a52a193a232a221a222,a196a49
a141a30a52a77a193a222a123a117a159,a38a30a52a24a199a192a220a123a135a73,ω12a52a243a14a145a45E a176a62a123a24
a39a135a73a42,a89a199a24a39a135a73a42a123a105a135a73a127a149a141a30a123a42a6,a89a199a121a210a128a210Gauss
a77a71a144a203Riemann a221a222a139a7,a79a143a7a52a139a234Riemanna221a222a210a139ω12,a0a143a183
a162a141a30a123a42a6a144a203Riemanna221a222a139a7,a89a52Gauss a104a35a105a122a63a123a24a199a136a42,
a203Guassa209a250a122a105a88a122a234a139a89a143a248a6a24a199a7a248,Gauss-Bonneta218a42a210a52a183
a162a7a70a141a30a89a199a42a6a123a232a73,a183a162a25a243a139a24a199a85a20a123a77a193,a199a52a189a53a123,
a7a70a141a30a123a232a73,a70a241a199a123a138a117,a0a143a104a30a183a15a139a24a171a70a216,a79a143a141a30a89
a199a42a6a17a108a52d a24a199a192a220a123a155,a243a24a199a85a20a77a193a222a123a232a73a97a148a520,a47a34
a222,a199a97a148a127a149ω12a215a248a89a199a77a193a123a30a142a123a232a73,a13a152a42a77a193a52a85a20a123,a199
a150a139a30a142,a196a49a97a148a520,a89a23a108a52a70a123,a143a31a143a199a88a127a1490? a183a162a175a108
a139dω12 =?Kω1 ∧ω2,a98a89a199a7a248a88a52a243a24a1992a145a56a45a222,a199a52a243Ea89a1993 a145
a56a45a222,a196a49a183a162a144a21a234a2433a145a56a45a188a126Stokes a189a174,a13a2433a145a56a45a123a155,a89
a199a77a193a2433a145a56a45a176a62a210a139a30a142a234,a28a7a178a89a199a77a193a12a1163a145a56a45a86,a14a143a12
a17? a210a52a154a154a7a201a24a199a252a89a154a41a198a154a123a35a37a222,a166a233a155a128,a89a210a52a196a162a123a37
6
a222a156.a196a49a89a199a77a193a137a7a139a199a37a222a156,a154a154a139a199a35a37a222,a13a89a199a35a37a222a52Ea176
a62a123a24a199a154,a210a178a89a199a77a193a12a116Ea176a62a86,a34a38a139a24a199a77a193,a52a88a52a24a189a139
a199a37a222a156? a89a88a59a92a234,a243a219a92a123a30a127,a104a108a105a59a92,a28a85a5a43a41,a177a34a85a74
a37a222,a210a139a234,a52a88a52a21a234a243a114a199a77a193a201a24a199a37a222a156,a89a52a1a91a176a62a196a162a114
a14a123a175a11,a202a47a95a192a210a180a89a199a175a11,a15a210a52a128,a219a92a23a108a44a49a85a37a222a123,a28
a139a219a92a43a41,a28a178a43a41a73a222a85a5a117,a104a108a210a139a199a37a222a156,a98a52a89a199a52a219a92a123,
a21a88a21a106a223a116a114a199a123a77a193,a88a24a189a44a21,a0a143,a183a222a39a46a178a89a44,a7a89a248a123
a155,a23a140a227a142a89a199a37a222a156a139a73a154(singularity),a7a48a128,a243a5a193a1a199a67a176a62a139
a1a199a37a222a156a123a190a6,( a67a68a63a210a161) a33a38a30a123a190a6,a199a123a73a154a210a243a198a154,a178a44
a89a199a198a154,a53a196a139a48a53a152a37a222,a248a234a198a154a131a105,a210a189a234a24a199a37a222a156,a98a52a198a154
a52a24a199a73a154,a199a52a196a139a121a22a241a117a123a241a21a154,a196a49a199a52a199a73a154,a145a19a199,a196a139a37
a222a209a53a198a154a18,a198a154a164a52a24a199a73a154,a198a154a210a35a196a24a199a188a5a86a123a24a199a154,a93a169
a119sink,a13a38a30a123a119source,a104a108a15a139a54a33a141a30a123a190a6,a44a89a74a190a6a44a49a23a241
a37a222a156a243a73a154a139a88a51a117a159,a152a91a195a85a199a123a88a51a123a117a159,a210a139a24a199a119a41a37a222a156
a123a141a41(index),a28a243a89a24a154,a34a38a139a199a241a193a123a73a154,a0a143a140a248a89a199a73a154a41a199
a66a203a87,a79a143a52a241a193a123a73a154,a196a49a243a66a203a87a222a123a154a123a37a222a52a113a92a104a189a123,a0
a143a25a243,a243a66a203a87a123a154a119a248a73a154a221a245a232a87a17? a152a42a221a24a87,a63a37a52a243a116a123a48
a53a221a24a87,a199a123a141a41a521,a152a42a53a139a123a48a53a210a52?1,a0a143,a243a183a123a222a193a190a6
a165,a195a88sink a164a52source,a141a41a209a521,a118a77a34a123a25a54a141a41a143?1,a73a154a105a133
a236,a79a36a141a41a44a49a82a127a91a138,a34a38a183a178a77a193a12a116a14a145a45a176a62,a12a116a203a177a45a176
a62,a63a37a227a142a139a73a154,a0a143a89a199a222a86a123a77a193a210a139a30a142,a89a199a30a142a210a35a104a149a89
a74a73a154,a196a49a202a226a218a42dω12 =?Kω1 ∧ ω2,a183a162a7a149Gaussa77a71a123a232a73a210a127
a149a73a154a123a141a41a90,a196a49a183a162a121a210a24a199a173a7a117a159,a88a88a28a82a127a91a24a199a37a222a156,
a34a38a199a144a139a139a33a199a73a154,a183a162a89a199a232a73a52a141a41a90,a253a52a178a154a199a73a154a123a141a41a28
a229a117a210a127a149a141a41a90,a89a176a105a7a155,a79a143a89a199a232a73a52a203a37a222a156a160a10a195a7a123,a196
a49a89a121a210a234a24a199a77a193a34a38a139a24a199a139a139a33a199a73a154a123a37a222a156,a243a73a154a123a141a41a90
a37a222a156a123a160a10a195a7,a199a127a149a0a199a232a73,a13a0a199a232a73a176a52a150a139a37a222a156,a196a49a210
a122a116a89a248a24a199a136a42,a183a242a128a24a40,a25a243a139a24a199a85a20a123a77a193,a82a24a199a37a222a156,a139
a139a33a199a73a154,a199a123a141a41a90a52a166a37a222a156a123a160a10a195a7a123,a89a52a79a143a199a127a149a141a30a123
7
a232a73,a13a141a30a232a73a202a253a150a139a37a222a156,a196a49a166a37a222a156a123a160a10a195a7,a143a31a143a89a199
a106a248a127a149Euler a43a117a106a17? a25a243a17a108a199a203a37a222a156a160a10a195a7,a28a210a127a82a24a199a37
a222a156,a7a48a128,a34a38a139a199a77a193,a28a178a77a193a73a191a234,a73a191a196a66a76,a154a199a66a76a52a174
a110a111,a233a149a89a174a110a111,a154a199a30a82a199a123a165a154,a174a110a111a82a199a123a173a101,a28a210a44a49a189
a24a199a37a222a156,a210a54a183a196a152a123,a44a186a154a241a86,a108a128a116a174a110a111a123a173a101a210a159a86,a233a149
a89a248a6a189a123a37a222a156a105a142a52a23a241a117,a166a28a243a30a222a123a89a171a154a123a141a41a127a149?1,a149
a52a199a243a186a154a123a141a41a521,a243a174a110a111a173a101a123a141a41a209a521,a98a52a243a30a222a154a199a154a141a41
a143?1,a196a49a178a89a141a41a28a229a117a123a155,a210a127a149a186a154a123a199a106+a193a123a199a106?a30a123a199a106,
a79a36a210a52Euler a43a117a106,a89a248a6a121a210a234Gauss-Bonnet a218a42.
4 Gauss-Bonneta218a42a123a77a18a249a97a126
Gauss-Bonneta218a42a93a116a139a126a123a30a127a52a77a193a139a30a142,a243a77a193a139a30a142a123a30a127,
Gauss-Bonnet a218a42a52a186a154+a186a154a123a105a110+a30a123a127a143a77a71(geodesic curva-
ture),a242a28a222a243a193a123Gaussa77a71,a5a193a52a24a196a123Gauss-Bonneta218a42
summationdisplay(pi?α) (a154) +summationdisplayintegraldisplay k
g(s)ds (a30) +
summationdisplayintegraldisplay integraldisplay KdA (a193) = 2piχ(M) (6.9)
a233a149a139a30a142a123a77a193,a62a24a92a73a52a30a142a186a154a123a154a77a71,a217a39a52a30a142a123a30a123a34
a77a71,a108a128a114a199a123a89a199a192a220a123a193a77a71,a196a49a28a139a24a199a139a30a142a123a77a193,a28a210
a82a30a142a123a154a77a71+a30a142a123a34a77a71+a193a77a71,a52Eulera43a117a106,a121a210a52a24a248a123.
a93a116Gauss-Bonneta218a42a33a139a126a123a52a139a30a142a123a60a89,a7a48a128,a243a24a199Euclida178
a193,a34a38a139a24a199a174a110a111,a89a199a174a110a111a132a134a34a196a196,a132a149a56a45a52Euclida56
a45,Gaussa77a71=0; a34a38a30a209a52a134a34,a196a49a127a143a77a71a15a520,a79a36a89a199a210a52
a128summationtext(pi?α)a127a1492pi,a89a52a79a143a7a52a174a110a111,Eulera43a117a106a521,a141a30a7a127a1492pi,
a196a49a89a210a128a210a174a110a111a174a110a131a90a243Euclida178a193a222a127a149pi,Gauss-Bonneta218a42a52
a174a110a111a174a110a90a218a42a243a24a196a60a111a123a77a18,a89a199a10a39a173a7a244a234,a199a210a52a114a199a14
a145a45a123a10a39,a183a128,a132a89a199a14a145a45,Maxwella48a199a210a52a89a199a60a89a123a77a18,a28a116
a212a174a222a97a126a123a30a127,a28a123a56a45a524a145,a523a145a56a45+1a145a30a45,a524a145a123a98a85a251a22
a111,a0a143a7a44Maxwella48a199a123a155,a28a7a126a24a199a203a177a45,a34a19a222a52a24a199a133a123a134
8
a34a45,a199a139a199a77a71,a183a162a123a77a71a52Gaussa77a71a198a49a193a232a195a163,a13a89a199a77a71a52
a1992a39a135a73a42,a178a44a43a89a199a77a71a52a85a20a123a30a71a85a241a117a210a52Maxwe lla48a199,a196
a49,Maxwella48a199a123a1a91a242a181a52a58a158a59a92a123,a210a52a79a143a45a142a524a145a123a56a45,a196
a49a52a442a145a56a45a106a223a1164a145,a0a143a89a199a77a71a79a143a52a24a1992a39a135a73a42,a164a52a39a233
a193a123,a79a36a2434a145a56a45a176a52a24a1994×4a48a106:
(Fαβ =
0 E1 E2 E3
E1 0?B0 B2
E2 B0 0?B1
E3?B2 B1 0
,(6.10)
a89a199a48a106a176a62E1,E2,E3 a52Electric Potential,B0,B1,B2a52Magnetic Po ten-
tial,a15a210a52a158a51a203a30a51,a89a74a209a52a48a106a176a62a123a60a106,a44a43a132a89a199a48a106a196a44a43
a1232a39a135a73a42a52a85a20a123,a253da89a199a42a6a123a135a73a1430,a210a52Maxwell a48a199
d(Fαβdxα ∧dxβ) = 0,(6.11)
Gauss-Bonneta218a42a52a24a199a187a36a222a105a245a220a35a123,a93a116a183a85a123a218a42a17a88a132
a149Gauss,a15a88a132a149Bonnet,Gaussa144a41a234a174a110a111a123a60a89,a132a127a143a174a110a111a41a116
a174a110a111,Bonneta15a150a139a41a95a192a123a97a126,Bonneta178a174a110a111a77a18a116a127a63a77a34,a198
a178a127a63a77a34a123a127a143a77a71a232a73a44a143Gauss-B onneta218a42a123a232a73,a104a35Bonneta52
a27a41a33a7a155a123a1a91a166a27,a198a243a135a73a1a91a41a234a58a158a228a253a123a224a26,a183a88a11a28a162
a234a137a245a232,a183a230a130a28a162a234a137a89a24a92a73a123a106a166a58a158a7a155,Maxwella48a199a52a199a123
a1a47a60a89,a89a199a58a158a139a126a255,a183a139a24a159a169a57a243a152a35a123a21a41a166a22a222a21a44,a11a248
a119a21Gauss- Bonneta218a42a166Maxwella48a199a22,a183a89a123a142a245a7a154a243a89a159a169a57a176a62
a139.
9
a9Gauss-Bonneta218a42(a153)
2001a3511a21930a134
1 a135a232a73a243a133a35a60a106a88a165a97a126a59a144
a183a164a97a148a242a89a220a39,a89a220a39a183a139a199a15a153,a189a247a89a24a154a133a35a60a106a88,a79a143a243a106
a166a165,a105a7a155a123a24a71a47a34,a51a30a243a106a166a36a222a15a52a58a158a7a155a123a24a71a47a60,a210a52
a139a133a106,a89a199a133a106a38a122a106a166a59a92,a133a60a106a139a142a245a164a224,a139a63a135a123a117a159,a79a36,
a89a38a122a89a74a60a106a243a97a126a222a1a47a139a126a255,a196a49,a183a189a247a89a24a89,a7a152a128,a133a35a60
a106a139a24a199a105a173a7a123a117a159,a127a63a123a83a106a48a199a243a133a35a60a106a131a165a24a189a139a137,a89a52a24
a199a88a122a234a123a47a60,a79a143a88a11a28a14a143a248a85a24a199a48a199,a28a7a52a227a142a137a52a133a106a123a155,
a199a24a189a139a137,a190a152,x2 + 1 = 0,a0a143a199a139a199a137a210a52√?1,a196a49√?1a210a89a143a248
a6a139a126a255,a88a98a152a36,a133a106a203a34a106a24a248,a44a49a28a62,a139a51a248a123a117a159,a196a49,a199
a44a49a228a174,a51a30a199a221a57a234a142a245a97a238a52a34a106a88a21a221a57a123,a183a46a183a123a49a243a44a199a165
a24a189a204a139a199a56a105,a243a56a105a123a30a127,a183a46a73a220a39a89a133a35a60a106,a183a189a247a89,a24a199a52
a183a166a98a89a123a83a106a123a228a253a189a174,a210a52a128a127a63a123a83a106a123a48a199a243a133a106a173a165a24a189a139
a137,a89a199a52a105a10a121a210a123,a137a7a106a166a222a99a123a10a39,a7a48a128,a149a76a106a166a27a152Euler,
a198a46a27a6a121a210,a98a150a139a21a196a213,a183a46Gauss a52a183a162a163a83a33a149a76a123a106a166a27,a198
a105a35a52a123a30a127a210a139a24a199a121a210,a15a210a52a133a106a137a7a24a74a1a91a123a117a159,a88a113a92a52a83
a106a123a175a11,a183a189a247a5a39a89a133a106a123a30a127a121a210a89a199a189a174;a51a30,a133a35a60a106a33a204a7
a123a24a199a189a174a52Picarda189a174,a210a52a128,a34a38a233a149a24a199a133a35a60a106,a82a199a123a60a106a138a243
a133a178a193a176a62a196a82a123a160a152,a199a178a114a199a133a178a193a209a152a212a234,a217a165a15a142a44a49a86a171a24a154,
a220a154,a89a52a88a122a234a123,a210a52a128,a60a106a152a42a52a24a199a92a24a60a106a123a155,a199a73a89a122a58a158
a131a254a225,a44a49a128a139a88a245a178a178a193a209a152a212a234,a139a63a135a123a24a71a47a60a52a89a199a189a174a52a133
a35a60a106a176a88a123a189a174,a44a49a188a126a183a162a25a243a7a89a123Gauss-Bonneta218a42a117a121a210,a89
a128a210a23a229a117a150a139a7a248a123a24a74a48a27a203a10a39,a136a42a52a139a7a248a123,a89a52a106a166a222a58a158
a7a155,a139a63a135a123a175a11.
1
2 a7a149a166a243a123a7a196a117
a89a199a49a79a136a101a234,a28a162a243a89a199a49a85a199a231a183,a33a80a52a7a196,a28a21a234a7a3a73a116a24
a199a175a11,a89a52a205a7a155a123,a183a46a28a162a209a52a76a166a9,a76a166a9a73a176a127a115a184a33a128a123a24
a227,a49a128a116a246a204a222a86,a253a38a243a166a68,a243a166a98a92a160a176a62,a33a7a155a123a24a189a7a7a196.
a88a7a52a127a144a19a119a28a41a31a143,a28a242a41a31a143,a89a199a33a160,a7a7a196,a7a7a3a21a73a175
a11,a7a7a3a21a234a73a133a7a3a73a123a175a11,a0a143,a104a108a28a73a123a175a11a88a24a189a92a55,a28a246
a30a15a88a24a189a21a234a122a116a73a153,a88a44,a28a165a45a178a44a24a74a107a52,a178a44a24a74a70a216,a44a49
a38a122a28a123a166a175a93a116a159a90,a13a38a122a28a93a116a159a90a123a210a52a7a178a44a89a248a123a67a153,a196a49
a183a243a185a76a27a7a7a196,a245a24a154a143a89a229a117,a28a162a4a150a7a21a234a28a132a24a199a76a14,a144a35
a231a183a73a175a11,a221a86a62a68a19a68a105a144a19,a51a166a117a41a231a183,a89a52a105a139a80a255a123,a7a3a7
a178a106a166a46a24a46,a221a86a233a127a63a123a166a175,a28a7a3a139a199a135a46,a250a122a139a199a31a143a248a123a217
a196,a233a149a28,a233a149a89a199a166a175a123a127a35a44a49a15a28,a51a30a28a233a166a175a123a21a197a15a44a49a15
a28,a196a49a89a52a105a138a122a213a63a123a24a71a47a60,a230a130a28a162a31a68a24a5a89a199a44a21a117.
3 Gauss-Bonneta218a42a123a121a210
a222a39,Gauss-Bonneta218a42a183a150a139a121a210a92,a196a49a183a11a178a121a210a128a92a234,a183a222a39a89
a123Gauss-Bo nneta218a42a210a52,a34a38a243a56a45a176a62a139a24a199a77a193,a199a52a24a199a114a199a123
a77a193,a63a37a34a38a89a199a77a193a52a189a53a123,a253a199a123a27a34a139a24a189a123a48a53,a149a52a89a248a6,
Gaussa77a71Ka210a52a77a193a222a123a24a199a60a106,a183a44a49a178a89a199a60a106a233a149a77a193a222a123a193a232
a221a222a70a232a73,a89a199a232a73a52a24a1992a173a232a73,a70a199a232a73a131a128,a136a42a89a199a232a73a127a149a24
a199a158a106(2pi)a198a49a77a193a123Euler a43a117a106,a253
integraldisplay integraldisplay
KdA = 2piχ(M) (6.1)
Eulera43a117a106a210a52a178a77a193a35a196a66a76a131a128,a55a149a24a154a7a108a123a30a71,a178a199a35a113a131
a128,a217a186a154a199a106?a30a123a199a106+a193a123a199a106,a89a2483a199a106a123a116a139a123a90a210a119a41a89a199a77
a193a123Eulera43a117a106,a104a77a193a52a69a193a123a155,a199a123Eulera43a117a106= 2,a152a42a199a52a199a162
a193,a199a123Eulera43a117a106a520,a28a162a44a49a65a24a65,a210a21a122a116a89a199.a152a42a77a193a52a199a189
2
a53a77a193,a89a52a199a123a141a24a123a95a192a88a35a42,a24a196a89a229a117,a34a38a69a222a28a1a199a162,a162a123
a199a106a210a203Eulera43a117a106a139a199a7a248,a89a162a123a199a106a202a47a119a77a193a123a90a194(genus),a89
a52a77a193a33a173a7a123a95a192a88a35a42,a139a63a135a123a52,a89a77a193a123a117a159,a77a193a222a62a60a106a123
a117a159a203a90a194a139a183a35a123a7a248,a196a49a90a194a52a95a192a88a35a42,a199a107a47a116a77a193a123a1a91a117
a159a90a137a219a117a159,a139a58a158a131a173a7a123a107a47,a196a49a114a199a89a74a7a248a52a105a253a163a123,a35a104
a253a163a123,a79a36,a15a52a58a158a7a155,a58a158a139a63a135a123,a183a222a39a121a210Gauss-Bonneta218
a42,a33a7a155a123a218a42a210a52
dω12 =?ω13 ∧ω23 =?Kω1 ∧ω2,(6.2)
a183a25a243a173a133a24a40,a7a207a196a77a193a88a123a155,a24a189a7a207a196a77a193a222a123a41a37,a34a38a82a89
a199a41a37,a38a41a37a1233a199a92a160a37a222a144a35a18a134,a63a37a183a162a34a189a199a52a199a141a67a248,a253
a243a220a199a131a45a160a10a24a199a141a67a221a86a38a67,a183a162a34a38a52a141a67a248,a0a143,a233a149a89a248a6
a41a37,a34a38a28a127a119a145a24a199a37a222a131a128,a217a199a220a199a37a222a210a104a189a234,a79a143a183a162a34a189
a145a174a199a52a77a193a123a92a160a27a37a222,a0a143a145a24a199,a145a174a199a189a234a123a155,a145a19a199a15a210a189
a234,a47a34a222,a183a89a52a24a199a92a160a41a37,a51a30a52a141a67a248(a141a67a41a37),a89a210a113a92a189a234.
a196a49a233a149a243a24a199a154a123a196a139a89a171a248a6a123a41a37,a24a225a89a171a41a37a139a92a107a106a248(one
parameter family),a52a202a226a234a24a199a35a106,a77a193a522a145a123,a242a28a222a89a154a123a41a37a139
a24a199a107a106,a196a49a77a193a196a139a41a37a52a24a1993a145a123a56a45,3a145a56a45a139xa89a199a186a154,a189a199
a243a77a193a123a160a152,a199a86a171a220a199a145,a108a128a242a82a24a199a35a34a48a53,a145a139a24a199a145,a79a143a35
a34a243a35a193a176a62a44a49a221,a196a49a145a245a234a24a145,a89a248a6a210a122a116a196a139a41a37a248a196a196a56a45
a1233a145a123a117a159,a139a234a41a37a248,a139a31a143a80a255a17? a79a143a139a234a37a222,a28a44a49a126a218a42a117
a44a43a241a117,a37a222a139a73a222,a89a73a222a210a139a106,a183a162a180a106a166a33a7a155a123a7a139a106,a28a7
a139a106a123a155,a195a85a52a239a104a123,a63a37a97a126a123a30a127a28a44a49a10a137a116a123a209a52a106,a243a236a171
a63a66a5a143a31a143a135a232a73a7a155? a183a46a106a166a204a7a123a248a123a52a207a196a60a106,a207a196a220a199a248a58
a123a7a248,a25a243a89a7a248a17,a60a106a88a80a180a234,a196a49a135a232a73a52a178a89a199a7a248a34a117a154,a79
a36a44a49a126a83a106,a37a222a44a49a28,a252a199a106a248a117a198,a196a49a135a232a73a204a7a123a196a210a52a178a56
a45a123a174a88,a178a60a106a123a174a88a34a117a154,a83a106a154,a139a234a83a106a49a128,a28a210a44a49a174,a196a49
a210a139a126,a79a36a15a173a7,a0a143a139a234a41a37a196a122a116a123a137a219a123a47a34a52a31a143a17? a183a178a89
3
a199a41a37a119a41xe1e2e3,
E = {xe1e2e3|Ma189a53 e3a52a27a37a222,e1 mbox4#%} (6.3)
e3a52a92a160a123a27a37a222,x,e1,e2,e3 a209a52a37a222,a196a49a199a162a123a135a73a15a52a24a199a37a222,a135
a73a131a128a52a24a39a135a73a42a37a222a138,a79a36,a199a162a44a49a44a143e1,e2,e3a123a24a199a34a117a28a92.
a183a178dxa44a143a34a117a28a92,a122a116a123a248a106a183a119a41ω1,ω2,
dx = ω1e1 + ω2e2,(6.4)
ω1 ∧ ω2a210a52a77a193a123a193a232a221a222,a52a24a1992a39a135a73a42,a199a104a108a44a49a126a117a41a199a173a232
a73a123a232a73a60a106(integral),a196a49a178a199a232a73a123a155,a210a122a116a89a199a77a193a123a193a232,a183a89
a123a7a149a77a193a123a174a88a123a89a74a136a42,a28a243a135a73a1a91a86a222a73a88a116,a152a42a28a88a21a113a92
a122a73,a88a21a113a92a195a123a155,a150a139a7a248,a79a143a89a74a19a142a76a150a52a202a47a135a73a1a91a44a49
a89a24a199a219,a183a89a24a220a39a210a178a199a89a113a234,a89a15a121a210a89a199a48a27a123a128a154,a199a123a128a154
a204a7a52a183a243a207a1963a145a56a45a123Euc lid a1a91,Euclida1a91a33a80a52a126a116a98a41a37,a79
a143a116a98a117a243Euclida1a91a88a35,a52a139a63a66a123,a196a49a33a80a126a116a98a41a37,a0a143,a24a196
a123a135a73a1a91a123a86a127a126a116a77a193a88a123a30a127,a199a88a126a116a98a41a37,a28a7a46a47a123a27a6.
a7a152a128,a178a193a137a219a1a91,a28a88a126a116a98a41a37,a28a123a220a199a43a41a88a18a134,a4a150a149a199a18
a123a48a53a88a52a92a160a123a48a53,a28a86a65a65a23a210a127a119a10a245a234,a102a38a245a234,a88a52a88a44a21
a41,a44a49a41a116,a210a52a102a38a245a234,a139a63a135a123a24a71a47a60,a104a108a183a162a209a127a119,a43a41a248
a58a52a27a41a123a83a166a27,a106a166a27a136a11a16a21a25a123,a198a62a24a39a126a43a41a123a30a127a88a52a116a98
a41a37,a209a52a127a63a123a41a37,a198a126a127a63a123a41a37a252a117a255a174a89a171a1a91a123a175a11,a88a127a119
a52a253a160a11a9a56a234a199a116a98a41a37,a49a128a28a243a86a222a23a116a123a209a52a116a98a41a37,a196a49,a183
a123a41a37a52xe1e2e3,a894a199a209a52a37a222,a199a162a123a135a73a15a122a116a37a222a138a123a24a39a135a73,a196
a49a154a24a199a44a49a44a143e1,e2,e3a123a34a117a28a92,a132a149a183a162a52a243a24a1993a145a123a56a45,a0a143
a89a210a52a222a193a85a123a89a199a218a42(6.4)a90a5a193a123a218a42:
dei = ωijej,(6.5)
a89a30a127,a79a143a52a24a39a135a73a42,a196a49a89a171a34a117a28a92a52e1,e2,e3a123a34a117a28a92,a199a123
a248a106a52a24a39a135a73a42,a88a242a52a60a106a234,a49a3a152a42a52a60a106a123a155,a199a34a117a28a92a123
4
a248a106a52a60a106,a25a243,a248a106a52a24a39a135a73a42,a89a74a24a39a135a73a42a173a7a122a105,a79a143a199
a195a85a24a199a41a37a203a199a248a163a41a37a123a7a248,a199a248a163a41a37a196a24a154a154,a203a198a117a35a139a245
a232? a35a139a52a24a199a135a73,a210a52a183a162a123ωia203ωij,a89a1a199a135a73a42a139a59a92a123a7a248,a33
a7a155a123a52ωij,a28a23a199a105a102a38,i,ja441a1163,a98a52a79a143a41a37a52a116a98a123a92a160a37a222,
a196a49ωija233a149i,ja52a39a233a193a123,a79a36,a28a178ωija85a196a24a1993 × 3 a123a48a106a123a155,a89a199
a48a106a52a39a233a193a123,a199a123a233a110a34a123a195a163a209a520,a63a37a233a149a233a110a34a199a52a39a233a193a123,
a196a49a144a1393a199a93a116a7a255a174a123a24a39a135a73a42,a28a7a126a41a37a117a207a196a1a91a123a89a171a60
a89,a243a197a166a105a7a108,a197a166a89a24a199a212a14a243a0a14a35a196,a0a143a199a123a160a152a210a52a30a45a123
a60a106,a79a36,a89a41a37a52a30a45a123a60a106,a89a171a60a106a243a197a166a222a52a24a199a35a106a123a60a106.
a79a143a243a197a166a222,a243a196a197a166a222,a93a116a123a35a106a52a30a45,a144a139a24a199,a98a52a7a207a196a1
a91a123a155,a60a89a117a122a133a236,a44a21a89a199a41a37a52a203a245a149a24a199a35a106a139a7a248,a44a49a52a245
a35a106a123a60a106,a79a36a89a131a45a210a139a74a7a248,a89a7a248a210a52a28a70d(dei),a183a89a44,a28a126
a222da123a155,da126a220a39a520,a196a49a28a178a89a199a7a248a85a241a117a123a155,a210a122a116dωij a52a24a199a42
a6,a44a49a126a217a198a123ωa117a44a43,a89a42a6a52
dωij = ωik ∧ωkj,(6.6)
a28a122a116a89a248a6a24a28a48a199,a89a52a139a63a66a123,a79a143ωija52a24a39a135a73a42,a28a178a199a135
a73a123a155a522a39a135a73a42,a13a243a141a30a52a220a199a24a39a135a73a42a35a198,a196a49a15a522a39a135a73
a42,a79a36a89a28a48a199a88a175a216,a89a28a48a199a58a158a7a155,a199a162a83a44a228a196a107a114a199a123a117
a159,a89a28a48a199a23a248a133a236,a217a34a58a158a59a92,a79a143a89a74ωij a52a39a233a193a123,a196a49a152
a42i negationslash= j a123a155,a190a152,a152a42i = 1,j = 2,a0a143k = 3,a89a52a79a143k a7a52a127a1491,a149
a52a139ω11 = 0,a13a7a52ka127a1492,a0a143a139ω22 = 0,a196a49a89a28a48a199a42a23a248a133a236,a141
a30a144a139a24a49,a105a59a92a143,a28a164a44a49a122a116a24a199a1a47a60a111,a210a122a116
dω12 = ω13 ∧ω32 =?ω13 ∧ω23,(6.7)
a89a199a218a42a7a155a244a234,a183a162a243a89a199a60a111a210a141a116a24a199a99a123a60a89,a51a248a28a162a39a135a232
a73a123a30a127,a24a196a144a139a24a199a56a45,a76a150a24a196a88a52a178a193a210a523a145a56a45,a44a52a183a162a25
a243a139a220a199a56a45,a24a199a52a41a37a158a106a196a123a56a45,a523a145a123;a12a24a199a52a183a1622a145a123a77
a193,a196a49a183a139a24a1992a145a77a193a164a139a24a1993a145a123a56a45,a893a145a56a45a52a199a41a37,a79a36a152
5
a42a24a199a41a37,a28a82a199a198a154a123a155,a183a162a128a199a210a61a107a116a77a193a222a86a234,a89a248a6a210a139
a199a61a107,a25a243a199a139a199a214a35a119a41a14a145a45,a25a243a52a203a177a45a234,a14a145a52a203a177,a139a24
a178a203a177,a13a114a199a123a203a177a196a196a123a56a45a210a52a183a198a117a123a77a193,a183a162a119a198a117a123a77a193a143
a142a56a45,a252a51a24a199a198a154a123a196a139a92a160a35a37a222a210a196a14a145,a149a52a232a196a14a145a45,a199a210
a54a183a162a31a113a145a123,a139a24a30a24a30a123a34,a33a59a92a123a14a145a45a52a199a123a14a145a52a134a34,a0
a143a199a52a134a34a45,a183a65a248a178a199a7a48a196a24a178a77a6,a28a139a80a245a77a6,a154a24a202a77a6a52
a30a134a34,a0a143a139a80a245a77a6,a114a199a77a6a196a24a199a56a45,a89a210a52a183a162a123a14a145a45,a89a52
a134a34a123a60a89,a183a162a25a243a41a123a60a89a52a203a177a45,a89a199a10a39a52a135a232a73a176a62a24a199a99a123
a10a39,a210a52a128,a28a88a52a255a88a24a199a56a45,a13a52a28a243a255a88a220a199a56a45,a63a37a89a220a199a56
a45a131a45a139a183a35a123a7a248,a24a199a52a203a177a196a196a123a56a45,a24a199a52a183a162a123a142a56a45,a15a210
a52a198a117a123a77a193,a89a220a199a56a45a131a45a139a183a196a128a123a89a199a7a248,a89a199a7a248a139a63a135a244a234,
a173a7a244a234,a79a143a139a5a193a123a7a248
dω12 =?Kω1 ∧ω2,(6.8)
a141a30a52a77a193a222a123a42a6,a89a52a79a143Ka52Gaussa77a71,ω1 ∧ ω2a52a193a232a221a222,a196a49
a141a30a52a77a193a222a123a117a159,a38a30a52a24a199a192a220a123a135a73,ω12a52a243a14a145a45E a176a62a123a24
a39a135a73a42,a89a199a24a39a135a73a42a123a105a135a73a127a149a141a30a123a42a6,a89a199a121a210a128a210Gauss
a77a71a144a203Riemann a221a222a139a7,a79a143a7a52a139a234Riemanna221a222a210a139ω12,a0a143a183
a162a141a30a123a42a6a144a203Riemanna221a222a139a7,a89a52Gauss a104a35a105a122a63a123a24a199a136a42,
a203Guassa209a250a122a105a88a122a234a139a89a143a248a6a24a199a7a248,Gauss-Bonneta218a42a210a52a183
a162a7a70a141a30a89a199a42a6a123a232a73,a183a162a25a243a139a24a199a85a20a123a77a193,a199a52a189a53a123,
a7a70a141a30a123a232a73,a70a241a199a123a138a117,a0a143a104a30a183a15a139a24a171a70a216,a79a143a141a30a89
a199a42a6a17a108a52d a24a199a192a220a123a155,a243a24a199a85a20a77a193a222a123a232a73a97a148a520,a47a34
a222,a199a97a148a127a149ω12a215a248a89a199a77a193a123a30a142a123a232a73,a13a152a42a77a193a52a85a20a123,a199
a150a139a30a142,a196a49a97a148a520,a89a23a108a52a70a123,a143a31a143a199a88a127a1490? a183a162a175a108
a139dω12 =?Kω1 ∧ω2,a98a89a199a7a248a88a52a243a24a1992a145a56a45a222,a199a52a243Ea89a1993 a145
a56a45a222,a196a49a183a162a144a21a234a2433a145a56a45a188a126Stokes a189a174,a13a2433a145a56a45a123a155,a89
a199a77a193a2433a145a56a45a176a62a210a139a30a142a234,a28a7a178a89a199a77a193a12a1163a145a56a45a86,a14a143a12
a17? a210a52a154a154a7a201a24a199a252a89a154a41a198a154a123a35a37a222,a166a233a155a128,a89a210a52a196a162a123a37
6
a222a156.a196a49a89a199a77a193a137a7a139a199a37a222a156,a154a154a139a199a35a37a222,a13a89a199a35a37a222a52Ea176
a62a123a24a199a154,a210a178a89a199a77a193a12a116Ea176a62a86,a34a38a139a24a199a77a193,a52a88a52a24a189a139
a199a37a222a156? a89a88a59a92a234,a243a219a92a123a30a127,a104a108a105a59a92,a28a85a5a43a41,a177a34a85a74
a37a222,a210a139a234,a52a88a52a21a234a243a114a199a77a193a201a24a199a37a222a156,a89a52a1a91a176a62a196a162a114
a14a123a175a11,a202a47a95a192a210a180a89a199a175a11,a15a210a52a128,a219a92a23a108a44a49a85a37a222a123,a28
a139a219a92a43a41,a28a178a43a41a73a222a85a5a117,a104a108a210a139a199a37a222a156,a98a52a89a199a52a219a92a123,
a21a88a21a106a223a116a114a199a123a77a193,a88a24a189a44a21,a0a143,a183a222a39a46a178a89a44,a7a89a248a123
a155,a23a140a227a142a89a199a37a222a156a139a73a154(singularity),a7a48a128,a243a5a193a1a199a67a176a62a139
a1a199a37a222a156a123a190a6,( a67a68a63a210a161) a33a38a30a123a190a6,a199a123a73a154a210a243a198a154,a178a44
a89a199a198a154,a53a196a139a48a53a152a37a222,a248a234a198a154a131a105,a210a189a234a24a199a37a222a156,a98a52a198a154
a52a24a199a73a154,a199a52a196a139a121a22a241a117a123a241a21a154,a196a49a199a52a199a73a154,a145a19a199,a196a139a37
a222a209a53a198a154a18,a198a154a164a52a24a199a73a154,a198a154a210a35a196a24a199a188a5a86a123a24a199a154,a93a169
a119sink,a13a38a30a123a119source,a104a108a15a139a54a33a141a30a123a190a6,a44a89a74a190a6a44a49a23a241
a37a222a156a243a73a154a139a88a51a117a159,a152a91a195a85a199a123a88a51a123a117a159,a210a139a24a199a119a41a37a222a156
a123a141a41(index),a28a243a89a24a154,a34a38a139a199a241a193a123a73a154,a0a143a140a248a89a199a73a154a41a199
a66a203a87,a79a143a52a241a193a123a73a154,a196a49a243a66a203a87a222a123a154a123a37a222a52a113a92a104a189a123,a0
a143a25a243,a243a66a203a87a123a154a119a248a73a154a221a245a232a87a17? a152a42a221a24a87,a63a37a52a243a116a123a48
a53a221a24a87,a199a123a141a41a521,a152a42a53a139a123a48a53a210a52?1,a0a143,a243a183a123a222a193a190a6
a165,a195a88sink a164a52source,a141a41a209a521,a118a77a34a123a25a54a141a41a143?1,a73a154a105a133
a236,a79a36a141a41a44a49a82a127a91a138,a34a38a183a178a77a193a12a116a14a145a45a176a62,a12a116a203a177a45a176
a62,a63a37a227a142a139a73a154,a0a143a89a199a222a86a123a77a193a210a139a30a142,a89a199a30a142a210a35a104a149a89
a74a73a154,a196a49a202a226a218a42dω12 =?Kω1 ∧ ω2,a183a162a7a149Gaussa77a71a123a232a73a210a127
a149a73a154a123a141a41a90,a196a49a183a162a121a210a24a199a173a7a117a159,a88a88a28a82a127a91a24a199a37a222a156,
a34a38a199a144a139a139a33a199a73a154,a183a162a89a199a232a73a52a141a41a90,a253a52a178a154a199a73a154a123a141a41a28
a229a117a210a127a149a141a41a90,a89a176a105a7a155,a79a143a89a199a232a73a52a203a37a222a156a160a10a195a7a123,a196
a49a89a121a210a234a24a199a77a193a34a38a139a24a199a139a139a33a199a73a154a123a37a222a156,a243a73a154a123a141a41a90
a37a222a156a123a160a10a195a7,a199a127a149a0a199a232a73,a13a0a199a232a73a176a52a150a139a37a222a156,a196a49a210
a122a116a89a248a24a199a136a42,a183a242a128a24a40,a25a243a139a24a199a85a20a123a77a193,a82a24a199a37a222a156,a139
a139a33a199a73a154,a199a123a141a41a90a52a166a37a222a156a123a160a10a195a7a123,a89a52a79a143a199a127a149a141a30a123
7
a232a73,a13a141a30a232a73a202a253a150a139a37a222a156,a196a49a166a37a222a156a123a160a10a195a7,a143a31a143a89a199
a106a248a127a149Euler a43a117a106a17? a25a243a17a108a199a203a37a222a156a160a10a195a7,a28a210a127a82a24a199a37
a222a156,a7a48a128,a34a38a139a199a77a193,a28a178a77a193a73a191a234,a73a191a196a66a76,a154a199a66a76a52a174
a110a111,a233a149a89a174a110a111,a154a199a30a82a199a123a165a154,a174a110a111a82a199a123a173a101,a28a210a44a49a189
a24a199a37a222a156,a210a54a183a196a152a123,a44a186a154a241a86,a108a128a116a174a110a111a123a173a101a210a159a86,a233a149
a89a248a6a189a123a37a222a156a105a142a52a23a241a117,a166a28a243a30a222a123a89a171a154a123a141a41a127a149?1,a149
a52a199a243a186a154a123a141a41a521,a243a174a110a111a173a101a123a141a41a209a521,a98a52a243a30a222a154a199a154a141a41
a143?1,a196a49a178a89a141a41a28a229a117a123a155,a210a127a149a186a154a123a199a106+a193a123a199a106?a30a123a199a106,
a79a36a210a52Euler a43a117a106,a89a248a6a121a210a234Gauss-Bonnet a218a42.
4 Gauss-Bonneta218a42a123a77a18a249a97a126
Gauss-Bonneta218a42a93a116a139a126a123a30a127a52a77a193a139a30a142,a243a77a193a139a30a142a123a30a127,
Gauss-Bonnet a218a42a52a186a154+a186a154a123a105a110+a30a123a127a143a77a71(geodesic curva-
ture),a242a28a222a243a193a123Gaussa77a71,a5a193a52a24a196a123Gauss-Bonneta218a42
summationdisplay(pi?α) (a154) +summationdisplayintegraldisplay k
g(s)ds (a30) +
summationdisplayintegraldisplay integraldisplay KdA (a193) = 2piχ(M) (6.9)
a233a149a139a30a142a123a77a193,a62a24a92a73a52a30a142a186a154a123a154a77a71,a217a39a52a30a142a123a30a123a34
a77a71,a108a128a114a199a123a89a199a192a220a123a193a77a71,a196a49a28a139a24a199a139a30a142a123a77a193,a28a210
a82a30a142a123a154a77a71+a30a142a123a34a77a71+a193a77a71,a52Eulera43a117a106,a121a210a52a24a248a123.
a93a116Gauss-Bonneta218a42a33a139a126a123a52a139a30a142a123a60a89,a7a48a128,a243a24a199Euclida178
a193,a34a38a139a24a199a174a110a111,a89a199a174a110a111a132a134a34a196a196,a132a149a56a45a52Euclida56
a45,Gaussa77a71=0; a34a38a30a209a52a134a34,a196a49a127a143a77a71a15a520,a79a36a89a199a210a52
a128summationtext(pi?α)a127a1492pi,a89a52a79a143a7a52a174a110a111,Eulera43a117a106a521,a141a30a7a127a1492pi,
a196a49a89a210a128a210a174a110a111a174a110a131a90a243Euclida178a193a222a127a149pi,Gauss-Bonneta218a42a52
a174a110a111a174a110a90a218a42a243a24a196a60a111a123a77a18,a89a199a10a39a173a7a244a234,a199a210a52a114a199a14
a145a45a123a10a39,a183a128,a132a89a199a14a145a45,Maxwella48a199a210a52a89a199a60a89a123a77a18,a28a116
a212a174a222a97a126a123a30a127,a28a123a56a45a524a145,a523a145a56a45+1a145a30a45,a524a145a123a98a85a251a22
a111,a0a143a7a44Maxwella48a199a123a155,a28a7a126a24a199a203a177a45,a34a19a222a52a24a199a133a123a134
8
a34a45,a199a139a199a77a71,a183a162a123a77a71a52Gaussa77a71a198a49a193a232a195a163,a13a89a199a77a71a52
a1992a39a135a73a42,a178a44a43a89a199a77a71a52a85a20a123a30a71a85a241a117a210a52Maxwe lla48a199,a196
a49,Maxwella48a199a123a1a91a242a181a52a58a158a59a92a123,a210a52a79a143a45a142a524a145a123a56a45,a196
a49a52a442a145a56a45a106a223a1164a145,a0a143a89a199a77a71a79a143a52a24a1992a39a135a73a42,a164a52a39a233
a193a123,a79a36a2434a145a56a45a176a52a24a1994×4a48a106:
(Fαβ =
0 E1 E2 E3
E1 0?B0 B2
E2 B0 0?B1
E3?B2 B1 0
,(6.10)
a89a199a48a106a176a62E1,E2,E3 a52Electric Potential,B0,B1,B2a52Magnetic Po ten-
tial,a15a210a52a158a51a203a30a51,a89a74a209a52a48a106a176a62a123a60a106,a44a43a132a89a199a48a106a196a44a43
a1232a39a135a73a42a52a85a20a123,a253da89a199a42a6a123a135a73a1430,a210a52Maxwell a48a199
d(Fαβdxα ∧dxβ) = 0,(6.11)
Gauss-Bonneta218a42a52a24a199a187a36a222a105a245a220a35a123,a93a116a183a85a123a218a42a17a88a132
a149Gauss,a15a88a132a149Bonnet,Gaussa144a41a234a174a110a111a123a60a89,a132a127a143a174a110a111a41a116
a174a110a111,Bonneta15a150a139a41a95a192a123a97a126,Bonneta178a174a110a111a77a18a116a127a63a77a34,a198
a178a127a63a77a34a123a127a143a77a71a232a73a44a143Gauss-B onneta218a42a123a232a73,a104a35Bonneta52
a27a41a33a7a155a123a1a91a166a27,a198a243a135a73a1a91a41a234a58a158a228a253a123a224a26,a183a88a11a28a162
a234a137a245a232,a183a230a130a28a162a234a137a89a24a92a73a123a106a166a58a158a7a155,Maxwella48a199a52a199a123
a1a47a60a89,a89a199a58a158a139a126a255,a183a139a24a159a169a57a243a152a35a123a21a41a166a22a222a21a44,a11a248
a119a21Gauss- Bonneta218a42a166Maxwella48a199a22,a183a89a123a142a245a7a154a243a89a159a169a57a176a62
a139.
9
