Sur le spectre du laplacien des fibrés en tores qui s'effondrent
Texte intégral
(2)
(3)
(4) ! #" $%& '. *( ),+-/. 02143654798;:4367=<?V >A1W@?@EBCH2@E7XD9F2H2>G7Y:43IH2O;H2D976:45J7954F2D91JKL79Z68M5JD65476D95NKL790/8MOED6F27I1N0[>C@\PQOE1 R2:4798;:SKLOE8THUF ]_^W.`a`[.!bdcfe#e#.-. g,hjilkfmonqpritsquwvfmyxzmo{}|~}/sqm 8M5S:SKG:SF2:NH27Xz@\:SM39z@\:SKL rFM765 fJ1SFM7I,?MKGo>L7=JU1 V @\[8M HC F2KL54547 .
(5)
(6) ' UBtH OEF 8MD95S:S0/KLP 76F2OED988M:S7J14H 7IH2P KG76HU8 5fFE@\ ;>A1SFM@\KA0276@\5S8;>A@E:S:4KLD95OEKL8M79V 8 5j36OED6@ ,79V 8;KL39:S54:S1W5W1S@\@\KL>L8; ;76:JFM5 765SH25 F27N21 79>C>L:fP 76@\5B 8[@\@\B KGOE:> U8M>C54D9PQ7JOE:SRKL5SOESF28M791}:N5N>LHH765 PP F2F288M@\7 1SEKL@E39@\5S:4:4361S7KL5}39>G76:4KG3X5S:S:j:4D639>COE1WPQ76@\5S:S:S0[F2KG@Ez147D9@\:47X:SdKLOE797688 5 14:SKL36OE5S8F2>G:WH2@\7:45,>A@ KG0214079OE1S:WKL@\Z9148r7 :45 E@\OE>L8r79: F21=798 02140[OE@\021S14:S7,KLD9HUF2F>GKL79>A1 @\0239>A:4@E3 D9KLOE798R2:4798rFM5 MD676OE7 8MV D679791=1S8[79@\8 8r:@ >AH2@ OE828MKG3,8MOEF21W8M@ 7 765S:SKGz@\:SKLOE8 798 B OE8MD9:SKLOE8 H P F28M7XD6OE8M5S:W@\8;:47XKL54OE0391SKG,39:S1SKL ;FM7 . . . . . . .
(7) . . . .
(8) !#"$&%$')(+*-,/. , G. <2=1I. ,9*. 8#N. n JKL1MG 1. ,98:.;8 , , * . * 01+24365-7 7 2=<5-11 2=11>2@? <BA 5? 2C? 11+2EDF2HGF2 ,/.PO * . . O ,X8:.;8 5@? 1QI 51 2 GF2TSU 2E29VF2:7 R GF2 53W5-7 A 5-7. (M, g). h. 1. Vol S , min(Vol M1 , Vol M2 ). h(M, g) = inf S. *-Y d 0. ,. *. ,X8:.;8 8 , ,/* A 5-7Z? 0 7 [ 2 1QI 2=<] 2GF2^I_I = 0I^`;3W5-7 I a 2=7^< b 2=IcGF2cG < 2=1QI 1 * , 8f.;8 9 . * .gO * . ./,/./,9* . 2=1MGF2=0D@I 0 I^`e365-7 I 2 G 1 h 2 ] 7 GH2^I i AQ5-7 1 1+2=1 1 2. S. .OZ\. O. *. M. F"nmWoprqtsvu:w x. jlk. .. M. kPy z{e| ')($,. h(M,g)2 4 * O. λ1 (M, g) ≥ ,. (M, g). . *Y ./,9*. n−1. M. ,X8:.;8. M. * O * 2~? 11>2ZDF2^v5 7I O * O O , 7E2365 2=0[7cA$7 A$7E2@G0 5^A 5? = 2 1. (M, g) 01+2}3657 .O ,9 λ1 (M, g) 2^I 5)A$7E2=< 1QIlGF2 M J. ?. *. GF2. SJ. <BA 5?. .. K OE8 @ SOEF2:47 F28M7 ;0OE:SMZ6547 5SF21 >A@ D6OEF21SR2F2147 H27 NKLD6D9K OE8_0/79F2:798MD @EHU14791>A@ 021479KLZ9147 E@\>L79F21J0214OE02147X798 B OE8MD9:SKLOE8 H27d>A@D6OE8M5S:W@\8;:47YH27 M767 V 791 5 V. II:51. I=07. 2^IPa. 1$?. 6. . 4. X. . h. F"nmWoprq u:w$s {| 'R(+*-,/.. jlk. ,9*. <2:1QI 1 nJ *-Y h(M, g)2 ) O. ,. ,98f.;8. *. .. *. ,. ? <BA 5? 2? 11+2ZD2GF2CG ` (M, g) 0[1+2H365-7 √ O* Ric(M, g) > −(n − 1)ag 5 7I λ1 (M, g) ≤ τ (n)( ah(M, g) + . * . . * ,/./, . * . . 8 . a 2=I 01>2? 1QI 5-1 2A I 3W22 τ (n) 0[1+2L? 1I 5-1 2L1+2LG AQ2:1$G5-1. (n,. ,9*. 8 M 7 \ @ 2 F S : 4 1 7 \ @ 2 0 2 0 4 1 % O 4 D M 7 6 7 S 5 : 2 H 7 G K M 8 E O 4 1 9 7 z 1 6 D 9 7 S : 4 : 7 E \ @ L > 9 7 2 F 1 2 0 4 1 E O 2 0 4 1 7 9 7 _ 8 t B E O M 8 9 D S : L K E O 8 2 H 7 R/OER2OE:41S798M8r76F5j5S0[F2@\1}1 >A@YD6OEF21S14R2OE,F214O 7 79:j>L7JHUKA@\,Z9:S147 H27N>A@ @\1SKL39:43 8 021479KL791j14365SF2>G:W@\: @X39:43 d. 0Q2RGF2. 54G. <2=1I. 1+J. . 4. . . n. F"nmWoprqCu:w 4mW. jlk. !. z {| ')(+*-,/.. .. 8. O. .. ,. .. .. *. ,/./,. .. a 2 d GF2=0DC7 2 II 7 ? 2=<2=1 A I a^I2 n 0[1 . O#O , . d 2=1 2=7fJ 2ZD I 20[1+2T? 1I 5-1 2 c(n, a, d) > 0 2 2 0Q2I (M, g) 2^I 0[1+2365-` ,X8:.;8 , , , ,/* * .O , :. .gO * , , 7 7 2=<5-11 2=1n1+2lG2hG <2=1QI 1 2RG 5-< 7Z2)2 5T? 07 ]=0[7Z2GF2c ?E? n G 1 8 ,N . . O* 3 7 2:1 diam(M, g) ≤ d 2 Ric(M, g) ≥ −ag 5 7^I ./,. O. ,. .. *. .. .. λ1 (M, g) ≥ c. .
(9) > 7 UKL5S:47?H P @\F2:S14765ID6OE8r:S1SKGR2F2:SKLOE8M5?<zD67=: %0/7 H27=KG8MOE1W@\:SKLOE8 79H2:=OE82>CPQ8M76795S8;:SKG: zHM@\@\:S8MKLOE5 8 HUF:SM36OE14Z9,7 @ 39:43 @ ,8M367 80[@\1S:SKLD9F2>GKL791 M798 V 79: aMOEF @ 76D O;H V 7 H27 N[@\@ V KL545W@\8;:J5SF21 >L765 B OE1S,765JHUK 3 X 7 S 5 0 6 7 9 D S : 4 1 I 7 U H F A > \ @ 2 0 A > E @ 9 D L K 9 7 T 8 2 H 7 14>L76795 8;:S:SKLM7936>G>LOE76145fZ9,@I3976:45 3N ,79OE: KG8M,5}54397 :SVFM39HU8MKL3 391W@\79>G:qKL54>A79@X8r r: FM@\76F 5S:SKLOEB 8OE1S54,7 760/5 O5W7 @\8[02@\0:S79F2>L14OE798M>G>L5 79, ;FM797I8r: 5SK H27N5W@ OEKG1}765S5S: K F28M7 E@\1SKLH239:4763j5 D6B OEOE821S,8M7 76U75NfHUD6K OE39140[798;@ED9:SKL:4797 >G>LOE761S5 KL79H28r7I:W@\H2R27 >LV 7 14\3 >L7j>A@\H2027 >A@ED9KL798?765S: H23 M82K%0[54F2@\11>CPQ798M5479=R2>L7 . . ;. . . . c=. . . .
(10). . . . π 2 − 1 cn √ad2 e 2 , d2. √ √ cn = max{ n − 1, 2}.. f. . . l. . . . . Q. . g. =. . . . Ωp (M ). (M, g). . . M 0 ≤ p ≤ n. p. ∆p = dδ + δd,. O5SF21 765S: >A@ UH K T@ 3914HU79KG8r,:SKL79798M>G>L5S7YKLOE7 8 U:4HU39F1SKL798MF2O 14r7X@\79F_: H27 54OE8 @E76H 5SS:?OEKG38rV : @\B >LOE7 1S,@\F 79> 0 KLOEZ9F2,1f7z>A@?8M8MOE=OE1SR2,1477 H27 79:S:SKH27 M79:N45 OE8 5S076D9:S147I765S:NHUKL54D91479: 8 8MOE:4791W@ . d Ωp (M ) M. . W. δ. !. ∆p. . . L2. p. =>L765F2>GE:S@\KGPQ02>L3979>G:SKLF2FMD914KGH25J:47J3 02HU14F,OEK 02>A@\1476025J>A@EH2D97 KLM79OE8 8 H2M147 7979:S 8 14O%OE14FH 39V0/7I7 39 :W>LH2@\7I7 8;>A:JJ@\02>L[76>A@\5@E D9KL@\79,8T>L79OEF2@ 8r14V :S5NKL1454027J5 14@\ ;OE8rFM02:J7 145S76>LF25J7N1J8MD6>LOEOE7685 0Bt8rOEOEF28M1S>GD9:4>L76:S795KL,OE58M79P 58rKG> : HUF@ >L5SHU760K 5 7639D9B OE14:S14798M78rD9:S:S0/KLKL79OEOE>G8MF2>L7651 5 E 8 9,0[@\7 1S@:SKLD976F2D}>GKL>C79P 1 76;5S0>L: 7 OEHU:S:SK MMZ6363954OE147f1479Z98r02,:N>GFM7HU5 F BtOE5S1S8M0:47767fD95WH2:S7 147j7 V D639HUOE8MF2F 391S1W>AR2@\@\F202>GKL14>A5W7 @E7 D954KL760[79D9@E8 :S5XKL@OE@\V82F KL8M54TB795 >G@\>L8rOE7 :N1SR,5SOEF2761 15 8MH2367 7 E@\1S8 KL397:4365 79: @EHU ,79 :S:WOE@\>GR8rOE: KL5jH27976:5 079:S KG:4OE76F25 1S:4OE@\KL>L5f79F2OE148r5z:j02H214OEOE82028M143 765 HMj@\8MDPQ5 765S: < HU KG147 H276 ;5}Fo7 PQ27OE8900279>LF2765: BC8M@\OEKG8147 8rEF2@\>G>L1S76KL795 1 :479>L798MF2H21 798;,: 39:S791S14KL r5 FM7 3914H2O 7%>L754OEHU1SKA:4@\7 , ;Z9FM:S7z147I>A@ 79: OE>AF_@=D6>L76OE5 F21S02R214F279147IKLZ954147676D95:SKLOE82@\8M>L7979F2>G>L14765 5 02141476OE5S:W02@\148;76:5 ROE1S8M367j5 :SM36OE14Z9,7 d021436D636H2798r:q5WOEF2>LZ 7}>A@ ;FM765S:SKLOE8?H27f5W@ OEKG1< ;FM79>G>L765D6OE8MHUKG:SKLOE8M5 F28M8 7 H239@\R21SF2KL39: :43H27 @E14HU39,0/79OE:I8M54F27 8M@Y7=390:4793 :SKGH2:4OE7 828M@\3 >L790[F2@\1 1 0214OE02 14OE7,>GR/< OEKLHU5 KA@\79,: Z9:S147, OE79:IF21SD6:4OEOEF2KL5 1SR2HMF2@\148M7 5 R OE1S 8M367 7954798 =,R2>AOE@\8;R2:S>L1W7 @\8r@\:zF ;:SFoM36PQOEOE8 14Z90/,797 F2:OE54R2K:4OE79882KG5417 0H2OEOEF28218M>L7765?F28MBtOE7 1S, %760/5OEHU:SMK Z6395W147,798r5SF2:SKL027902>G>L>L76395z,F2798_8r:W14@\36KG145S7F2>G5S:WF2@\:1 >L7I1W@ OE8 H P KG8 S76D9:SK ;KG:43XH27X>A@ E@\1SKL39:43 0 = λp,0 (M, g) < λp,1 (M, g) ≤ λp,2 (M, g) ≤ · · · ≤ λp,k (M, g) ≤ · · ·. . ∆p . . p=0. . _. . . Q. !. _. . 1 ≤ p ≤ n−1. !. . . $. . "#. . . ". . . . . L). . . %. . !. T. ). '&. _. Q. . '. . :. ). . ). . . . L. %. . . . jhk. n 7. ,. -. ,#O. 2ZD. , I. 2:<511. . ,. 2l01>2h? *. 2=11+2?. |K(M, g)| ≤ a. A. 7E2^IXA2Z?. *. 07. ./,. .X*. 3W2=<2=1. 0. R. q)(>'+**. "nmWop. .. F. . .O. 2. *. 0[7 1QI. .X* .. <BA 5? . 5T?. 51 .. .. 2 *. O. .. a d c2 r 0 c (n, a, d, r) > 0. 0I7. 24?. 8. . 2. I. 11+2ZD2@G2G. injrad(M, g) ≥ r *. 0[7f]=0[7E2RIf2E?. ./,/*. 11+2. O#O. ,. I. . . 2. , ?. O#O. ,/*. 2. .gO. 2. 7Z5 ,. ;. .. 2=<2=1 d. 02. 1. λp,1 (M, g) ≥ c0 , pJ. .. n K(M, g). < 2=1QI *-Y 2)2. 7. *. I 8. 3 2. 1MG. ,. .. 7. A. * I. ,/./,. a^I@2 .. .B.X* 0. .. 2=1. ./,. 2=7 ,X8:.;8. (M, g) 2^I 0[1+2l365-7 . 51 diam(M, g) ≤ d 8 , . injrad(M, g) G I V-1+2=1 \, ./, ,/.;8 O* 1.- 2Z? 3 GF2 M g5 7^I ,N.
(11) 7 14365SF2>G:W@\: @39:43 @\,39>GKLOE143 0[@\1 [@\82KG>G>LO79: 14Z 765 } ;F2KJH2OE828M798;: F28M7 E>L7d@\>L179EF2>L7X1N7HUU0F 21W>GKL@ D9KG:4OE7X8 H2H 7dP KG>A8 @ 76D6D9OE:S8MK %5SKG:W:4@\3X8;:4HM7 @\8M5JHUD6F79:S:4:SM7Y36D6OEOE148MZ95S,:W@\7 8r:47279 :J 798 , 79 :S:W@\8r:J798 3 %KLH2798MD67 8zM@\78282D6KLOE798M82548M3676 ;5 FMH2798M7 D6HU7dKG,KG79,8M5S36KLOEHU8KA@\:47IH2767 5S: HU rKA@\FM,7dZ95S:SK 14765 79:fD6OEF21S76R25S:F214F2768M5f7dF25S82F2K BKG:4OE7d1S,H2739,@\798r1SKL: 39:4R36OE5 1S1S8MKL367 5 :479>G>L7 ;FM7 :4798MH @\FM:454795SK8MH2797 1457914395 14O 39 1414O 79z0@\OE14F2 ;1 FMF27 8_M5SD6Ko791S>L7=:W@\HUKG8 KA@\,Z9:S147=79:N>A@zD6OEF21S@\R2>LF2OE141475 58M476OED914:S79KL1OE82>L7 8M79>GOE>L7,>GF254,OE8;7 :I R/qOE>G1SFM8M5 36554KG02KG>L8M79OE,1479791Y8r: >L7=5S1WK @ OE8 OE8547=H H2P KG8OES82768MD97Y:SK ;F2KG8M:473,E76@\54:d1SKL3936:4 ;3?F2KD6OE@\>L790[8r@E:=D9<:47 K 79: ;Fo:4PQ79OE8M8 H27BC@\KG: 7914E5 @\1SKL3979141dO a5W@ @\>L,OE14395J:S1S54KLOE ;8 FM71W@ <,OEHU8 KA@\H ,P KGZ98 :S14767?D9:S79K :%KGD6:4OE3IF2:41S79R28MF2H 147=79R/14OE5 1S8M3936147=O MH27YDPQ76545SOE: 1S :4< 7= HU rKGFM1477 :Wr@\Fo8rPQ79:476>G>L5 7 5 PQ8z7 0[OE@\8M1SHU:S14KLD97 F2>GKL 79791 :S%:45S7 K D6OE8MHUKG:SKLOE:48w798M@TH H2076OE5F21qD6OE>A@X8M5WHU36KL r5S:WFM@\798M8MD6D67 765 H27 :4OE014OEOE>LO,V O KL rFM 765,@\FMKG54540/H2OEOE11 B D4M7914O5d5476F25 8Mj7 HM@\E8M@\1S5 KL392F:43 H2 7R HUKG79,: 792F8M5SKLOE8 ;KG8UFMB 7 391SKL79F20/147 O 545W Z6H2 7J2F F2
(12) 8M@ 7dr5S@ :S1S@ FMD9,:SF2OE148r7d:S14H23 7 M798;R2:S14143d7zH2@\OEF28r:S: 1476>A@ 5 R[@E547I765S: >A@ @\1SKL39:43d>GKGKG:47. . . f. -. c0. . L. λp,1 (M, g) ≥ c00 (n, a, d) · r 4n. . :. . . . :. . 2 +8n−2. . .. B. (Min , gi ). n. l. &. @. λp,1 (Mi , gi ) injrad(Mi , gi ). &. . p 1 ≤ p ≤ n. T. . . =. -. . . . h. M. _. &. . λp,1. . Q. . $. . -. . . ). . . . . . . . F"nmWoprq. 'h(>*-,#.. y. . M. (Mi , gi ). ,X8:.;8. (N, h) i. ,. M. . (N, h). . i. !. f. . ,#. 01>2BI^0 2 GF2c3657 , * . 2 0[1+2h365-7 7 2:<511 2=11+2R? <BA 5F? .X* . O * ./,/* O#O 8 ,N 0 5r? 0[7 ]=07E2CIf2E? 11+2 2tGF2 i 3 7 * * O , . * ? 13W2:7/VF23W2=7^I A 0[7 5
(13) G I 5-1$?^2)GF2 L7 . , O , . N ./,9* I=0 If5-<)<2=1 V-7i51$G 2ZD I 2@0[1+2 ]=7Z5 1 , , O ,98f.;8 01+2 1 a=7i5-1 365-7 J .. Q. . (M, gi ). ). jlk. &.
(14) . . ,X8:.;8. ,9* <2=1QI 1 (n, * 2GF2G <2=1I 1 J A 0[7 . , 2 h2 I i i * * O* * .X* . < 3 ` l5-0FIfG 7 L5 7^IA 07 0 * . O N . G 1 5 ]:7Z2H2=I i i. *. .. ,. n m<n |K(M, g )| ≤ 1 (M, g ) I. ?. <BA 5?. ,. ,9*. . 2^IlGF2G. . π : M →N. 790/798MHM@\8;: KG>M8oPQ7654:q0[@E5}5SF z5 @\8r:} ;FM7 >L7 1W@ OE8 H P KG8 S76D9:SK ;KG:43 :4798MH27 79145 3914O 0/39OE14F2O 1I0OE8 F21Y079 ;F2FM:?7 0[>A@ @\102147 79279KLZ902147>L7 D6@\OE>L798MF25S1XKLH20239141479OE1 0214>L77 8M02OE14O%8HU8;F2F2KG: >G>L7z1SKL79HUzF@\>A82@\8202KL>A79@E8 D9KL79H 8P F2:48M797 8MH2E7 @\1SKL3979:41435 :4z798M@\KLHU5 14754OE ;>L8 7 FMHU795S0>LKAD6@\76OE,D98M:SZ9 r14:S7FM1477 7654Y:f79F2>A14@?825 KL147I3939F214H O 82P F2KLEOE8M>L8 77 1WH2,@ 76395J:SOE1S5S8zKL0/ ;H76FMD9P 7KG:S8 147676U35 D96:SH27IK 7 %@ KG:43N76D HUF2F8 027914:4:NO%OE14HUHU7 F2F KG02::4>AOE@\:41479:N7 8MH22H OE79:N8r:79H214OEOE5 88MDd39BC14@\8MO KG7: D6OE8r:SKL798r: 0[@E5NH27I079:SKG:4765 @\>L79F2145N0214OE0214765 7d>A@,OE:SK 7I>A@ . . . -. . &. . . &. . Q. . . T. . (N, h). &. _. . . . -. . (N, h). . . *. O. ,. '. d. O#O. 02. 2^Ih? ./,/. A 2 Q 2^I. *. ,/./,/*. 1$G. O. Q. . @\1J@\KG>G>L79F2145 U>A@?KG8MOE1W@\:SKLOE8 j8oPQ765S: q f!#"+. &. . . 1QI. 0[1+2 365-7 * 2=0[7IA+7 A+7Z2^I. ,98f.;8 d. 0. ,. \ I. 2. *. 1+G-7Z2R5G<2. .. `i2. O#O. 2l0[1+2. 798 D67I rF2KoD6OE8MD6791S8M7I>CPQ7 U0O5W@\8r:JHUF 1W@ OE8 H P KG8 S76D9:SK ;0[KG:4@E3 5 OE 02>o:S54KG7 z0@\O>L7 54M7I79H28 OE8M0[DX@\1S@\:SFMKLD954F25S>GK KL79>A@ 1 0)A. 0FI. 2=0[7^I. 365. !. . . q f!#"+. *. O#O. . * ' O. 2=0 O. . `. ,. * 1. 5cA+7 -. . 5-<. 8=O ,/*. . *. 7E2=7. O. 5@<. , 1. *. 7i5. ,9*. 7. ,. . . ./,/*. !. 1MGF2. O. 5A+7Z2=<. ,X. 7E2R365. O. 2=0[7A$7. *. A$7E2. d 6 7 j 5 4 1 9 3 0 E O M 8 4 5 6 7 5 2 0 4 1 6 3 9 D L K 4 5 6 7 5 \ @ F r M F 6 7 S 5 S : L K E O M 8 5 9 7 : ? E O r 8 : 9 3 4 : I 3 \ @ 2 0 / 0 E O S 1 4 : 6 3 6 7 5 M H \ @ M 8 5 2 H 6 7 5 S 5 K :SF[@\:SKLOE8M5J:4OE0OE>LO V KL rFM765 OEF V 36OE,39:S1SKL ;FM765J0[@\1S:SKLD9F2>GKLZ914765 PQ765S: >L7XD @E5 H2765N>GKGKG:4765 1. 1C1n0 . 2RG-0. 5^A. 5?. 2=1. GF2. G VF2=`iGF2_U[5-<. . . . !. XX.
(15) ER2@ HUF2KA:S@\KLOER[8 @\:SH2KL ;7JFM5476OE5 FM5 5S76K 5S0[@ED6765jH2767 5S:F28M7 79: @\1SKL39:4>A@X3 HU1SKLKL7954z:S1S@\KGR282F282:SKLKL79OE828 8M7JOE1SD6:SOEMO V 0[OE@E8[D9@\:4>L7 7d< F28Mr7OE8 HUKL05S:S791SF2K : 365SF2D91S1 KG147 >A79@ : ,39:S141S76KL5S ;0FM767 D9:SK 5479OE,FM5 798r>A@ : [BtOE791S: ,H27 3 M82KG1J5SF21 >A@O B @\KG79>G>L: 7XH27d5W,OE8r39:q:S1SH2KL ;76FM576,5 39:S1SKL ;FM765 . L. (M, g). TM. . B. . A. A. B g = gA ⊕gB gA M. g. A. . gB. gt = t 2 gA ⊕ g B .. L> 7 79@ 1SD :S>G@EKLKGD5,@\KG>LO :47 7H27 765S:F28 >A@_M ;R2F[HU14@\KL3 8M5S:SH 1S1SKLKG79R2zF2:S@\KL82OE828KL7679M5S8 :XOEq1S@\K>A02@ OE08;HU79:WKL>L5S@\36:S7>L1S7 KG,>R2F2GKG:SKLOEKG8 :47,79@E: HU76KA5SOE@\: 8R[@\>A@ BC:S@\KL ;KGHU: FMKL5S7 :4:S791S 8MKGR2HUF2d14:S7 @\KLOE8M>L8 75 H:4U79KA>@\,7 Z9OE:S8M147 HU14H2797,79>A@ 8;: MR2 147 @ 797614O=5 79: 3914O 790[>G14@\O1,547 H2OE768;5: M,OE,OE8rOE:S:S14M3 39:S KL765 H27 1W}@\02 r0/FMOE7 1S>L: 7 8MOE YOER2F2141 7IF2H28 7 >CP @\@\>LKL79H2F27 145 H27,0214>AOE@ 025S14F276KG5,:47 H27 5S0>C76PQOED914:SHU1W@\14>L7 7,H2H27 7 79 r1W@ F[@\8MHUH F MR2:479148M3 H 8MOE79:414OE5 8M5=3914 ;O FM7 079MOF2E:1Sz547 @\8 D @\>LJD9F2> >L79@\1z147 < 79MO: JOE1479H : % F
(16) O OE8;: V 398M790391W79@\8M>GHMKL5W@\3 8r: D6M7THM14@\368M545F2>GF2:W8M@\:7d5S@\KGF :SF[@\:SBt79KLOEF28 KG>G>LH279:W7 @ >GV KG765 KG:41S7XKL79@EzHUKA@\@\82R[82@\KL:S79KL8* ;FMHM7 @\28M>A@ 5 :4D6 ;OEOEFM14F27d7 1S0[R2@\F21N14P @\7MF2OE8o:S,14PQ767 54OE0[:d:SM@\791S398 : :SEKLV7YOE398?H28M7X0391W79>A@\@F2>:M0[79R2@E8 145d7 V R39OE8M1S398M1W@\367>M7 798 OE8M0[HU@\141S79:S1KLD9>L7F2>GMKL79R21Y143f5SK54F2>A@ 15M@NR2R[147 @E548o7 PQ76@\5SF2:d:S140[79@E,5I79F28;8 : @\1I@\KG>G>L79F2145 OE>GROEKL5 79: OEF21S:4OEKL5 OE8r: 39:SFMHUKL3=HM@\8M5 >L7XD @E5 O @\FM765454H2:jOEF21 8M7 79@\141S5jKL39F2:48M3d7 H27 @\1SHUKLKG39,:43 798M54KLOE8 H27 HUKG ;,F279Ka8M:45S79KL8MOE8H 0OE79F2:f1 H2>AOE@Y82HU8MKL795S8;:W@\:f8MH2D6767I5fH2767 5S:SKGz14OE@\,:SKLOEO 8M5 02>A@ >GFM:45fOE0/0214OE36>LOD9KLV 54KL767 5fH2H27 7J>A@=OE79F :=>LH2765}702>A@1479V36KLOEZ9,1476395 :S1SKL@\7 >L79H2F27z145j>CPQ027 14OEOE028M14HU76145f798M,OE798 8;:8rF2>G >L7976>L5 OE7988z>LBt7OE8M:SMD936:SKLOEOE148 Z9,H277 2OE8 5W@\KG: ;FM7 765S: 8M36D676545 @\KG1479,798r:NF28 MR2143I798 D67914D9>L7I5SF21 U79: >CP %0/OE:SMZ65W7 HMR2PQOE141S3 KL79768;5S:W:j@\DR2@\KG>G1WKG@E:4D93 :4395S1SF2KL15W367 0[@\@E1 5W5S5WF2@Y147D9>A@E r54FM547 7JHD67 P qMF2R2>L79141 3 765S: 021SKG8MD9KG0[@\> D@ B :4OE0 OE>LO V KL;7 0 H27 E D6 7 0OE8TF28M1NOE>A:W@?@\,8r: 39:S1SKL ;FM7 >L7I145S79F2021 1436547928;OE:W8T@\8r@:d@\[>LOE@\141S5,OE82KL rFM7?H27I>A@D9>A@E5W547XH27YD6OEMOE,OE>LO V KL7 . . . (M, gt ). M TV M B. . %& &. . T HM . &. 4. . t2 . . . . . &. . t . . . =. . . . . -. . . . . H. . . . ). c. M. &. . . %. . t. ) %. . . A. &. . . . t→0. . . . n+1 (N, h). _. %. !. n. . . M M. . . . M. . . . h. "nmWop. 0[1+2. 365-7. qr% z ,X8:.;8 7. 'h(>*-,#.. N. ,. 2=<5-1n1. . ,. a. X. . N Z. [e] ∈ H 2 (N, Z). e(M ). jhk. . F. . . . [e]. h. . 2. 2:11+2 . 2. 8 O GF2=0D 7 2 I , ,/* GF2 G <2=1QI 1. d. I. . 7. , ?. .. . * ,/./, 2=<2=1 A I aI O , . 2ED I 2 G 2^I F * A 0[7 , ,/* G <2=1QI 1. nJ Ci (n, a, d, (N, h)) > 0. ?. 2. . *. (N, h). . 5-1 2^I . O O 2 2 I ^ 8 ,N . 3 7 5-1 O* 7^I 0 5 1QI. .. ε0 (n, a, d, (N, h)) > 0 i = 1, 2, 3 , . ,X8:.;8 , , d (M, g) 2^I 01+2 3W5-7 n+1 02 I 7 2=<5-1 2=11+2 GF2 . diam(M, g) ≤ d |K(M, g)| ≤ a 2 ε = dGH ((M, g), (N, h)) ≤ ε * A 07 1 ≤ p ≤ n . .
(17) J. J [J. J. J J. λp,mp +1 (M, g) ≥ C1 (n,. (n,. e 6= 0 C2 ke(M )k22 ε2 dim H 2 (N, R) = 1 . ≤ λ1,1 (M, g) ≤ C3 ke(M )k22 ε2. C2 ke(M )k22 ε2 ≤ λp,k (M, g) ≤ C3 ke(M )k22 ε2 pour 1 ≤ k ≤ mp . 5-3-2Z?. =. mp = bp (N ) + bp−1 (N ) − bp (M ) J. .
(18) K o0[@\1X7 U7902>L7,54K >L7 MR2143z765S:I:S1SK ;KA@\> OE8 @ 0/OEF21N:4OEF2: 7X:SM36OE765S14:Z9,H2OE7=8M54DI7I>A14@?36HU5479F2F2KG:d>L7d@\0/>LOE7914:S5 KG:4< 7 @\>LG79 F21 ?02P @\14F2OE02:S14147Y7I0[0@\OE1SF2: 1 M5S>L76K 5 BtOE1S,@\>L76OE5 145 OE>GR/OEKL5N79: OEF21S:4OEKL5 ,OE8;:S14798r: @\FM5W5SK HM@\8M5 r FM7 5S rK F2Ko54OEKG: KG8MH2390798MHM@\8r:47?/OEH28 7d8M>A@ 7Y:40/OE790F2OE: >LO 0[V @EKL5N7X:SH2147 OEF 791IH27=D6OE8M5S:W@\8r:47 391SK a@\8r: 7 S : M 6 3 E O 4 1 9 Z , 7 L K W 5 E O L > = 7 L > 6 7 d 5 1 E>L765IH27?>A@ :4OE0OE>LO V KL7 0[@\1 >CP KG8;:4791S,3 H:4U79KA8M@\KGH 147 79H2147 5 3914OrX=HM79@\: 8M5?HUF >A@ 1W@KGOE8M8 OE1WH @\P:SKGKL8 OES8_76D9H2:SK 7 %KG>A:4@ 3 02 14 ;79F2K KL76Z95S14: 7 H27 @\>L>C79PQF2OE1=14HU0214147 OE02H2147 7 8M rOEF[8 @\8r8MF2H >G>L7 HU5 PQF 7 >AOE@\8M02HU>A@E14D97 KL7985SK}>L7z8 >A@\02>AOE@EKGD9: KL79H28 OE8M@EDTHU, rFM79:=7TF2HM8M@\7z8M5,0/79>A@:SKG:4547 KG:SF[E@\@\>L:S79KLF2OE81=02H 14P OEF2028 147 MR27914>G3T>L7 795487,D6D679OE14D9>L07TOE r1S:4F27 K @E5 ;02:4OE:SKL ;FM79,798r: D6OE,7I>L7XD @\1S143XHUF 1W@ OE8TH P KG8 S76D9:SK ;KG:43 V V V I \ @ M 8 Y 5 2 H 6 7 Y 5 4 5 G K S : [ F \ @ S : L K E O M 8 Y 5 4 : E O 0 E O L > O L K r M F 6 7 ? 5 9 7 : 6 3 E O , 9 3 S : S 1 L K ; M F 6 7 = 5 2 0 G > M F 5 9 3 M 8 9 3 W 1 \ @ L > 6 7 5 E O 8 M 8 7 D6OE828[@\KGOE:q:S0[:N@E@?5qH2@\027 0/14OE365S1SF2:43 >G:WF2@\8M:j7 5479D6YOE8rR2:S>A1S@\KGR2R2>LF27 :SKLOE8 790/KG798M0HMOE@\1S8;:W: @\r8r79:48z7ID6HM7 @\8M ;5F2K[D6OEO 8MD679R 1S8M797 : >A@I ;O FM765S@ :SKLOE7988 H2H23OEM8r82:KL>L547j5W@\540/8r:?76D9F2:S148_7 OE765S0/: 39>A1W@N@\:4>G79KGF21?KG:47>GKGHUF KG:475S00/76D9OE:SF2141=7f>LH27z7j>A>A@\@\0202>A>A@E@ED9D9KLKL79798 8,H2D7 PQ76 5S: O% H< V HU7 KG 14H27 7 F2N8[@\OE
(19) 039 r1W@\F[:4@\798MF2H 1 >A@\@ >LOEE145X@\1SH2KL39OE:4823T8M35 PQ7 0[@\OE1X8M>AHU@ 147 =qF2>L7>G:SKG8M02OE>GKL=D9KGR2:43 147TH2H27 7 >A@ E@\>L@\79>LF279F2145z1I02021414OEOE02021414767,5z8;0F279>G>L:S7,KG:476HM5z@\8MOE5IF>L7,8rF254>G0/>L76765D9:S39147:W@\H28r7: >COEPQ0/OE0/391W39@\1W:4@\79:4F279F21 1j>GKG>GKGKG:4KG7 :47a@\IKG8M@\5S8MK5q ;FM>L7N7IDW>L[76@\5N02D6KGOE:S8M147YHUKG :SKLOE8M8MOE5 FM5j8M367 D6%07654O5W@\5479KG141476OE5 8M5< >A>C@IPQ7 D6UKOEL8M5S:45479:S8M1SFMD67YD9:SH2KLOE7I8 0H2797N:SKGD6:47679:5 E@\>L79F2145 0214OE0214765 rFM 7 OE:S:J798 H236HUF2KG: HH PP F2F28M8 7,76V 5 5S14:SOED41SF2[FM0/@\D97 :S02F2KGH2:S141477765 MOEKL7,7914O 36: V 5WZ9OE8M>GF2 7 54R279 14>L7OE8rD6: OE8MD60[OE5S@\8M:S1=1S5WF2@EF2KGD98:4145X361450236>G54<FM7 5Y@\>CFPQ023914:SD636FMO%D9H2D6KL7OE5439q,00[79OE@E8rF2D9:?1: D6H2OE765 ,Mo7,R2HU14F r36FM5,5SOE0=:S76KLF279D98;82:S14:4KL755 HUHUFK >A39@\1402798;>A@E:SD9KL79KL79>G>L8 765jKG8 E@\1S14KA76@\5S8;:S:41476795KG8r>L:NOE14<=5j>CH PQ76PQ5S7 0[@EOED68M7dHUH214797 ,HU79KG8r,:4795 8M0[54@\KLOE1 8 H2M765q82KL,7 39:S1SKL ;FM765fMH2OE76,5}O B VOEZ91S8M,76765 5 @EM@\>L8 79F2H2147X5 ,021479OE:S02:S14147=765,798 798_3 ;B KLOEH28M79D98M:SD6KLOE7=8 D6OEH27 ,>A@ 798r:4: OE0/EOE@\>L1SO KL7YV KL027 14HU36D9F KL54M39,R214793 8r: 79:>L7IH28M7 OE>A=@ R2V 14367=OEH2,7I39:S01S79KL:S7 KG:4H27675 >CEPQ@\7 >L79OEF28M14HU5 14027914,OE02798;1476: 5NH28,7 7 79U: 79E:N>L76HM5@\08M795 :SKG>L:47I765D @EE5 @\>LO 79F214>L5q7 02V 1414OEOE02F2140767 5qH27 7654: 82KG>G5W0/OEOE8r:4:j798;@\:FM5454K%079OE:S:SKG:4:J76@ 5 ,1W@\OE,8rZ9:S8M143?7Y14<?36D9>CPQKG3902:S14FMO;H2 ;7YFM79H2,7 798r:X ;FM7=>A@z1476DWM7914D4M7H27?079:SKG:4765 @\>L79F2145 0214OE0214765IH27 547 . @q[p. Qm . qv%$%n'. p. . e = 0. . . m1 = 1 λ1,1 (M, g) @q[p. Qm . qt% s. Qm . q. % . . ). Q. C2. M. f. -. . %. . ε. T.
(20) . . . . @. Z. . . -. . . . . . . ε. &. . . mp = 0 e 6= 0 ! 1. '}. e(M ). r. X. . c. '. dim H 2 (N, R) ≥ 1 . @q[p. . $. . . . . . . . . L. . . . C. 4. . . . . . . . . . . . . G. ∆pinv. . . . . . Γ Ωp (M )G. . :. . . . . . . . . . ∆pinv. . . ∆p . G. Q. B. ∆pinv.
(21) m6" ,. .. . 6 !#"+&%u:w " .. *. n kRk. ,/./,. z. .. s. ∆. . * . . . 0 GF2=Iv? 1 I 5-1 Q 2^I 2 . , , O X , : 8 ; . 8 , ,9* d I 7 ? 2:<2:1 A I 3-2^I 2 2=I 02I 2=I 01+2 1 a=7i51. 365-7 GF2tG <2:1QI 1 , \ 8:. , d * * * O d O O 2 0 *Y <R0[1 2TG 0[1+2< 7 02U < V 1+2 A 0[7 5 02 2 ∞ .O * . * . , . * * 51 7^<2hG-0 2=1If2=0[7hGF2R? 0[7 ]=07E2^2 I 2^I 0[1+2a 7=<2A+7 A+7Z2)G-0 ∞ 2^I O O , * .PO O * 8 ,N −2 5 A ^ 5? 2=1tI=07 G 1 53W5 2:07_A$7 A$7Z2 3 7 2 ∞ O* .P, , . 5 7I 2^I 1365-7 51 2 J .. P". . O#O. {| '. . O. 2ZD ,. ,. I. .. a(n) a (n). 2. c(n). M. M. λ. kRk diam(M ) ≤ a α λ < a diam(M ) − ckRk. α. 790/798MHM@\8;: %D67 14365SF2>G:W@\:f8M7 5W7 V 398M391W@\>GKL 547 0[ @E5 <X:4OEFM5j>L765 V 14OEF20/765f143654OE>GF2R2>L765 8 798 791S1W@ F28 7 27902>L7Y@\F 0[@\1W@ V 1W@\02M7 ! . .
(22) . . .
(23) I7@\8MBC5z@\KG:>L7 rD4Fo[P F2@\8M027KG:S147 @\1S KL39:48M3dOE rFMF25,K 539:SPQ7 FMHUOEKL8M7914HUOE148M7I5 5SF2H21 765 F28 MR2D6147936145zD9>L7XH27 @EHUM,R214797 :S:47dF28M547IF215S:S1S>A@ FMD9R[:SF2@E145477 H:S2KL7,OE8 5WOE7 > U0E2@\1S>GKLKLD939KG:4:437I76HM5S@\:X8MH25 3 F24< 8 D6D OE@E825 8;5SF KG 02 >L7 7 %>L765 MoR2F 1436 5 D6OE8M5SOEKLH2FM395Y1479368 5N54Bt79791414OEOE8;8M:N5IH2F23 8MM7z82KLD65NOE8MD6OE5S:S1SFM,D 7 5S39FM>L3954,0/79798M8;5S: KLOE8_H P F28 HUK 36OE,dOE7 1S02022>GFMKL545 ,78MOE>GFMKG8M53 @\Bt79KG14147OE8MH257 >CP>A @ ;M0R2OE14:S7MZ6547 5414KG7902021436>GK 54[79D 8r@\:4:S3 1SKL0[D67@\1 ;F2FM8 7 @\>G>LOE@E8MHU5J,,79:J79:SF2:S8T147X>LO 79V 8 @\1S3 KG%:SKL2H2,798M7XD6147X365479OE> 8r: 76H25NOE02821S8MKG368M76D95NKG0[0[@\@\>L1765 >L7 0214OE021SKL39:4365 HUFT5S0/76D9:S147= rFM7X8MOEFM5 . . . . S1 . c. . . . . . . . . %. . =. '. . . . . A ∈ SLn (Z). . A. Tn. . Tn. . . . X. :. qr%(. "nmWop. jhk O. d. 5. , G. <2=1I . O. 'h(>*-,#.. ,9* 1. G-0I ,. n ≥ 2 A ∈ SLn (Z) *. .;8. 0FI=`i2^I;A 5?2t?E5-7Z5? ,9* * *. 2. 7. ,. I. .. B ∈ Mn (R). ./, d O. *. 025WII , .. .. O. d. A = exp(B) * 5M3W5 2:07RA$7 A$7E2 0 G = G(B) ⊂ 0 AQ2 N 8 7XAnUQ2l5-0 ]:7 M G2. 2 I O. ,X8. ?. *. 0Q2. O. d0 5G <2=1I 1 GF2HI 1 1 ,5-0QJL 2ZD I 2H0[1vV-7 . 8 . O *-,/. * 8Z* * U < < GLn+2 (R) 2 017 I 2Z5-0 Γ ⊂ G 2 d 02 Γ\G I N N ./,9* * . ,#. ,9* , 8E* * , n GF2 1 ? 1QI 7^0 ]:7Z2 T ]=7Z5 1 p : M → S A 57PI^0IXA2=1QI 1G-0)G < 7XA$U I=<2 (n, O * * O f 8 . , * . * * . . O O d d GF2BA 0I 1 I^0!AA I 2 02 2^IR< 7 02^IhI=07 M I 1 U < V 1+2=I2 2 2^I AJ *-,/. ,9* , , * 8:. , d *O d 02 p I 01+2cI^0]=<2=7^I 1H7 2=<5-1n1 2:11+2A 07c0[1+2c< 7 0Q2lGF2c3 0[<2 1 I=07 O* S 1 g5 7^I B. GF2. 2. . J. . J [J. J. dim Ker ∆1inv = d0 + 1 ,. G. ./,. I. ** .X*. 0[7 . 0. .. 1+?. *. 2^I. .X*. 0 8 . f. 2c<. *. 0
(24) 1. . 7. ,. 02.
(25). |K(M, g)| < a . . I. J J. J $J. (n,. O*. ∆1inv. .. 5FG-<2. n − d0. 1+J ,#O. a > 0. , d. . 2. *. 13W5-7. ,. 15 *. ,. 2ZD. .. .. I. *. 2C01>2C?. .. 3W5. O. .. *. A$7E2^I1. *. 1C1n0. O#O. 2=I. . . O O * d 5-1 2 2 2 0Q2A 0 7 [ * .O * ./,9* O#O 8 ,N G 1 5? 0 7f]=0[7E2cI 2Z? [ 11+2 2B3 7 2 * .X* . >A 0[7 0 J. 1QI. c(B, a) > 0. 5-1 2 g I=07 M λinv 1,i (M, g) < c .X*. 2:07^ILA$7. i. ,#O. ,. .. *. .. .. 0 a > 0 2ZD I 20[1+2T? 1I 5-1 2 c (B, a) > 0 O * ./,/* O#O 8 ,N d 2 2 0Q2cA 0[7 0 2< 7 02 g I^0[7 M 5H? 0[7f]:07E2If2Z? 1n1+2 23 7 2 * 0 inv |K(M, g)| < a 15 λ1,d−d0 +1 (M, g) > c J (n, . O* . , O *. . .l,#O , . ,#. 2ZD I 2C0[1+2HI^0 2GF2 d = n 2 d0 6= n l5 7^I G 2^I 1 A 2=1 R2 . O O O * * # , . , * 8 . * 8 . 8f. , d d < 7 02^I gε I^0[7 M 2 2 02 5
(26) ? 0[7f]=0[7E2)I 0[1 a 7=< <2=1 ] 7=1 22 * .X* . d 0 λinv 0 < i ≤ n − d0 J 1,i (M, gε ) → 0 0Q5-1$G ε → 0 A 0[7 (n, O* . .X* .bO * * , d d = d0 = n 5 7I G = Rn M 2^I 0[1 7E2l2 2^Iba 7^<2^I_U 5-7^< 1 02^I .. O O. *. **. 1. d 6= n B5. 7IcA .X*. *. .. 0[7. .. 2ZD5? .X*. . 0. *. 2=<2=1. 07. .. .LO. 8:.. *. 2^Ia. GF24? 0[7f]:07E22 * . . 00 ? 1QI 5 1 2 . inv L2. G. ,. :.. 5-<. , d. ,. 7=<2^I. k ≤ d − d0 .. 0. ,#O. 2ZD ,. 7E2@0[1 a . O#O 2 2. c (B, k) > 0 λ1,k+1 (M, gεk ) > c00. ε→0. 1n365-7 , *. ,. 5-1. .. 2^IfJ. ,#O#O 8:. , d 20[1+2la:5< 2 t GF2< 7 0 2^I Q 8 . * 8 * . 7=< <2=1 ] 7 1 = I A 577Z5^AA 7 * d inv k 0Q2 A 0[7 ε 1,i , I J I. .. k. gε. ε i ≤ k. λ (M, g ) → 0 k<n. M. I^0[7 . 2 01+2 d 0Q5-1$G. 02:S1S14KLOE7?02H214767Y5J>CPQ0/7 79F2OE: 8MMHU ;14F[79,@\8M79H 8rT7 : >A@ 0/ :4OEOEKG798;0:S: :4OE7?>LO 5SV KG:SKL7IF[@\>L,7d:SKLOEOE08r8 79:S1S14HU,7 K 79 ;: Z9UFM147=Bt7 OEH21S>L:47TOE798M,8MD=OE79HUY8;F : R214D 7 @\@E1S5 H2KL79O 7T1 T0/@ >A79@ :S76KGDXM:476R2>A5@ 147=V7636@\5SOE>L79:d,F2H2143 75 HUKG,798M5SKLOE8 239:SFMHUKL3I0[@\1 OE>GR/OEKL5 79: OEF21S:4OEKL5 HM@\8M5 >L7 D @E5 H P F28 7 OE8MHU14@=79,H23979,8r: OE8M0[54@\:S1 1W@\M:SKLOEOE,8 OEH2:S7=M39 :SKL765 ,H27T79: >A@798 M3 R2;14KL7 H279 8M5SKGD6:S7dF[>L@\7 :SKLBCOE@\8 KG:fH2 ;7TFM7d>GKGHM@\KG8M:475 Tqp. .
(27) . Qm q~% '. . . T. . . . . 1. Tqp. Qm qt%. y. '. . . . B). . . . :. . . . f. . %. .
(28) E@ HUKA@\R[@\:SKL rFM7 [KG> 8oP @ 0[@E5JH27d0/79:SKG:4765 E@\>L79F2145 0214OE0214765 >L7I5S0/76D9:S147IH27 8M:4797N8MD6H OE8M54793614 ;5 FM79398M14D6O 7 [@\KG>LOE,145 36>LHU7 KA@\V :4147 OEF2HU0F 7 14365SF276>G5S:W:@\:d82KG >G0OEX:476795S8r:f: rFM7 5SK[:4OEF2: P \ @ 2 F S : 4 1 I 7 [ 0 \ @ S 1 : [ 6 D d 7 S : M 6 3 E O 4 1 9 Z , Y 7 / 0 9 7 S 1 , 9 7 J : 2 H Y 7 2 H E O 2 8 M 8 9 7 J 1 2 F M 8 X 7 6 D E O M 8 U H G K S : L K E O T 8 M 8 6 3 6 D 6 7 4 5 W 5 \ @ G K 4 1 X 7 9 7 : 5SF z5W@\8r:47 5SF21 0OEF21 >CPQ7 UKL5S:4798MD67 H27j0/79:SKG:4765 @\>L79F2145 0214OE02147650OEF21>L765 BtOE1S,765 =. @q[p. Qm . . qC%[. . !. '. . &. H. ∆1inv. Q. . G. . . . . mWqM%. xl"nmW". . B. 'R(+*. O. 0I. . *.. 2^IU ^ , A. . U. .. If2^I
(29) G-0 O#O d I^0[7 2 2 0Q2 , O 8 ,/. I 5 7 R G 0 2 GF2. 8Z* U. 7. . <2. L. 1. , O . ,. 2ZD. . I. ,/.. 2T0[1+2@I=0. 24GF2 , . 2 . * 1 1. K Q7 6 7 5 4 1 6 3 S 5 2 F G > W : \ @ 4 : } 5 9 7 q : , E O r 8 S : 4 1 9 7 ; 8 : ; M F 7 L > q 7 6 D E O 0 E O S 1 4 : 9 7 , 9 7 r 8 : E @ 5 ; 0 :482OEKG>G:S0/KL rOEFM:4797 8;HU:47,FH25S7076>AD9@ :S14147z36HUH2F27 KG:47,H27 76OE5S14:=HM76@\5W8 5479H28r7:SKL79>G>L79,8798;7:?>G79KL3: < >L7 >A@ R2>L8[O%@\D :SF28214KG>G7 0/OEH2:4797z8;>A:Y@ 760[5S:X@\1SD :S@ KL7 1W@ED9:4391SKL543I0[@\1 >L765NKG8 E@\1SKA @\8;:45 79: H27 0[@\F@\1S:S0[IKL@\D9@\1WF28M@ >GKLV5 791W1Y>L@\7 02>L7zMD @E7 D65 OE 14H2OE 76>GG5>A@\KG14 r7TFoB OEP KG1S>},0/8o76795 PQF2765S:Y:=7 0[UK@EL5S5 :479%1X1W>A@T@\F2Kq8M540/7zKG:SOEF[0/F2@\791 :S:SKLKGOE:478 79E: 76@\5S>L:=79F2 ;021=FM>GFM79025>LD614OEOED602OE8M14 r7,FM027 0>L7 OE27F2OE81X>L7976:795 1S791W8@ BtF2OE8M1S7N,D676OE5?8M@\HU>LKGOE:S14KLOE5=8 ;8MFM367 D6765W5W8o@\KGP 14@ 7 0[<d@E>C5=PQ7 H2UK7,L5S:4R279>L8MO%D6D,7 82H2KG7>G00/OE79:4:S79KG8r:4:76 5 E@\>L879F2014795}F202:=14OED6027901476795 8M;HM<X@\5W8r@ : H2OEKGOE1f82 r8MFM7971 8oPQ765S:N0[@E5J5479K 5SKG02>L7 8T7 79: MOE8 @>L7 8f.. <. If2=0 10. O. . , d. 7. O. 02^I. U. .. 2:<2:1. *. <. J. @q[p. Qm . . V. gε. 1+2^I. d 6= d0 . ,. I. *. q. s. '. z. X . .. M. d λinv 1,1 (M, gε ) → 0 * 7ZG5-1GF2 B 5h01]. . .. E. B B. . . p. n. . %n'R(+*. ,. .. O. 0I. . r. p. 2. . . . 2^ITU ,^A .. . *.. c (B, a) > 0. O. . . . U. . . .. If2G-0 * d 00 5 7I 2ED I 2 2 02A 0[7 * .O * ./,9* O#O 8 ,N G 1 5T? 0[7f]:07E2RI 2Z? 11+2 2 3 7 2 ,#O. . =. B. F"nmWoprqts. . . . O*. . . p ≥ 2. .. B. jlk. d0. d. . I. . ∆1inv. . ε→0. 0Q5-1$G O* , O *. ? 1 A 2:1. U. 8Z* .X*. 7. 0. .. . <2. 8:.. 2<. 7. , d. cI. B. 0Q2U. *. |K(M, g)| < a . ,. 2^I. *. *. .. f. ,. I 2=< . ,. `;I. O. <BA. 2=. g I^0[7 M λinv (M, g) > c00 J p,1 <. V. 1+2. S5@EF2545417 I>L7 @\5S8M:4KGOE5=1402>L7 7 >L7 DW[@\Y802KGF2:S5W1482@\7 KGK :XH 79P 8F28M8MOE7 7 FM795,5S:I:S@\1S rFM>G>LFoD9OE:SP8MF2F25?8 147 8MMOEH2R2FM7 145 3MKG028rR21S:414KG393 8M14D97602KG5W0[1S54KG@\798M1,>jD9KG79<T0[8 @\H2> :476OE5 147M79F2R25SF21,14361d:45OE>L7 0798 OE:4OE>LO:414OEV 7 14KL77 0/7976F25S:: 9:W@\:S14:X7J02=>GFMF25I8202K14H 36P D9F2KL8M5Y7d795S8:S1SF2FM:SD9KG:S>GKLF25W14@\7d8r:=H27 >L7 82BKG@\> KG:X@\1S ;KL39FM:473 >A @ R[ @E 5W 7, 765S:X8 H27,@XHUKGKL,D9K/79,8M5SOEKL8rOE:S8 1479 1 oF28D67 1436M5SR2F214> 3 765S:HUK 36OE,OE1S02M7 @\F 0214O%HUF2KG:,H P F28 :4OE147 79:H P F28M7 82KG> @\1SKL39:43 H27 HUKG,798M5SKLOE8 F2KL5dOE8 D @\>LD9F2>L791W@ >L7 5S076D9:S147?HUF >A@\02>A@ED9KL798 147654:S1479KG8r:Y@\F B OE1S,765IHUK 3914798;:SKL79>G>L765 KG8 @\1SKA8 @\8rOE:4R276:S5 KL79798r8 : 5S>LF27I02140/36O5SF25 @\>G:W8r@\:N:J>L75SF2MK R2@\148r3d: =F282K H P F28M7I,39:S1SKL ;FM7IMOE,O V Z98M7 . 1M5. . . . T2. &. . . . ). #. X. . . = . Tk. . 4. . . . . . . F. . . jlk. . F"nmWoprqts$s>'R(+*-,/.. T2 J . J. . J [J. J. O*. M. 0[1. N. h. ]:7. 8. A$7. ,. 1+?. ,. AQ5. O 1. *. . 1. 7. , 3. , 5. O. 2=1. .X*. 7I 2^I. N. 01+21 . 2^I 0[1+2. ,#O. 3657 ,#O. ,X8:.;8. 7E2. Tn. . I^0[7. O. . , . * 8Z* * 2 L I 2^I U < < 7;AnU 2 *-Y ,X8:.;8 , ,9* 1 3657 GF2RG <2:1QI 1 J O , . . ./, O . . O N n . 2 O d 02hI , 2ZD I 2h013W2Z? 2=07 3W2=7 ?E5 5-1V2=1 5 ]=7E2 , \ 8:. , d * * O* * .X* . <)01 G 01>24< 7 02TU < V 1+2=h5 7IlA 0[7 0 . , d O * * O#O ( O./, O , ,/.;8 . 5G-<2 01>20[1 0Q2 365 2=0[7A$7 A$7E2l1 110 2 J 5<)0 A ? 2=I . .O *-O O N 8 −2 2 *-Y 2 2^I 2 3 0<2)GF2 54]Z5WI 2)G-0 ]=7 O 8:. , d , ,#. 5< 7 0Q2 1$G-0 2!J. M. .. 3. . n≥2 M. . V. 3. . . λ. λ = Vol(B). |V |. Vol(B). . X. .X* 2. 7E2. N × T n−1 .. T M 2=I p ∈ [1, n + 1] ∆pinv Cnp−1 A. *. 0[7.
(30) G 8oPQ765S: 0[@E5 8M36D676545W@\KG1479,798r:z1SKL79z@\8282KL798 7 2 0 4 1 % O U H 2 F G K : U H F 01476OED9F2:41X79,>L76795I8;:,H239:S7d1SKL>A ;@=FMB 76OE5Y1SYD6OEF28M>L7=5SKLH2H273914 3676285 28M797,:S 5S076D9:S147 8M7079F2:YH2OE8MD0[@E5X5W7H236HUF2KG147,HUK 7 ;FMOE7 8MHU14767954,: 79@\8r>LOE:j140[5 @\021q14MOE8 0OE,OE1SOEOE:SKG:SKL:OEM ;8239FM8M:SKL7 7976>G5}D6>L7?OEH28r@\7 :SF 1W>A@\@ D KGM14@\791SR2,14143Y7 79HU8;02F:f14O%< HUHUKA>AF2@\@ KG,:q5SKGZ9F2:S:SF[8M14@\77Y:S0KLH2OE797I8,:SKG>AHU:4@ 7 FM:SR2@\M14>L36797 OEF2[141< Z902,:414OE7IOE002 OE14>L7 O ErV F279KL8 7: 2367 1M436R25S14F236>G5 :W@\79:48=5D6H2797 14D9>L765 5SOEF2I>G1 R@\H2OE8M76KL55=579R[>L:7 @E54D 76@E5 5? rOOEFMF279>L1SD6:4OEOE8MKL5 ; ;FMF276KU5 >A3979@ :S: FM145WHU79@\KLz8M795o8r@\:q141476 ;HM54FM@\:S7 1S8MKL5 D902:S14KL 36OED68M365 H25S79F28;1>L:47f7 >A@ 54140/,7 76D939OE:S:SKG141S8;KL7f:? ;H2FM>L76767 55. 8MOE@\F KL5 7 54@\KNF>A@@\02HU0[KG@\,1W@ 79L8M: 5SKL[OEKG8 >7 H2UK7 L5S:4>A@7=HMM@\R28M1475dD6767?5S: D @E025 >GFMH25 765dV 1W7 @\8MOEH28M7 HU14 79,798r:45X HU}F F28wMR20214M3=39:48M79OE>L,5 ;Z98MFM77 8M7 :4798MH27z0[@E5}>L7T79148M5 OEY39R214O 147 H2OE7TFM05=7979:S8_KG:476H25 OE82@\8M>L797914F2OE148M5 5 02H21476OE5 02147 76U75 98M7T02>L76H25 390@\79F8MHy0[@\H21W@OEV8M1WDT@\020[M@E75 K F2D6OE82KL ;,FM797X,797988r:? H27 >A@ :4OE0/OE>LO V KL7 790/798MHM@\8;: OE8@ 0[@E5XH27 >GKGR791S:43,5SF21XD67 8MOE=R2147 I \ @ M 8 5 2 F M 8 d 7 2 H 9 7 F UK L 9 Z , d 7 [ 0 \ @ S 1 S : L K 7 2 U H F W D [ \ @ 2 0 G K S : 4 1 7 , \ @ F W D [ \ @ 2 0 G K S : 4 1 7 % M 8 E O M F N 5 \ @ G > L > E O M 8 5 M 8 E O M F 5 KG>A8;@ :4M3914R27614547z5479761=5S:X< H2>A@7,HU ;FMKG,765S79:S8MKLOE5SKL8 OE8 23767:SM3 V36OE@\14>L7Z9,< 7 OE8M8 70D679OEF28M:Y5SKL54H27Z914H27z79z rFM@\78MH2H279761?5 HMM@\R28M145X365X ;FMH279OE>G8;>L7: ,KLD9K 765S8MF2OE14FM7 5 KG>IKG8r54:47391476V 5W39548M79391 1W@\02>G>GKLFM5475 02@\F1436D9KLM5439R2,1476795 8r:zH27 @\F HU_5KG,SKG79:S8MF[5S@\KL:SOEKL8 OE8 02H2>GFM7 5 MV R21W1W@\@\8M:SH2KLOE7 8M5,02OE1SFMKG8M5 D9@\KG0[>G>L@\OE>L8M7655 5KG8UPQ7 Bt1W@\OE828MKGHU> E1W@\@\1S8rKL:39:43654F25 1 0>LO7954F2541 36HMR[@\@E8r54: 7 fF2>A8M@ 7 M5SR2:S1S14FM7 D9:S39F2:W@\147T8;: H2@\7 >LOEV 14145 OEF2F208y7 :4OE147 OE:S 14D67 7TR25WOEF28r: : HM>L@\768M5 5 5479D6F279:S>L:47675 5476D6OE8MH27X0[@\1S:SKL7X765S:NH P @\1S1SK 791 @\F 14365SF2>G:W@\:J5SF2K @\8r: . Tqp. Qm q. s '. . . !. . Tqp. . Qm qsQ'. . . . . . λ. . . L. . . Tqp. (. Qm qs. '. . . n = 1. . P). . %. . . . . Q. . . &. R. λ . n ≥ 2. . =. . . Q. n ≥ 2. @. . . . . . . . . . . . . . 1. . . . . . . . . =. . "nmWop. 'h(>*-,#. ,X8:.;8. (N, h). a. ,. 2. .. d ,. GF2=0Dr7. 8. l. . , . . * ,/./, ./, . I
(31) I 7 ? = 2 <2=1 A I a^I 0[1 2=1 2=7C2 , ,/* . , . ., 8 , 0[1+243657 7 2=<5-11 2=11>24GF2HG <2=1QI 1 I 7 ? 2:<2:1 1 a 7 2=0[7Z2 J O , . * . . . 2ZD I 2 G2^I ? 1QI 5-1 2^I 2 0 . # O O , . ,X8:.;8 , , , ,9* 8 ,N . d 2 2^I 02I 2=I 01+2@3W5-7 7 2:<511 2=11+2@GF2TG <2:1QI 1 3 7 5-1 . , . N ./,9* 2 I 2^I 01+2 ]=7Z5 1 , , O N , * # , . * , / . 9 , * * k d A 7 1$? A 5 2BG2 $ ]:7Z2 0 I 01+2 `i5^AA$7 D <5 14G2 l5-0I:G 7 5-3W2Z? 0 O* 5 7^I jhk. qs. . O. 2. n. . n ε (n, a, d, (N, h)) > 0 C(n, a, d, (N, h)) > 0 (M, g) n diam(M, g) ≤ d |K(M, g)| ≤ a π : (M, g) → (N, h) T ε ε<ε. . . λ1,1 (M, g) ≥ C · Vol 2 (M, g).. 8 E O 2 R S : L K 9 7 r 8 : 2 F M 8 7 G K M 8 E O W 1 \ @ S : L K E O 8 9 7 8 B E O M 8 9 D S : L K E O 8 U H F E O G > 2 F , 7 2 H 7 A > @ < @\ 1SKL239KG:48M3dH27939:f0790[8M@EHM5 @\798 ,B OE798M8rD9:d:SKLH2OE7I8 >AHU@ F HU1WKG,@ 79OE8M8 5SKLHOE8TP KG8 H2S767ID9>A:S@ K ;EKG:4@\3 1S%KL39z:43 @\KL5N@ 76DdF28 7 U0/O5 @\8r: 3 V @\> 7X14365SF2>G:W@\:J54OEF2>LZ 7XH279F ;FM765S:SKLOE8M5N ;F2Ko14765S:4798r: OEF 791S:4765 . Tqp. Qm qs. y. . '. . . f. -. . . . q f!#"$&s'* . - 2Z?. ./,. A. . *. . 3. ,/.;8. . 07. 0. .. p. . '*. . 2=0. * `. *. 5-3W2Z?R0[1M2ED^A. q f!#"$&s .X*. .. 2=0. .. * 1. `. If5-1 *. 1vV. .. ] . , . 8. Q. 2:1 1+G. 1. . f. . . :. . . R. 8. , 8. 7Z5. 701+2@< AQ2=1+G5-1 O ,. , .. GF2. I 2=7C?2H7. . XX. * 1. 8. l. ./,9* * ./,/* 7Z5 1v2:1Ta 1$? 1vG-0 O , ,9* 5TG <2=1QI 1. 7i5 ,. . I^0. O . 5. .. 5-0D. p` a. *. 7=<2^I
(32) G. ,. 8. *. 7E2=1. \,. 1G. ./, 2. O#O. 1`. 2=I.
(33) MV 39R28M1436391W5qOE@\02FM>GKL1S5I5WKG@\8MD68rD9OE:?KG0[>A@\@,F z779D98M>A@E9D685W795414:47,OEOE8MH145=7 P rHMHMF2@\@\>L8M798M15I5}>LH2>L7 7 765 D4R2[F2M@\:jR20214H2KG36:S7 5?147H279398_:4790[D61S79@\141XD9KG8M>LHU7z79KL1f5W79D9F2:=F28,:4 ;79F2KG1X8 K}H20@\7=1SOEKAF2>A@\@ 1S8r1W:4:j@ OE:409OEOE:S0/14>LOEO7 >LV O F2KLV 7:SKLKG ;>GH2KLFM7654735 0/OEF2I1N@\D6OE8M8;5q:S>L17NEDW>L79[1J@\02>L7XKG:S5S1407 76D9:S148M7 OEFM5j39:SFMHUKL7914OE8M5jD6OE,798;: ;HM@\8M5}>L7ND @E5}H P F28 MR2143N021SKG8 D9H2KG760[5 @\0>%79798=:SKG:4:476OE5 147 @\>L79F2=145dF28202K%14HOE02P F2148M765I7j,HUF 39:S>A1S@\KL02 ;>AFM@E7jD9KLKG798 8E@\< 1SKA@\D6798;>G:4>L77EHUOEF8=05S079F276D9: :S54147}7=1WHU@\F ,79>A@\8M0279>A1q@E<D9KL>C79PQ8 39:SFM1476H25 7 :4H27976KG5 8r:zBtOE@\1SF,B765XOEHU1SK ,3976145798rHU:SK KL7939>G>L1476795?8r:SH KL79P F2>G>L8 765,MKGR28 14E3,@\1S79KA8@\8;D6:47976145,D9>L0[7 @\1 >COEP @E>GRD9OE:SKLKLOE5X8 79: H27 OEF21S:4OEIKL5X@\8MH25OE82>L78M79D 8;@E: 5 @>CPQOE761SD:SMD6O 79V 1SOE:W8[@\@\KG8M> 76H25 765 %B OE0/1SOE,:SM76Z65 5WKG768 5z@\5SF21SKA1@\8r>A@ :4765,39:S1SKL ;FM7 F28M7 KG8MOE1W@\:SKLOE8HUFw5S076D9:S147 5SF21 . . . . F. . . #. . . . . . . Tk. . c. . . T k . . . ). f. . F. F. F"nmWoprq . jlk. u:w xx. z. . . R. znzQ{| ')(+*,#.. ,X8:.;8. *. .. (N n , h) 0[1+2 3W5-7 ? <BA 5F? 2^ S 1 ,→ N 8 , , O . 8f. , d , , . 1 M n+1 → N 0[1 S 1 ` ]=7 A+7 1$? A 5 B2 gε 0[1+2< 7 02 S ` 13W5-7 5-1 2)I^0[7 M . O O . d O O* N *-,/. d 2 2 02 |K(M, gε )| < a diam(M, gε ) < d 2 0Q2 5 1V-02=0[7GF2^I ]:7Z2^IhI , 8 , O , . * . . . ε JL 2ED I 2HGF2=IH? 1QI 51 2^I C = C(n, a, d, (N, h)) 2 ρ = ρ(n) 1 a 7 2:07E2 . O O X . * . * * O * . , , . d C 2 2^I 02 0 2Pa 7=<2LA+7 A+7Z2)GF2 365 2=0[7A$7 A$7E2 λ ≤ 2^I S 1 ` 1n365-7 5-1 2 J 1. EzO @\FMKL5z5N5W@\@\>G>L8MOE5 8M5NF2:SH2KG39>GKL,5479OE1 8;:SH 14P 79 1J;0F28OE:SM1436Z65S54F27T>G:W5S@\F2:N1z5W79>L7 YHUR2KA>A@\@\,R2>LZ97Y:S14< 7 D67979>G:zF2Ko>A@H27 D6OE F2OE1S>GR2RF2OE14KL7T5N79HU: F MOER2F2141S3 :4}OEKL795 : >C79PQ8 OE1SH2:SMOEO82V 8[OE@\8[8;@\:X> F2H28M767?5 B OEKG1S8M,OE761W5@\:SKGKL8 OE8@\1S02KA@\>GFM8r5d:476025 1436D9KL547HUF 5S076D9:S147 HUF >A@\02>A@ED9KL79814765S:S1479KG8r:Y< ε 2ρ. . . . F. F"nmWoprq %n'R(+*-,/.. jlk. S1 `. ,. 1365-7 (+*-,/.. *. 5WII. ?. ,X8. ,. 51. λ 2^I. .. 2 JgKL11. I. 0[1+2 * 1. 365 .. O. N. (M, g) 0[1 O 2 l0 2l<5 D. 2=0[7A$7 ,. S1 `. R. *.. *. 1n365-7. ,. .. ,. 8. ]:7. A$7. <R0[<. ,. ,. 1+? GF2=I. (n,. ∆p J. A$7Z2hGF2 5-1. . AQ5 O*. λ<. 2^IfJ. O. . , \ 2c<R0[1 G 01>2< N 1V0Q2=0[7^IRGF2^I ]:7Z2^I J 2=1
(34) ?2=7Z?. 2π l0. 2. O. O*. 5. O. 7^I. 2^Iba. *. 8:.. 7=<2^IgA$7. *. 7. , d. 0Q2. A$7Z2=I. HUF :SM36OE14Z9,7I765S:fOE02:SKGz@\>L7 2798 D67d54798M5 ?@ D6OE8M5S:W@\8r:47 r< FoF2P KG8M>27 7 UKBtLOE5S1S:4,7 H27 76025 14MOER2021414367 58M0/OEOE8 F21KG8>L76E ;@\FM1S79KA@\>G>L8;76:45q7 79>G>L7 8 765S7 :3 79V : @\>L7N< F28M7 76E5S@\:>L79>A@F210202141479OE02KL14Z97J147 @E5WE54@\O;>LD979KLF23671 02Y14F2OE8202K 147H P HUF2F8M7 D679,14D939>L:S7,1SKL rH2FM7 7 >LOE028 14VO%FMHU79F2F2KG:1 ;>L765jIB @\OE8M1S,5I>L7675 DHU@EK 5I39H 1479P F28;8 :SKL79M>GR2>L76145N3 H2:S7J1SK >A;@YKA@\B > OE1S,765 O 7H2765q65SB :dOE1SF2,8M7765f02 Bt14OEOE1S02,14767=5f[H2@\7 1S,OE82%KLH2 r7JFM79H2,7 7 E@\79>L: 79F21fF2028M14OE7=02B 14OE7d8M rD9:SFMKL7OE8 02 14OEOE02147 ,H27 7 >A@IB OE8M5W79D914:SOEKLOE8r8 : 8oPQ7654:q0[@E5}D6OE8M5S:W@\8;:47 r>L765qBtOE1S, 765}0214OE0214765f@E5W54O;D9KL365f<d>A@ @\>L79F21}0214OE02147 8M7 54OE8r:N0[@E5J:4OEF2:4765JKG8 E@\1SKA@\8;:4765 OE,7 765S:N>L7 HUFTD67914D9>L7=H27I>LOE8 V FM79F21 }>A@D6OE8M5S:W@\8;:47 HUFy:SM36OE14Z9,7 UT079F2: 5 P KG8;:4791S021439:4791 D6OE,7 >A@R/OE1S8M7 KG@ 8UV BtKL39541S5WKL@\798rF2: 147d5SF25S1NF21 >L76>C5NPQ79Bt8MOE548M79D9=:SKLR2OE>L8M7I5 H2H2767I5 M>A@ R214M76R25 14H27 7 I>A@=@\8M025N1479>L7XKLD Z9@E145 7 H276@\5 >L79MF2R21f140236145 OE79028 147d:4OEHU14F 7 [>A@\OE028 >A@E0/D979KL79F28 : ,HUF OE8r>A@\:S021479>A1Y@ED9F2KL798 8 143654F2>G14:W76@\5S:Y:S14547979KGY8;: R2@\>A@\F R2:4>L7 OEo147 B @\KL5 @\8r:XKG8;:4791 7982KG1Y>A@ 021479KLZ9147 @\>L79F21I0214OE02147 . @q[p. Qm . q
(35) $s. . '. . . . 2π l0. . 2. . l. . . L. . p. . N. "#. ∆p . l. +. . α. . 2π 2. . . f. l. 2π l0. 2π 2 l. c. 2π 2 l. λ0,1. . c. . . $. . ∆0. M = N × S1 α ∧ f S1. f. . 2. . B. . f. H. . Z. . . . .
(36) "nmWop. jhk. qt n 'h(>*-,#.. N. π. 8. .X*. k ¯ k ∈ N∗ T k ,→ M → N 0[1 ]=7 2=1 7Z2 T g 01+2 . * ./,/* . , . . * ,#./, k < 7 0Q2 1365-7 51 24I=07 T 2 1I^0[7 N I 7 ? 2=<2=1 A I 3W2 J f 0[1+2 a 1+? ( * * . , \ 8f. , d , , . . O#O * d d k 0!AA I 1QI 02 M 2^I <)01 G 01+24< .X* . O . , ./,9* O N 7 02 T−1 ` 1365-8 7 5-, N 1 2 g 2 2 02hA 0[7 0 1 5 ]=7E2 π x ∈ N 57Z2^I 7 ? g¯x GF2 g (x) 3 7 2 g¯x ≤ f (x) · g¯ J (>*-,#. O * O O , , . O * , 8 λ 01>24365 2=0[7hA$7 A$7Z2G-0 5^A 5? 2=1 5 V I^I:5-1 I=07 2^IBa 7=<2=I4G 7E2=1` ./, O#O (n, O* O * * 2 2^IGF2 M J λ < (sup f (x))−1 · λ0,1 (T k , g¯) _5 7I 2=Ia 7=<2=IA$7 A$7E2^I5WI^` 8:.. , d. ,. ,. .. . *. ,X8. *. .. ,. Tk`. x∈N. ,. .. OE,7 0OEF21 >L7f:SM36OE14Z9,7 U \>A@ D6OE8M54:W@\8r:47 765S:OE02:SKGz@\>L7 HM@\8M5 >L7XD @E5JH P F28 MR2143I:S1SK %KA@\> YF282K H P F28M7X,39:S1SKL rFM7I0214O%HUF2KG: H,276395?:S1SHUKL ;8KA@\FM,0/7 79Z9F2:S14:N76H25 836H2HU76F2579KG1S14M1W7=@ R2HU1479F 768 5 :S7 M 3679OE79:N140/Z9 ;79,8MFMHM77 @\>A 8;@? : H2F2OEOE8M8287d8MKG36H28M7IOE3 KGHV :?@\P F2@ >GKG8MS:4OE7d3=F2R79:4879OE1?1SBt8MOEF27d8M8MD95S7,F2:SKL1 OE%8T>L70HUOEHUF:SKAM@\z,Z654@ Z97 %K:S1454G=7YF21YF2H276>A@ 5 MR214765J8M7d0/791S,79:N0[@E5JH27Iz@ OE14791J>A@=BtOE8MD9:SKLOE8 HUF :SM36OE14Z9,7 I. ?. 2^I. I. 1. . Tqp. 13W5-7. 5-1. 2=IfJ. Qm qR n'. . . . . . . . g . . . 6. 6. (. #mWqC . F. . 4. . xl"$m6". . . f. 'h(>*-,#.. N. π. 8. .X*. .. k k ∈ N∗ T k ,→ M → N 0[1 g¯ 01+2 ]=7 2=1 7E2 T 2 8:. , d , , . ( * * . , \ 8:. , d d k k < 7 0Q2 1365-7 51 2cI=07 T J 0!AA I 1QI 02 M 2^I <)0[1 G 0[1+2 < 7 , , . . O#O * .X* . O . , ./,9* O 8f. , d @O 02 N T ` d 1365-7 5-1 2 2 2 02A 07 0 5 ]:7Z2 *-,/. * * ./,9* O O x ∈ N 5T7Z2^I 7 ? 1rGF2 5@< 7 02 π −1 (x) I g¯ J A$7 A 7 11+2 2 2 (>*-,#. O * O O , (n, *-Y .O , π λ 0[1+2 365 2=0[7LA$7 A$7E2)G-0 5=A 5F? 2:1+J λ< d0 2^I 2 <5 D ` d , :. N * O 8:. , d , ,/. 0 O* O * <)0[< GF2^IG 5-< 7E2^IBGF2^I ]:7Z2^IbA 07 5l< 7 02 1$G0 2AQ5-7 g 5 7^I 2^I a 7^<2^I , , . * * ,98 * . T k ` 1365-7 5-1 2^I J A$7 A$7E2^IR5-II ? 2^I I 1. @ 2 H 9 3 , E O M 8 S 5 S : W 1 \ @ S : L K E O 8 2 H 6 7 = 5 2 H 9 7 F : S M 6 3 E O 4 1 9 Z , 6 7 Y 5 , 9 7 = : 9 7 8 3 % L K 2 H 9 7 M 8 6 D , 7 L > 7 BC02@\14KGOE:q02 r14FM7Y7 @E5S5WK254O;>A@ D9KL=3IF2D6>GOE:S8;KG02:SKL>G79KLD98rKG:d:43JH2H 765P F2B 8MOE7 1S,@\76>L5 79F2KG8 102@\141SOEKA@\028r14:47 76765 5S:KG0[@\KG147 r@\>LOE145>L7 54OEFM5 765S0[@ED67 @\F21W@X0/OEF21jR2F2:jH27J,OE8r:S14791 rFM7 0OEF21jOER2:47982KG1f>L7 :SM36OE14Z9,7 J 7 4 D [ \ @ 2 0 G K S : 4 1 7 OEOE88 0,79OEF28;:N:S54147791W1W@@\, ;Fo798MP F2798M1 7 <=,F28M39:S7I1SKL5S ;KG:SFMF[7 @\:SH2KL7OE8 D6OEV F2361SOER2,F23914:S7 1SKL79 ;:zFM7IHUKA02@\>G,FM5NZ9:S5S14KG702R/>LOE7 1S8Mq3658 5S0[F2@\11S:S>LKL7 D9F2M>GKLR279141 3 7V65S36:fO%H20236145SO%KLD4 ;MFM7d765H P F28M7 ,39:S1SKL rFM7 KG8 @\1SKA@\8r:47d0OEF21 >A@E rFM79>G>L7 >L765 MR214765 54OE8r: :4OE:W@\>L79,798;: . Tqp. Qm q . . '_. :. . . F. . . . . . . . . F. . "nmWop. jhk . 2. (N, h) O. ,. qt y. 'h(>*-,. 01>2
(37) 3657 .. .. 2 1 : ,X8:.;8. *. 7. .. ,. a .. 2. .. d. 2=<5-11. . , . . * ,#./, II 7 ? = 2 <2=1 A I aI^ , , ,/* . , . . 2=1n1+2HGF2tG <2=1QI 1 I 7 ? 2: < 2:1 GF2=0D. 8. 7. 2. O. ./, 0[1 2=1 2=7 , 8 , 1 a 7 = 2 0[7Z2. n. . .. n J 2ZD I 2tGF2=I? 1QI 5-1 2^I ε0 (n, a, d, (N, h)) > 0 τ (n, a, d, (N, h)) > 0 2 . O#O , . ,X8:.;8 , , 7 2=<5-11 2=11+2TGF2 τ 0 (n, a, d, (N, h)) > 0 2 2=I d 02TI (M, g) 2^I 0[1+2T3W5-7 , ,/* 8 ,N . . G <2=1QI 1 n 3 7 5-1 |K(N, h)| ≤ a |K(M, g)| ≤ a diam(M, g) ≤ d 2 , N ./,/* , , O N , * / , . k d π : (M, g) → (N, h) 0[1+2 ]=7i5 I 1tA+7 1$? A 5 24GF2 ]=7E2 T 0 I 0[1+2 ε ` * , ./,9* * O* ,#O , . 8:. , d . ˜ 5^AFA$7 D <5 1MGF2 5-0I:G 7 ~5-3W2E? ε < ε0 g5 7I 2ED I 2RGF2^I < 7 02^I g ˜ 2 h . ./, . . N ./,9* , , O 0 ˜ N 7E2^IXA2Z? 3W2=<2=1 2 0[1+2 ]:7i5 ˜) → (N, h) I=07 M 2 14A+7 1$? A 5 2 π : (M, g .. . 2. O#O. . 2^I J. [J. d. . 02. l\. ./,9*. 5?. 2^I. N. 1G2. ]:7Z2^IhG2. Tk O 5. N. I^0[7 ]=7Z5. (M, g˜) ./,9* 1. π0. .g,. 2^I I. * 1. I. * <. .g.X*.. 8:.. 5. O. 7. , d. 02. 2=<2=1. .. V. 8Z* G. 8 I. , d. 02^I.
(38)
(39). J. $J. 1 g ≤ g˜ ≤ τ g τ 57E2^I . * X . 0. .. 7. ,. ?. ./,/*. 1 ˜ ≤ τh h≤h τ CO N ˜ 1GF2 g 5 ]=7E2H2^I 2. .. .B. 2. O#O 2. d. 0Q2. diam(π 0−1 (x)) ≤ τ 0 ε A. *. 07. 765SF25S: 1 >A@ D6KGOE8 F2E1S@\R21SF2KA@\148;7Y:4H27 o7 @\8 >LOE14795I1S1WOE@Y8 0[@\0@\FM1N7954F25S>CP:dKa r%14FM7907NOE:S5402MKM>A@EZ6>CPQ54D6OE7I798 1X025SHM>GF2FM@\025 8M0BC5d@\O>LKG547?R27N>L:S7 ;MFM367NOE14>A@IZ9,,7 39:S 1S KL r>CFMP 7 ;0OE5SF2:SM1 Z6547 U8MOE FM5NH239,OE8;:S147914OE8M5J>L7d:SM36OE14Z9,7 798 F2:SKG>GKL5 @\8r: 8 M 8 M M H \ @ M 8 5 L > I 7 W D [ \ @ 2 0 G K S : 4 1 7 >L02761S5IKG,1436795S1jF2>A>G@=:W@\D6:4OE5Y8MH25S:W76@\5Y8rD4:4[7 @\02KG:SHU14F 765 :SMT36OE< 14Z9,o797 :X 8M OE79FM8,5=B HUOEKL8M54D9D9:SF2KL:4OE798 14OEH 8MP KG5Y8 H2@\71SKA>A@\@ 8r0:45 O54V 5S36KGR2OE,KG>GKG39:4:S3z1SKLH ;PQFM7 765 H27 x ∈ NJ. . @q[p. Qm . Tk. . q $. '. . @. g. . F. (M, g). . . K(M, g) ≥ −a . . . . . F. C. (N, h). M. . .
(40)
(41)
(42) .
(43) ! .
(44) !"$#
(45) !"%$'&!)(*%,+-&"./0.21436578&"908;:<=+->?1@A1'&CB D8'90&"EGF. @DE@\HU1 ,KG79>oOE:SKG:WFM>G>G@\5FM8r5S: @\:S>G14F2>L7IOE8M8MF27 58 0/D679D6OE:S79KG1S:4:W7 ,@\KGE798T@\8M>L8MD67979F2OE1Y1f0[02R2@\1414OE17X02H202147I147 360254 79M8;7939:4:f8M79OE7 127,F29Z98_8M02767 5 >L277 98MOE02FM>L5f7 5WH2791W7 @XEF2@\:S1SKGKL>L397 :43 0[@\D61jOE>A@=0[5S@EF2D9KG:4:477 8 D6OE8M5SKLH2Z9147X>L7 V 14OEF207XH P 79KL54798rR791 V H27XHUKG,798M54KLOE8 G PQ765S: F28 V 14OEF207Y82KG>G0OE:4798r: HUK 36OE,OE1S02M7< /79: 5WOE8TD6798r:S147 391SK [7 V 14OEF207JH27 B OE1S,3JH2765jz@\:S1SKLD6765 8 M 8 E O 4 : 7 L > 7 4 5 E O M F 5 <H2T3 D6MO%827 KGz:j>AD9@=KL79828rKG:4> 5z@\791S8rKL39:S:4KL793 14H5 P 79PQKL76545S79: 8rRF2798 1 V 54OEH2FM7 5 HUV KG14,OE79F28M0547 KLOEHU8 KL54YD9140[79@\:,1 D6O%D6OE0[@ED9:, H2@Y7 :4OE0/OE79>L:O V OEKL8 7 0/79F2: 9:S147XH236D91SKG:47YH27I:S14OEKL5 z@\82KLZ914765 H 0[V 14@\OE1 F20/D6OE7X8M825SKG:S>G1S0/FMOED9:4:S79KL8;OE:J80[D@\PQ1N765SF2:_8 F2548MOE7FM5 82V KG14> OE@\F21S0/KL397X:4D63 O%D6DOEPQ765S0[: @E< D9 : HUKG147 >L7 ;FMOE:SKL798r:wH P F28 H 0[DPQ@\761W5S@\:,@\39FM:S541S5SKLKf5W@\F2:S8 KLOE8TMR2H214OE3 82798M8 367d:40[OE14@\7 1 5S F2 1?G >L7 U>LD676795 145WD9OE>L7 FM5 H2 EOE@\8;1S: KL39>L:47636 5 5JMH R2PQ361476 r5F[@\54:SOEKL8;OE: 8 HM@\8M5?>A@ 765q54OEFM5 @\1SKL39:4365qH27 54OE8;:qHUK 36OE,OE1S02M765f< E79:>L79F2145q rFMOE:SKL798;:45qHM@\8M5 5798 MW8 OE8r2:NDPQH2765S76: 5NF2:4OE8 14M765 R2143d021SKG8MD9KG0[@\>o798 D67914D9>L7I5SF21 >L7 :4OE147 U>L765 MR214765 39:W@\8r:JH23 H 82KL765JD6OE,7I>L765JOE1SR2KG:4765JH27I>CP @ED9:SKLOE8 HUFTD6798r:S147 OEKG: 79: >L765 D4[@\0M5JH27 76D9:479F2145 KG 8 E@\1SKA@\8;:45 < V @\5SFMF21D4M7I54F21 798 V 798 HU5SF214361 5=798a79: 391SK [798;0[: @\1 765?D4[@\0M5?0[8 @E54@5W79a8r<,:>C@\P @\FKLH2 ;7YFMH2OE7=:SKL79D68r76:5 79: 79: . B. T. . . !. . 1 x z 0 1 y , x, y, z ∈ R . G= 0 0 1 . . [G, G] = { . . 10z 010 001. , z ∈ R}. R3 . . Γ. . . ). Z(G) =. G. . G. . N = Γ\G . #. l. . . . . . . N. . . =. . . G. . x = cte . R2 . . . '. Z(G). . N. B. N. . X Y Z (0, 0, 0) ∂/∂x ∂/∂y ∂/∂z [X, Y ] = Z [X, Z] = [Y, Z] = 0 . . . N. C .
(46) WDD6[OEF2@\1SR20MF25 147 D654OE798M14OE5S:S8r1S:qF2F2KG14827zK B F2OE8M1S,739BC,@\798rKG>G:f>L7,RH2OE1S78M,365 39:S1S54KLOE ;KGFM: 7,0OEF2791X: >A@E ;:SFM14OE79KL>G5q>L7,1436>L7,79>L5HUKA0@\O,5SKGZ9:S:SK 14B 57 79U3:Y6>A5 @ KG 8 OEEF2@\11SKA:4@\OE8;F2:4:7 < V @\FMD4M%7 OE8:479H2>G>L37 79M: ;82FoKG: PQ795S8 F21 :4OEF2: >A@Y0/,5WOEOEKG398;KG:j:S: 1SOEKL r1S>A:S@TFMM7 OER[8M@EOE5471SD6,OE367 ,7 76395j:W@\D98r14O%: D4>AMH2@I793 ,:4M5f8239H2KL:S7 1S7 KL ;0[FM@\KL771 798;:S147X>L765NDW[@\0M5JH27 76D9:479F2145 H27ID679:S:47IR[@E547X54OE8;: 79: OE,7 >L765 0[@\1W@\,Z9:S14765 79: 54OE8r: 0O5SKG:SK Bt5 >L765?8MOE1S,765,H27 79: RKG8 OEE1S0@\8M1SOE3 KAF2@\18; ;:4F[>A5 @@\8M>L,H 7z39:4:S791S8MEKL ;54@\FM791SF2KL7 7 1 H27zP54@\OED6F2OE8;:SF2: 141S7 KGR28UF20[B 1439@\7 1S1SKL: 790fF27914D6F276OE:?5 5 <w,PQ367D9 1SfKG>L1476H27 5 OE8M79D48[DT@\Bt>LOE7T8M0MHUD95 KA:S@\KLOE,8_Z9:SH214767 5 H2D6797 O;: 7 zD914KL547976OE8;5S8;:4:47:5 H2765 D914O%D4M79:45 H27 KL7 279H28;7X:S14547 OE1SD6:4767I5 ;DWFM[7I@\>A@ 0MD65 OEF21S8R2F2KG147X014O76545S7J:47IH279OE>G8M>L7=DI@\< FM545SK R79OE: 1S8M367 H27 391SK [791 OE,7X>A@Y,39:S1SKL ;FM7I5SF21 7654:fKG8 E@\1SKA@\8;:47=< V @\FMD4M7 M>CPQ765S0[@ED67IH2765 BtOE1S,765 HU5S0K 7639D914:S14797X8r:S79KL879>G>L14767655S:SKG1S8 KLD9:S@\KL1SOEKA8 @\8r< :476D6579:J765S76:5S0[5S@E:WD6@\7 R2>L7_8 0[@\0179F2>L:N7 H2>A36@\HU02F2>A@EKG14D97YKL79H28 7 8 r@yFM7D @\>LD9F2>L79154OE8 79: O H2365SK V 8M7I>A@ B OE1S,7XHUF[@\>L7XH27 0OEF21 >A@ ,39:S1SKL rFM7 279:JH2OE8MDX ;FM7 ; 79: M8[8 @\>L147976,5S:S791S8;KLD9: :SKLOE8 @\F BtOE1S,765?KG8 E@\1SKA@\8;:4765 >CPQOE0/391W@\:479F21 765S:Y8;F2> 8 @ H2OE8MD 79: @ 8 BtOE1SOE,KG:Y7j rHUFMK 7z395S14K 798;:SKL79>G>L7 0[@\761=5S: 7 27H29OE8MDq02>LF27,8M5S7K B OE1S,7q0214OE02147fH27 79: @\>L79F210214OE02D67914:S7 :47 E@\>L79F21 0201479OEF202: 148M7 OE:4:479798M1dH ;Fo79PQ79148T5 147 @\@8MD4M@\7 1SKL395S:4K 3 @EHUa,0[79@\:X1 H27 27OE8M9D 02F2>L8M7=7?540/K 79:SKG:47 @\>L79F21d7902: 14OE02147 a>A8@ H2@\7I>L79zF2@\1 8202KLZ914OE147X0214027 >GFM5N8M027 1436:4D979KL8M54H7 0[@E5 79145 3914O @? rFM7654:SKLOE8 H2OEKG: H2OE8MD 9:S147JB OE1SYF2>L367 . . . ε ∈]0, 1] . . N. Xε = ε−α X Yε = ε−β Y. [Xε , Yε ] = Zεγ−α−β. Z. ε. . . . . α β. . N. . dZε[ = −ετ Xε[ ∧ Yε[ ,. . . δ(Xε[ ∧ Zε[ ) = δ(Yε[ ∧ Zε[ ) = 0.. . 1. . . . . τ >0. . . . . . λ. . V. q f!#"$}%n' 8:. <. . . . Q . . . α=β=1. γ=2. #. . . . 8E*.
(47) &. . γ =3. . ). N τ = 0. λ = ε2τ . α=1 β =1. . . . . . T. . ∆Zε[ = ε2τ Zε[ . P. Zε[. :. X. dδ. ∆Xε[ = ∆Yε[ = 0. . . gε . U. δ(Xε[ ∧ Yε[ ) = −ετ Zε[. ~. . dXε[ = dYε[ = 0. 1. 1. . . . . . . γ. . . U[. . . F. . X Y N Zε. X ε Yε. . . . . g. τ =γ −α−β ≥0. . !. γ. gε. . . [Xε , Zε ] = [Yε , Zε ] = 0.. α β. !. . (Xε , Yε , Zε ). Zε = ε−γ Z. . . γ. gε. . . . α β. . . 7. ,. . 2)GF2. ' OE\. 2. S. * *. <)<2=1. .. 1$G-7E2=<2=1. ,. 365-7 .. . O 2. 21. *. <]:7Z2RGF2LAQ2. ./,/.. 2^I. 365. O. 2=0[7I_A$7. *. A$7Z2=Ih5-3W2Z?. O 5. 4.=+ % / + !"&D& /0 .
(48) ! . 0
(49) @ OErF2@ 1>LHM7 @\@\8r>A8M@\:Y5d02H >A>L@E7P @\D9RDKL79@EOE8z5I14H2H2H2797 761Y5 >AB @O;OEH 8MD6V D9OE7 :S8MKL5SOEH2:S8M7 1S5FMJD9[I:SKL@\OE@\ 88M5EB 1W@\@\2FKG:4027,0/790[ >L@@\OE1 8M5q2F>L 7
(50) 14@OE36r:S5S:=@F2>GH ,:W@\P F2OE:q88rOE:SR2OE14:470/79398r>C1WPQF7 @\UK:40[L795S@\F2:411=798M >GD6KG7 2FKG :47 545SF2OEFM1 5 >L76D65 791SBt:WOE@\8MKGD98M:S76KLOE5J8MD65 OE8MHUKG:SKLOE8M5 H P F28 OE0/391W@\:479F21 >GKGKG:47d0OEF21 >L7 >A@\02>A@ED9KL798T@ V KL545W@\8;: . l. _. f. . . . H. . . . . . . f. .
(51) OEKG: :,7948r79:>G>L765d rFM7?>L79F21I>CD6PQ79OE8MF254791S79R2: =F214R27>L7 54H276D976:S5 KLOEE82@\8M1SKL7939>G:4>L3675 M791S:79KL79: >Lz79F2@\1I8282HUKLKA79@\82,8MZ976>A:S5@ 147 KL39Z91S,K [7 798rEH2:X@\7j>L1479HU76F2KG5S,1N0027679D98M14:SOE54K 02KLOE147 8 7 H:4U36F 765X>A@\5 02P KG>A>@ED9KL79@ 8 Y@ V F2KL>G54:S5 KG@\028r>GKL: D9KG54:4F23 1 >L7675 02B OE148MOED9R2:S>LKLZ9OE,8M5J7,H27675S:XH PQ239:4>L767958MHU14@\7 >L79D6F2OE148r5N:S02KG8r14F2OE02,1479768;5 :=39:W0@\OE8rF2:J1X14:439OE0F23 : H2H2767d>A5 @ >A@ 76B 5SOE:40[OE8M@E0/D9D6:SOE76KL>L5OEO 8 V ,KL7Y39:SH21S7KL ;FM76145IOE,D6OEO 0[ @E@\D9FM:45 54H2OE< 1 PQ>C76P 5S@E[: HUzMKG@\39KL140/5 79O8M2F54D6 547,KG
(52) R2@ H2>Lr7=7 @z5SKD6OEOE8M8T5SKL=H2Z9F214827IHMKG:X@\>CPQ8M79D65I8M79:54>C79PQ7979=8M8MR2545W7979>L7Y=YH2R2R276>L>L775 76H25S7Y0[>A@E@zD676:45}OE0/,OE39>L:SO 1SV KLKL ;7FMH2765j7 , 76@\5SFMF25414H236OE5 1 ,765SF2Y1436F27 82KH2H3 MP F2828MKL7 7ND6,OE7654F2,1477 H25SF27JKG:0214aOE54R[OE@\KGR2: KG>GKG:43 YF2F28M827 K 5S@\F202KG02:4>G7zKLD H @\PQ:S76KL5SOE0[8M@E5 D676,5=76,5SF2391W:S@\1SR2KL ;>L76FM5 765Y,765SF214365 >G>L7 79D6: OEF28 8M797 1 V 5S7F2KG:47 79145S5 :S1SKLD9:479,798r5 P:KG>j07 OUK5SLKG5S:S:4K 7,7 H2765 391SK a@\8r: P OEKL54KG8[@ V 7XH27 7654: OEF21J:4OEF2: HM@\8M5 2OE8 @ b. M(n, d). n. diam(M, g) ≤ d . |K(M, g)| ≤ 1 . λk (M, g) M _. . . k. (M n , g). (M, g) → λk (M, g) h. b. . ,. . . . k. M(n, d). . f. J#2 J. . ). . !. F. @. (Xi , µi ) . (X, µ). . Ψi : X i → X. . . . ). . εi. lim εi = 0. i→∞ . . εi. Ψi (Xi ). . . p, q. Xi. X. . |d(Ψi (p), Ψi (q)) − d(p, q)| < εi ;. OEF21J:4OEF2:47dB OE8MD9:SKLOE8 5SF21 MOE8 @ . X Z. f. lim. BCD @\PQ76KGR25S: >L7 < HUKG147? rFM7Y>A@,765SF2147 >oOER2:SKL798;: @\>LOE145. i→∞. ∗. F"nmWoprq. jlk. . J. [J. , *. A$7 J. 8. *. (. .X*. 0[7. IIf5-1. 5GWU. 02^IR<2^I=07 ^ I _2=1M<)0[1 , * O , 8 2:11+2 1 7=<5 I 2!J. a. **. OZ\. M(n, d). ./,9*. λk M(n, d)\{(point, 1)} J 5. 5 V.
(53). 'R(+*-,/.. , d. 2^IXA 5F?2^I)< 7 , I^0[7Z2l7 2=<5-1n1. D6OE8 791 V 7 79145 0/OEF21 >A@:4OE0OE>LO V KL7 F. .. 1$?. 0. .. A$7E2)GF2. 1. I. \ 8:.. 8. ,. R. µ. M(n, d). OE\. 7Z2=1+?2
(54) GF2 G5-1QI . ,X8:.;8 d II:51 ?=U[5 02)3W5-7. 2=1$G. ,#O. ,. ?. .. * 1. (X, µ) ∈ M(n, d) 2ZD I 2l0[1 . O I=07 L2 (X, µ) 2 d 0Q2 λk (X, µ) P(X,µ) J. * * d 0!AA I 1 I Q 02 i→∞ i i * O , 8 * ,X8 RO O * 7=<5 I 2 5WI^I ? 2 5R365 2=0[7gA$7 A$7E2 O* 2 J 7I k (X,µ). lim. 1. f dµ,. l. %$' y. 8f.. f ◦ Ψi dµi =. (Ψi )∗ (µi ). Z. ./,. *. 102=<2=1. (M n , g). .. OE\. . (+*-,/.. k. *. O. 24GF2^I. GF2hI:5T<2=`. 2=1If2=<]. . .X* *-, . 7 5 Z 2 0[7R5-0 = `i5G1 *,#.@8 O vO ,9 I V5 5 ` <2 A. 8. 2:1QIf2=<]. ./,/*. O 2. P(X,µ) O. 365. 2=0[7. *. ϕk,i 01+2ga 1$? 1A$7 A$7E2 (M , g ) = (X, µ) J . λk (Mi , gi ) 2 Λk,i = {ϕ ◦ Ψi , ϕ ∈ limi→∞ d(Λk,i , ϕk,i ) = 0 J ϕ = λ (X, µ)ϕ} . I \ @ M 8 I 5 L > 7 D E @ X 5 O C > Q P 6 7 S 5 [ 0 E @ 6 D 7 G > G K G K 4 : 7 7 6 S 5 I : 2 F M 8 7 E \ @ S 1 L K 9 3 4 : , 3 S 1 L K 7 z@\8282KL79828M7 >A@ ,7654F2147,>GKGKG:47 79:=>CPQOE0/391W@\:479F21=>GKGKG:47 M8 7 54OE8r:=0[@E5X8M3 L (X, µ), P. . @q[p. Qm . qv%$' . . '. . . µ. (X, µ) P(X,µ). . ).
(55) ,D676OE5458;@\:SKG14147X79,>CPQ7 79278r9:Y>A@02>L,7X5S76F25SF2K 14@\78r1S: KL79z@\82KL79828M7 82Kq>L7>A@\02>A@ED9KL798H27,>A@ E@\1SKL39:43 D6OE,7,>L7 8 D6OE8M5SKLH2Z9147J>L7N:4OE147 B OE8MD9:SKLOE8 0O5SKG:SK 7 279:N>A@=BC@\KG>G>L7XH27d,39:S1SKL ;FM765 H23 M82KL7X5SF21 0[@\1 F28M7 K 2OE8T@@\>LOE145 4. . qp. . #q%n'. . '. T 2 = {(s, t), s, t ∈ S 1 } c : S 1 → R gε (c) T2. C∞ . . . l. gε (c) = ds2 ⊕ ε2 c(s)2 dt2 .. f ∈ C ∞ (T 2 ) . ∆(T 2 ,gε (c)) f (s, t) = −c(s)−1. K OE8 8MOE:47. ∂ ∂s. . c(s). ∂ ∂2 f (s, t) − ε2 c(s)−2 c(s)−2 2 f (s, t). ∂s ∂t. >A@ KLZ9,7 E@\>L79F21 02M14OE@\02>LOE14147Y5 H27I>CPQOE0391W@\:479F21 H23 M82K 5SF21 0[@\1 Pc . l λk (c) k d d Pc (f )(s) = −c(s)−1 ds c(s) ds f (s) . S1. lim λk (T 2 , gε (c)) = λk (c),. 79: @ 76D D6OE8MH25S:W3 @\M8;82:4KL765W5=5479548rF2: 1XH2D476[5 @EMD9R2F2761W85N@\:SH2B KLOEOE768M58MD95 :S%KLOE r8MFM5d7 D6>L76OE5q8rB:4OE798M8;FMD9:S76KL5IOE8MHM 5f@\798M02>A514@ OE025SK 14V 76825 KOE[8r7 : >AHM@54@\OE028M8;145I: OE>L@\027021SKL02D 39@E14:4O5X3 %KOHG,P 936:S>L761476755 0[@\1JH2765 BtOE8MD9:SKLOE8M5 D6OE8M5S:W@\8r:4765 5SF21 >L765 MR214765 ε→0. lim (T 2 , gε (c)) = (S 1 , µ). ε→0. . Tqp. µ = c · ds . Qm q%n'/% z. 'P. Ψi . . Ψ−1 i (x). . . x∈X. . . . . #. . . . .. ϕk,i. .
(56) . . Λk,i. . . !. V 3 6 6 D 9 7 , 9 7 r 8 : E O S : , : @ 9 3 M 8 9 3 W 1 \ @ G > L K 4 5 3 L > 7 4 1 6 3 S 5 2 F G > W : \ @ : 2 H 7 MF
(57) E@ r@ @\F BtOE1S,765HUK B B>C39PQOE140798r391W:SKL@\79:4>G79>L76F25 1N>G KGO KG:47I R 798 8MOEFMO 5 14@ 765S :S1479K V OE8[FM@\5 8;@\:N>G>L0[OE@\8M1N5 54KLOED9FMK D902KL5 1436H2547d798;D9:4>A79@\11S:4>A3Y@ @\D6F OE8MD @E5S:S5 1SHFMD9P F2:S8KLOE8wMR2H21437 4 5 2 F } 1 2 F M 8 7 E \ @ S 1 L K 9 3 4 : 3 S 1 L K 9 7 z \ @ 2 8 2 8 L K 9 7 2 8 M 8 I 7 2 H E O r 8 f : A > @ M 2 R 4 1 7 6 7 S 5 j : 2 F M 8 J 7 2 8 G K > R[< @\@EV 1S54@\KL7 39FM:4DW3 MOE7YFM545}OE8MKL79OE8;:4:N79140[OE[@\8M791W5:N@\>G r>LZ9F2>LK765 :479>A8M@IH D6OE082OE8MF271 UK>AL@ OE8 HUKL5S5SF2:W@\1 8MD67X:479H2>G7>L7J ;14FMOE7 ,>L76O 5fD4[@\@\FM540MH25jOE1KG8 @\1S79KA14@\5N8r5W:4@5 >C:SPQMOE360OE39141W@_Z9@\,:402797X14F2791 >GKLKGZ9@ 14V 7KGKG:4:7 HU5SK F2zd1 D9@\F28M>G5 :43>A@ 7654D6: DOEPQ8M76H25S5S7 ::S1SH2< FM39D9HU:4:S79KGKL14OE1S7 85SKGF28MH21797 1 F22F8F2 876
(58) 5S@ 0[76r@E5S@ 0[D67 @E>CD6H2PQ7OE7f0B5439OEF21W8M1@\D9:4:S>L7976KLF2OE r1 8MFM5q79> < E@ @\>L79@ F2VHU14KGF 15 14H2367 79>GB >LOE761S5 ,5S76F25N1jHU>A@XK R[39@E145W7978r:SKLI79>G@\>L7X8M5j5SF2>L7 1 D >A@?@E5 R[H2@E765W5j7 UB OEz1S,@\KL5J765f< HUEK @\>L397914F2791N8r:SHMKL@\79>G8M>L765 5 F228 OE8 7654F20[:S@EKG>GD6KL7d547 02F2>GFM8 5 76V 5S1W0[@\@E8MD6H 7 5:4S ;F279FM> 1 7 ;>AFM@ 7 R[>L@E76>G54FM5 7=5 H202OE148r3654OED9:dKLKL5479D4798;[,:N@E ;79OEFM8r1S:S:7 MEMOOER2V8=OE14D67=8[OE@\768MF 5S54:NKLH2Y79Z98;14F27j:S8214F2KL7I7?8 79H FMP R2F2148 3 0214768MO%D9OE:4HU:4OEF23 1SKGKL:d79> 54VDU1W@\79@E>A: @\HUKGOEFM148 73 YV 1WF2@E82HUKGFM:N3 D67 MR2143 H P F28M7X54F20/7914D6OE828M7 UKLOE8 H27IH27 V 143z 2DPQ765S: < HUKG147YH P F28 OE0391W@\:479F21JH27d>A@=B OE1S,7 . . . . . . . . . . =. . . B. F ,→ (M, g) → (N, h) F = Γ\G ∇af f. l. . !. M F. F. . . . . L. . f . L2 (N ) . . . [. . . R. P(X,µ). !. h. +. . . E=. . Ei. . . A0. . . A0 = A0[0] + A0[1] + A0[2] . hE. . Lm. j=0 , E. . j. J 2 J.
(59) O. . H. 67 5S:JF28M7ID6OE828M7 %KLOE8 5SF21 ;F2Ko02143654791 7I>A@ V 1W@EHUF[@\:SKLOE8 85SF2,1 H0/>CPQ7976F25S0[:j@E39D6:47798MHU147JD679:S:47JH25476F25?0/79B OE14D61S,OE82768M5,7 UKHULK OE8z39140[79@\8r1q:SKL>A79@X>G>L7614Z5,V 5S>L7NF21 H27 79< KGR282@\K >L79F2<d1,F2HM8z@\OE8M05 391W@\:479F2@ 1 ,54D 39@\:S>A@\1SKLKG r147NFM7 5SF21 79: >L7?0214rO%79HU:}F2H2KGOE:I8M54DJD @\H2>A7N@\KGH21437 M82KG1}0/F2798z1S,OE07939:S1W:4@\79:48;79:XF2H21j7 @EH D6SOEOE8MKG8r5S:S: 1SF2< KG147 F28M8 OE:4023 14O;HUF2KG: 0[@\1 8 8MOE:4791W@ >A@ 714976:J5SF2:S1S8 KLD9>A:S@\KLOE028 >A@ED9H2KL7 798 54F21 < OEF21=D6OE8M5S:S1SF2KG147>L7 MR2143,5SF21d>L76 ;FM79>}@ V KG:I>CPQOE0/391W@\:479F21X>GKGKG:47 E O S : Y : 4 5 ? 7 W 1 @ ,zZ9@\8M827?82KLH798 P @\ROE14H 798 F2:SKG>GKL5W@\8r: @\F D @E5IO 765S:dF28 MR2143,@ ,8M7 1SKL7 H. A0[0] ∈ C ∞ (N, Hom(E ∗ , E ∗+1 )) A0[1] E 0 2 ∗ ∗−1 A[2] ∈ Ω (N, Hom(E , E )) . . Z. . . Ω(N, E). &. . . N. hN hE W. Ω(N, E) Ω(N, A0 (A0 )∗ + (A0 )∗ A0 ! L E) a ∆E = b ∆E a+b=p Ω (N, E ) . . . . . . E. (A0 )∗ ∆pE. A0. . . . . (M, g). . r. .
(60) !#"$&%$'/%n%n'. . . 5 v1+2. ]=7. 8 7. O. Aff(F ). . 2=<5-11. ,. F ,→ (M, g) → (N, h). 2=1. . . F. M. . ,. , , 2=<5-11 2=1CI * . . N 8 . * . 2lV-7 0!AQ24GF2I 7=0 ? 07E24G-0 ]=7 2^I ? 1 2 1n0 = G5-1QI , 8Z* * , , 8 . d af f G < 7;AnU I^<2^IhGF2 0 A$7 f I 2=7=3W2=1 . , \ , . , ./,9* * , * . O H 2^I <)01 G 0[1+2RG I 7 ]=0 1HU 7 1 5 2 7. ,. N. . . 1. . . N. g. ∇. ∇. G5-1I N . , \ 8 . , d : d ?=U[5 0Q2 ]:7Z2R2^I <R0[1 2 G 01>2c< 7 0 2 @O * ,9* af f I=07 O 5 N ]=E 7 2 5T? 1 1+2ZD n 1 5 v1+2 . , \ 8:. , d 2^I <)01 G 0[1+2l< 7 02 N O 8f. , d \8 ,/. 5R< 7 0 2 I^0[7 I ? 7 : N. T M. h g = h ⊕ g Fb. M. T HM J. d. g Fb. O. 7Z2. ,. O. 0. ./, 5. .. 2^I. *. 2lV-7 * G. A 5-7Z5. 3W2=<2=1. O#O:O. RO .. *-O* U. N. 0[1+2. * 1. , <. GF2^I. , I. . 7. ,. ]=0. .. 2h2=I *. 2LA 5-7c7Z5^AA. 5G. 8. ]:7. Aff(F ). 0!AQ2. .OZ\ 1. .. 2^I. . 7. ./,9* 1. 79:=8yMH2OE361SKD6OEOE8;:W0@\O>L5W7 @\q8;OE: 8_>L760/5z79F2B OE:1S,36D9761S5KG147 HUK 3914798r:SKL79>G>L765 H27 798wO >L79F2145 0[76@\5S:?1S:SKLF2768 5 M79R21S14:S3 KLD 54@\F2>L71 2 H E O r 8 : A > @ M 2 R 4 1 7 6 7 S 5 : L K 4 5 E O , E O S 1 2 0 M 7 < d \ @ M 8 5 L >. 7 D E @ 5 H P 2 F 8 M 2 R 4 1 3 @ , M 8 7 L > 7 VH2147OEF2D67 0/7 V 14H2OE7 F20/5S7z:S1SFM5SF2D9:S1 F2147 HUF M02R2143614354791 7z02>L71436545WOE791FM5 7 765S0[@ED67,H27976:?5X0[Bt@\OE11S,D6OE768M5X54KG368 ;FM@\791S8rKA@\:8r:4>CP76@E5 D9:SKLOE88 0/79F2: H2OE8MD 5SF2H21J3 M>L7682 ;KG1FM79>L>7 5W@zOEFM@ 5 V KGM1 R2>C14PQOE3 0/391W@\H2:47 79F21 >GKGH27 KG:4M7 R2147 8 @,H27Y02>GFM@\KG5J8MF254K8T ;02FM>L7OE8 >CV PQ76795S,0[@E798rD67: r 2 F K 0 9 7 S 1 , 9 7 X : H P L K 2 H 9 7 r 8 S : K [ 9 7 ? 1 4 D [ E @ ; M F , 7 9 3 L > 9 3 , 9 7 r 8 ? : 2 H 7 < F28M7 BtOE1S,7XHUK 3914798r:SKL79>G>L7=5SF21 0[@\1W@\>G>LZ9>L7I0/OEF21J>A@ D6OE828M7 %KLOE8 9 7 L > E O 8 L > 6 7 5 4 1 6 3 S 5 2 F G > W : \ @ 4 : 5 2 H E O 2 8 M 8 6 3 5 M H \ @ M 8 5 S 5 2 F 1 L > 6 7 5 S 5 2 F 0 9 7 4 1 6 D E O 2 8 M 8 7 UK L E O M 8 5 9 7 : 9 7 8 F2H2:S7,KG>G>AKL@ 5W@\M8rR2:,14>L7 7 76BC5S@\:=KG:,54:W ;@\FMR27 >L7z5SF20[1@\1=F2>C8 P @E:4D979:S>KLOEM8 R2H2143 7z>A>C@ PQ76HU5S0[K @E39D6147 798rH2:S76KL579>G>LBt7OE1S7,%:7645,391SKL0[79@\F21W14@\7 >G>LZ9D6>L76795,:S:47>L7 HU>LK OE8 3 V 145S79F28r0/:S79KL1479D6>G>LOE7T828MKG8M7 HUUKF2LOEKG:8 0[5SF2@\11 >CP KG8r:479:4>G79>L1S7I, ;36FMHU7 KA@\KG147 HU54OEFwKG: 02>A>L@?OE8 HUV K 79,391479798;8;: :SKL79>G>L7Y5SF21 F254OE8MKG7: >A@YD6OE828M7 UKLOE8 5SF21f>L7 MR2143 KG8MHUF2KG:47d0[@\1 79: 5WOEKG:f>L7 0214O%HUF2KG:fKG8;:4391SKL79F21 ,39:S0[1S@\KL r1zFM>A7 @ BtOE1S,KG8M7THUF2H2KG:j7 F2D68,OEF2021S14R2O%F2HU14F27 KG: 54D H2@\7>A@\KG>A14@7 HUKL5S:S1S5SKGF2R21}F2>L:S76KL5OE8 MR214765jHUF MR2P 14@\3 F2:S14767 D9:40[OE@\1SKL1S79: > q>A@ . . &. . . N. . . Ω∗ (F ). M Z. . ' ∗ Ω (F ) Z ∇af f . E. . W. . . Ω∗ (N, E). T. M. Ω∗ (N, W ) . Ω∗ (M ). . M. .
(61) . F. . . iT. . . Λ∗ (n∗ ) . . ∇af f . . . Ω∗ (N, E) . T HM. E. T. g Fb. hE. . . Ω(N, E) ,→ Ω(M ) Λ∗ (n∗ ) A0[1]. . A0[0]. E. . . . . . . . Ω∗ (N, E) ,→ Ω∗ (M ). . W. A0[2] . T H M P . . _. E.
(62) @ S 5 S : S 1 M F 9 D S : 2 F 4 1 , 7 2 H 7 M 2 R 4 1 z 3 @ , M 8 , 7 S 1 L K 9 7 z \ @ 2 8 2 8 L K 9 7 8 G K M 8 U H 2 F G K Y : 2 H E O M 8 , D < A > @ B E O L K d 5 2 F M 8 , 7 S 5 2 F 0 9 7 1 DF26OE8 82>A8M@\7 02UK>AL@EOED98 KL79879: F28 025SF214O%1 HUF2KG:J54D @\>A@\KG147I8 5SF28M1OE:W@\8r2: 79: 0791S,79>L: 7I0[5S@\01J76D9D6:SOE148M7d54HU36 ;F FM>A79@\8r02: >A@EH2D97IKL79H28 3 M82KG1 E O S : : , E O r 8 S : 4 1 d 7 ; M F 7 M H \ @ M 8 5 6 D 9 7 S : 4 : 7 S 5 G K S : [ F \ @ S : L K E O 8 % L > 6 7 f 5 / 0 9 7 S : G K 4 : 6 7 5 \ @ L > 9 7 2 F 4 1 } 5 2 0 4 1 E O 2 0 4 1 6 7 5 U H F A > \ @ 2 0 A > @ D9KL798 54F21 54OE8;:JD679>G>L7XH27 O R U:SM36OE14Z9,7z . . . . Ω(N, E) . ∆E. . M. "nmWop. qr%n'/% s. .. R. O. R. .. g. M. .. 2^I. N. . . ∆pE . . . . . , , . ,L* 5 v1+2R7 2=<5-1n1 2:1I^0[7 2 I ./, . . 2=1QI 2=0[7IlGF2h? 0[7f]:07E2^Il7E2^IXA2Z? 3W2=<2=1 I^0[7 2 I 07 = M 2 F 2^I b A 8:. , d O * 8 , , :. N < 7 0Q2 5t] 7=1+2I^0!A 7 2:07E24GF2^IG 5-< 7E2^IGF2^I ]=7E2^I@2 Fb * * * . O N O* ,#O , . * . . I 2Z? 1$G2ga 7=<2ba 1$G5<2:1 5 2lGF2=I ]=7Z2=Ig5 7^I 2ZD I 2RGF2^Il? 1QI 5 1 2^I , 8 . d . O#O , d d 2 0 1+2lG AQ2=1+GF2=1 02hG2 2 2=I 0Q2I F ∞ * .X* . A 07 0 jhk. 'R(n,. σ(∆pE ). . ∆pE. ∆M. . E. 0[1. *. ]=7. 8. N. M. diam(F ). c. dim(M ). kR k. p ≤ dim(M ). F. 1. 1 *. *.. 07. .. Π a a0 O* diam(F ) ≤ a0 5. 2 O. O. 5 5 2. .. 7^I. σ(∆pM ) ∩ [0, a · diam(F )−2 − c(kRM k∞ + kΠk2∞ + kT k2∞ )[ =. σ(∆pE ) ∩ [0, a · diam(F )−2 − c(kRM k∞ + kΠk2∞ + kT k2∞ )[.. 8 _ 6 7 S 5 ? : 2 H E O M 8 z D W 1 \ @ , 9 7 M 8 3 < A > @ 4 1 6 7 4 D M 9 7 4 1 W D M 7 H P 2 F 8 E O 0 9 3 W 1 \ @ 4 : 9 7 2 F ? 1 G > G K G K 4 : 7 S 5 2 F 1 ,765S839:I:SD68M1SKLOE7 ;8MFM0/5S7:W79@\F28;: :4D67zOE@\8MF5S8 KLH2D67 39OE14F2797914:1N5X54H2390[7?@\8M>C14PQ7Y739,H2OE39798M08;HU:79148M>A79@XH ,>G0[79KG8r@E:5 KG:4H27d797=: H2>A7 @>A,@YH25439F23:SV1S0/KL39 ;798M14FMZ9D67=14OE7,825SF28MH217 OEUK8MLDOE8 5W79@ : >GH2KGOE798M:fKGD=:4H277 79>G8M>A>L@77 0H27d791SD6,OE798M:J5SKL0[H2@E39145N79H21 7Y>CPQH2798M3 M547982=KG1JR2F2>L7 8 OE0391W@\:479F21 %@EOH SOEKG8r: D676OE5S:j>C,PQ76547 0[@ED67 H2765 5SP KLF2H20367Y7914H2D6OE7 828MOE7 :SUK: LOE768M5S5: 79H2H2:d77 H24H2@\7=7 F V ;V 14FM7z3OE54:S F2KL791 8r5SF2:4791 1I ;D6F279Kq:d79:0279148M3654547979Y1 >CR279PQ76>L8;7?5S:Y0[0[@E>A@\@ D61 7 V >L1W7H2@E76VHU5 14F[OE02@\F2:S140KLO;OE7HU8 F2KG:4 5>jH2OE5476R2D 5 @\:SKL>A79@\8rKG14:765 @\>L79OE FM14:S5XD91W>G@\KL>L8MHU75KL7914B OE368M5S1S5 F2z>G5S:W@\F2@\:S1 :?KLOEH28M75 D6OE0[@ED9KG:43=5SF2K E@\8;: O R M:SM36OE14Z9,7 . . Ω(N, E) A0. . hE. . . . A0. M. . . . E. Z. . . . GE. GL(E). . . . . ,X8:.;8 , , N 8 01>23657 7 = 2 <5-1n1 2:11+2 D . O , . ./, * . 2 JP 2ED I 201+2 AQ5-7 24? <BA 5F? 2 E E E , . ,X8:.;8 , , , ,9* . O O d d n 02I 2^I 0[1+23W5-7 7 2=<5-11 2=n 1 1+2GF2G <2:1QI 1 2 2 0 2 Q . O* O O \8 d , O * O 2 5 7I 5 ? W 5 II 2 G 0 3W5 2:1$ ? 2G-0 ? 0!A 2 ∞ GH , ,/. ./, . 1$G-0 A 5-7 5=AA 5-7 2:1 J E. jhk. qr%n'/%[ 'R(+*-,/.. . E. HE. . "nmWop. (A0 )∗ . (SE × HE ) SE. E. c. hE. W. (N, hN ). n > dim(N ) K(n, a, ε) ⊂ (S × H )/G n. a>0. ε>0 . O#O 2 2 (M , g) kRM k ≤ a d (M, N ) ≤ ε 0 (A , h ) g K. q W : \ @ r 8 J : 2 H E O 2 8 M 8 6 3 d 7 2 F M 8 d 7 S 5 2 F G K 4 : d 7 2 H d 7 , 9 3 S : S 1 L K ; M F 6 7 5 ; 2 F K 7 E O M 8 U H 4 1 7 S 5 2 F 1 W 5 = @ [ R E @ 4 5 7 F2OED6OE88M8M7 5S0:S54791SOEF2F2FM:KG5147J7 54U:F2F2SKG8z:41W7@\>AKG@\14 ;027F2>A@EKfH2D9D67 KL79OE88>A@ >G79KG541 F2V KGKG7 :4:47 7 79145?F2H28MOE7 8;54:}F2>L0/7N7954140/D676OEH D982:SPQ398M14>L77 39UK,76L5SOE79:q8_8;>A:4@d>G5TKG>GKGH2KG7 :4KG7 :47 ;HUF2FzKf0/5S079761S,D9:S79147J:HUH2F 7 >A@\02>A@ED9 KL798 5SF21 V , @ S 5 2 F G K 4 : 7 M 8 X 7 6 D E O 8 9 7 1 = 7 [ 0 E @ 5 M 8 6 3 6 D 6 7 4 5 5 \ @ G K 4 1 9 7 , 9 7 r 8 : 8 079F2F28M:X7d0[Bt@\OE1X8MD97 :S27KL9OE8 02 r>LF27zK @\OF54D90[KG>G@\>L7X1W@ 79V 8r1W:S@\14027 M7 79 : G,021479@ 8MHU021414779KLZ9147 @\>L79F21J0214OE7902:X14D47 MEOE@KL5SKGO1=54D90/KGOE>G>LF27911 79H28;7 :S147 79: 5W@\8M5D6OE8 791 V 791 PQOE0/391W@\:479F21 8M7 D6OE8 791 V 7NH2OE8MD 0[@E5 \82K2>A@ D9>A@E54547 OEF21J>CPQ39:SFMH27IHUF 540/76D9:S147d5SF21 >L765jBtOE8MD9:SKLOE8M5 2OE8 079F2: 547 14765 :479KG8MHU147J@\F MR2143 ;F2KM765S:F28 MR2143 :S1SK ;KA@\>a798,HU14OEKG:47 143679>G>L7 5SF21 790798MHM@\8r: . . Tqp. M. Qm q. [(A0i , hE,i )]i∈N. 4. %$'#%['. γ. F. Qm q%n'/%(. . E. . 1. . 2. α = 0 β = 1. . . ∆E. '. !. F. [(A0i , hE,i )] . 1 t (A0t , hE,t ) . M (SE × HE )/GE. ∆∞ . . Tqp. . (gi )i∈N. l. . . !. 0. . . . N. .
(63) L>G> KG7j02KG14:4O%7XHUHMF2@\KG:8M554D >L@\7I>A@\:SKG1W14@ 7 E@\KG>H28o7 PQMF765S :
(64) E0[@ r@E@5 :S1SK ;KA@\> >UD6OE1S14765S0/OE8MH,< >A@ ,765SF2147 5SF21>CPQ765S0[@ED67 . . . . hE 0. . X. .
(65)
Documents relatifs
Sur le plan projectif, nous montrons que deux fibr´es logarithmiques g´en´eralis´es sont isomorphes si et seulement s’ils sont associ´es au mˆeme ensemble de points ou bien si
L’´ ev` enement C S (j, k, l) correspond au fait que le marcheur partant de la position j passe par la position k sans ˆ etre pass´ e pr´ ec´ edemment par la position l.... Donner
En effet, quel que soit le domaine (connexe) que l’on étudie, le lemme de McGowan (tout comme le lemme classique de minoration des valeurs propres du problème de Neumann en fonction
Dans [C-C1] nous examinions le cas d'une suite de surfaces de Riemann convergeant vers le bord de l'espace de Teichmùller et dans [C-C2] le cas d'une suite de variétés
— Soient (X, g) une variété riemannienne compacte et Y une sous-variété de X de codimension d^2, le spectre (v,,(£))^i du laplacien sur X^X—TLWY avec condition de Neumann au
assure déjà que, pour une fe-compactification lisse X d'un tore défini sur un corps de nombres k. Le corps de classes en donne la valeur exacte, mais, pour la finitude, il suffit
symboles de Christoffel aussi.. ainsi, la base d’une fibration de type I-L est naturellement munie d’une structure affine entière et d’un champ d’endomorphismes J
Si l'ensemble des ye Uo, qui sont tangents à des géodésiques passant par un point du réseau différent de o, n'est pas partout dense dans Uo, il existe un élément de Uo qui n'est