Quantifications naturelles projectivement équivariantes

Texte intégral

(1)

(2) !"#$&%'('*)+("&-,%'./#0&21 ./34576./(98/3 (:4;%<&2&= >"?< 1 .3 *#A@B5C6/./D%. "5#E#$FG&.34'H3JI(.&*: -5K&;5I(5L&(./NM 4@O<P1 &#EQ&./34&R#E5*8/5S&3 : T)+('&&UV5S= ./4'W54WXY3 Y%''&&+= >'& @B53 5L*1 3Z6/.(*I([".3 M \"#$&%'(\;'8F5I#$F^]45L5=`_a5 &.3 I01b./3 cIc:4(&%'3 &"(./4cde&3 %gf 3 '3 &'aXY3 ? ./34a5C6./ 3 'a' &'#ih IM j+4klmY["i&'#E'&%0Iinl./ : o1 ./34Ip5Kde./&#E5L./cEIp5rq%s= '&%s= K:l5L It@Ouv : 3 w"(R'Q: 5 xIy@Bza8/&%3 I{3 "+XY3 5Gh ('|6/.3 I(3i#J@B5%%'./&: 'x3 49hb./3 &" :@B5&1 5FM. }.

(3) Hab T. a4 1 &.H%;': 3 & :4\XY3l5F{kl%C5L&(./ 1 '34. '&& : ;<kl \8/&.&&>"#$F % ./#$#$V;'5F 3 h4 ["%<./ I( ;C5L(&J: Iy@B'&1l5%'c: ' ./h &'"6L5h I %'Ip5

(4) f &XH342:l5423 ^&1 5%K:@O./1b;5S34&2:

(5) N;'&F&(I258/&&5F2&3 2:42de./ %gf ./ 9:N@B./4: :43 #E./ 9: 5 DI%C5:4&: oIp50XY3l5F&kl%5L./r8/;'./#E;<"(XY3 m 6/./<M oD4."D1 .(FQ: 634m/Iy@B'&1l5%' :4+./h "w65Lh I(' y5141 'I(;x53 "& S(M )"F(5h I"3 I(Qklh4&;+%.Ls5 8F. w % / . 4 " { . 3 + ; 4 : Q e d / .. ' % & ( / . 4 . : (. ; !"$#&%' :@B34 D6L5&;'; XH34 &./FT1b./(I )H .#E(5I('WIDI(. 8: 'Qklh &'MY,+@O&1 5% T M : ' ./1b;&5L'3 M&:

(6) N;'&'Y&('I(wQ5L&&#EI(;xGIy@B'&1l5%'T: W./1b;5S34& D : ( (M ;")FI*58/&5LYG"3 : λfv: ' &{;G53Hfv: '&&34a: M M uvI/'"%. Y3|XY3@O(I/N@BgP4"&+1l5: TXY3l5F{kl%C5S(.l5L&3 &'I(I(LMCj+-:@B534" '&#$mI(x&1 5%': ** )H#ih .I(x'T:@O./1b;5S34&T: ( ;"FIQ *&.YT1 5 ("./#E.&1 = '*Zs5FXY3 ?&'1 &;'&F5L./ : Diff(M ) M ,+@B: ;0: iIp,5 +.-/103254 6879/12:4 0<;9+.-=4?>$/1@45/1032 mb(F".: 34?1l5) M,'%./#0& ' A0M Bo6"(' C.K:l54D

(7) E=Fw": 2&;': 3 ("2I(% 8&./3 1b2: 2:

(8) N;'./#$./&1 =4(&#$ I(.H%C53HPr: -I(5K#E5 >"&3 {6L5FM G 0534&3 "|./F?;'&3 : ;$IK%C5?: 3 8/&./341 K1 ".S["%<d 58/&5LYKI.%5I('#E'FK&3 KIp5c6L5"(;'&; M = R 1l5K: '$P&5GL(m wde./&#E+5L&1,(./R) I( ;C5&'ade5L%'./ l5L(&'M uvI(2#$./F&>'&'Y?XY3 KI('2"1l5%'|:4$* )H#ih .I(2< :@O./1 ;'5L&3 "-: ( ;"FI(o&./F?%C54./ (XY3 '#E'F|&./#$./"1 = o^s5F-XY3 &'1 &;'&F5L./ 0:4 P GL(m + 1, R) e./3 :4 &. 5I(8/>'h &$: , sl(m + gM4j+Z:@B534&&*#$.&m4I(R1 &.346/>'&FGXY3@O(INgP("&23 23 4(XY3 ! +.-/1032:4?H 1,79R) 68 /12:4 " 0!IJ@ "KL% 7254?> %M% 032N;9+.-=4?>$/1@45/1032 % M :@O./1 ;'5L&3 " 54O QPtmI?534&3 "-./F|;<3 :4(;!I('-&1 5%' : ( ;"FIa&5 wde./&#E5F2: ' λf : "&;2 :4 Dµfv:4(R ");''-I3 "

(9) f 1l5%'8/&5: 3 ;'25"&.H%(;' M uvIo#$./F&>'&'YIt@O<P(w4%i<It@O3 %{;-:@O3 XY3l5F{kl%C5L&(./ 1 ".S["%<6S#$Fr;'XY3 6L5"p5FLmT:l5 EI($%C5L S. R Ip5 65LI(34 @ 5L1 1l5w('F1l5G!3 A' &'#ih I?:4-6L5I3 "*%"XY3 'M δ =µ−λ ∗. λ. m. λµ. δ. T. m.

(10) .

(11) . . aV1 &'#EG<P#$1 I(|: 0XY3l5LY&kl%5L./V14&.S["'%'{6/'#E'Y;XY3 6L5"p5F& 1 .3 9: 'D./1b;&5L'3 &: ( ;"FI(958/&5LY&34D: 9%s=l5#$1 D: o& "3 "5 ;'&;?:4./ ;-:l54D

(12) EStM 3 "XH34 # 5F 5FCmTI(' &;'&3 I{s5L& : .Y ./3 E6/4./ : A1l5&I$./F &5i^:4!6L5&;';'!1 .3 "6H3 '$:N@B3 4Jw"3 %'&3 &Z1 ".S["%<6Z1 Ip5L&M a;C5Lf #E.( 'mN:l54M ymW] M 9./3 5&"./3 :S[#E.Y&5 XY3 $Ip5rde./&#03 I(!: KI(5ZXY3l5Lf {kl%C5L&(./E1 &. ["%'&6#$FG;XY3 6L5"p5F&1 .3 D: '*./1 ;'5L&3 "*: ( ;"FI :@O./&: "-: 3PJ<&.(1 '34 '&&-<P1 "(#$;-A34I((&5F*3 -%'./ gP4./VI{f ;C5L(&05 &./&"(./NmN: i&I(I0#E5 >&|XY3 KIp5 XY3l5F&kl%5L./ !: ;'1 ' : XY3 -: -Ip5 7 # / ' % IJ@ "KL% 7254?> % : ?Ip5i%. <P./M ]3 R%<&&oIp5L %;'mF:l54

(13) E ymH) MH,%.#0G%./ ["%'&3 5?It@O<P(w' %G:@O3 XY3l5F{kl%C5L&(./ . . . Q : Sδ (M ) → Dλµ (M ). : ;1b4:l5FR:@B34 *%./4 <P(.$I( ;5("9&5 Q.&&./mLXH34 &'5{+l5L&3 &'I(I(9 ./34a"5L&8/3 #$F&a<oXY3 &'"&5La( %s=l5L 8/;?&.3 oIy@B N<o:@O3 A%s=l5 8/gf #E'F1 &.S["'%'&d : -%./ 4<P(./NM ,+@BgP("& %'-:@B34 2II( +.-/103254 6879/12:4 " 0 0J/12:-=@ %#?#&% IJ@ "KL% 72:4?> %M% 032N;9+.-=4?H >$/1@ 45/1032 5;'&;1 "./346;91 5 U M 9./&:4#E5 ?:l5 H|34I(5FI(54../ : %. % <P./ :4 *= ./#E5wf = &= 5: 5&".%'(;Z 3 4J%'Ip5"& 1 ".S["%<{6/ : K%./4 <P(. O e6/.(Q E +1b./3 -3 K: &%34&&./ &3 -I(' %./4 <P(. T: *= ./#E5wf = &= 5: gMY]45-%./ w"3 %'&(./$52;'&;a5L:l514&;*1 I3 s5":c1l5] M_a5 "./3 Ix: 5

(14) EE :l54I(|h 342: |5L'aI0%5: './1b;5 f '3 &T:

(15) N;'&F&(IT58/&5LYT"3 R: +de./"#E'm/8/I./hl5I(&5FT5(4& I+&;'&3 I{s5L& : ! (ESyM q;%'#E#$Fm4:l54*50=4>&Lmb] M _a5 "./3 I

(16) E 3N#E.Y&5i%'./#E#$F98/; ;gf 5I("Ip5$#E;<= .H: ?: ./ ;'i1l5U M 9./&:4#E5 5Lk V: i";"./3 : "iI(?1 &.Sf h I(>'#E : AIy@BgP("& %'Z:@B34 ZXH3 5Fk %C5L&(./ l5L&3 &'I(I 'K1 ".S["%<6#$F ;XY3 {65L&p5LY&|1b./3 G: './1b;5S34&o:4( ;&'F('I(a58/&5LYo&3 a: o%s=l5#$1 : 2' &'3 &a5&h4&5("'M G '"2=4>&-"?"&3 2:l5 *I1 "./I(./48/#$F*: 2&./34'*%'G"%s= '&%s= 'M j+I(I(K'"!I(5A&.#E#$E:4 &576L53HP XY3 951 1 ".de./ :4(&"Fi%'i:4./# 5L( '|:@O(Hf 6/'"8F5L&(./ <aXY3 - .3 o5LI(I(. &./34G:@B5h .&:Z&;'&3 #$h "(>'6#$FCM 54 tma ./3 ^5C6./ ^5l5L(I )H&;\I 1 "./h I>#$ : Iy@BgP4"& %'\:@O3 XY3l5F{kl%C5L&(./ l5L&3 &'I(I($'i1 &.S["'%'&6/'#E'F!;XY3 {65L&p5LY& 1b./3 |: i.1 ;gf 5L&3 "0:

(17) N;'&'Y&('I(|58/("5F0F"r: 4&&;i'\34&(I(5F?Ip5Z= ;'./&E:4 %./4 <P(. a:4 G 5ws5J1 &. ["%'&6M a./34o5C6/. o./hY3c34 |de./"#i3 I2<PYf 1 I(%{01b./3 -I(5rXH3 5Fk %C5L&(./^l5L&3 &'I(Ii'?1 ".S["%<6#$F|;XY3 6L5"p5F& &&#$E: cI(5 %. <P./ : G 5ws5 ./&#E5IV5"&.H%; 34 c%I(5&" 1 &. ["%'&6?:4?%'./ gP(./ I( ;5("M . . !". '. . . $#$. . %. &. . (. $. )(. *.

(18) .

(19) . z 4."A;<./ 4#$FCm(I51 1 5&34 XY3 V%''&&Vde./"#i3 I(ZgP414I((%'&JN@B'" &!:@B534&&XY3 Ip5ode./"#i3 I(9: ./ ;':l5 QP 1 ./34+I(D%5T1 Ip5Lm/53K"#$1 Ip5Sf %'#E'Y01 &>'0: '0: ;'&{6/;'i1l5wI(I21l5L|I?: ( ;"F(5L./ -(F6L5&(5F' 1l5!&51 1b./"E Ip5 %. <P./: G 5ws5 ./"# 5LI(Mxj+1l5w%3 I('mx .3 5C6/./ $#$./F&;ZXY3 AIp5 XY3l5F{kl%C5L&(./ l5S3 "I(IZ'$1 &. ["%'&6#$FrY6L5Sf &(5FQgP4"&R"F'"3 I#$F&FI(5DXH3 5Fk %C5L&(./ sl(m+1, R)fv;XY3 {6L5&(5F <P(w2:l5L *I(a%C5*1 I(5LCM a./34576./ 5 &N51 1b./"&;23 5E4#$FG53Z&;gf &3 I{s5Lx: U M 9./&:4#E5 iXY3 N@B;<s5{Q1l5x%C51l5Lh I(D:49&5 %s= '+Ip5oXY3 '"./ : KIt@O<P(w' %E: KIp5ZXH3 5Fk %C5L&(./^l5L&3 &'I(I!1 &.S["'%'&6/'#E'F0Y6L5"p5F& :l5 I*&{3l5S(. DS. R δ ;'5;'8F5IK3 26L5I3 *%'&&(XY3 LM ]3 !%''"ZIp54%;LmQ 34&(I(5F0I(!";"3 I5L!: Q D'K: QF3ymQ .3 5C6/./ K"./346;J: 5 ES3 de./"#i3 I( gP1 I(%&r '&#$K:@O./1 ;'5L&3 " l5L&3 &'I(i"3 !Ip5A6L5"(;<;r: h 5& M 1b./3 iI(5VXY3l5F{kl%C5L&(./ 5L3 "II( < 1 &. ["%'&6#$FT;XY3 {65L&p5LY&M G <&&de.&#i3 ID8/;' ;&5I(&DIQde./"#i3 I('Q: ./Hf ;' 1 5 9./3l5L&&.3 :S[:l5 3Y< : 5 Y1b./3 I./1b;5S34&: ( ;"FI :@O./&: ": '3PJ'*&&./M , 1 "./h I>#$ : EIt@O3 (%'&;E: Ip5AXY3l5LY&kl%5L./ 5L3 "II($1 &.S["'%'&6/gf #E'FQ;XY3 {6L5&(5FR@ 5C6L5{x1l5LQ %'./&9;';D5h .&: ;LSM Bi&576L5{xXY3 9Ip5aXY3l5Lf {kl%C5L&(./ sl(m + 1, R)fv;XY3 {6L5&(5F*;'5R3 4(XY3 G:l5 RI('R"&3l5L./ + . %"XY3 'M G 'Ip5r@O(#$1 I(XY3l5*&./34&'de./o1l5oIt@O3 (%'&;|: !I(5EXY3l5F&kl%5L./ l5L&3 &'I(I(!14&.S["'%'{6/'#E'Y0;XY3 {65L&p5LY&M 5L M Qtm ./34|1 "./346./ 0XY3 Lm %./F&5&#$F-Ip52XY3l5F{kl%C5L&(./ sl(m + 1, R)f ;XY3 6L5"p5F&m/I(5?XY3l5F&k4f %C5L&(./El5L&3 &'I(I(a'*14&.S["'%'{6/'#E'Y;'XY3 6L5"p5Fa@O"1 5D3 XY3 !# #$ :l5 ZI &{3l5S(. ./f %"XY3 'M a./34A";C5I(&. r%'Ip5 34I(5F I(5 = ;'./&0: '%. <P./ 2: G 5L"s5L '2' 1l5"&(%'3 I(I(?4&342: )HI XY3 1 '34 '&&-(F".:43 D&&>'G 5L3 "II(#$F!1l5L"*: -%'I(I(gfv%'tM j+4klmY:l5L

(20) E=PymF ./3495 5(I )H&. I(52XH34"&(./E: GIt@O<P(w4%G: aXY3l5Lf {kl%C5L&(./4|l5S3 "I(I|'!1 ".S["%<{6/#$FK;'XH346L5&(5F'i1 .3 0: '!./1b;5 f '3 & : ( ;"FI( 58/&&5Fx"3 x: 'x%s=l5L#E1 : & "3 "+5"h &5&'M G '" XY3 w(.K576L5{x:4;

(21) [&;'&;*5hb./&:4;D1l5LT] M/_a5 "./3 I4: 5

(22) E 4!34&(I(5FWI(' %./4 <P(. : *= ./#E5s5L : (NXH34R4./3 It@B5hb./&: . 8/%T53HP-%./4 <P(. : G 5L"s5Lc1 &. ["%'&6M,T@B5C6L5Fs58/|: 04."i#E;<= .H: ?w2XY3@B'I(I0: ./4 I(3 r3 4$8; ;'5I(5S(.V: &%<!: '2de./&#i34I(oXH34Q1 '346/'F <&!;'%"' :l5 0I( %'./FgP& : rI(5VXY3l5F{kl%C5L&(./ 1 &. ["%'&6#$FK;XY3 {65L&p5LY& "3 :l54aIG&&3l5L&(./ *4./fv%'&{(XY3 '' gml&(#$1 I#$F1l5Ip5!"3 h "&&34./ R : 9: ( ;"F(5L./ R(F6L5&(5F1l5D&51 1b./"D03 o%./ 4<P(./ :4 G 5"5 53P : ;"6;!1 5"&(II('MW,5V#E;<= .H: E#E<K53 "&9' <P"8/3 3 I('\: {f &'%'F"aIy@BgP4"& %'a:@O3 aXY3l5F{kl%C5L&(./ sl(m + 1, R)f ;XY3 {65L&p5LY&*&3 'Iy@BgP4"& %': 2Ip5|XH3 5Fk %C5L&(./r 5L3 "II(o<*1 ".S["%<6#$F M = R ;XY3 {65L&p5LY&2%./"&'&1b./ :l5F&?"3 3 -6L5&;';-5L&h {5L(&LM . . )(. . . . '. . (. . m. m.

(23) .

(24) . 54oI(?1 "#$(%s=l5L1 &&?: 0%'|&5C65L(Itm4./c5L1 1 'I(I2I(a .&(./ *de./Hf :l5#$Fs5LI( ;%'"5&R3 4*h ./4 *%./#$1 ";= ' &./!: *%<T./346H5L8/M G .&(./ W&.YQ .s5L#E#$F &I(5L{6/'Q53P?klh4&;+14&(4%(1 53P mS53P0./1 ;'5L&3 " : ( ;"FI2'| )H#ihb./I('05"&.H%;?'053P : ;'&6;'0%.76L5"p5F'MW) .3 ?kf ('m ./ 5141 'I(I(?Ip5$1 .&&(./A: 3 1 "./h I(>'#E?: !I(5EXY3l5F{kl%C5S(.cl5L&3 &'I(I( 1 &. ["%'&6#$FG(F6L5&(5FM 54iI(E&%'./ : %s=l514"m ./\5L1 1 'I(IEIi .L(. |:4rhl5" &I(5L{6/' 53PV%./4 <P(. -: G 5ws5^'253Pck h &;'?:4 G 5"5MNza1 &>'-5C6/./: ;'k 3 7 " 030 % 4 " 0 % /1@ 2 /10!'E&./348/;' ;&5I({;Lm/./$<P1 I(XY3 %GXH3N@B'"D34 2:@ -372:-=@ % IJ@ "KL% 7254?> % 'a./V:4;#$./F&?XY3 0Iy@B' &'#ih I|: 'aw"3 %'&3 &'1 &.Sf ["%<6?w0' %.&&'&1b./ :l5L %Eh (3 46/.HXY3 E576%EIt@O "#ih IK: ?%I(5&" 1 &. ["%'&6V:4 %. <P./ JI((4;C5&Z&5 A&./&"(./NM B2 #E.Y&& "3 & XY3@ 3 :4 %'c%I(5&"'ma./ 1 '34c5L&&.H%A: #E5 >&^l5S3 "I(I 34 %./4 <P(.E: G 5ws5$1 &. ["%'&6m51 1bI(;' 7 " 030 % 4 " 0DIJ@ "KL% 72:4?> % 0 " @ / #&% 0@B534"1l5"mFI('Tde. %'&(./ T;'XH346L5&(5F'R"3 TI(*klh ";a:4R&1b>"R:@O./&: " m 1 '346/'F '"D&'I(<6/;'Q!:4xde./4%'./ W;XY3 {65L&p5LY& &3 Q34 9w&34%'34& 1 &. ["%'&6M B 1b34d 5L(&058/a"3 2%'de./ %'&(./4o3 !.1 ;'5L&(./V51 1bI(;' 14 ;@ % 032:45/1254 " 0 4?03>$/1@ 45/1032 % XY3@O./ %./ w"3 0A1l5L"-: EI(5J%'./ gP(./ : G 5"5Z ./"# 5IM 54 I(V&&./&(>'#EJ%s=l514"mD. 5l5

(25) I )"cIp5 XH34"&(./ : Iy@BgP("& %' :@O3 XY3l5LY&kl%5L./ 5L3 "II(^1 &. ["%'&6#$FcY6L5"p5F& :l5L VI %5 :@O./1 ;'5L&3 "A:4( ;&'F('I(Z58(&&5FJF" : ' &{;'M B2 #E.Y&&^XY3@O3 'I(I(9XH3 5Fk %C5L&(./!gP4"&a"l'T&'3 I('#E'F"lIp52XY3l5F{kl%C5S(.!%./"&"1 .f :l5FKgP4"&$: 5 -I(!%5|14Ip5LmN : ./ 5F|34 !de./&#i34I(igP414I((%'&! '&#$G:4aI(5?%'./ gP4./ :4 G 5ws5E ./"# 5IM G <&&Gde./"#i3 IG'"%./4""3 & 1l5w(x: *Ip5:

(26) N;'&'Y&p5L&(./0(F6L5&(5Fm ∇ m/<T:@O3 K./1 ;'5L&3 +XY3 .3 5C6/./ |51 1bI; Div M,5Z%. t %: %'KF"$de./"#i3 I

(27) 14Ip5L& !<

(28) %.3 &hb 5rI(3 &xIt@O./ &3 h4"{3 ∇ rI(5r: ;"6L5L./c%'. 6L5"p5F&i1l52&51 1b./"- I(5 %./4 <P(.|%5 ./ XY3 T1 Ip5L&+: R < Div *I(5:46/'&8/' %R%.76L5"p5FT1l5L 5141 ./w-r%<&&$!# #$0%'./ gP(./M,+@O<P45%<&3 : !: !I(5Ede./"#i3 I(

(29) %.3 &hb &J: ;#$./F"A' #$./F5LY$XY3 A%II(gfv%D&54"de./"#EZ: '$de./4%'./ $;XY3 f 6L5&(5F0 : 'ide./ %'&(./4i;XY3 {65L&p5LY&'MQ) .3 !%rd 5&Lm ./ %5I(%'3 I( 53 1 &;5Ip5Lh I(*Iy@B;'%C5w9?It@O;XY3 {65L&p5L %*: 'de./4%'./ ∇ f ' Div S mYS. R f ' "1 ";&'FFo&"1 '%'&6/'#E'Fo3 4?:4 ";-<a34A* )#0h ./IM S ,ZXH3 5L"(>#$Z%s=l51 {"J"A"&3 Z:l5 KI(Z1 "./I(. 8/#$FK: 3 %s=l514" 1 &;'%;': FCM B )Z5l5L(I )H&2Iy@BgP4"& %'?:N@B3 4?XY3l5LY&kl%5L./ l5L&3 &'I(I1 &.Sf ["%<6#$FZ(F6L5"p5FV:l5 I %C5 :@O./1b;&5L'3 &Z:

(30) N;'&F&(I 58/&&5F F"0%s=l5L#E1 a: -4&34&-5"h &5&'M Bc5L1 1 'I(I./34o:@B5h .&:V'V: ;gf s5I(DI*./3(I*'GIp5i#$;'&= .H: -34I((";: 5 GI(2%C5L: &-: ?Ip5iXY3l5F&kl%5L./ f ;XY3 {65L&p5LY&: 5 R M/,x )H#ihb./I('Q'+I ./1 ;'5L&3 "Q:

(31) N;gf sl(m+1, R) &'Y&('I(x;<s5FT: '+&'1 &;'&F5L./ +: sl(m+1, R) mL./!1b34+%'./ w&34(&9I( . . . . . ω. ω.

(32) . . . . . . . ω. m. ω. ωl s. m. ωl.

(33) .

(34) . ./1b;5S34&Q: G 5&#Ex5L&&.H%; ' "3 Q%'+"1l5%'MF,5oXY3l5LY&kl%5L./ F6/.(5I(./"3 |6%'&3 1 &.1 &R: C C "C3 W3 |6%'&3 1 &.1 &R: C :4#! #$ 6L5I('3 x1 "./1 "D'+XY3 5o1 .3 Q )H#ihb./I91 "( %'(1l5IHI( )H#ihb./I(: *: ;'1l5"M/) 5 5l5I./8/mb./ :4;'kl {?:l5 ?Ip5 &{3l5S(. %./3 "h $: ' v./1 ;'5L&3 "-: G 5& f #E a58(&&5F*&34*: Dde./4%'./ D;'XY3 6L5"p5F'M4,W5iXY3l5LY&kl%5L./ %.3 &hb w%'./ w&34: G#E5 (>'&*gP 5L%''#E'F9"(#$(Ip5L(&2Ip52XY3l5F{kl%C5S(.$14Ip5L& ' ./0#$./F"XY3@B5 &ym7I(I+5L "de./"#ER: ' de./ %<./ W;XY3 6L5"p5F&W0:4 de./ %<(. -;XY3 6L5"p5F&M,5Z%./4%I(34&(. "3 &1 "l5LY& '"05I(./"2XY3 :l5L I(' &&3l5L&(./ ./f %&{XH34m9(IogP("&^3 ^XY3l5F{kl%C5L&(./ %'./3 &hb &2< &'3 I('#E'YD&I(50XY3l5F{kl%C5L&(./ %./"&"1 . :l5F-1 I(5La<P(wLm I('9de./&#03 I(' : ./ 5FI(XY3l5F&kl%5L./ *%.3 &hb@O./h4&l5LYaJ&34h "&3 5Fa: 5 *I(' de./&#03 I('a: . l5FIaXH3 5Fk %C5L&(./c14Ip5L&GIp5E:4( ;&'Fp5S(.AY6L5"p5F& 53PZ:4;&{6/;'G1l5"&('I(I('M 54DI(G% XY3 (>'#EG%s=l51 {"mH./r: ./4 3 Gde./&#03 I(G<P1 I(%{G1 .3 9I(5 XY3l5F{kl%C5L&(./2 5L3 "II(Q1 ".S["%<{6/#$F ;XY3 {65L&p5LY&+: ./F(I"WXY3 '"./ 53J%s=l51 {&4Mb)W./3 %-d 5("ml./J: ;<6/I./1 1b|IG./1b;5S34& ∇ ' Div ^'&#$2:@O./1b;5S34&?l5L34&I2&3 -I(5 6L5"(;<;!: Khl5" M ' 34&(I(5LY : ./3(ID<P1b./&;'a: 5 D FyM j+4klm*: 5 I(V&{P>#$c%s=l51 {"mD./ #$./F"cXY3 Îp5 XY3l5F&kl%5L./ l5L&3 &'I(I(!14&.S["'%'{6/'#E'Y|;XY3 {6L5&(5FK: 3 %s= 51 &&JN@B'"i1 5|34 (XY3 Lm # '#E+: 5 I('W"&3l5L./ .fv%'&&(XY3 'mC'|%'./ "&&3 5F : 51 14I((%5L./ O l5L&3 &'I(I('D1 ".S["%<6#$FG;XY3 {65L&p5LY&*'Y&&-"1l5%'G: - )H#ihb./I('M B &;5I(&R%I(58/ L%D5L3 2 % 0 % -=@ %!%$# XY3 *Iy@B./01 '34+(F&&.H: 3 ("R&&>'+l5L3Hf &'I(I('#E'F1l5I(Th (5(:4I(5%'./3 "h 3 &R: RIp5*%. <P./0: G 5ws5? ./"# 5IM. . . ωl s. ωl.

(35) . o 4 . Gq '#E'&%'(#$F& uvF&.H: 3 %<./ G =l51 {" M a.L(. Dde./ :l5L#E'Y5I(' M nxh &;'*1 & %1l53P M B1b;5S34&: ( ;"FID'G )H#ihb./I 4M o;'&{6/;'a%'.765L&p5LY& M )W./&{(. : 3A14&./h I>#$ G =l51 {" M nxh &;'G<%./4 <P(. a:4 G 5L"s5L M G ./4 <P(. : G 5"5 M ,28/"./3 1b?14&.S["'%'{dx<G&./A5I(8>h "2: ?, 4M ]H&&3 %<3 "G1 &. ["%'&6 M G ./4 <P(. 1 &.S["'%'&6/' M o

(36) N;'&'Y&p5L&(./ (F6L5&(5F M qI>'6/'#E'Fa:4*de./ %<(. *;XY3 {65L&p5LY& G =l51 {" 4M ,-%C5: 'a: ' &{;' M ,+@O./1 ;'5L&3 *:4?:46/'&8/' % M ,2&;'&3 I{s5LG1 "( %'(1l5LI G =l51 {" M ,-%C58/;' ;&5I M ,2%C5LG14Ip5L M B34&(I(9: 3A%5%./34&hb 4M G ./4""3 %'&(./Z: -I(5KXY3l5LY&kl%5L./ G =l51 {" M a4-de.&#i3 I2<P1 I(%{ M ,' &'3 G: -: ;'de./"# 5S(. M G 5I%3 IN: 3J& "3 G: -: ;<de./&#E5L./ <a:4 Div 4M o;<6/'I(./1 1b#$F&a: M ,5ide./&#03 I(gP1 I(%& ∇ G =l51 {" M G./f 3 (%'&; :4cXY3l5F&kl%5L./ V 5L3 "II( 1 ".S["%<6#$Fa(F6L5&(5F M ,' &'3 G: )I. .

(37) . . ωl. . ωl. . . . . . . . .

(38) ./4""3 %'&(./ :@B51 14I((%5L./ i 5L3 "II(!14&.S["'%'{6/'#E'Y ( F6L5"p5F' 9(h4I((.8/514= ( . M.

(39) . . . G. .

(40)

(41) + T o'T . R . a.3 2<P1b./&./42:l5 2%'!%s=l51 {&0I(o ../ a ;%'&&5("?rIp5Eh ./4 %./#$1 ";= ' &./ : !%<?./346H58M 5L ./3-I(0&5C6L5(Iymb4./3 -: ;"(8/4&. 1l5 M 3 6L5&;'&;rXY3 A4./3 K&3 141 ./""./ C mx&;1 5&;'mT%./4 <PJ'E hl5" : ;' ./#ih &5h IM a./340&341 1 .&"./ !: E1 I(340XY3 I(5A: #E' &./ m : %<&26L5"(;'&;2wG&./3S["./34&"&&(%<'#E'Y"3 1b;&3 "0 M . ∞. . E "!# $%&!')(+*. 54W%'+%s=l51 {"mC .3 .&&. P m534dH#$F./2<P1 I(%'x: 3-%./F&5("m 3 Ak h &;?1 & %1l5I: |8/&.3 1 -: |"&&3 %<3 " G &34a3 4|6L5&;'&; M M uvI&'5 ;8F5LI(#$F!"./3

(42) fvF& : 3 XY3 π :4;&8/ '5 Ip5V14&.S["'%'./ P → M ' R It@B5%'&(./ : G &3 P M jQs5F*:4./ ;34 o6L5&;'&; m3 r<P'#E1 Iode./ : 5#E'Fs5Ib: oklh4&;1 "(f %1l5IH&3 M w+I( 6 @'; %' @ % MI),@ %' - " @ 1@ % k . .; P M m/: ./FTIQ1b./(F& &./F2I"1b>"-:@O./&: " : '^&./3421b./Yo: M)W./3 2&51 1bItmb3 &'1 >'&D:@O./": & k K3 !1b./k(F xM: M '"TI["'Q:@B.&: M" k !Iy@B./"(8/ : :N@B3 V: ( ;./#$./"1 = ("#E2I.%5I: :l5L 51 1 I(XY3l5F E"3 Mb, R klh "; P M '"G3 klh ";21 & %1l5I : 2R8/&.3 1 2:4M?w"3 %'&3 & G m4I(28/".x/3 1b : ["<*:@O./": & k 'ZIt@O./&8/( i: R : '*: ( ;./#$./"1 = ("#E'9I.%53P : : 5 oI3 {f O# '#E51 1 I(XY3l5F E&34 4Mb,+@ 5L%'./J:@B34c;I;#$F j (g) : R &3 3 J&'1 >'& : P M '"G: ./4 ;?: -Ip5i#E5 >&2"3 6L5F&0 / G j (f ) . . . m. . k. m. k. k m. m. m. k m. k 0. k 0. k. j0k (f ).j0k (g) = j0k (f ◦ g).. +j J%'..&: ./4 ;'ml& j (f ) &'&1M j (f )9'""1 ";&'F;?1l5 (f , f ) ""1M '" e&'&1M w&'1 &;'&F&;1l5L e&'&1M gm ) ) (f , f , f j (g) j (g) (g ) (g , g 5I(.& i. i j. i jk. 1 0. 1 0. 2 0. 2 0. i. i j. i j. i j. i jk. (f i , fji ).(gji ) = (f i , fki gjk ) i i r i r s (resp. (f i , fji , fjk ).(gji , gjk ) = (f i , fki gjk , fri gjk + frs gj gk )).. ] ' &./F-: '3Pc6L5"(;'&;2<-& waM:4;'kl N-: -Ip5i#E5 (>'&2"3 6L5F0/. φ ∈ C ∞ (M, N ). mIy@ 5141 I(%C5L&(./. P k φ : P k M → P k N : j0k f 7→ j0k (φ ◦ f ).. 1. P kφ.

(43) }

(44) } ,0klh4&;!14&(4%(1 53P 5: #$'"F!:4!%s=l5#$1 i:4r6/%<'3 &K1l5w%3 I("M uvIT@B58/?: ' 7 / I % > % 72 % - @ L" 0 / M% 032 /1- . XY3 R&./F05&".%'(;'i53P ;I;#$F*: -Iy@ 5I8/>h4&2: ?,( g : G M ] h ∈ g mID7 / I L" 0 / M% 032 / # / ' " 74 ; -'"TI(9%s=l5#$1!: D6/'%''3 & &3 P 4.; h 'G: ;'k Z./341b./Y1 5 . . . . . .

(45) . . . . . .

(46) . . ∗. h∗u =. d Rexp(th) u|t=0 . dt. qa5L1 1 'I(./4253 "&TIp5r .L(. : K&;:43 %'&(./^:@O3 ^klh &;K1 "( %'(1l5LItM]H./('Y 3 klh ";?14&(4%(1 5IN: -8/"./3 1b2: -"&&3 %<3 " G ' P 34Jklh4&;-1 & %1l5I : P 8/&.3 1 2:4?w"3 %'&3 & G Ma -#E#$"&(. χ : P : 5 P 6;&{k 5F 0. 0. 0. .SR Φ ' "34 i(#$#$χ(u.g) &"(./Z=<χ(u).Φ(g) 3 c=4./#E.#E∀u./"1 ∈= P,&#$∀g2: ∈0G, 8/"./3 1b'F& G ' w 2 . 3. O@ ; 1-7254 " 0 1- 1@ " - I % % 2:@ -372:-=@ % . : V 3 l k h4&;014&(4%(1 5I P G " G &'%./346H"|Iy@ 5141 I(%C5L&(./E(: 'Y&&;2&3 Ip5ih 5& M M G χ ] w-3 f klh ";i1 "( %'(1l5IW' 3 i6L5&;'&;i#i3 0:@O3 K5L%'./ !8F53 %sP= ρ : GGm 5LI(./"DI(8 6 @ ; / ' " 74 V; % 6 @ % 2 I % V . .&; . 0. . . 0. . waI2XY3 .F: . 0. . P × V, l 1 L 5 *I p ! 5 &'Ip5L&(./r:N@B;'XH346L5I(' % ∼ : ;<kl ?1 5 P ×V G. , 5i%Ip5L&&2:N@B3 Z%'./3 1 I (u, v) 1 ./34D%<&2"I(5L./r:@O;XY3 {65LI(4%2w .&; M,k h &; '"3 klh ";2&3 iklh " )H1 V m4Ip5i14&.S["'%'./ [u, :@O3 v] -%I(5&" [u,Pv]×;<s5VFG: ./ 4;-1l5 π(u)MM uvIY<P(wR: R ./#ih "3P2<P#$1 I(' :4R&I(klh4&;'M) 5&#$F%'3PYfv%'tm ./?1 '34 %{9I(klh ";2: : ' &{;'m F (M ) m4'I(k h &; S (M ) XY3 53 "./F3 Z /I (#$1 ."s5LY9:l5 I(5K"3 &Ml,-1 "#$(*"-: ;%'&:4?I(5!# 54(>"&346L5F / (u, v) ∼ (u0 , v 0 ) ⇔ ∃g ∈ G tel que (u0 , v 0 ) = (Rg (u), ρ(g −1 )(v)). G. . k δ. λ. Fλ (M ) = P 1 M ×GL(m,R) ∆λ (Rm ),. .SRZIp5i"1 ";&'Fs5L&(./ ρ :4. GL(m, R). &34. ∆λ (Rm ). '"G: ./ 4;-1l5. ,2: 3P>#$?'@B: F&kl-XY3l5Fa!I(3 ρ(g)A = | det g|−λ A,. .SR. ∀g ∈ GL(m, R), ∀A ∈ ∆λ (Rm ).. P 1 M ×GL(m,R) Sδk (Rm ),. wiIy@B'&1l5%' S R ⊗ ∆ (R ) '0.SR I(5J"1 ";"Fs5L&(./ ρ : &3 S (R ) "G: . ;-1l5L. Sδk (Rm ) GL(m, R). k. k δ. m. δ. m. m. & g ∈ GL(m, R) m A ∈ ∆ (R ) ' X , . . . X ∈ R M B .&&5 It@O"1l5%+:4&'%'./ : F (M ) < S (M ) Iy@B'&1l5%'+: "%'&(./4: . ρ(g) A ⊗ X1 ∨ . . . ∨ Xk = | det g|−δ A ⊗ gX1 ∨ . . . ∨ gXk , δ. λ. m. 1. k δ. k. m. M. Fλ (M ) Sδk (M ).

(47) }

(48) }v} 54aI(5$"3 &ml ./34a: ;'&8/ "./ G1l5 ""1M *I(2de./ %<'3 G5&".Lf %(5FGK34 265L&(;<; M : -: (#$4&(. m I(Ek h &;|5&"E.%'(; . . .

(49) . 1. 2. E1 (M ) = P 1 M ×GL(m,R) V1 ,. "&1NM E (M ) = P M × <mF.1R ' V "./F:4T&'1 &;'&F5L./ V V : -: (#$4&(. 9kl : GL(m, R) M ] '"*34 ftklh ";o1 "( %'(1l5I "3 ' 3 rk h &;25L&&.H%; : ak h &PG )1b V mYGI ) 5|h4 ["%<./ %5 ./ MXY3 o'PY&×&aI(V&'%'./ : P × VP '+IQ;'I(;'#E'Y& f : C (P, V ) XY3 "./F Gfv;XY3 {6L5&(5FmStMOMLXY3 5S(wde./F. 2. 1. GL(m,R). 2. 1. 2. G. G. ∞. f (Rg (u)) = ρ(g −1 )f (u),. ∀g ∈ G, ∀u ∈ P.. ] wD3 o&%<./ : mH5I./&RIp52de./ %<(.EI3 %./"&"1 . :l5F*1l5L %<&T-h ["%<(. "GIy@B3 4(XYP3 ×2de./V %<(. f ∈ C (P, V ) 6;&{k 5F G. ∞. T. T (π(u)) = [u, fT (u)].. zo &ym+1l5$Ip5^&3 {V4./3 4V: " 8/3 '&./4$14I(3 EI($&'%'&(./ E: $klh "; 5&".%'(;'G%'./ "(: ;'&;: 'Gde. %'&(./ *;'XY3 6L5"p5F'*%./"&'&1b./ :l5F&M '"a3 nx P4. 3 2de./(1b./3 D&./34'aI(5! .s5S(.J"3 6L5F&"/l" 6L5&;';G#i3 4(G:@B34 25%<(. 0:4&./{ R :@B34r8/&.3 1bo: o, (P,G R) '9& (V, ρ) w|34 $6L5"(;<;!#i3 4(!:@O3 E5%'&(./ Z8F53 %s=4 : m5LI(./"2It@O "#ih I : de. %'&(./ Gfv;XY3 {6L5&(5Fa: !%I(5&" C 'Yρ&& (P,G R) < (V, ρ) "5 .&; ∞. ,od 5{9: o o%'./ &: ;"DXY3 2:49klh4&;5L&&.H%;9 6L504./3 D1b

(50) f #E<&&&R: 1b./346./(W: ;<kl (W3 : ;"6;D: ,(R&3 I(' "P%'&(./M4x: %Wklh "; :l5 xIp5o: &'%'./i: '+%s=l5#$1 Q: D6%<34&T&3 M MY] X 5L1 1l5w('F+It@OHf &'#ih I( Vect(M ) : 'a%s= 5#E14G:4-6%<34& C &34 M m 5I(./"*(I5L: #E<3 &'I(>'6#$F%.#E1 I' :l5 M ]H '"3 2"1 ";"Fs5Sf ./A:4 GL(m, R) mbIp5KXde./ %'&(./AVect(P ;XY3 {65L&Mp5LY)&|%'./(V,&"ρ)"1 ./4:l5F- I(5$: ;'&{6/; : r, 1l5|5141 ./wi X :@O3 r"%'&(./ T :4 P M × '"!1l5L : ;'k &(./EIp5|:4;&{6/;': I(5|de./ %<(. ;XY3 {6L5&(5FoI3 %./"&'&1b./ V:l5F:l5L Ip52: &%<./K: X M]l./ "a:4./ G: R%.H./": ./ ;''m./$1b34%./4&(:4;&' $3 $1 ./F : I*&1b>"a%5 ./4(XY3 %./"&"1 . :l5F9?%'G* )Hwf >'#E?: 0%'..x&: ./4 M;'MN] .c51 1bII( f Ip5!(xde./, δ%'&)(./A5"&.H%p5LYE3 A1 .(F : I('%'./#$1 ./&5F': :l549%G&'1 >'&LmI(5-de./4%'./E5&".%'p5F x I('D%./M#$1b./5F&: (L T ) T:l5 *I&'1 >'& (x , δ ) wG5I(.&D: . ;-1l5Lx C ∞ (P, V )G .. 1. ∞. 1. C. 1. GL(m,R). C. i. i j. x. X. .SR DX wTIp5#E5L"(%'D: I(.H%C5Io: X M. i. x. X.f − ρ∗ (DX)f, gl(m, R). i j. :4T: ;"6;T1l5L"II(x: *It@O<P1 "&"(./.

(51) 5} T a.L./ -XY3 $& '"|3 :4( ;./#$./&14= ("#E|F"E: '3P^6L5&;'&; < mN Y./E: ;'k It@B5%'&(φ./$: φ "3 934 ;I;#$F [u, v] : 3$klh &;o5&".%'(; EM(M ) %./#$#$&3 { / T. . . . . . . . . . .

(52) . . . 1. φ.[u, v] := [P 1 φ(u), v].. Gq 51 1bI(. *53 "&I(5i .&(./r:4de./&#$o: M7 " 030 % 4 " 0E&3 *34rklh &;-1 "(f %1l5Itm ./3 7 " 030 % 4 " 0 - =@ %' /1030 a4 f L" @ M% % 7 " 030 % 4 " 0 &3 $3 f klh ";1 & %1l5I P '"3 f de./"#E Υ &3 P 26L5I('3 &T:l54Iy@ 5I8/>h4&*: G ,( g : G 5L&(wd 5(&5F 1 ./34D./34 h ∈ g Υ(h ) = h 1b./3 D.34 a ∈ G M R Υ = Ad(a ) ◦ Υ . . . . ∗. . . . . . . . ∗ a. −1. + !

(53) ' (

(54) $ . ] wA34 6L5"(;<; 'J" m &./FAI(rk h &;'J6%'&.Lf &I($l5LM&3 &'I(E&3 M :4Aklh "r E)H1b(M r")&E1b%<(M {6/) V ' V :4./Fr(IGw XY3 w(. 53 1l5&58/&51 = A1 &;'%;:4FCm*3 " I3;@ /12 % - @ 14 ;@ % 032:4 %# D 'Y&& It@O"1l5% Γ (E (M )) : '!&%<./ i: %Ip5L&& C : E (M ) '!Iy@B'&1l5L% '"T3 5L1 1 I(%C5L&(./ fvI( ;5(": .YQIt@O<P1 &'&"(./KI(.H%C5LI(R:l5 Γ 3 !(E ./346(M wT)): *%C5w*<+: D"6Hp5LI((&R5L./ &#i3 I{s5 ;' : E (M ) < E (M ) wa:4?I(50de./&#$ 1. 2. 1. ∞. ∞. ∞. 1. 2. 1. 2. 1. (Df )(x) =. X. 2. . αm f )(x)), Cα (x)((∂1α1 . . . ∂m. .SR k ∈ N m |α| = P α < C '"T3 4a5L1 1 I(%C5L&(./|: %I(5&" C 5"&(8l5F 0%s=l5XY3 -1b./(F9: 2It@O./346"D: %C5L"234 -51 1 I(%5L./EI( ;5("a:4 V :l5L 1b./3 &./34-#i3 I{ fv(4: (%' α M,!l5L34&I k 5L1 1l5&5("5F-:l54?Iy@BgP1 &'wf V &./ 9'"GIt@ " @ 1@ % % 14 ;@ % 032:45/1254 " 0 ./3Z.34&#E1 I#$F*Iy@B./": &2:4 D M ,+@B' &'#ih I(-: -%'./1b;5S34&: ( ;"FI*"G .L; |α|≤k. i. i. ∞. α. 1. 2. . . . D(E1 (M ), E2 (M )),. /. 3Z1 I(34*&(#$1 I#$F D(M ) M ,+@B.&: " : EIt@O<P1 &'&"(./\I.%5I(K:@O3 \.1 ;'5L&3 |: ( ;"FIT;'5F!(f : ;1b4:l5F: ?Ip5K%C5L"|%s= ./&(Lml%<"1l5%'0woklI{";|1l5LaIy@B./": &-: ?:

(55) N;gf &'Y&p5L&(./ / D(M ) =. [. D k (M ).. +, @B' &'#ih I( (M ), E (M )) .3 S(M )G: ' $!"$#&%' IJ@ 4?0 74I/1- . ./3 &#E1 I#$FQS(E :4 5&".%'(;%<R&1 5%:@O./1b;5S34&T: ( ;"FI w Iy@B'&1l5L%T8/5L: 3 ;R!5&""$.#&%'%' (.;R*%<&+klI{&5L./M B|55( "/1b./3 ./3 k ∈ N k∈N. 1. 2. . 0. S k (M ) = D k (M )/D k−1 (M ),.

(56) T. S 0 (M ). . . . wG(&.#E./"1 = -. . . . . D 0 (M ). '. S(M ) =. G. =l5XY3 . . . }.

(57) . . M. . S k (M ).. '"G("./#$./&1 =42Zs5FGXY3@B'&1l5L%-6/%<./"('I$Iy@B' &'#ih I( S (M ) k∈N. k. Γ∞ (E1∗ (M ) ⊗ E2 (M ) ⊗ S k T M ). : +%s= 5#E14Q: 9& "3 "R*)H#E;<&XY3 +%'./F5C6L5"p5Fx: D: '8/&; k o6L5I3 " :l5 *IG51 14I((%5L./ I((4;C5&F"|IDklh4&* )H1b V ' V : E (M ) ' E (M ) M ] "o3 V.1 ;'5L&3 :

(58) N;'&F&(I:@O./": & m 5$1l5"&(-: ?1 I(34G= 534 ./&:4&rD5J3 r"(8/4kl%5L./ F& &>'XY3 M )+I(3 |1 &;k'%(";#$FCm I('|'&#$|: 1 I(34 =l534J: '8/&;^: Ît@O<P1 ""&(. I.%5I(c:4 D "./FAIy@BgP1 &'&&./ I(.H%C5I :@O3 Z )H#ihb./I(:4: 8&; k m4'It@B51 14I((%5L./$5&".%'p5FG D %2* )H#ih .I('" 51 1bI; IJ@ 4?0 74I/ # ' wx4.;' MSj+I(I+wx: . ;R|%.H./&:4./ ;' σ I(.H%C5ID1 5!"$#&% 1. 2. 1. 2. σ(. X. αm Cα ∂1α1 . . . ∂m )=. X. αm Cα ⊗ ∂1α1 ∨ . . . ∨ ∂m ,. .SR ∂ , . . . , ∂ '" Ip5*hl5"I.%5I(+:43?klh4&;Ts5 8FCMS, 5%'&(./4W:4 Vect(M ) ' Diff(M ) &3 D(M ) "./FD(4: 3 &R1l5RI('D5%<(. %'./&""1 ./4:l5F'D&3 I('*&%<./ : E (M ) ' E (M ) / .V5i.34:@B5h .&: |α|≤k. 1. |α|=k. m. 1. 2. (φ.D)(ϕ) = φ.D(φ−1 .(ϕ)), ∀ϕ ∈ Γ∞ (E1 (M )), D ∈ D(M ), φ ∈ Diff(M ).. j+\: ( ;"F(5F?%'" 5L%'./m./ ./h4&(F0It@O<P1 &'&"(./ : I(5J: ;'&{6/; : ,(:@O3 A./1 ;'5L&3 *:4( ;&'F('I /. 1 .3 D./34 ϕ ∈ Γ (E (M )), D ∈ D(M ), X ∈ Vect(M ) M oÎp5 #O '#Ec#E5 >"m9I 5L%'./ : &34 I( "%<(. : Diff(M )3 ("FK3 Z5%<./ : < ' K " . 3 !I ( ' i & ' ' % / .. K . : ( . 4 : E (M ) &3 E (M ))H#ihb./I('MHzo3 {6/53r(kl T&M 9I ( ;"(#E5ItmFI(5|:

(59) N;'&F&p5L&(./$: Diff(M %<&?5%<)./Z1 '&#$': -%C5LI(%34I(*I(5i: ;"6;?: ?,2:@O3 A*)H#ih .I(2:l54aI(5 : ("%<(. :@B34A%s=l5L#E1J: 26/'%''3 & X XH34I(%'./ XY3 LM ,J"1l5%' S(M ) < D(M ) &.YA("./#$./&1 =4 s5FJXY3@O&1 5%' 6/'%'.&('I('M ./3'de.(-(I2 EIK&./F|1l5? 5F?XH34 &'1 &;'&'Y5L./ ?: 3 8/&.3 1 2:4: ( ;.#E./"1 = &#$DI.%53P M j+ %./4&;XY3 ' %Lm+(IT@O<P(wr1 50: EXY3l5F{kl%C5L&(./ %C5 . (XY3 El5L3Hf &'I(I(-5L3A" aS. RA'I(I2%./#$#i34&|5C6%0I:

(60) N;'./#E.&1 = &#$DI.%53P mlS. RJ./ F' :V1 5*XY3l5F{kl%C5L&(./Z3 4|5L1 1 I(%C5L&(./ I((4;C5& (LX D)(ϕ) = LX D(ϕ) − D(LX ϕ). ∞. 1. 1. 2. ". Q : S(M ) → D(M ).

(61) } 1 &;'&'"6L5F0IK )H#ihb./I(K1 "( %'(1l5IymtMOM&5L"d 55F-"3 |%s= 5XY3 S (M ) (I 5 %./4: &(./ ]./3 I(8/4./ i%'1b :l5LYEXY3@O(I*σw$◦1 Q./"=&h Id. I( : Z%'./ "&&3 &Z%C5L ./ XH34#$F 3 XY3l5F{kl%C5S(.E-It@B5: D: %s=l5XY3 G: ;'&6;G%.76L5&(5F ∇ &3 M mY%''&& %./4""3 %'&(./Z;<s5FG: ;%'&{-:l5 I(5i&%<./ M 54Q%<Q./346H&58/mL./i: ;'&8/ &5o1l5 Iy@B'&1l5%'D: 'Q./1b;5S34& D (M ) : ( ;"FI|5L8/("5F|F" λ ' µfv: ' &{;'m&%<./ 0:4Eklh ";i 5L3 "I :@O./&: " (F".: 34?:l540Ip5J&'%'./ M B\ .L&5 S (M ) It@O&1 5%E: )H#ihb./I('*XY3 I(3 N"o5"&.H%(;Lm S. R δ = µ − λ M . . . . '

(62) . . . k. λ,µ. δ. Z: ;"(8/ -1l5L %C5L&(./ h4(I( ;C5L(& B. + . . =;@4?>$/1254. 5L&(wd 5(&5F*1 .3 D./3. % . . ' #')$ . " 0 7 " >$/1@ 45/1032 %. &3 3 4-6L5"(;<;. M. 3 ?51 1 I{f. ∇ : Vect(M ) × Vect(M ) → Vect(M ). . f ∈ C ∞ (M ). ∇f X Y = f ∇X Y, ∇X f Y = f ∇X Y + (X.f )Y.. vu I: ;'%./34I(: 2%''"2: ;<kl &(./ XH3N@B'J%'..&: ./4 ;'GI.%5I('34 2: ;"6; %.76L5&(5F2'"G: -Ip5ide./"#E ∇X Y = X i ∂i Y + Γkij X i Y j ∂k .. ,|de. %'&(./ Γ 51 1 55&&5F0:l5L |%''" de./"#i3 I!&./F!5L1 1 'I(;' $ H "$#&%' % =@ 4 2 " %# : ∇ <* "./F: ;'kl4(*XY3@OZ%'../": ./ 4;*I.%5I('M ,5 2 " @ 4 " 0 :@O3 -: ;'&6;?%.76L5&(5F ∇ '"aI(' &'3 T : ;<kl 1l5 . . k ij. 5 x%9&5C6L5(IymS4./3 Q 9%./ "(: ;'&'&./ xXH34D: +:4;&{6/;'+%.76L5"p5F'Q5L ./"&./ &3 M ' ./34G4.'&./ C Iy@B' &'#ih I(2:4?%'%. <P./ M uvIb"91 .&&h I(G: G: ;'k (&3 It@O "#ih Ia:49:4;&{6/;'D%'. 6L5"p5F&934 &'Ip5L&(./2:@O;XY3 {6L5I(' %LmL5L1 1 'I(;' ;9+.-=4?>$/ #&% 0 7 % IJ@ "KL% 72:4?> % . :4I(5#E5 >"+&3 f 6L5FMW]H ∇ < ∇ &./F|: 3P %'./ gP(./ ?I( ;C5L(&'5L ?&./&"(./^"3 M m 5I(.&RI(I&./F*: {' IJ@ "KL% 72:4?> %M% 032 ;9+.- 4?>$/ #&% 032 %' '@BI gP("&3 f de./"#E &3 M 'I(IXY3 α T (X, Y ) = ∇X Y − ∇Y X − [X, Y ],. . ∀X, Y ∈ Vect(M ).. M. 0. ∇0X Y = ∇X Y + α(X)Y + α(Y )X,. |_ M )I-: ;'#E.Y&5 :l54 F2XY3 ^: 3P :4;&{6/;'Z%.76L5&(5F'J"./F 1 &. ["%'&6#$Fi;'XH346L5I('F?&R'|&'3 I('#E'Fi&RII('?:4;'kl &"F!&34 M I('*O# '#E'*8/;.H: ;"(XY3 'o!&'1l5&5#E;<&58/21 ">M . ∀X, Y ∈ Vect(M )..

(63) . . . . '

(64) . }. . . a4r%I(5&"r:@O;XY3 6L5I %'r1b./3 i%''&&r&'Ip5L&(./ wK51 1bI; 25@ -725-=@ % ./3 7 # / ' % IJ@ "KL% 7254?> % "3 M M jQs5F0: ./ ;$: '3P k h &;'|6/'%'&./&I( ' E (M ) &3 M m 4./3 5C6/./ T: {+:l54RI(5: 3HP4>#$*&'%'./KXY3 ItE@O (M "#i) h I D(M '"T&./#$./&14= ) V5FXY3@O"1l5%i6%<./"(I "./c"1l5%'i: 0* )H#ih .I(o5&".%'(; ) M uvI @O<P(w1l5L9: h4 ["%<./ %5 ./ XY3 ol5S3 "I(Ia'F&2%'D:43P S(M "#ih4I('m # 5L(DINwo1b./"&(h4I(:@B5"&(8 GK%s=l5XY3 ∇ ∈ C 3 ?h ["%'&(./ I( ;5(". IJ@ "KL% 72:4?> % .. 1. 2. M. L∇ : S(M ) → D(M ).. ' "+h ["%'&(./2'"%'./ "&&3 {+: +I(5D# 5L (>'&x"3 6L5F&M G../4./34:N@ 5hb./": XY3 T%./#$#E E (M )⊗S T M wW3 ?k h &;R5"&.H%(;T P M m7./-1b34W;'& :4& Ip5Z: ;"6L5L&(./ %.76L5&(5F ∇ 53P "%'&(./40: r%'Eklh4&;MxjQs5Fi:4./ ;r34 &'%'./ γ ∈ Γ (E (M ) ⊗ S T M ) ml./Z%. "&&3 {o5LI(./" J1 ./&5F ∇ γ ∈ Γ (E (M ) ⊗ S T M ) G. . l. 1. ∞. ∞. s. ∗. l. 1. l+1. 1. 1. (∇s γ)(X1 , . . . , Xl+1 ) =. ∗. ∗. X 1 (∇Xν(1) γ)(Xν(2) , . . . , Xν(l+1) ). (l + 1)! ν. z 1l5w9: 2I(4m.J%'./ w&34*3 42XH3 5Fk %C5L&(./ 'J1b./&5F*1 .3 D./3 S ∈ S k (M ). .SR. (L∇ S)(f ) = hS, ∇ks f i,. S . ∀f ∈ Γ∞ (E1 (M )),. 4: ;&8/ -Ip5i%'./F&5%'&(./Z: -: '3PJ%s=l5#$1 : 2' &'3 &'M ,5ide./ %'&(./. h., .i. ) 7→ D(M ) : ;'k (G1l5 S+w*51 1bLI;: C+.-/1×0325S(M 4 6879/1254 " 0 2 /10 /1@ IJ@ %' 7@ 4IJ254 " 0 - " @ 1@ % . 2 /10 /1@ . ./3^ %'./& +.-/1032:4 6879/1254 " 0 % 4 7 =0 ;@ " 4 7 G <&&E5L1 1 I(%C5L&(./ wa 5L3 "II(< L wG3 -XY3l5F{kl%C5L&(./Z1b./3 D&./34 ∇ ∈ C M ) .3 x'&#$( 'x%''&&D&'%'&(./mL5141 'I(./ Ip5o:4;'kl {./0: 9XH34I(XY3 'Q'f &'3 &R: G%./3 "h 3 "G%'Ip5"&XH34M B$: ;"(8/ G1l52 % 0 % - @ % 7 " -=@ -=@ % :@O3 : ;"6L5L&(./K%.76L5"p5F ∇ ∈ C Ia%s=l5L#E1E: G& "3 " R "3 M : ;'k 1 5 M. . . . . . . . . . ∇. M. . M. R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,. 1 .3 E&./34 X, Y, Z ∈ Vect(M ) MD,5 25@ /=7 % trR : c%''&&c%.3 &h 34&c<rI( 2 % 0 % -=@ % 4 7'74 &./F*ID%s=l5L#E1 9: o' &'3 &"3 M : ;<kl (*""1 '%<f Ric {6/#$F1l5L . '1l5L. trR(X, Y ) = tr(. 7→ R(X, Y ).), Ric(X, Y ) = tr(. 7→ R(Y, .)X)..

(65) . . . . . . . }. . . . $

(66) ( ! . . a4+.-/103254 6879/1254 " 0 0J/125- @ %#?#&% IJ@ "KL% 7254?> %M% 032 ;9+.-=4?>$/1@45/1032 % 1b./3 : 3HP klh "; E (M ) ' E (M ) '"*34J.1 ;'5L&3 *5L&&.H%(5Fi%s=l5XY3 6L5"(;<; M 3 ?51 14I((%5L./ . 1. 'I(I(oXH34. 2. QM : CM × S(E1 (M ), E2 (M )) → D(E1 (M ), E2 (M )),. "8 0J/12:-=@ tMOMSXY3 *1b./3 &./34x: ( ;./#$./"1 = ("#ETI.%5I φ : M → lm (IN6H('F %# . = φ (Q (S)), 1b./3 *./34 ∇ ∈ CQ < S(φ∈ S) (N ' " IJ@ "KL% 72:4?> %M% 032 ;9+.- 4?S(E >$/1@ 45/1(N 032),p./E 3 IJ @ ")). 7254?> 032 4?03>$/1@ 45/1032 Q tMOM4XH34 Q = Q & ∇ ' ∇ "./FG1 &.S["'%'&KL% 6/'#E'%FMa% ;'XY3 6L5IF'M . '"!3 4rXY3l5F{k4f Q : S(E (M ), E (M )) → D(E (M ), E (M )) %C5S(.J1b./3 D&./34 ∇ ∈ C M W./3S["./3 "1b./3 25II(;'8/GI(';%'&&3 &'m ./3 N@BgP414I((%'&&. 1l5o1l5I(5 &3 {2I(5i: ;1b : 5 %|: Q !Ip506L5"(;'&; M M G &5C6L5(I-51b./3 Jh 34V: \";1b./ : "\ 34 \XY3 w./ 1b./&;' (I ) 5 XY3 IXH34a5 ;'1l5G) M ,'%./#0&M . QM N. ∗. M (φ∗ ∇). ∗. 1. N. . M ∇. . M ∇. 1. N ∇. 2. 0. M ∇0. 2. 1. 2. M. . M. 2 /1032 " 3 0 0 ; % - 6 '@ ; > % 7 2 " @ 4 %# 0J/12:-=@ %# +.-/103254 6879/1254 " , 0 0J/12:-=@ % #?#&% IJ@ "LK % 72:4?> % M% 032 4?03>$/1@ 45/103E2 1%. %2 % 4 2 % H 25H 4 # -=0 % I " -E@ 2 . 7 '% 6 @ ; . z It@O./&8/(4: -%<&?XY3 w./m ./Z&.346/?I('G";"3 I5L:@O<P(w' %|: ` .h4'H34?"3 R M G./3 ?5I(I./ ah &>'6/gf #E'FZ51 1bI ' XY3 ./2%'./ "("&FA%Z";&34Is5SZ'Z: ^XY3 'I(I( #E5 >" It@O<P(w' %K:@O3 KXY3l5F{kl%C5S(. 5L3 "II(i<|14&.S["'%'{6/'#E'Y?Y6L5"p5F& J%./4&(w3 -3 4?8; ;'5I(5S(.M ,21 I(. 8/#$F*14&.S["'%'{d : + 1, R) w3 -&./34wf

(67) 5LI(8/>'h & : o%s=l5#$1 : ?6/%<'3 &-&34 slR MNjTII(?sl(m w2 8 : ";!1l5LoI('o%s=l5L#E1 o: 6/'%''3 & ∂ , x ∂ , et x x ∂ . j+I(I( w &./#$./"1 = Q sl(m+1, ''"#E5SP(#E5I(W:l5 Iy@ 5LI(8/>'h & Vect (R : %s=l5L#E1 : 26%'&3 "o1b./(I )H R).#E(53PE&3 R M Bc: ;'&(8 01l5L +.-/1032:4 6879/1254 0 H 4?03>$/1@ 45/1032 % 1 ./34oI('aklh4&;G6/'%<f ./"('I( E < E 3 -XY3l5F{kl%C5S" (. sl +.-/103254 6879/1254 " 0 H 4?03>$/1@45/1032 %' slm+1. m . m+1 m. i. i. j i. i. i. ∗. m. m+1. 1. 5L&(wd 5(&5F. 2. Q : S(Rm ) → D(Rm ) LX Q = 0,. ∀X ∈ slm+1 .. m. ).

(68) } UA./F"./ oE1 ";&'F2XY3 iIt@O<P"' %|:@O3 0XY3l5LY&kl%5L./V 5L3 "II(-1 &.Sf ["%<6#$FG(F6L5&(5F2#E1 I(XY3 aIt@O<P(w4%?:@B34 -XH3 5Fk %C5L&(./ sl f (F6L5&(5FLM*] Q : ;"(8/4 34 XY3l5F{kl%C5L&(./ l5S3 "I(IV1 ".S["%<{6/#$F (F6L5&(5FG'9& φ &'1 &;'&F&2I( .D:@O3 %s=l5L#E1r:4a6/'%'&3 " X 51 1l5wgf l5FG sl mlIN6H('YG'A N< / . . . . . . m+1. m+1. 1 .3 E&./34 t ∈ R mR1 ./34 ./3r )H#ihb./I S m" ∇ :4;&8/ cI(5 : ;'&{65S(. %.76L5&(5FJ1 Ip5SJ%C5 . (XY3 A: R M9) 5 : ;<kl {(.: AIp5^: ;"6/;'c:@O3 %s=l5#$1-: +4&34&m "FIt@O./-1l5"6HFW*: ;#$./F"XY3 φ 1 &;'&w6/Ip5D%I(5&" 1 &. ["%'&6: ∇ 1 .3 R&./34 t m4./ 5L3 5?;8F5I#$FD14&./346;XY3 Q(∇ ) w 3 GXY3l5F{kl%C5L&(./ sl fv;'XH346L5&(5FLm/1l5TIt@O(F6L5"p5 %'*1 &. ["%'&6a: Q M ,d 59XY3 -Ip5i%I(5&"21 &. ["%'&6?:4 &./{*1 ";&'"6;?1l5*It@B5%<(.r: 3 : ( ;.#E./"1 = &#$ φ wG: V!I(5K14&./1b./1 ∇"(;'&;&346L5F / φ∗t Q(∇0 )(S) = Q(φ∗t ∇0 )(φ∗t S) 0. m. ∗ t. 0. 0. m+1. 0. t. & X @O;%'& A ∂ + A x ∂ + α(x)x ∂ , .1RcI(' A <I '.SR α ∈ R M j+J 'ml& Y, Z ∈ Vect(R ) ml(I 6F LX ∇0 = α ∨ id. i. i j i j. i. m∗. i. i. i. m. Aij. "./F2:l5 . R. (LX ∇0 )(Y, Z) = [X, ∇0Y Z] − ∇0[X,Y ] Z − ∇0Y [X, Z] = Y i Z r ∂ir X j ∂j = α(Y )Z + α(Z)Y. 51 ">: ;'6I./1 1b#$F'M a>GI(./"m d ∗ 0 (φ ∇ )x = (φ∗s (α ∨ id))x ds s x ∈ Rm. ∇0 β Z s βx (Yx ) := (φ∗u α)x (Yx ) du,. 1 .3 0.34 s ∈ R mQ1 .3 i./3 M 54$%'K%. : &(./4m w 1 &. ["%'&6#$F;'XH346L5I('F2 6Hp5$Ip5 f de/. "#E : ;<kl (|: 0I(5$φ#E∇5 >" &3 {6L5F0/ .SR. m. Y ∈ Vect(Rm ) x ∈ Rm. M. 0. ∗ s. 0.

(69) .

(70) . N T o a R a a a.3 +5141 'I(./ +%I(+14&(4%(1 53P|d 5&Q%./4%"l5FRI('Q%./4 <P(. T: G 5"5M4)T"#$(>'&#$Fm ./3 *'Z: ./ . 3 2: ;<kl &(./ 8; ;'5I''f &3 {* .3 +: ./ 4./ T1 I(34+: *: ;'5(IQ&34RI+%'./ gP4./ T: G 5ws5K1 ".S["%gf {6/+<TI('3 &+I(4+5C6/%GI('+"&&3 %<3 "T1 &. ["%'&6MH)W./3 +3 4de./&#E5L&(./ 1 I(34*: ;'5(II(;Lm4 .3 &F6. )./ aI(2I('%'&3 G (E 3yM . . . E). . $$ &* $

(71). . ' ' $. ] ./F G 3 V8&./3 1b?: 0,?< H 3 V"./3

(72) fv8/"./3 1b?de'&#$;?:4 G Mo;"{f 8/ ./4a1l5L g < h I('o5LI(8/>'h &'a: 0,(-%./"&'&1b./ :l5F&M]./{ P → M 3 f k h &;1 "( %'(1l5I4&3 M mY'IlXH34 dim M dim G/H MSa G%./4 <P(. : H G 5"5r"3 w*3 f de./"#E ω &34 P |6L5I('3 &:l5 g XH34N5S(wd 5RI 1 &.1 &;';'*&P3 {6L5F / 1b./3 D&./34 a ∈ H m R ω = Ad(a )ω.34 b 1 / .. 3 D m ω(h ) = h h∈ H wa34 -h ["'%'./ I( ;C5L(&LM ∀u ∈ P, ω : T P → g ,5|%. <P./r: G 5"5 1b"#E<9: . %o: G&5 "de.&#$RI(;I;#$F: It@B5I(8>h "R:4*,( g !: 'Q%sω=l5#$1 : 6%'&3 "T&3 P M a.&./ XH34D& h ∈ h m '"r;<6H(: '#E#$FE;8F5IG 8 %'V Ip5^: 3P>#$V1 "./1 &;'&;J:4 ω %./4 (h)<P(. a:4 G 5L"s5LJM 0@ 5L34"|1lh5L"CmlIy@B'&1l5L%-6/%<./"('I g &?: ;'%./#$1 .& %./#$#$&3 { /. . ∗ a. . −1. ∗. . u. u. . −1. ∗. g = g−1 ⊕ h,. '; 5F+3 K&./34wfv'&1l5L%D6/'%'&./&I : g &./#$./"1 = D g/h M G I(5;'5F+: m ./3 o1 ./36/./ o(F".: 34(&?Ip5E&#E5"XH340&346L5FiXY3 & '5 ">234&(I|:l5L Ip5i"3 &0/. g−1.

(73) 4 h ∈ h % 2. 4 X ∈ g . / #&" @ −1. [h∗ , ω −1 (X)] = ω −1 ([h, X]).. }.

(74) T .

(75) . . . .

(76) . . . . . .

(77) . . o -d 5{Cm4IN6H('Yml&Iy@B./ 4. ϕ I( l.L: . h∗. t. /. }1. d Rexp(−th)∗u exp(th) ω −1 (X)u exp(th) |t=0 dt = ω −1 ([h, X])(u). [h∗ , ω −1 (X)](u) =. 1l5 Adfv(F6L5"p5 %': ω M +. . ( ! ! % $ '

(78) . ./ "(: ;'&./4iIt@B5%'&(./ : 3 8/"./3 1b "3 iIt@O"1l5%' 1 &. ["%'&d RP Ml,-"s5Lh (I(5S34 H : -GIt@O=;I(;'P#EGL(m 'F [e + 1,] :lR)54 RP w uv: S A 0 { : A ∈ GL(m, R), ξ ∈ R , a 6= 0}/R ξ a 'J(I2'@B' &34AXH34 H wVI 1 &.H: 3 {A"#${f : ("%' G o G mS. R G w ("./#E.&1 = Q GL(m, R) 'WS. R G w &./#$./"1 = T R $M B|5Ip5*1 ".S["%<./ &3 {6L5F G. m. m. m+1. m∗. 0. 0 m∗. 1. π : H → GL(m, R) :. . A 0 ξ a. . 7→. 1. 0. A a. +, @ 5LI(8/>'h &-: 0,-: P GL(m + 1, R) " gl(m + 1, R)/RId M j+I(I-" : ./ %G("./#$./&1 =4a + 1, R) '9I(I&o:4;%.#E1b./&G%.#E#$3 o"./#$#E : ("%<2: -&.3 wfv5I(8/sl(m >'h &' g−1 ⊕ g0 ⊕ g1 ∼ = Rm ⊕ gl(m, R) ⊕ Rm∗ .. ,+@B&./#$./&14= ("#Ea"G: . ;-1l5 . A v ξ a. . 7→ (v, A − a Id, ξ).. ' "!%.&&'&1b./ :l5L %! : 3 {3 !w"3 %'&3 &K:@B5I(8>h "i: !,(0&3 R ⊕ M4,'5I(8>h "D: -,o%.&&'&1b./ :l5LYa m ' "./F gl(m, &'&1b%'R)&6⊕#$RF g m g m9' g ⊕ g MD,5 w"3 %'&3 & :@B5IG8/>h "GV: c,HV"3 "a: ./4 ;-1l5I*%".%s=4'G&346L5F / R ⊕ gl(m, R) ⊕ R G. m. m∗. 0. 0. m. 1 m∗. [v, v 0 ] = 0,. .SR. < v. 0. [ξ, ξ 0 ] = 0,. [U, v] = U v,. [U, U 0 ] = U U 0 − U 0 U,. & ./FJ: Z;'I(;#$F& : R m U ' v'.1R < ξ 5L1 1l5w(' 'YG R M gl(m, R) ξ [U, ξ] = −ξU, 0. &./FJ: Z;'I(;#$F& : . [v, ξ] = v ⊗ ξ + hξ, viId,. m. 0. 1. 1. m∗. U0.

(79) #(% ( 0! %

(80) . . . . . T.

(81) . . a ;'I(;'#E'Yo: 3 8/"./3 1b H 1b34 <"06H3 %./#$#E?3 i&54"de./"# 5L&(./ : KIt@O"1l5%$1 ".S["%<d RP XY3 +I(5("&K&./ ./"(8/ [e ] k4P;'Mx]+./ I( %<&?&5 "de.&#E5L./Z:l5 aIp5E%5"&|%5 ./4(XY3 0:4 RP 53J6/./&( 58/?: m./ ./h4&('Y-34 !%./"&'&1b./ :l54%KF"KI(2;'I(;'#E'Y&2: '-:4 [e ;I;#$]F*: G Mb]It@O./Z .&2XY3 -I(D&54"de./"# 5L&(./4D: RP H : 3 {' 1l5I;I;#$F: &./FW3 {6/.HXH34#$Fx:4;''&#$( ;'1l5I3 "W:4;&{6/;' ["3 &XY3@BÎt@O./&: " cH 4m H wE:4./ %Z1 I(. 8/; :l54 G 6Hp5 Iy@ 5141 I(%C5L&(./ &3 {6L5F0/ . m. m+1 m. m+1. 2 m. m. 2 m. ι:H →. G2m. :. . . A 0 ξ a. 7→ (. . Aij Aij ξk + Aik ξj ,− ). a a2. a 25@ -725- @ % IJ@ "KL% 72:4?> % P -=@ M w 5I(./"3 T&;': 3 %<(.?:43?klh4&;T: W&gf 1 >'&'x:@O./&:4& P M 5L308/"./3 1b H M$o9&II(W"&&3 %<3 "QgP("&FCmL%./#$#$ I(?1 &.346/!I0&;'&3 I{s5Lo&3 {6L5F

(82) E m)+&./1 M $1M GXY3 x'"-I|1b./(Fo: : ;1 5": - .L&-#$;'&= .:40 /

(83) a.hlL5 )F5"= {f o58F5L . # % 4 2 % - 0 % 7 " @ @ %' I " 0 /10 7 % . 2. . . . . 4 KL% 7 2:4?> 0 /125-=@ % #?#&% % 0325@ %N#&%' 7 # / ' '% J I @ "KL% 7254?> '% % 7 " 030 % 4 " 0 # 4?0 ;9/14?@ %' % %2 J / 10 2 " @ 4 " 0 - @ 2% &# %' 2:@ -37 2:-=@ %' J I @ " KL% 7254?> '% -=@ M M.

(84) j+^d 5m I(5r%./"&"1 . :l5 %'K"?%'./ "&&3 {!: KI(5 # 5L (>'&i"3 6L5F /ZIp5r%'Ip5"&K:@O3 $%./4 <P(. &5 2./"&./ ∇ m./ 5

(85) f &.H% Ip5\"&&3 %<3 " 14&.S["'%'{6/ : ./FJIp5 k h & 53 : "&3 A: 3 1 ./FZ: %.H./&:4./ ;'I(.H%C5I x '"G: ./ P4;-1l5*Iy@B' &'#ih I(2:4&'1 >'&G:@O./&:4& &3 {6L5F / i. S. RcI &.YI(a )H#ihb./I(': G =4&(w1. IW: M B 1b34a6/;"kl'XY3 % <&*%'./Γ&""1 ./4:l5 %'G'"Rh K: ;'k (*<Th ["'%'{6/∇D8 %'-I(5I(./H: *&5

(86) f de./&#E5L&(./i:4 Γ "./3 RIt@O 'T:@O3 $%s=l5 8/'#E'F:4G%'..&: ./4 ;'<T8/ %' Ip5EI(./&'I((5FI(' )H#ihb./Io:4 G = &". N'Ix: $%./ 4<P(./421 &.S["'%'&6/gf #E'F;XY3 6L5IF'M (xi , δji , −Γijk ).H,. i jk. i jk. . . . . $$ &* $0! % . +j 8/;' ;&5ItmF" ω '"3 o%'./ gP(./$: G 5"5E:4;'kl a"3 R3 H ftklh "; 1 & %1l5I P ml5LI(./"*5!%./34&h 3 " Ω wG: ;'kl4(-%./#$#$:@O=l5h {3 : -1l5L 1 Ω = dω + [ω, ω], 2 SR [ω, ω]. .SRZ:A'"GIp5i:4( ;&'F('I(I'G. ZI(2%'&.H%s= '. [ω, ω](X, Y ) = 2[ω(X), ω(Y )],. '"G: ;'k 1l5. ∀X, Y ∈ T P.. .

(87) .

(88) . . . . &} T. .

(89) . ] ω = ω e + ω e + ω ml.SR e wo34 |h 5&?: g m e 3 ?hl5"|: g ' 3 0h 5&|: m I-d 5LGXY3 '@ 5L Y3 I(|&3 GI('a%s=l5L#E1 a: -6/%<'3 & 6/'"%C53HP #E14I((XY3 *g XY3 a%G4&34ΩD1 '34D'@B;'%"(&a&./349Ip52de./"#EG&3 {6L5F / i. i j j i. i. i. i. i. i j. −1. i. 0. 1. X. Kkl ω k ∧ ω l ,. .SRJI(*de./4%'./ K &./FoK6L5I3 ":l5 g MJ5 GIp5K&34Lml ./3 G: ;'%./#!f 1 .&"./ G53 "& Ω J&G1 ".S["%<(. &I./ g m g < g / Ω=. kl. −1. Ωij eji. 0. 1. a./341 ./36/./ '|.34"+: Ω;'kl=4(Ω*e1l5+wN: +Ω Ω3 +.de./ %<./ : -Ip5i#E5 (>'&"3 6L5F&" / g ⊗ g) i. i. i. i. . ∗ −1. κ ∈ C ∞ (P, g∗−1 ⊗. '"Zde./ %'&(./ &'5 ;8F5I#$F$: ;%'./#$1 ./";J "$%./#$1b./5F& "I./ m g ' g / g κ=κe +κ e +κ . 0@B534"V1l5L"CmT" " r . 3 V%'./ gP(./ : G 5ws5 &3 E3 klh4&;V14&(4%{f 1l5I P mN"-1 ".S["%<ω(. 2&'I(./ g ' g : ./{6/F?5S(wd 5("iI('%. : &(./4 &3 {6L5F / mF1. R h "Ip521 ".S["%<./ ω (h ) = 0, ω (h ) = h ∀h ∈ g + g &'I(./ g : h 5L: a )(ω + ω ) ∀a ∈ H mS. R 5: a w (R ) (ω + ω ) = ( It@B51 14I((%5L./ :4 g/g : 5 I(3 fv!# #$a : 3 {21l5L*It@B5%'&(./ 5L:S["./(F& 5: a : g :l5 g ] ω (X) = 0 ml5I(.& X wa6"&(%5ItM ]./F$I('$8&./3 1b G < H :4./FEIDw XY3 '"&(./ :l5 $Ip5 "%'&(./ : ^%^%s=l514"M, = ;'./&>'#E^"3 6L5FJ; . %; :l5

(90) E -1 58/ w de./ :l5L#E'Y5I / κ(u)(X, Y ) := Ω(u)(ω −1 (X), ω −1 (Y )).. G. −1. 0. 1. i. i j j i. i. −1. . ∗. −1. . a. ∗. i. 0. 0. 0. 1. 0. . 0. −1. −1. . ∗. 0. i. −1. 0. −1. −1. 0. 1. . −1. . % " 030 % - 0 %. " 4?2 P - 0. H6. @'; IJ@ 4?0 74I/. dω i = −. X. ωki ∧ ω k ,. . -=@!-=0 % $> /1@ 4 ;2 ; 4 M 7 "$ I " /1032 %' i +.- 4 /1254 L" 032 #&ω%' - I I # ; M% 032 /14?@ %. # H $> / = 0" H L" @ M% @ 1 / 0 &# % g % ω/10 −1 2% N-=0 % L" @ M% >$/ &# % - @ −1 7 I g % "$ " /1032 %' ω i ω / 7 " 0 14?254 " j0 7 " 0 14?2:4 " 0 M% 003254 " 30 0 ; %' IJ@ ;'7'; % 0 M% 032 % 2 # !. - 4?>$/1032 % . / &# " @ 4 # % 4 2 % =- 0 % -=0345+.- % 7 " 030 % 4 " 0 % 1/ @2 /10 " 032 ω = ω + ω + ω −1 0 1 # /D7 " - @ =- @ % Ω % 7 $" I " /1032 %' (0; Ωi ; Ω ) 1/ 2:4 /14?2 # /8IJ@ " IJ@ 4 ;2 ; -=4?>$/1032 % j j X. "

(91) Ωij =. i Kjil = 0,. X1 2. i Kjkl ωk ∧ ωl..

(92) . . . . TT. . . . ] '"K3 1 .(F!: '$& '"$3 6/'%''3 o5 8/'Yo P Mu 5L=3c1 j.f(F u m I(5 L" @ M% 79/10 P" 0345M+.- % θ : XP M w Ip5 f de.&#$G65LI(34&W:l5 R ⊕gl(m, R) :4;'kl D53|1 .(F u : Ip5# 5L (>'& &3 {6L5F0 / S. R π '"Ip521 &. ["%'&(./$%C5Lθ (X) ./ XH34=a(P: Pf )M(π"3 X),P M 'R.SR e '"I(&'1 >'& JIt@O./&8/(4a: R &'1 &;'&F&;|1l5LGI(5i# 5S&%uv:NM %'-%'XH34 w:4+:l5L (E 3tmF.$1b3+6/./+XH34GI(5 f de.&#$9&34R34 "&&3 %<3 "o1 ".S["%<6 P ?6L5I3 ": 5 R ⊕ gl(m, R) : ;'k (a1l5 i θ mYS. R waI21 I(. 8/#$F%5 ./ XY3 2: :l54 M mb&5L"d 5{DI*%. : &(./4 i: 3c=4;./">#$ 4M |@ 514&>2%'!&;'&3 I{s5LPmN(IW<PP "03 !3 XY3 !%./4 <P(. : G 5"5i"3 : ./FQIQ1 ".S["%<./ x"3 g < g %. %'(: 'Y+576%*%'I(I : P < H X 4 3 F L 5 & ( " d 5LI(5D%./4: &(./2"3 1 1 I;#$Fs5&+"Ip5S6+53P2%./#$1 .5F' i: o θ &5i%./34&h 3 "M a./3 6/4./ *: ./4%: #$./F"9I(5|1 "./1b./&{(. &346L5F /. 2 0. 2. 2. . 2. m. 1. u. −1 ∗e. ∗. 2. 1. m. . ∗. m. 2. −1. ∗. . 0. .