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(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/; :,(:@O3A./1;' 5L&3 *:4(; &'F('I /
(LXD)(ϕ) = LXD(ϕ) − D(LXϕ)1.3 D./34
ϕ ∈ Γ∞(E1(M)), D ∈ D(M), X ∈ Vect(M)M
o^Ip5 #O'#Ec#E5> "m9I 5L%'./ :Diff(M)
&34 I( "%<(. :
E1(M)<
E2(M)'K"3 !I('i&'%'./K:
TM(:43("FK3Z5%<./ :
Diff(M)&3 9I( )H#ihb./I('MHzo3 6/53r(kl&;"(#E5ItmFI(5|:N;' &F&p5L&(./$:
%<&?5%<./Z1' &#$':-%C5LI(%34I( *I(5i:; "6;?:?,2:@O3A*)H#ih.I(2:l54aI(5:( "%<(. :@B34A%s=l5L#E1J:26/'%''3 &
XXH34I(%'./XY3LM
,J"1l5%'S(M)
<D(M)
&.YA("./#$./ &1=4 s5FJXY3@O&15%'6/'%'. &('I('M" ./3'de.(-(I2EIK&./F|1l5? 5F?XH34 &'1 &;'&'Y5L./?:38/ &.312:4:(;.#E./ "1=&#$DI.%53PM
j+ %./4&;XY3'%Lm+(IT@O<P(wr150:EXY3l5Fkl%C5L&(./ %C5.(XY3El5L3Hf &'I(I(-5L3A"a.SRA'I(I2%./#$#i34&|5C6%0I:N;'./#E. &1=&#$DI.%53Pml.SRJ./F':V15 *XY3l5Fkl%C5L&(./Z34|5L11I(%C5L&(./ I((4;C5 &
Q : S(M) → D(M)
' 1 &;'&' "6L5F0IK )H#ihb./I(K1 "(%'(1l5IymtMOM&5L"d 55F-"3 |%s=5XY3
Sk(M)I(5
%./4:&(./
σ Q = Id.]./3I(8/4./i%'1b:l5LYEXY3@O(I*w$1./"&hI( :Z%'./"& &3 &Z%C5L./XH34#$F3XY3l5Fkl%C5S(.E-It@B5:D:%s=l5XY3G:;' &6;G%.76L5 &(5F
∇&3
MmY%''&&
%./4" "3%'&(./Z;<s5FG:;%' &-:l5I(5i&%<./ M54Q%<Q./346H &58/mL./i:;'&8/ &5o1l5
Dλ,µ(M)Iy@B'&1l5%'D:'Q./1b; 5S34 &
:(; "FI|5L8/("5F|F "λ'
µfv:'&;'m&%<./0:4Eklh ";i5L3 "I
:@O./ &: " (F ".:34?:l540Ip5J&'%'./ M B\.L &5Sδ(M)
It@O&15%E: )H#ihb./I('*XY3I(3N"o5"&.H%(;Lm.SR
δ = µ − λM
+ % ' #')$ BZ:;"(8/-1l5L =;@4?>$/1254 " 0 7 " >$/1@ 45/1032 % &3 34-6L5 "(;<; M
3?511If%C5L&(./ h4(I(;C5L( &
∇ : Vect(M) × Vect(M) → Vect(M)5L&(wd 5(&5F*1.3 D./3
f ∈ C∞(M)
∇fXY = f∇XY, ∇XfY = f∇XY + (X.f)Y.uvI:;'%./34I(:2%''"2:;<kl&(./ XH3N@B'J%'.. &:./4;'GI.%5I('342:; "6;
%.76L5 &(5F2'"G:-Ip5ide./ "#E
∇XY = X i∂iY + ΓkijX
iY j∂k.,|de.%'&(./
Γkij
5115 5&&5F0:l5L|%''" de./ "#i3I!&./F!5L11'I(;' $ H "$#&%' % =@ 4 2 " %# : ∇ <*"./F:;'kl4(*XY3@OZ%'../ ":./4;*I.%5I('M
,5 2 " @ 4 " 0 :@O3-:;' &6;?%.76L5 &(5F ∇ '"aI('&'3 T:;<kl1l5
T (X, Y ) = ∇XY −∇Y X − [X, Y ], ∀X, Y ∈ Vect(M).5x%9& 5C6L5(IymS4./3Q9%./"(:;' &' &./xXH34D:+:4; &6/;'+%.76L5 "p5F'Q5L./ "&./ &3
M'./34G4.' &./
CM
Iy@B'&'#ihI(2:4?%'%.<P./MuvIb"91.&&hI(G:G:;'k( &3 It@O"#ihIa:49:4; &6/;'D%'. 6L5 "p5F&934
&'Ip5L&(./2:@O;XY36L5I('%LmL5L11'I(;' ;9+.-=4?>$/ #&% 0 7 % IJ@ "KL% 72:4?> % . :4I(5#E5> "+&3 f6L5FMW]H∇
<∇′ &./F|:3P %'./gP(./?I(;C5L( &'5L?&./ &"(./^"3
Mm
5I(. &RI(I&./F*:' IJ@ "KL% 72:4?> %M% 032 ;9+.- 4?>$/ #&% 032 %' '@BIgP("&3 f de./ "#Eα&3
M'I(IXY3
∇′XY = ∇XY + α(X)Y + α(Y )X, ∀X, Y ∈ Vect(M).
_|M )I-:;'#E.Y& 5 :l54 F2XY3^:3P :4; &6/;'Z%.76L5 &(5F'J"./F1 &. ["%'&6#$Fi;'XH346L5I('F?&R'|&'3I('#E'Fi&RII('?:4;'kl&"F!&34
MI('*#O'#E'*8/;.H:;"(XY3'o! &'1l5 &5#E;< &58/21 ">M
' a4r%I(5&"r:@O;XY36L5I%'r1b./3 i%''&&r &'Ip5L&(./ wK511bI; 25@ -725-=@ %
IJ@ "KL% 72:4?> % . ./3 7 # / ' % IJ@ "KL% 7254?> % "3 MM
jQs5F0:./;$:'3P kh &;'|6/'%'&./ &I(E1(M)
'E2(M)
&3 Mm 4./3
5C6/./T:+:l54RI(5:3HP4>#$*&'%'./KXY3It@O"#ihID(M)
'"T&./#$./ &14=V5FXY3@O"1l5%i6%<./ "(I "./c"1l5%'i:0*)H#ih.I(o5&".%'(;
S(M)MuvI
@O<P(w1l5L9:h4 ["%<./ %5./XY3ol5S3 "I(Ia'F &2%'D:43P "#ih4I('m# 5L(DINwo1b./"&(h4I(:@B5"&(8 GK%s=l5XY3
∇ ∈ CM
3?h ["%'&(./ I(;5( "
L∇ : S(M) → D(M).G '"+h ["%'&(./2'"%'./"& &3+:+I(5D# 5L(>' &x"36L5F&M G../4./34:N@ 5hb./ ":XY3T%./#$#E
E1(M)⊗SlT ∗MwW3?kh &;R5"&.H%(;T
P 1Mm7./-1b34W;'&:4 &
Ip5Z:; "6L5L&(./ %.76L5 &(5F∇
53P "%'&(./40:r%'Eklh4 &;MxjQs5Fi:4./;r34&'%'./
γ ∈ Γ∞(E1(M) ⊗ SlT ∗M)ml./Z%."& &3o5LI(./ "
∇sγ ∈ Γ∞(E1(M) ⊗ Sl+1T ∗M)J1./&5F
(∇sγ)(X1, . . . , Xl+1) =1
(l + 1)!
∑
ν
(∇Xν(1)γ)(Xν(2), . . . , Xν(l+1)).
z 1l5 w 9:2I(4m.J%'./w &34*342XH35Fk%C5L&(./ 'J1b./&5F*1.3 D./3
S ∈ Sk(M)
(L∇S)(f) = 〈S,∇ksf〉, ∀f ∈ Γ∞(E1(M)),
S.SR
〈., .〉:4;&8/-Ip5i%'./F &5%'&(./Z:-:'3PJ%s=l5#$1:2'&'3 &'M
,5ide./%'&(./
L : CM × S(M) 7→ D(M):;'k(G1l5 S+w*511bI; +.-/103254 6879/1254 " 0 2 /10 /1@ . IJ@ %' 7@ 4IJ254 " 0 - " @1@ % 2 /10 /1@ . ./3^%'./ & +.-/1032:4 6879/1254 " 0 % 4 7 =0 ;@ " 4 7 G <&&E5L11I(%C5L&(./wa5L3 "II(<
L∇
wG3-XY3l5Fkl%C5L&(./Z1b./3 D&./34∇ ∈ CM
M) .3 x' &#$(' x%''&&D&'%'&(./mL 5141'I(./ Ip5o:4;'kl./0:9XH34I(XY3'Q'f
&'3 &R:G%./3 "h3 "G%'Ip5"&XH34M B$:;"(8/G1l5 2 % 0 % - @ % 7 " -=@ -=@ % :@O3:; "6L5L&(./K%.76L5 "p5F
∇ ∈ CM
Ia%s=l5L#E1E:G&"3 "R"3
M:;'k15
R(X, Y )Z = ∇X∇Y Z −∇Y ∇XZ −∇[X,Y ]Z,1.3 E&./34
X, Y, Z ∈ Vect(M)MD,5 25@ /=7 % trR
:c%''&&c%.3 &h34 &c<rI(2 % 0 % -=@ % 4 7'74 Ric
&./F*ID%s=l5L#E19:o'&'3 &"3 M
:;<kl(* ""1'%<f6/#$F1l5L
trR(X, Y ) = tr(. 7→ R(X, Y ).),'1l5L
Ric(X, Y ) = tr(. 7→ R(Y, .)X).
$ ( !
a4+.-/103254 6879/1254 " 0 0J/125- @ %#?#&% IJ@ "KL% 7254?> %M% 032 ;9+.-=4?>$/1@45/1032 % 1b./3 :3HPklh ";
E1(M)'
E2(M)'"*34J.1;' 5L&3 *5L&&.H%(5Fi%s=l5XY36L5 "(;<;
M3?5114I((%5L./
QM : CM × S(E1(M), E2(M)) → D(E1(M), E2(M)),'I(I(oXH34
QM"80J/12:-=@ %# . tMOMSXY3*1b./3 &./34x:(;./#$./ "1=("#ETI.%5I φ : M →
Nml(IN6H('F
QM(φ∗∇)(φ
∗S) = φ∗(QN∇(S)),
1b./3 *./34∇ ∈ CN
<S ∈ S(E1(N), E2(N)).
Q'" IJ@ "KL% 72:4?> %M% 032 ;9+.- 4?>$/1@ 45/1032p./3 IJ@ "KL% 7254?> %M% 032 4?03>$/1@ 45/1032 .tMOM4XH34
QM∇ = QM
∇′
&∇'
∇′ "./FG1 &.S["'%'&6/'#E'Fa;'XY36L5IF'M
QM∇ : S(E1(M), E2(M)) → D(E1(M), E2(M))
'"!34rXY3l5Fk4f%C5S(.J1b./3 D&./34
∇ ∈ CM
M
W./3S["./3 "1b./3 25II(;'8/ GI(';%' &&3 &'m./3N@BgP414I((%'& &.1l5o1l5 I(5&32I(5i:;1b:5%|:
QM
!Ip506L5 "(;'&;MM
G & 5C6L5(I-51b./3 Jh34V:\ ";1b./: "\ 34\XY3w./ 1b./&;' (I ) 5XY3IXH34a5;'1l5 G) M ,'%./#0&M
2 /1032 " 030 ; % - 6 @'; > % 72 " @ 4 %# 0J/12:-=@ %# E1% 2 E2 . %
4 2 % H 25H 4 # -=0 %+.-/103254 6879/1254 " 0,0J/12:-=@ %#?#&% IJ@ "KL% 72:4?> %M% 032 4?03>$/1@ 45/1032 % I " - @ 7 %' 6 @ ;
z`It@O./ &8/(4:-%<&?XY3w./m ./Z &.346/?I('G ";"3I5L:@O<P(w'%|:+.-/103254 6879/1254 " 0 slm+1
H 4?03>$/1@45/1032 %' .h4'H34?"3 Rm
MG./3?5I(I./ah &>'6/gf#E'FZ 511bI ' XY3./2%'./"("&FA%Z ";&34Is5SZ'Z:^XY3'I(I( #E5> "It@O<P(w'%K:@O3KXY3l5Fkl%C5S(. 5L3 "II(i<|14 &.S["'%'6/'#E'Y?Y6L5 "p5F&J%./4&(w3-34?8;;' 5I(5S(.M
,21I(.8/#$F*14 &.S["'%'dslm+1
:sl(m + 1, R)
w3-&./34wf5LI(8/>'h &:o%s=l5#$1:?6/%<'3 &-&34
RmMNjTII(?w28: ";!1l5L oI('o%s=l5L#E1o:
6/'%''3 &
∂i, xi∂i, et xjxi∂i.j+I(I( w &./#$./ "1=Qsl(m+1, R)
''"#E5SP(#E5I(W:l5Iy@ 5LI(8/>'h &Vect∗(R
m):%s=l5L#E1:26%'&3 "o1b./I()H.#E(53PE&3 Rm
MBc:;'&(801l5L +.-/1032:4 6879/1254 " 0 slm+1
H 4?03>$/1@ 45/1032 % 1./34 oI('aklh4 &;G6/'%<f./ "('I(
E1
<E2
3-XY3l5Fkl%C5S(.
Q : S(Rm) → D(Rm)5L&(wd 5(&5F
LXQ = 0, ∀X ∈ slm+1.
UA./F "./oE1 ";&'F2XY3iIt@O<P"'%|:@O30XY3l5LY&kl%5L./V5L3 "II(-1 &.Sf["%<6#$FG(F6L5 &(5F2#E1I(XY3aIt@O<P(w4%?:@B34-XH35Fk%C5L&(./
slm+1
f(F6L5 &(5FLM*]
Q:;"(8/4 34 XY3l5Fkl%C5L&(./ l5S3 "I(IV1 ".S["%<6/#$F
(F6L5 &(5FG'9&φ &'1 &;'&F&2I(.D:@O3 %s=l5L#E1r:4a6/'%'&3 "
X511l5 wgf
l5FGslm+1
mlIN6H('YG'AN< /
φ∗t Q(∇0)(S) = Q(φ∗
t∇0)(φ∗
t S)1.3 E&./34
t ∈ RmR1./34 ./3r )H#ihb./I
Sm"
∇0 :4;&8/cI(5 :;' &65S(.%.76L5 &(5FJ1Ip5SJ%C5.(XY3A:
RmM9) 5 :;<kl(.:AIp5^:; "6/;'c:@O3
%s=l5#$1-:+4&34 &m "FIt@O./-1l5 "6HFW*:;#$./F " XY3φ∗
t
1 &;'& w6/Ip5D%I(5&"1 &. ["%'&6:
∇0 1.3 R&./34 tm4./ 5L3 5?;8F5I#$FD14 &./346;XY3
Q(∇0)w
3GXY3l5Fkl%C5L&(./slm+1
fv;'XH346L5 &(5FLm/1l5 TIt@O(F6L5 "p5%'*1 &. ["%'&6a:QM
,d 59XY3-Ip5i%I(5&"21 &. ["%'&6?:4∇0 &./*1 ";&' "6;?1l5 *It@B5%<(.r:3
:(;.#E./ "1=&#$φt
wG:V!I(5K14 &./1b./1 "(;'&;&346L5F /
LX∇0 = α ∨ id
&X
@O;%' &Ai∂i + Ai
jxj∂i + α(x)xi∂i,
.1RcI('Ai
<IAi
j
"./F2:l5R'.SR
α ∈ Rm∗ M j+J 'ml& Y, Z ∈ Vect(Rm)
ml(I6F
(LX∇0)(Y, Z) = [X,∇0
Y Z] −∇0[X,Y ]Z −∇0
Y [X, Z]
= Y iZr∂irXj∂j
= α(Y )Z + α(Z)Y
51 ">:;'6I./11b#$F'M a>GI(./ "m
d
ds(φ∗
s∇0)x = (φ∗
s(α ∨ id))x
1.3 0.34s ∈ R
mQ1.3 i./3x ∈ Rm
M 54$%'K%.:&(./4mφ∗
s∇0 w
1 &. ["%'&6#$F;'XH346L5I('F2∇0 6Hp5$Ip5 f de./ "#E β
:;<kl(|:0I(5$#E5> "&36L5F0/
βx(Yx) :=
∫ s
0
(φ∗uα)x(Yx) du,
.SRY ∈ Vect(Rm)
mx ∈ Rm
M
N T o aR a a
a.3+ 5141'I(./+%I(+14 &(4%(153P|d 5&Q%./4% "l5FRI('Q%./4<P(.T:G 5 "5M4)T "#$(>' &#$Fm./3*'Z:./.32:;<kl&(./ 8;;' 5I''f&3*.3+:./4./T1I(34+:*:;'5(IQ&34 RI+%'./gP4./T: G 5 ws5K1 ".S["%gf6/+<TI('3 &+I(4+5C6/%GI('+"& &3%<3 "T1 &. ["%'&6MH)W./3 +34de./ &#E5L&(./1I(34*:;'5(II(;Lm4.3 &F6. )./aI(2I('%'&3 G (E 3yM
E) $$&* $ ' ' $]./F
G3V8 &./31b?:0,?<
H3V"./3fv8/ "./31b?de' &#$;?:4
GMo;"f
8/./4a1l5L g<
hI('o5LI(8/>'h &'a:0,(-%./ " &'&1b./:l5F&M]./
P → M3
Hf kh &;1 "(%'(1l5I4&3
MmY'IlXH34
dim M dim G/HMSaG%./4<P(. :
G 5 "5r"3 Pw*3 f de./ "#E
ω&34
P|6L5I('3 &:l5
gXH34N5S(wd 5RI
1 &.1 &;';'*&36L5F / R∗
aω = Ad(a−1)ω1b./3 D&./34
a ∈ Hm
ω(h∗) = h1b./3 D.34
h ∈ Hm
∀u ∈ P, ωu : TuP → gwa34-h ["'%'./ I(;C5L( &LM
,5|%.<P./r: G 5 "5ω1b "#E<9:.%o:G &5"de. &#$ RI(;I;#$F:
It@B5I(8>h "R:4*,(g!:'Q%s=l5#$1 :6%'&3 "T&3
PM a.&./ XH34D&
h ∈ hm
ω−1(h)'"r;<6H(:'#E#$FE;8F5IG
h∗ 8 %'V Ip5^:3P>#$V1 "./1 &;'&;J:4%./4<P(.a:4 G 5L "s5LMJ0@ 5L34 "|1l5L "CmlIy@B'&1l5L%-6/%<./ "('I
g&?:;'%./#$1.&
%./#$#$&3 /
g = g−1 ⊕ h,
g−1
;'5F+3K&./34wfv'&1l5L%D6/'%'&./ &I:g&./#$./ "1=D
g/hM G I(5;'5F+:m
./3o1./36/./o(F ".:34( &?Ip5E &#E5 "XH340&346L5FiXY3 &' 5 ">234&(I|:l5LIp5i"3&0/
4h ∈ h % 2 4 X ∈ g−1 .
/ #&" @
[h∗, ω−1(X)] = ω−1([h, X]).
T 1 o-d 5Cm4IN6H('Yml&Iy@B./ 4.
ϕt
I( l.L:h∗ /
[h∗, ω−1(X)](u) =d
dtRexp(−th)∗u exp(th)ω
−1(X)u exp(th)|t=0
= ω−1([h, X])(u)
1l5 Ad
fv(F6L5 "p5%':ωM
+ ( ! ! % $ ' G ./"(:;' &./4iIt@B5%'&(./ :3 8/ "./31b
G = PGL(m + 1, R)"3 iIt@O"1l5%'
1 &. ["%'&dRP m
Ml,-"s5Lh(I(5S34 H
:-It@O;I(;'#E'F[em+1]
:l54RP m
w
(
A 0ξ a
)
: A ∈ GL(m, R), ξ ∈ Rm∗, a 6= 0/R0
uv: S
'J(I2'@B'&34AXH34H
wVI 1 &.H:3A"#$f :( "%'G0 o G1
m.SRG0
w("./#E. &1=Q
GL(m, R)'W.SR
G1w &./#$./ "1=T
Rm∗ M$B|5Ip5*1 ".S["%<./
&36L5F
π : H → GL(m, R) :
[(
A 0ξ a
)]
7→A
a,+@ 5LI(8/>'h &-:0,-:
PGL(m + 1, R)"
gl(m + 1, R)/RIdM j+I(I-"
:./%G("./#$./ &1=4asl(m+1, R)
'9I(I&o:4;%.#E1b./&G%.#E#$3o"./#$#E:( "%<2:-&.3wfv5I(8/>'h &'
g−1 ⊕ g0 ⊕ g1∼= R
m ⊕ gl(m, R) ⊕ Rm∗.
,+@B&./#$./ &14=("#Ea"G:.;-1l5
[(
A vξ a
)]
7→ (v, A − a Id, ξ).
G '"!%. & &'&1b./:l5L%!:33!w "3%'&3 &K:@B5I(8>h "i:!,(0&3 R
m ⊕gl(m, R)⊕ Rm∗ M4,'5I(8>h "D:-,o%. & &'&1b./:l5LYa G0
mG1
'H
"./F &'&1b%'&6#$F
g0
mg1
m9'g0 ⊕ g1
MD,5 w "3%'&3 & :@B5I8/>h "V:c,V"3
Rm ⊕ gl(m, R) ⊕ Rm∗ "a:./4;-1l5 I*% ".%s=4'G&346L5F /
[v, v′] = 0, [ξ, ξ′] = 0, [U, v] = Uv,
[U, ξ] = −ξU, [U, U ′] = UU ′ − U ′U, [v, ξ] = v ⊗ ξ + 〈ξ, v〉Id,.SR
v<
v′ &./FJ:Z;'I(;#$F& : Rmm
U'
U ′ &./FJ:Z;'I(;#$F& :
gl(m, R)'.1R
ξ<
ξ′5L11l5 w(''YG
Rm∗ M
T #(% ( 0! %
a ;'I(;'#E'Yo:3 8/ "./31bH
1b34 < "06H3 %./#$#E?3i& 54"de./ "# 5L&(./:KIt@O"1l5%$1 ".S["%<d
RP mXY3+I(5("&K&./ ./ "(8/
[em+1]k4P;'Mx]+./ I(
%<&? &5"de. &#E5L./Z:l5aIp5E%5 "&|%5./4(XY30:4RP m
53J6/./&(58/?:
[em+1]m./ ./h4&('Y-34!%./ " &'&1b./:l54%KF "KI(2;'I(;'#E'Y&2:
H'-:4
;I;#$F*:G2
m
Mb]It@O./Z.&2XY3-I(D& 54"de./ "# 5L&(./4D:RP m
:3'1l5 I;I;#$F:
H&./FW36/.HXH34#$Fx:4;'' &#$(;'1l5 I3 "W:4; &6/;'
["3&XY3@B^It@O./ &: " c 4mH
wE:4./%Z1I(.8/; :l54G2
m
6Hp5 Iy@ 5141I(%C5L&(./&36L5F0/
ι : H → G2m :
[(
A 0ξ a
)]
7→ (Ai
j
a,−
Aijξk + Ai
kξj
a2).
a 25@ -725- @ % IJ@ "KL% 72:4?> % P -=@ Mw 5I(./ "3T &;':3%<(.?:43?klh4 &;T:W &gf
1>' &'x:@O./ &:4 & P 2M
5L308/ "./31bHM$o9&II(W"& &3%<3 "QgP("&FCmL%./#$#$
I(?1 &.346/!I0 &;'&3Is5Lo&36L5F E m)+ &./1 M $1M GXY3x'"-I|1b./(Fo::;15 ":-.L &-#$;'&=.:40/
a.hl5L)F5"=f o58F5L. # % 4 2 % - 0 % 7 " @ @ %' I " 0 /10 7 % 4 KL% 72:4?> %% 2 0J/125-=@ %#?#&% % 0325@ %N#&%' 7 # / ' %' IJ@ "KL% 7254?> %' % 7 " 030 % 4 " 0 # 4?0 ;9/14?@ %' /10 2 " @ 4 " 0 - @ M % 2 #&%' 2:@ -372:-=@ %' IJ@ "KL% 7254?> %' -=@ M
j+^d 5mI(5r%./ " &"1.:l5%'K"?%'./"& &3!:KI(5# 5L(>' &i"36L5F /ZIp5r%'Ip5"&K:@O3$%./4<P(. &52./ "&./
∇m./ 5f
&.H% Ip5\"& &3%<3 " 14 &.S["'%'6/P
:./FJIp5 kh & 53 :"&3A:3 1./FZ:%.H./ &:4./;'I(.H%C5I
xi'"G:./4;-1l5 *Iy@B'&'#ihI(2:4 &'1>' &G:@O./ &:4 &
&36L5F /
(xi, δij,−Γi
jk).H,.SRcI
Γijk
&.YI(a )H#ihb./I(': G =4 &(w.1IW:∇M B 1b34a6/; "kl' XY3
%<&*%'./ & ""1./4:l5%'G'"RhK:;'k(*<Th ["'%'6/D8 %'-I(5I(./H:* &5fde./ &#E5L&(./i:4
Γijk
"./3RIt@O 'T:@O3$%s=l58/'#E'F:4G%'.. &:./4;'<T8/ %' Ip5EI(./ &'I((5FI(' )H#ihb./Io:4 G = &".N'Ix: $%./4<P(./421 &.S["'%'&6/gf#E'F;XY36L5IF'M
$$&* $0! % j+ 8/;'; &5ItmF"
ω'"3o%'./gP(./$: G 5 "5E:4;'kla"3 R3
Hftklh ";
1 &%1l5IPml5LI(./ "*5!%./34 &h3 "
ΩwG:;'kl4(-%./#$#$:@O=l5h3:-1l5L
Ω = dω +1
2[ω, ω],
.SRZ:A'"GIp5i:4(; &'F('I(I'G.SRZI(2%' &.H%s='[ω, ω]
'"G:;'k1l5
[ω, ω](X, Y ) = 2[ω(X), ω(Y )], ∀X, Y ∈ TP.
T&]
ω = ωiei + ωije
ji + ωiε
iml.SR
ei
wo34|h5&?:g−1
mei
j
3?hl5"|:g0'
εi30h5&|:
g1
m I-d 5LGXY3Ω'@ 5LY3I(|&3 GI('a%s=l5L#E1a:-6/%<'3 &
6/' "%C53HP #E14I((XY3*XY3a%G4&34 D1'34D'@B;'% "( &a&./349Ip52de./ "#EG&36L5F /
Ω =∑
Kkl ωk ∧ ωl,.SRJI(*de./4%'./
Kkl
&./FoK6L5I3 ":l5gMJ5GIp5K&34Lml./3G:;'%./#!f
1.& "./G53"&ΩJ&G1 ".S["%<(.&I./
g−1
mg0
<g1
/
Ω = Ωiei + Ωije
ji + Ωiε
i.a./341./36/./'|.34 "+:;'kl4( *1l5 w N:
Ω3+de./%<./
κ ∈ C∞(P, g∗−1⊗
g∗−1 ⊗ g)
:-Ip5i#E5(>' &"36L5F&"/
κ(u)(X, Y ) := Ω(u)(ω−1(X), ω−1(Y )).G '"Zde./%'&(./ &' 5 ;8F5I#$F$:;%'./#$1./";J "$%./#$1b./5F& "I./
g−1
mg0
'g1
/
κ = κiei + κije
ji + κiε
i.0@B534 "V1l5L "CmT"ω"r3V%'./gP(./ : G 5 ws5 &3 E3klh4 &;V14 &(4%f
1l5IPmN"-1 ".S["%<(.2&'I(./
g−1
'g0
:./6/F?5S(wd 5( "iI('%.:&(./4&36L5F /
ω−1(h∗) = 0, ω0(h
∗) = h0 ∀h ∈ g0 +g1
mF.1Rh0
"Ip521 ".S["%<./&'I(./
g0:
h (Ra)
∗(ω−1 + ω0) = (5L:
a−1)(ω−1 + ω0) ∀a ∈ Hm.SR 5:
a−1 wIt@B5114I((%5L./ :4
g/g1
:5I(3 fv#!#$a:321l5L *It@B5%'&(./ 5L:S["./(F&5:
a−1 : g:l5
g ]
ω−1(X) = 0ml5I(. &
Xwa6 "&(%5ItM
]./F$I('$8 &./31bG
<H
:4./FEIDw XY3'"&(./ :l5$Ip5 "%'&(./ :^%^%s=l514 "M, =;'./ &>'#E^"36L5FJ;.%; :l5 E -158/ wde./:l5L#E'Y5I /
" 4?2 P- 0
HH 6 @'; IJ@ 4?0 74I/ # -=@!-=0 % >$/1@ 4 ;2 ; M 4
" 0 % " 030 % - 0 % H L" @ M% ω−1
>$/ #&% -=@ /10 g−1
% 7 "$ I " /1032 %' ωi
% 2N-=0 % L" @ M% ω0
>$/ #&% - @ /10 g0
% 7 "$ I " /1032 %' ωij
+.- 4 /1254 L" 032 #&%' 7 " 0 14?2:4 " 0 M% 03254 " 030 ; %' IJ@ ;'7'; % M% 032 % 2 # /!7 " 0 14?254 " 0 - I I # ; M% 032 /14?@ % - 4?>$/1032 %
dωi = −∑
ωik ∧ ωk,
/ #&" @ 4 # % 4 2 % -=0 % -=0345+.- % 7 " 030 % 4 " 0 % /1@2 /10 ω = ω−1 +ω0 +ω1 " 032
# /D7 " - @ -=@ % Ω % 7 "$ I " /1032 %' (0; Ωi
j; Ωj) /12:4 /14?2 # /8IJ@ " IJ@ 4 ;2 ; -=4?>$/1032 %
∑
Kijil = 0,
"Ωi
j =∑ 1
2Ki
jkl ωk ∧ ωl.
T T ]
u = j20f
'"K3 1.(F!:P 2M
'$&X
'"$36/'%''3 o58/'Yo
P 2M5L3c1.(F
um I(5 L" @ M% 79/10 " 0345+.- % θ
:P 2M
wIp5 f de. &#$G65LI(34 &W:l5
Rm⊕gl(m, R)
:4;'klD53|1.(Fu:Ip5# 5L(>' &
&36L5F0/
θu(X) = (P 1f)−1∗e (π∗X),
.SRπ'"Ip521 &. ["%'&(./$%C5L./XH34a:
P 2M"3
P 1M'R.SR
e'"I( &'1>' &
JIt@O./ &8/(4a:Rm
&'1 &;'&F&;|1l5L GI(5i# 5S &%uv:NM %'-%'XH34w:4+:l5L (E 3tmF.$1b3+6/./ +XH34GI(5 f de. &#$9&34 R34
"& &3%<3 "o1 ".S["%<6P?6L5I3 ":5
Rm ⊕ gl(m, R)
:;'k(a1l5 i∗θ
mY.SRiwaI21I(.8/#$F%5./XY32:
P:l54
P 2Mmb&5L"d 5DI*%.:&(./4
:3c=4;./ ">#$ 4M |@ 514 &>2%'! &;'&3Is5LmN(IW<P"03!3XY3!%./4<P(. :G 5 "5i"3
P:./FQIQ1 ".S["%<./x"3
g−1
<g0
%. %'(:'Y+576%*%'I(I :
i∗θ<XH34F5L&("d 5LI(5D%./4:&(./2"311I;#$Fs5 &+ "Ip5S6+53P2%./#$1.5F'
:o&5i%./34 &h3 "M a./36/4./*:./4%:#$./F " 9I(5|1 "./1b./&(. &346L5F / 0MI % -=2/ ' " 74 % @ 7 / +.- % 25@ -725- @ % IJ@ "KL% 72:4?> % P- 0 % 7 " 030 % 4 " 0 % /1@ 2 /10 M>$/ #&% -=@ /10 # - / # , @ % % 4 % sl(m + 1, R) .7 % 2:2 % / ' " 745/1254 " 0 ;2 /1032 0J/125- @ %#?#&%,5G5L3 &5I(;+:RIt@B5"&.H%p5S(.21 &.76H('Y | ' :RIp5l5L34 5I;x:
i∗θ'9:*1 "./1 "(;'&;D#$F&(./4;9:l5DIa&=;./ ">#$ i5&bXY32:Iy@B34(%';:!Ip5r%'./gP(./^XY3+5S(wd 5oIp5r%'./:(. &3114I(;#$F5( "i &'Ip5L&6/K I(5%./34 &h3 "M5WIp5&34Lm ./0:;'&8/ &5a"(#$1I('#E'YW1l5L
θIp5 &'" "(%<(.
i∗θ:*Ip5de./ "#E9%C5.(XY3:P 2M
PM B$.& 5
θ−1
'θ0
"T1 ".S["%<(.&'I(./
Rm'
gl(m, R)M
,5 %./4<P(./ 5"&.H%(;'$Z34Ew "3%'&3 &K1 ".S["%<6P"|5141'I(;KI(5
%./4<P(. 14 &.S["'%'6/ . &#E5I(A: Ip5\w &34%'34 &^1 &. ["%'&6M G @O"J%''&&%./4<P(.J:4 G 5L "s5LAXY3-./34a34&(I(&' &./4D%./4"s5L#E#$F:l54GI(5i&3LM
+ $' $ $ ' #')$a ./3(IGde./:l5L#E'Y5I "I(;c53HP %./4<P(. : G 5 "5 &34 JIXY3'I
./3Thl5L& "./R./34&T./R%."& &3%<(.'"I(*%'./%'14D:4G:4(; &'Fp5S(.(F6L5 &(5F:;'6I(.11;*:5QF""#=tM/]./
P34" "3%'&3 &*14 &.S["'%'6/*<T&./
ωIp5!%./4<P(.A1 ".S["%<6/-./ "# 5I25"&.H%(;'M ]./
(V, ρ)34+ &'1 &;'&F5L./|:4
HM7]
f ∈ C∞(P, V )m
5I(. &DI(5i:(; "F(5L./ (F6L5 &(5F:4f1l5L * 511b./ wG
ωwGIp5ide./%<(.
∇ωf ∈ C∞(P, Rm∗ ⊗ V ):;'kl4(-1l5
∇ωf(u)(X) = Lω−1(X)f(u) ∀u ∈ P, ∀X ∈ Rm.
a.3234I(& "./a53"&Q34i6/ "&./^;' &;'i:iI(5 :(; "F(5L./cY6L5Sf &(5F0/
T ]
f ∈ C∞(P, V )5I./ &
∇ωk
f ∈ C∞(P,⊗kRm∗ ⊗ V )wa:4;'kl15
∇ωk
f(u)(X1, . . . , Xk) = Lω−1(Xk) . . . Lω−1(X1)f(u)1.3
X1, . . . , Xk ∈ RmM
].3 )H#E;< "(&.%'"26/' &"(./J&; &;'ml.3G5h.34&&.$I(5 ]
f ∈ C∞(P, V )5I./ &
∇ωk
s f ∈ C∞(P, SkRm∗ ⊗ V )wa:4;'kl15
∇ωk
s f(u)(X1, . . . , Xk) =1
k!
∑
ν
∇ωk
f(u)(Xν1, . . . , Xνk)
1.3 X1, . . . , Xk ∈ R
mM
F $ $% $0 ( ' #')$ ,ode.%'&(./G;XY36L5 &(5Fa&3
P 1M1' &#$'&&F:40%."& &3 &0:4
de./%<(.2;XY365L &p5LY&2"3 PM a$14I(3'mI(2"%<(.-:@B34 klh4 &;Kl5L&3 &'I
:3$1 &'#E +./ ": &&.YR' %. & &'&1b./:l5L%Gh(346.XY3G5C6/'%a%'Tde./%'&(./4:;'k('*&3 *I(50"& &3%<3 "?14 &.S["'%'6/LMl, &;'&3Is5L&D&36L5F*&.Y"(8/5I(;':l5 QFm1M yM
](V, ρ)
w23| &14 &;"Fs5S(.c:GL(m, R)
m5I(. &./c1b34a:;<kl( 3- &'1 &;'&'Y5L./
(V, ρ′):
H:-Ip5i#E5> &2"36L5F&"/
ρ′ : H → GL(V ) :
[(
A 0ξ a
)]
7→ ρ π(
[(
A 0ξ a
)]
) = ρ(A
a)
1.3 D./34A ∈ GL(m, R), ξ ∈ R
m∗, a 6= 0M
UV5F5FCm'A34&(I(5LY9Ip50 "1 ";"Fs5L&(./ρ′ m4./31b./346/.:./'
Ip5J &I(5L./ F "rI('0de./%<(.|;'XY36L5 "p5F'0"3 P 1M
'!I?de./%'&(./4;XY365L &p5LY&"3
P/N&
P'"-30" "3%'&3 &K1 ".S["%<6/!&34
MmI(5 1 ".Lf
["%<(.J5L3 "II(P 2M → P 1M
:33-1 &. ["%'&(./p : P → P 1M
<./3G5C6/. /
4(V, ρ) %' 2 -=0 % @ % IJ@ ; % 032 /12:4 " 0 % GL(m, R) ./ #&" @ # - /I I # 4 79/12:4 " 0
p∗ : C∞(P 1M, V ) → C∞(P, V ) : f 7→ f p=; 6 034?2 - 0 % 4 KL% 7254 " 0 % 0325@ % C∞(P 1M, V )GL(m,R)
% 2 C∞(P, V )H
,9 ";"3I5LW" ">d 5%I(Ta;<s5h4I(( / Iy@ 5141I(%C5L&(./w./3:@B5hb./ &:ch4(V:;'kl4(i' 5&./A:iIp5E%./ " &"1.:l5%' GF& &I('D;I;#$F:
H'I ["<*:@O./ &:4 & iXH3N@BI(D "1 ";&'FFMN,+@O(S["'%'6H;
:iIy@ 5141I(%C5L&(./Zw-;'6H(:4Fi5:XY3!5r"3 e["'%'6H;|:;%'./3I!:3cd 5XY3@B34Dde./%<(.
Hfv;'XY36L5 "p5FR&34
P'"T%."5F*I(I(./48:@O3*./ "h&
:G1
M
T z 1 &;'&'YmY%./#$#E
Rm<
Rm∗ "./F:4T &'1 &;'&F5L./Rl5L34 &II('+:
GL(m, R)m II('a:<6H(4F2:'o "1 ";&'Fs5L&(./o:
H'a./3a1./36/./
;'5hI( 934|1 "./1 "(;'&;2(#$1. "s5LY&:-Ip5i:4(; &'Fp5S(. (F6L5 &(5F /
4f/I I/1@254 % 032 C∞(P, V )G0 .
/ #&" @ ∇ωf/I I/1@ H
2:4 % 032 C∞(P, Rm∗ ⊗ V )G0
,+ &;'&3Is5L'"34R%'./";XY34%R:+It@Oz*:fvF65L &p5L%:-Ip5i%'./gP4./J: G 5 "5
ωMlj+AN< /
(∇ωf)(ug) = ρ′(g)−1(∇ωf)(u) ∀u ∈ P, ∀g ∈ G0
⇐⇒
(∇ωf)(ug)(X) = [ρ′(g)−1(∇ωf)(u)](X) ∀u ∈ P, ∀g ∈ G0, ∀X ∈ g−1
⇐⇒
(Lω−1(X)f)(ug) = ρ′(g−1)(Lω−1(ρ′(g)X)f)(u) ∀u ∈ P, ∀g ∈ G0, ∀X ∈ g−1.]lIt@O./E511bII(
ϕt
I( l.R:ω−1(X)
'ϕ′
t
I l.R:ω−1(ρ′(g)X)
m/Il&3 K:./%-:26; &kl XY3
d
dtf(ϕt(ug))|t=0 = ρ′(g−1)
d
dtf(ϕ′
t(u))|t=0 ∀u ∈ P, ∀g ∈ G0,./3Z%'./ &-XY3
ϕt(ug) = ϕ′t(u)g ∀u ∈ P, ∀g ∈ G0.
G '"r1 "./1 "(;'&;rw!h\6/;' &kl; / : d 5CmI('i%s=l5#$1ω−1(ρ′(g)X)
'
ω−1(X)&.Y
Rg
fvI(;'D1l5 Ad
fv(F6L5 "p5%':ωM
a4* &#E5 "XH34Dde./:l5#$F5I("TXY3%'* &;"3I5L+@OwT1l5Q6H 5LK8/;gf; &5I/1b./3 Ide./4%'./
Hfv;XY36L5 &(5F/C1b./3 3Qde.%'&(./
Hf ;XY365L &p5LY&
fmI(50de./%'&(./
∇ωf@OwGJ8/;;' 5I15
G1
f ;XY365L &p5LY&M54?I(5 &34Lm./34234I((" "./oI(5r &'1 &;'&F5L./
ρ′∗
:4$Iy@ 5LI(8/>'h &|:,(!:
H&3
VMW]+./3- &511bI(.2XY3K%'"E5I(8>h "!"?&./#$./ &14=K
gl(m, R) ⊕ Rm∗ m.3G576./a5I(. &
ρ′∗(A, ξ) = ρ∗(A), ∀A ∈ gl(m, R), ξ ∈ R
m∗.
5x./Q%5I(%'3I('mL.3Q34&(I(&' &./WIp56/' &&./!(kl&;"(#E5I(T:DI(5a "Ip5S(.<P1 "(#E5FIt@O;XY36L5 "p54% /"
f ∈ C∞(P, V )H
ml5I./ &*.A5
Lh∗f(u) + ρ′∗(h)f(u) = 0, ∀h ∈ gl(m, R) ⊕ R
m∗, ∀u ∈ P.
zaklK:@B5l5LI()H&' +It@O;XY36L5 "p54%D:'+de./%'&(./4m/.3+:("1.&./+:43E ";&34Is5Sd 5%I("36L5F /
T 4
(V, ρ) %' 28- 0 % @ % IJ@ ; % 032 /1254 " 0 % G0% 2 % > 4 % 032
- 0 % @ % IJ@ ; % 032 /12:4 " 0 % H7 "$ M%% I # 45+.-3;N/1-=; - 2 % 7 % 2:2 % % 72:4 " 0 . / #&" @ - 0 % L" 0 72:4 " 0 f ∈ C∞(P, V ) %' 2 H−
;9+.- 4?>$/1@ 45/1032 % 4 % 2 % - #&%M% 032 4
f %' 2 G0−;9+.-=4?>$/1@ 45/1032 .
Lh∗f = 0I " -=@ 2 " -=2 h
/10 g1
jQ6H(:'#E#$FmCIp5H−
;'XH346L5 &(5%'Qw;XY36L5IFDIp5D%'./S["./%<./2:+Ip5
G0−':+I(5
G1−;XY36L5 &(5%LM7,5
G1−;XY36L5 &(5%
w+;XY365LI(F&DoI(5g1−
;XY365L &p5L%D143("XH34G1
'"+3!'&1l5L%D6/'%'&./ &ItM,2 &;"3I5LG@O"3G134(&XY3
G1
5L8/D& &6(5I('#E'FD&34 VM
,: "(' + &;"3I5LTXY3./3R5II(./4x;'5hI( Q:l5LT%%s=l51 "*@B5113&3 34?14 &./1b./&./r;4./%;'|:5DF1l5L8/ /
" 4?2 ω # /7 " 030 % 4 " 0 % /1@ 2 /10 0 " @ / #&% 7 " 0 2:@ -=4?2 % - @ # / 2:@ -372:-=@ % IJ@ "KL% 72:4?> % P
/ ' " 74 ; % O-=0 % 7 " 030 % 4 " 0 /10 2 " @ 4 " 0 ∇ 4
(V, ρ) %' 2 - 0 % @ % IJ@ ; % 032 /1254 " 0 % GL(m, R) . 0 " - /1> " 0 M# / L" @ - #&% - 4?>$/1032 % I " -=@2 " - X ∈ g−1 . f ∈ C∞(P 1M, V )GL(m,R)
% 2 u ∈ P
(∇ω p∗ − p∗ ∇)f(u)(X) = ρ′∗([X, τ(u)])(f(p(u))).
,5ide./%'&(./τ:.YGINwGXH34"&(./J:l54G%'- &;"3I5L*'"G32de./%<./
&3 Pi6L5I3 "*:l5
g1
:;'kl4(-:l5 QF31l58/ 4M,50&'%'&(./ I.%5I(G:
[email protected]%'./"(:>' &:l5LDIp5|:;#$./w 5S(. :2I(5
1 &.1./"&(./o1' &#$'+:@O(:4Fkl QI.%5I('#E'FP
U×H.SR
U"T34$:4.Lf
# 5L(*:a%C5 wG:M
'(:34R:./%G3r*)H">'#Ea:o%'.. &:./4;'9I(.H%C5LI(&3
PM354G%'G%./4:&(./'m4./J1b34G;4./%' Ip5!1 &.1./"&(./r"36L5F&"/
" 4?2 ω # /7 " 030 % 4 " 0 % /1@ 2 /10 0 " @ / #&% 7 " 0 2:@ -=4?2 % - @ # / 2:@ -372:-=@ % IJ@ "KL% 72:4?> % P
/ ' " 74 ; % O-=0 % 7 " 030 % 4 " 0 /10 2 " @ 4 " 0 ∇ " 4 % 032 (V, ρ)
-=0 % @ % IJ@ ; % 032 /1254 " 0 % GL(m, R) % 2
E = P 1M ×GL(m,R) V#&% 6 @ ;/ ' " 74 ; P 1M
7 " @ @ %' I " 0 /1032 4 f ∈ C∞(P 1M, V )GL(m,R)%' 2
# / L" 0 7254 " 0 ;9+.- 4?>$/1@ 45/1032 % @ % IJ@ ; % 032 /1032 -=0 % % 72:4 " 0 ϕ % E . / #&" @
(∇ωk
s p∗)f = (p∗ ∇ks)f + g,
" g %' 2 -=0 %L" 0 7254 " 0 -=@ P " 032 # - % IJ@ %'' 4 " 0 #&" 79/ #&% 0 % 7 " 032:4 % 032 %'
=;@ 4?>1; %' %' 7 "$ I " /1032 %' % ϕ+.- % K - +.- - # - " @1@ % k − 1
]k = 1
mIV &;'&3Is5L wr6H 5G8/ L% I(5 1 ".Lf1.&&(./ M G '"K#!#$!1 &./1b./"./c1b "#E<?5(";#$F-:K:;#$./F " 2I &;'&3Is5L1b./3
kXY3I%./4XH34?15 (:34%'./M
i <a o oH
54i%r%s=51& &Lm .3!5l5LI()H&./40Ip5JXY3w(. :rIt@O<P"'%E:@O3XY3l5Fkl%C5L&(./ l5S3 "I(I^1 &.S["'%'&6/'#E'Fc(F6L5 "p5F :5VI^%C5J1l5 wf%3I(' r: It@O&15%^:'A./1b; &5L'3 &Z:N;' &'Y&('I(Z58/&&5FA'F &
λ<
µf
:4&&;'mDλµ(M)
M a.& &c#$;'&=.:4V%.:3 \:'r%./4%I(34&(.r&34 &1 "<fl5F' /&./34G:@B5hb./ &:ml34?&II(XY3l5F&kl%5L./AgP("&-&'G&'3I('#E'Fo"Ip5KXY3l5Fkl%C5L&(./
sl(m + 1, R)fv;XY36L5 &(5F-%./ " &'&1b./:l5F&0<P"|:l5
RmM o$1I3m.3-./h4'./?3Kde./ "#i3I(0<P1I(%!1b./3 -Ip5 XY3l5Fkl%C5 f
./Z;% "&?'V34&(I(5LYI(5!%./4<P(./A: G 5 "5A4./ &#E5I(-5L&&.H%;|$34"& &3%<3 "E1 ".S["%<6/M G <&Kde./ "#i3I(!'"|<P45%<#$F0Ip5 #O'#E!XY3$%'I(I:./5F!Ip5JXH35Fk%C5L&(./ ;XY36L5 "p5F& :l54
Rm&It@O./\"3h"&&3 53P
:(; "F(5L./DF65L &p5LY&DI:;' &6;'a1l5L "II('M) .3 |%'./#$#E'%' m 5141'I(./?Ip5Z:;<kl&(./ &346L5F$: (E #mW)T "./1 m
1M /
a.3:;'kl4(&"./GI('*./#ih "
γ2k−l =m + 2k − l − (m + 1)δ
m + 1.
a!6L5I3 -:δw 7@ 4?2545+.- % '@BI+<P(w k, l ∈ N
&I?XY31 ≤ l ≤ k
<
γ2k−l = 0M
aJ:'G ";"3I5L:!E#='"a5I(. & /
4δ0 - %' 2 I/ 7@ 4?2:45+.- % . / #&" @ 4 #% 4 2 % - 0 % - 0345+.- %
+.-/103254 6879/1254 " 0 sl(m + 1, R)H9;9+.- 4?>$/1@ 45/1032 %
z 1 ";&'FCm-(F ".:43("./cIy@B./1b; &5L'3 c: :46/' &8/'% 5"&.H%; 34%./4<P(. : G 5 ws5NM G < ./1b; 5S34 r&' 5 IV1 "(%'(1l5I./3(I:V. "%./4" "3%'&(./NM
E) ! #' ( $% ]./
(e1, . . . , em)I(5Ghl5L&D%5./XY39:
Rm'x&./
(ε1, . . . , εm)&5ohl5"
:3l5I2:l5R
m∗ MT
T ,+@ " I3;@ /12 % - @ % 14?> % @ % 0 7 % 1l5 G 511b./ w$Ip5K%./4<P(./A: G 5L "s5L ωwo5LI(./ "D:;'k1l5
Divω : C∞(P, Skδ (Rm)) → C∞(P, Sk−1
δ (Rm)) : S 7→
m∑
j=1
i(εj)∇ωej
S,
.SRi:;"(8/-I21 &.H:3*(F;' &3 'MG 'T.1;' 5L&3 +1b34 < "*%./"(:;' &;%.#E#$934G8;;' 5I(5S(.0%.3 &hb
:^It@O./1b; 5S34 :^:6 &8%' 34&(I(&;A:l5 E FyM,Z1 &.1./"&(./ &3 f6L5F#$./F "FG&G1 "./1 &;'&;*I('1I(3*#E1b./ ws5F'M
4S ∈ C∞(P, Sk
δ (Rm))G0 ./ #&" @ DivωS
/I I/1@ 2:4 % 032 C∞(P, Sk−1
δ (Rm))G0
,+ &;'&3Is5L1b3 ' "+%./F& /I;Q:( "%<#$FW*1l5 f
i:JIp5 :;'k&(./M B2 1b34$;8/5I('#E'YK "#E5 &XY3' KXH34DivωS
w I(5%./F& 5%<(. :2Ip50de./4%'./ (F6L5 "p5F
∇ωS 6/./ D1 "./1.&&(./ 9':-I(5
de./%<(. (F6L5 &(5F2'%'./ws5F
ID : P → Rm ⊗ R
m∗, u 7→m
∑
j=1
εj ⊗ ej.
,oh39:'9 ";&34Is5SR&365LY&9'"D:4o#$&34 & RIt@O;%5 "9|I(5G1
fx;XY365 f &(5%0:KIp5r:N;' &F&p5L&(./cF65L &p5LY&i'-:[email protected];' 5L&3 :4$:46/' &8/'%LMzo3E6C53E(4kl4;'&# 5Iym5L3E6H3r:'9;XY3l5S(. Q' gmY%I(5?%'./:3453%C5I%3IR:3\%'./#$#i34s5S34 |: %0./1;' 5L&3 "!5C6/'% Ip5A:4; &6/;'r: ,(
Lh∗
m1.3
h ∈ g1
MDjxP45#$(./4E.34r:@B5hb./ &: %'cXY3 :./4c%V%C5I%3Ia1./34 It@O./1b; 5S34 D:-:6 "8/%'/
" -=@D2 " - 2 S ∈ C∞(P, Skδ (Rm))G0 .
0 " - /1> " 0
Lh∗DivωS − DivωLh∗S = (m + 1)γ2k−1i(h)S,I " - @ 2 " -=2 h ∈ Rm∗ ∼= g1
)+ &'#E> "#$FCm "#E5 &XY3.cXY3 Ip5 :4; &6/;' :
,(J1l5 $ 511b./ wr 34 %s=l5#$1 :J6/%<'3 & %'./#E#034A5C6/%V3c;<65LI(3l5Sf./ /l&
η1, . . . , ηk−1 ∈ Rm∗ m4./3G5C6/./
(Lh∗DivωS)(η1, . . . , ηk−1) = Lh∗(DivωS(η1, . . . , ηk−1))=
∑m
j=1(Lh∗Lω−1(ej)S(εj, η1, . . . , ηk−1)).0@B534 "-1l5 "m:@B51 ">G3A ";"3I5L:3J%s=l51 &Hm
[h∗, ω−1(X)] = ω−1([h, X]), ∀h ∈ gl(m, R) ⊕ Rm∗, X ∈ R
m,
T .SR I( % ".%s=4'E:l5KI(Z#$#ih4 &Z:Z: &. '"E%'I(3*:
sl(m + 1, R)MQuvI
@O"3GXY3-It@O<P1 ""&(.AXY3?./3G5C6/.o!%5I(%'3I(' *"G;'8F5I(-
m∑
j=1
(Lω−1(ej)Lh∗S(εj, η1, . . . , ηk−1) + (L[h,ej ]∗S)(εj, η1, . . . , ηk−1)).
) ./34 k( 'mJ34I(5FDI(5! &I(5L./ <m.A./h('F
Divω(Lh∗S) − (ρ′∗([h, ej])S)(εj, η1, . . . , ηk−1)
= Divω(Lh∗S) + (ρ∗(h ⊗ ej + 〈h, ej〉Id)S)(εj, η1, . . . , ηk−1).,| ";&34Is5Sa14 &.76H(F5I(./ "Dd 5%I('#E'YG:?Ip5K:;<kl&(./J:
ρ"3
Skδ (Rm)
M
a.3./h4'./'&3
" - @ 2 " -=2 S ∈ C∞(P, Skδ (Rm))G0 .
0 " - /1> " 0
Lh∗(Divω)lS − (Divω)lLh∗S = (m + 1)lγ2k−li(h)(Divω)l−1S,I " - @ 2 " -=2 h ∈ Rm∗ ∼= g1
]
l = 1m Ip5K1 &./1b./"./Z%.t%:|576%iI-I(#$#$
MY, &;'&3Is5L&G:;#$./F "G5I(./ "T1l5 T(:43%'&(./mF' 34&(I(5LYQI(RI('#E#$ ' MuvI+6H('F|' N<Cm&+./ &3411.& I! &;'&3Is5L-6H 5["34&XY3
l − 1m
XY3
Lh∗Div(Divl−1) − DivlLh∗
wa;'8F5I
(m+2(k− l +1)− 1− (m+1)δ)i(h)Divl−1 +DivLh∗Divl−1−DivlLh∗,tMOMl
l(m + 2k − l − (m + 1)δ)i(h)Divl−1.
UV5F5FCmT5l5LI()H&./4!I(r:;<d 534$:@O(F6L5 &(5%'r: Ip5c:4(; &'Fp5S(.(F6L5 &(5F;' &; /
4f ∈ C∞(P, ∆λRm)G0 .
/ #&" @
Lh∗∇ωk
s f −∇ωk
s Lh∗f = −k((m + 1)λ + k − 1)(∇ωk−1
s f ∨ h),I " - @ 2 " -=2 h ∈ Rm∗ ∼= g1
]
k = 0m/5LI(./ "QIp5ade. &#i3I9'"T;'6H:#$#E'F+6 &5(LM
BV1 ".%'>:!5LI(./ "a1l5L a:3%<(.M zo3V6H3c:0Ip5K )H#E;< "(0:4gP414 &"&./XY39.3+5C6/./4+o%.#E1l5L & 'm (IYw+&3 $&5F+:96; &kl XY3@B'I(Ix%'. 4%(:4FXY3l5:A./VIG;'6L5I(34|"3 o3
kft31I
(X, . . . , X)1b./3 .34
X ∈ RmMb,5
T T 11 &'346/-'"G&#EIp5 &Gi%'I(I(o:3ZI(#$#$ /4&./34*:@B5hb./ &:NmIt@O;'6L5I3l5L./ <Ip5i:;' &6/;-:|,(%'./#$#i34'F /
(Lh∗(∇ωk
s f))(X, . . . , X) = Lh∗((∇ωk
s f)(X, . . . , X)).j+&3Lm./ 3(I("iIp5r:;<kl(. :4$I(5 :(; "FI(Ii(F6L5 "p5F0;' &;'$<./3Dd 5L(&.D%.#E#i3 9I(*.1;' 5L&3 "
Lh∗
'Lω−1(X)
:4II(a&./ w2XY3Ip5i:' &> "?gP1 &'&&./V:4'6H('Y
Lω−1(X)Lh∗(∇ωk−1
s f)(X, . . . , X) + (L[h,X]∗(∇ωk−1
s f))(X, . . . , X).)x5 *Iy@B= )H1b.=>'&?:- &;'%3 " &4%mbI(o1 &'#E D' &#$2wo;'8F5I
∇ωk
s Lh∗f(X, . . . , X)−(k−1)((m+1)λ+k−2)(∇ωk−1
s f ∨h)(X, . . . , X).j+A%'?XY3%'./%' &4|I:3HP4>#$2 "#ELm./J34I(&oI(5i1 &.1./"&(./ 'GI(5 &'Ip5L&(./ 9'G./J./h4&('Y
(ρ∗((h ⊗ X) + 〈h, X〉Id)(∇ωk−1
s f))(X, . . . , X).,2 &;"3I5LG1 &.76H('Fo5LI(./ "D:-Ip5i:;<kl(.J:4
ρ∗
M
( ' !# $%&!' 54Q%<&91l5 "&(LmL.3Q:./4./ 39de./ "#i3IgP414I((%'&1b./3 Ip5aXY3l5Lf
kl%C5L&(./|l5L&3 &'I(I('+14 &.S["'%'6/'#E'YTF65L &p5LY&9'$3(I(&5FxIx14 &./1 "(;gf;' :RIp5*:N;' &'Y&p5L&(./-Y6L5 "p5F&R;' &;T<x:RIt@O./1b; &5L'3 ::46/' &8/'%LM
4δ0 - %' 2 I/ 7@ 4?2:45+.- % . / #&" @ # / 7 "$#?#&% 7254 " 0 - /I I # 4?H
79/12:4 " 0 QM : CM × Sδ(M) → Dλ,µ(M)
=; 6 034 %' I/1@
QM(∇, S)(f) = p∗−1
(k
∑
l=0
Ck,l〈Divωl
p∗S,∇ωk−l
s p∗f〉), ∀S ∈ Skδ (M)
=; 6 034 -=0 % +.-/103254 6879/1254 " 0,0J/12:-=@ %#?#&% IJ@ "KL% 72:4?> %M% 032 4?03>$/1@ 45/1032 % 4
Ck,l =(λ + k−1
m+1) · · · (λ + k−l
m+1)
γ2k−1 · · ·γ2k−l
(
kl
)
, ∀l ≥ 1, Ck,0 = 1.
)+ &'#E> "#$FCm./34A:<6/./4A6/;' &kl JXY3 %''&&de./ I(25!3J&' /lIp50de./%<./
k∑
l=0
Ck,l〈Divωl
p∗S,∇ωk−l
s p∗f〉
:./ '& &Hfv;XY36L5 &(5FMLj+I(I9w+de./ &%';#$F
G0fv;'XH346L5 &(5F1l5 QI(5o1 &.Sf
1.&&(./ '1l5 I0I('#E#$ MuvI w-:./%i"3 $5F?:i%'./F /I(' Ip5g1
f;XY365L &p5L%LM4j+I(I1 ". 6HF*: &%<'#E'Y:4*1 "./1.&&(./4 i< <:
T Ip5i "Ip5S(.
Ck,ll(m+2k− l− (m+1)δ) = Ck,l−1(k− l+1)((m+1)λ+k− l).
j+&3Lm+ 34I((&5F!I('K &;"3I5LK:3 %s=l514 " m+./ 1'34K6/./ !XH34AI )H#ihb./I(!14 &(4%(15Ix:
QM (∇, S)w|gP45%'&#$F
Sm<?:./%KIp5rde./ "#i3I
x:4;'klR3GXH35Fk%C5L&(./mXH34b'"R:G1I(34R14 &.S["'%'6/'#E'YRY6L5 "p5F&1l5 G:;'k&(./J:
ωM j+4kmNI(5$l5S3 &5I(;2:|Ip5KXH35Fk%C5L&(./J:;'k(|:
%<&# 5L(>' &R"Q5&"D;<6H(:'Y& /SI(IT1 &.76H('Yx:DI(5Gl5S3 &5I(;T:9Iy@ 5L&&.Sf%(5L./!:@O3G" "3%'&3 &G1 ".S["%<6
P → M1./34 "6H3:@O3G%./4<P(./E:
G 5 "5c. &#E5I(ω 3!%'Ip5"&i:!%'./gP(./o5&./ &"(./V1 &.S["'%'&6/gf
#E'Fx;'XH346L5I('F &3 M
'x:9I(5Gl5S3 &5I(;T:3i "I(><6/'#E'Yx: de./%'&(./4;XY365L &p5LY&*"3
P 1MZde./%<./*;XY36L5 &(5F"3
PM
' ( ,c=;'./ &>'#E \'Z%^XY3'"J: Ip5 k :3 1 "#$(' %s=l514 "(#$1I(XY3Fo:( "%'&#$F2XY3LmXH35:
M"?Iy@B'&1l5%'!3%'I(:('
Rm
'-XY3l5L:∇
'"?Ip5E%./4<P(./ 1I(5Li%5./4(XY3!:R
mmNIp5 de./ "#i3I(
:./o%'. 4%(:4 5C6%!%'I(I: (E=F de./ I(' M < M *'QP ede. &#i3I M <m53!#$./(xXH35:
δ@OwT1l5T% "XY3M,*14=;.#E>'
&34 &1 "l5F+'"+XY39./x%'. $%('FCk,l
%'. 4%(:4FQ5C6%*%'3P0: QP ede. &#i3IM ' 4M <mZ34 %'. E%'('Y?%'./#ihl5L.( &|1 &>'mXY3Tw:J!32:;<kl(.rI(;'8/> "#$F:(; "F2:42It@O./1;' 5L&3 9:2:6 f8/'%LM j+ 1l5 w(%'3I( 'm.L &ide. &#i3Ii1b34 '& &K<P1 "(#$;Lm%./#$#$%'I(I2:!PymlZ "#E':2de./%'&(./4*= )H1b "8/;./#$;'& &XH34M
,501 "346:3 &=;./ ">#$o1 ";%';:'Y1b "#E<D5L3&"N:@B5l5I)H& I(G1 &.SfhI>#$|:|It@O<P(w'%|XH35:
δw3|65LI(34 a%' &&(XY3 /b&311b./"./
XY3@O(IDgP4"&k ∈ N
<r ∈ N
&I(0XH341 ≤ r ≤ k
'γ2k−r = 0
M54%'%./4:&(./'m @O(I<P(w
i ∈ 1, . . . , rIWXY3
λ = − k−im+1
,5I./ &W./i1b34x &'#E1I(5%' I %.H $%FCk,i, . . . , Ck,k
15 '; &.o'+I(5de./%<./ Dwo'%./ "
Hf ;XY365L &p5LY&m%'|XY3"(8/kl-XY3|I(5$%'./If
I('%'&(./QM
:4;'klT%'./ &34*XH35Fk%C5L&(./Kl5L&3 &'I(ID'T1 ".S["%< f6/'#E'F9Y6L5 "p5F&MY]ym/1l5 T%'./F &Lm
λ@B511l5 wFR1l5R?Iy@B'&'#ihI
− k−1m+1
, . . . ,− k−rm+1
mR5I./ &KIGN@ ) 5 15E:4V&.I(34&(./ 5314 &./hI>#$
13&XY3Iy@B. 5LXH3N@BI@O<P"a1l5D:4oXY3l5Fkl%C5S(.sl(m + 1, R)
f;XY36L5 &(5F5L3r&'D: (E=F% E#=yM4)W./3 D ";"3#E' mH./3D576./D#$./f ";-I(2 &;'&3Is5L"36L5F /
# % 4 2 % -=0 % +.-/103254 6879/12:4 " 0 0J/125- @ %#?#&% % 2 IJ@ "KL% 7254?> % HM% 0328;9+.- 4?>$/1@ 45/1032 % 4 % 2 % - #&%M% 032 - 4 # % 4 2 % -=0 % +.-/103254 6879/12:4 " 0 sl(m +1, R)
H9;9+.-=4?>$/1@ 45/1032 % /1- % 0 % (E=F -=@ M = Rm
i
54Q%%s=l51 &Lm.3x1 "./346./QIy@BgP4"&%'D:9XY3l5LY&kl%5L./ l5L3Hf &'I(I('14 &.S["'%'6/'#E'YF65L &p5LY&1./34 9:'*./1;' 5L&3 "D:N;' &F&(ID58(wf5FT'Y& &*"%'&(./4+:@ 5L34 "Qklh ";TXY3*I('xklh ";T:*:'&;'M G./3T5II(./;'5hI( E34 &'Ip5L&(./ ;< "./&VF& & It@O<P"'% :4cII('rXY3l5Fkl%C5L&(./4'Iy@BgP4"&%'?:4*XY3l5Fkl%C5S(.
sl(m + 1, R)fv;'XY36L5 "p5F'9%'./ & "&1b./f
:l5F'2&3 Rm
Mj+cd 5CmbI(i1=4;./#$>4i&3 "1 &'l5F?./h" "6;K53 %s=l514 "1 &;'%;':F-:l54I(|%C5a:2:'&;'&i8;;' 5I(&/IGde./ &#i34I(G6L5Ip5h4I(:l5aI('o&3l5S(../f % &XH34o"3
Rm1.3 oIGXH35Fk%C5L&(./a;XY3 f
6L5 &(5FT1b346F< &G: &%<'#E'Y8/;'; &5I(&;'9-36L5 &;'&;a5L &h 5L( & &3hw3l5FE&#E1I#$F$53P :4; &6/;' 15 "&(II('!I($:N;' &F&p5L&(./(F6L5 &(5F'D1l5 5L11. "G!Ip5i%'./gP4./J: G 5 "5A4./ &#E5I(LM
540./30%' %s=l514 "mWI('?klh4 &;E1(M)
<E2(M)
:;%' &0:50I%s=l51 " ./F1b./3 kh & )H1b23 "1 ";&'Fs5L&(./Z( " &;':3%'&(hI
(V, ρ):
GL(m, R):;'k(Z:ZIp5c#E5> "r&36L5F /Q&.
(V, ρD)3Z "1 ";&'f
s5L&(./ :GL(m, R)
%./ " &'&1b./:l5FKJ3 :(58/ 5L#E#$!:R./348YD
:1 &.Lde./:'3
n < mMJB k4P
λ ∈ R'
z ∈ Z<a.J1b./"
ρ(A)u = | det(A)|λ(det(A))zρD(A)u,1.3 D./3
A ∈ GL(m, R)ml1b./3 D&./34
u ∈ VM
E % ') ! '54|%''&&K&%<./m./3405II(./4h &>'6#$F? 511bI 2I-.&(./2<
I('!#E;<=.H:!F w6/'l5F :5KI( %5: &Z: I(5 XH35Fk%C5L&(./ 1 ".S["%< f6/'#E'Y ;XY365L &p5LY&Z&3
Rm&I(IKXY3@OI(I$&./F <P1.&;'r:l54 Qa<
yM G '1':l5FCmT.3K5I(I./|1 &;'&F& KI('!./34I(|: %'i 5C6L53P\:@O3# 5L(>' &1I3*(F& &(4&>XY3-<o5LI(8/;'h &XH34M
E)5E ' ! $ ( ! #' (# $ G o.hH["'&o./F;<;|:;'kl4(G:l5aI(?1 &'#E %s=l51 "mb# 5XH35:M
'"iIy@B'&1l5L%$3%'I((:4(Rm
m./ 1'34iN'%'&3 |I(-:Fkl%C5S(.-&3 f6L5F /
E1(Rm) ∼= C∞(Rm, V1),
Sk(E1(Rm), E2(R
m)) ∼= C∞(Rm, SkV1,V2
),.SR
SkV1,V2
"1 ";&'F?It@O&15%V ∗
1 ⊗ V2 ⊗ Sk(Rm)M
T,+@ 5LI(8/>'h &-:|,(
Vect(Rm)58/&3 a%a"1l5%'o:@O3|# 5L(>' &-h('
%./4H34 e6./( R%s=l514 " /H&X ∈ Vect(Rm)
'9&S:;"(8/4o3r*)H#ih.I(m
./J5
(LXS)(x) = X.S(x) − ρ∗(DxX)S(x)
.SRρ'"GIt@B5%<(. l5L34 &II(:4
GL(m, R)&34 GI(50klh "M
,+@B'&1l5L%D(E1(R
m), E2(Rm))
:4T./1b; 5S34 &T:(; "FIQwT;XY3(1b;:-Ip5i:;' &6/;?:?,(
L:./4;-1l5 I%./#$#i345L'3 M
E)(+ ' ! % % ' ! % ( -,R8/ "./31b
G = PGL(m + 1, R)58/-&3
RP mM G ./#$#E
Rm1'34 < &E5"&(#$(I;K
It@O./346 " :RP m
:@O;XY3l5L&(./xm+1 = 1
m ./|:("1./"9:@O395%'&(./?I(.H%C5LI(T:
G&34
RmM ,'a%s=l5L#E1:-6/'%'&3 "5"&.H%(;'oK%<&&|5L%'./J&./FG:./4;
1l5
Xhx = −h
&h ∈ g−1
Xhx = −[h, x]
&h ∈ g0
Xhx = −1
2[[h, x], x]
&h ∈ g1
, 1
.SRx ∈ g−1
∼= RmM G 'D%s=l5#$1D:a6/%<'3 &*:;<kl&&'F3"./3wfv5I(8>h "
:Vect(Rm)
XH34"a("./#$./ &1=42sl(m + 1, R)
MuvI'"R(F;' &'&5LYT1b./3 +4./+%C5I%3IQ3I; "('3 &Q:4G &511bI QXY3*I(5&./3f
5I(8>h "g0
'"G &;':3%'&6|<&?:;%'./#E1b./"?%'./#$#E
g0 = h0 ⊕ RE
.SRh0
" &./#$./ "1=$ sl(m, R)
'|.SR It@?; # ; M% 032 - - #&% @ Ew0:;<kl
1l5 ad(E)|g−1 = −Id
M
E) + ' ( ')$ % ' $ ' $ -uvIH<P(w934Dh ["%'&(./?hi%'./Y3:iIy@B'&1l5%'!:o )H#ihb./I(':l52Iy@B'&1l5L%i:'2./1b; 5S34 &2:N;' &F&(Io"3 R
m/Ip5 +.-/103254 6879/12:4 " 0 / 0 % XY324./3D5LI(I(.4.' QAff
M4]b3 )H#ihb./IS ∈ Sk(E1(R
m), E2(Rm))
@O;% "
S(x, ξ) =∑
|α|=k
Cα(x)ξα,
.SRαwa34A#i34I fv:(%'m
ξ ∈ Rm∗ ' Cα(x) ∈ V ∗1 ⊗ V2
m 5LI(./ "D./A5
QAff (S) =∑
|α|=k
Cα(x) (∂
∂x)α.
,5rXH35Fk%C5L&(./QAff
'"-3KXY3l5F&kl%5L./^5 $#$F?;XY36L5 &(5FMj+0 'm (IFwxd 5%'(I+:R6./( WXY3@OI(IT;%s=l548/9I 5%'&(./:RIt@B5I8/>h "R5 $ %'./"&&3;?:G%s=l5#$1G:-6/%<'3 &%'./ws5Fa'aI(;C5 &''9&34 aIy@B'&1l5%':*)#0h./I':4G./1;' 5L&3 ":(; "FI('M
z 1 ";&'FCm./34?1b./346./234&(I(&' oI(5 de./ &#i34I( Sa:5-I(ih434-:@O<PYf1 &#E' D%<&&?XY3l5LY&kl%5L./Z:-# 5L(>' &Y& &&>'XH34"/
4h1, · · · , hk ∈ Rm ∼= g−1 . A ∈ V ∗
1 ⊗ V2 . s ∈C∞(Rm) % 2
S(x) = s(x) A ⊗ h1 ∨ · · · ∨ hk," 0 /
QAff (S) = (−1)ks(x) A LXh1 · · · LXhk . ,521 &'346/'"T: &%<LM/uvI&3 $R:* "#E5 &XY3' QXY3
I('*%s=l5#$1Xhi
"./Fo%'./"5F'M
E) ' !! % ' $γ-jT-34&(I(5LYNIp5DXY3l5F&kl%5L./25 $Lm7./21'34
#i3 NIy@B'&1l5%'T:*)#0h./I:@O3Tw &34%'34 &+:T &'1 &;'&'Y5L./-:Vect(Rm)("./#E. &1=
D(Rm)M jQP1I(%''#E'FCm./Z1b./&
LXS = Q−1Aff LX QAff (S),
.SRS ∈ S(Rm)
'X ∈ Vect(Rm)
M GKXH35Fk%C5L&(./^;XY36L5 &(5F!w5I(. &?3
sl(m + 1, R)−&./#$./ &14=("#EK:EIp5A "1 ";"Fs5L&(./
(S(Rm), L):l5I(5i &14 &;"Fs5S(.(S(Rm),L)
Mzak$:4D#E'&3 " +I(5o:(; "%'*F "*%'+ &14 &;"Fs5S(.mFIy@ 5141I(%C5L&(./
γ : g → gl(S(Rm),S(Rm)) : h 7→ γ(h) = LXh − LXh
5i;'&;?F &.H:3a:l5QtM G <&&|5L11I(%C5L&(./E1'34 '& &d 5%'(I('#E'FD%C5LI(%3HfI(;'?'A%'.. &:./4;'a<o.3a &511bI./*(%'&'a1 "./1 "(;'&;GI('G14I(3(#$1b./ wfs5F& /
- /I I # 4 79/1254 " 0 γ %' 2 - 0 H97 " 7 7 #&% % % >$/ #?#&% H 4 #&% 0 % @ % 2 - /10303- #&% -=@ g−1 ⊕ g0
4
h ∈ g1% 2 k ∈ N . # / @ %' 25@ 4 72:4 " 0 % γ(h)
Sk(Rm)
/ %' >$/ #&% - @ /10 Sk−1(Rm) % 2 %' 28-=0 " I3;@9/12 % -=@ 14 ;@ % 032:4 %# - " @1@ % 7 " % 74 % 032 7 " 0 2 /1032
4h, h′ ∈ g1 . "
0 /[γ(h), γ(h′)] = 0
' ( G ./#E#$γ(h)
'"*3 .1;' 5L&3 :(; "FIb:@B. &: " i%.H $%F*%'./ws5F'm(Iw%./#$1I(><'#E'Y9:;'& "#E;1l5 95i &'"& &(%<./53PZ*)H#ih.I(*%'./ws5F'M
) .3 oI(5E&34|:04./o:;<6/'I(./11b#$F&mIW'"o(F;' &"5F2:N@ 5C6/.( o34<P1 "&"(./J(F& &(4&>XY3-:
γM&G./3G5C6/./
4h1, · · · , hk ∈ Rm ∼= g−1 . A ∈ V ∗
1 ⊗ V2% 2
h ∈ g1∼= Rm∗ . 0 " - /1> " 0
γ(h)(h1 ∨ · · · ∨ hk ⊗ A) =∑k
i=1 h1 ∨ · · · (i) · · · ∨ hk ⊗ (A ρ1∗([hi, h]))
+∑k
i=1
∑
j<i h1 ∨ · · · (i, j) · · · ∨ hk ∨ [hj, [hi, h]] ⊗ A,
" [hi, h]/I I/1@ 2:4 % 032 gl(m, R)
1@=7 % # - 4 "$M" @ I =4 M% " 030 ; /1- 7 /1HIJ4?2:@ %
) 5 :;'kl4./Z:γmIt@O<P1 ""&(.
QAff (γ(h)(h1 ∨ · · · ∨ hk ⊗ A))
wa;'8F5I-
LXh QAff (h1 ∨ · · · ∨ hk ⊗ A) − QAff (LXh(h1 ∨ . . . ∨ hk ⊗ A)).
G '" <P1 ""&(. '"K3 ./1b; &5L'3 !:N;' &F&(I:@O./ &: " 53 1I3kMQ]./
' &#$|:@O./ ": &k'@ 54H34I("/I&3 K1b./3 GI(-6/.( :@B511I(XY3 GIt@O./1b; &5L'3
σ <MNuvI+./3- &'"&$:4./%$J"./#$#E' 2I(& "#E'?:N@B./ ": &K4de; "(34 ./3
;8F5LIk−1
:5QI1 "#$(' W& &#$9:49It@O<P1 ""&(. gM G D: "(' & &#$@O;%' &
(−1)k[LXh A LXh1 · · · LXhk− A LXh1 · · · LXhk LXh ].
,21 &'#E D' &#$2wo:@O./ ": &k'
k + 1Ml,-&%'./:J "#E2'"
−(−1)kA LXh LXh1 · · · LXhk
−(−1)k∑k
i=1 A LX[hi,h] LXh1 · · · (i) · · · LXhk
−(−1)k∑k
i=1
∑
j<i A LX
[hj [hi,h]] LXh1 · · · (i, j) · · · LXhk ,
%./#$#$[hj[hi, h]]
"93r%s=l5L#E1E:a6%'&3 "*%./4"s5LYM4,G1 &'#E T' &#$w*:.346/53Z:@B. &: "
k'
k + 1M4z 14 &;"FCm53r6H3 :Ip5?de./ &#i34I( gm
I(& &#$2:@O./ &:4 &k − 1
:l5
−(−1)k
k∑
i=1
A LX[hi,h] LXh1 · · · (i) · · · LXhk
wagP45%'&#$F
(−1)k
k∑
i=1
A ρ1∗(DX [hi,h]) LXh1 · · · (i) · · · LXhk ,
'I2 &;"3I5L@O4&3o134(&XY3DX [hi,h] = −ad([hi, h]).
,+@ 5L11I(%C5L&(./γ1b./&">:?3?(#$1b./ "5F1 "./1 "(;'&;2:@O(F6L5 &(5%0/
4a ∈ GL(m, R) . h ∈ g1
% 2 4 S %' 2 -=0 $! "$#&% ." 0 /
ρ(a)(γ(h)S) = γ(Ad(a)h)(ρ(a)S)
,5-1 &'346/a"R3$%C5LI(%34Il:( "%'RXY3l34&(I(&9Ip521 &.Sf1.&&(./ 4M
E) + ! ' ( ') 54 %!tm9Ip5 %'./"& &3%<./ :Ip5JXH35Fk%C5L&(./\'"!h5&;' &3 iI(5V%.#E1l5L 5&./ :3 &1b%< & <!:'06/'%<f'3 &R1 &.1 &'9:a% ws5R./1;' 5L&3 ":4 G 5"(#$( :43 &'%./4:r./ &:4 & gM G ./1b; 5S34 &R&./FD:N@B34o1l5 [email protected];' 5L&3 T: G 5&#E
C5L&&.H%;G|Ip5- &'1 &;gf
&'Y5L./(S(Rm), L)
<2:@B534 "!1l5 w2It@O./1b; 5S34 : G 5"(#$( C5&".%'(;
iIp5i "1 ";&'Fs5L&(./(S(Rm),L)
MG./35II(./h &>'6#$Fa5:l514& *I* &;gf&3Is5L&*:!:l5LI(2h34G:-%5I(%'3I(' D%'a./1b; &5L'3 &'M
z 15 "&( R:o#E5(F&l5F["3&XY3@BiIp52kl :a%<&o&%<./mH./3%s=./("(f&./4D32hl5L&
(er, hs, E , εt):
sl(m + 1, R):l5*I(5XY3II(GI(9hl5&'
ei
<
εj:
g−1
<g1
"./F 2(II((48Lfv:35I(''x:l5QI(5XY3II(hj
'"x3Dhl5L&D:h0
MuvIQ5r;<;!5I./ &o1 &./36/;!:l54 QXY3!I(5 hl5"!:3l5I0'@B;'% &
(εr, h∗s,
12m
E , et)'XY3?:-1I(34*./V5m
∑
r=1
[er, εr] = −
1
2E .
a./34G1b./"./G53"&
N = 2∑
i
γ(εi)LXei .
,? ";"3I5LG&365LYaw5LI(./ "G34|8;;' 5I(5S(. :( "%'&|:4|%'I(34<P1.&;:l5 Q/
%' % - " I3;@ /12 % -=@ % / 4 4?@ " 032 @ %# 4 ; % # / /1034 ,@ % -=4?>$/1032 %
C = C + N.
,5r1 &.H%s=l5!;'51b!"-:N@ 55I()H&' 2I(i14 &./hI>#$i:'65LI(34 &21 "./1 &':ZIt@O./1b; 5S34
CMT) .3 E%'I(54m+./3E5II(./i&./34$:@B5hb./ &: k4P $XY3IXH34
.5L./ /%'./#E#$! &'1 &;'&'Y5L./ :4h0
∼= sl(m, R)mSk
V1V2= SkRm ⊗
V ∗1 ⊗ V2
"?:4;%.#E1b./&-'A34|"./#E#$:- "1 ";&'Fs5L&(./G( " &;':3%'&(hI
SkV1V2
= ⊕nk
s=1Ik,s.
) ./34 0%s=l5XY3 &14 &;"Fs5S(.\( " &;:43%'&(hIIk,s
mW.\.&Ek,s
It@O&15%E:&'%'./%. & &'&1b./:l5LYmltMOM
Ek,s = C∞(Rm, Ik,s).o-1I(34m:l5L
sl(m, C)ml./Z%'./&:> "|I(5i&./34wf5LI(8/>'h &2: G 5 ws5J3"3II(
C%./4"3;'r:'0#E5L "(%'|:4p58/.l5I('-:E& 5%' Y3I(IM B\%.&:> " I('
;I;#$F*:C∗ :;'kl4(1l5
δi(diag(a1, · · · , am)) = ai.uvIiw^%./4H3 XY3@O3` )H"&>#$ "(#$1I( : 5%'('cw :./; 15
δi −δi+1, (i = 1, . . . , m− 1)
M,o6%<34 G:4 )Iw:;'kl4%./#$#$o;'5FIp5
#E.;o:-I(5i&./#$#E2:4 5%'('1./"&6/'*'GwG:./;-1l5L
ρS =∑
i
(m − i)δi.
,5Ede./ &#$0:4 2(I(I(8K:sl(m, C)
w2It@O<PH'&./ :iI(5Ede./ "#E|: 2(II((48:
sl(m, R)<o:3a34V1 ".:34a&%C5LIp5 &
(, )"3 oIy@B'&1l5%'|6%'&./ &IW &;'I
8: ";-1l5 I('* 5%'('M G -1 &.H:3&%5Ip5 &&5L"d 5
(δi, δj) =1
2m2(mδij − 1)
5:(δi, 2ρS) =
m − 2i + 1
2m,
S
1.3 D./3i = 1, . . . , m
M) .3 |%s=5XY3 "1 ";&'Fs5L&(./ ( & ";:34%'hI(
Ik,s
:sl(m, R)
mI(5Z "1 ";<f&'Y5L./J%./#$1I(gPkl;'
Ik,s ⊗ C:4
sl(m, C)w5L3&" & ";:3%<hI(2<a.
.& 5µIk,s
&./J1I3*=l534G1b./(:4Mzo3i6H3K:+:4;'kl./x:4+ &'1 &;'&'Y5L./
V1
'V2
mIgP4"&*:+.#ih &' &;'I(
a1
'a2
&IGXY3
ρ∗(Id)|Vi= ai Id.
zaklZ:@ '& &-%.=; "FG1l5 * 5L11. "!Ip5i:4;'kl./ :3 "=d :.;-:l5L% yml./Z:;<klGI &=4d o:@O3J%./314I(
(V1, V2)1l5
δ =1
m(a1 − a2).
B wKA14 &;"FK #E'&3 " :@O;'5hI( -I($ &;"3I5Li14 &(4%(15I+%./4% "l5FIt@O./1b; 5S34
C/
- %' I/=7 % %' $! "$#&%' S(Rm) %' 2 # / "$ M% 14?@ % 72 % %' %' I/=7 %' IJ@ " IJ@ %' % C # - IJ@';'74 ; M% 032 . 4 k ∈ N . # /!@ %' 25@ 4 72:4 " 0 %C
Ek,s
%' 2; / #&% αk,s
Ek,s
. "
αk,s =1
2m(mδ − k)(m(δ − 1) − k) +
m
m + 1(µIk,s
, µIk,s+ 2ρS).
S
z6/'%.& &%s=./ P$:ah5&':35I('mYIt@O./1b; 5S34 C@O;%' &
∑
i
(LXεi LXei + LXei L
Xεi ) +1
2m(LXE )2 +
∑
j
LX
hj LX
h∗j,
tMOMmJ34&(I(5F9Ip5! &'Ip5L&(./ gm
2∑
i
(LXεi LXei ) −
1
2LXE +
1
2m(LXE )2 +
∑
j
LX
hj LX
h∗j.
G ./#$#EC%'./#E#0340576%
LXh
1b./3 G./34h ∈ g−1
∼= RmmNI :. ' "K
%.H $%FG%."5FM uvI&3 $a:./4%?:4|1 ":4 &?3(XY3'#E'Fo%' "#E'
c%'./"(:;' 5L&(./Mbj+c34&(I(5LYGIy@BgP414 &"&./c:iI(5$:4; &6/;'i:!,(M <It@O<P1 "&"(./J:'G%s=5#E14G:4-6%<34 &
Xhm./3./h4./
C = −1
2ρ∗(ad(E)) +
1
2m(ρ∗(ad(E)))2 +
∑
j
ρ∗(ad(hj)) ρ∗(ad(hj)),
.SRad
'"?Ip5r "1 ";&'Fs5L&(./ 5:S[".(F!:g0
&3 g−1
M,'-& "#E'd 5(&5F(F' "6 EIy@B;'I(;#$FK:@Bj+3I(' E1b36/F < "Ad 5%'(I('#E'FK%C5I%3I;K13&XY3ad(E)|g−1 = −Id
M ,5K &w "(%'&(./Z:-I(5!de. &#$2: 2I(I(80:4sl(m + 1, R)JI(5 &./3f5I8/>h4 &
sl(m, R)"|;8/5I(K m+1
m
de./-Ip5rde./ &#$!: 2(I(I(8 :
sl(m, R) I( hl5"
(hj)'
(m+1m
h∗j)"./FT5(4&H:3l5IQ1l5L Q 5141./ w+%''&&
: "(>' &LMJB2J5K5&∑
j
ρ∗(ad(hj)) ρ∗(ad(hj)) =m
m + 1C ′,
.SRC ′ "1 ";&'FZIt@O./1b; 5S34 |: G 5&#E |:
sl(m, R)58(&&5Fi"3
Ik,s./3ZIt@O./1b; 5S34 D: G 5&#E 9:sl(m, C)
5L8/("5F&34 Ik,s ⊗ C
MuvI'"Gh%./4H3 e6./( *15 *<P#$1IOEml1M $XY3?%<a.1;' 5L&3 wG;8F5I
(µIk,s, µIk,s
+ 2ρS)de./(9It@O(:'Y&&;&3
Ik,s
M
E)(F+ ( ' $ % # ( o) ./34 r5l5LI()H&' EIJ"1'%'& &:GIt@O./1;' 5L&3
CmY./3R(F ".:43("./mF%'./#$#E:l54 Q" ymYIt@B5 &h4 &
Tγ(Ik,s)5&".%'(;-!3? &'1 &;'&'Y5L./J( & ";:34%'hI(Ik,s ⊂ Sk
V1V2
/./34G1b./"./
Tγ(Ik,s) =⊕
l∈N
T lγ (Ik,s),
.SRT 0
γ (Ik,s) = Ik,s
'T l+1
γ (Ik,s) = γ(g1)(Tl
γ (Ik,s))ml1./34 G&./34
l ∈ NMb,
"1l5%'T l
γ (Ek,s)"./Fa:4;'kl:-Ip5!#O'#Eo# 54(> "0/
T lγ (Ek,s) = C∞(Rm, T l
γ (Ik,s))1.3 D./3
l ∈ NM ,"1l5%'
Tγ(Ek,s)./F34|1 "./1 "(;'&;2(#$1. "s5LY& /
- %' I/=7 % Tγ(Ek,s)%' 2 2 / .#&% " - !#&%' /=7254 " 0 %
LXh% 2 % LXh . I " - @2 " - 2 h ∈ sl(m + 1, R)
,5 1 &.1./"&(./ 1b "#E<Z: 1 "./346/' A15 (f:3%<(. XY3
Tγ(Ek,s)"|"5hI$"./3|It@B5%'&(./^:
ρ∗(A)1./34 -&./34
A ∈gl(m, R)
MYuvIbw5I(. &Tde./ &%';#$FDws5hIo"./39Iy@ 5L%'./$:Ip5?:; "6;:o,(
LXh
1b./3 *&./34h ∈ sl(m + 1, R)
$%53&?:?It@O<P1 "&"(./ :LXh
M , &;'&3Is5L'@B'&3G13("XY3
LXh = LXh + γ(h).
,5r:;'k&(./c"36L5F&!"-34!8/;4; 5LI((&5L./V:4( &'%'&!:!%'I(I(':./Hf;':l5D% /
G %'./31I : &'1 &;'&'Y5L./(V1, V2)
'",7@ 4?2545+.- %@O(Ib<P(w
k, s&I9XY32Ip5?65LI(34 91 "./1 &
αk,s
511l5L "F53 "1'%'& &2:2I(5 &'" "(%<(. :
C ⊕
l≥1 Tl
γ (Ek,s)M
E) # $ (% $ ' (')$ % ' $ ,? &;'&3Is5LawoI-&3 f6L5F /
4 #&% 7 " - I #&% (V1, V2)0 - %' 2 I/ 7@ 4?2:45+.- % . 4 # % 4 2 % -=0 %+.-/103254 6879/1254 " 0 IJ@ "KL% 7254?> %M% 032 ;9+.-=4?>$/1@ 45/1032 % % S(E1(R
m), E2(Rm))
/10 D(E1(R
m), E2(Rm))
,5o1 "346Dw+55I(./839G%'I(I(' :4./;'Q:l5L8Q' QyM a.3?:./.(%I(:;'W14 &(4%(15I('1l5 &.3%F:R%.#E1I;'&3:M
)+ &'#E> &'#E'FCm* &'# 5L &XY3./ZXY3 1.3 ./34S ∈ Ek,s
I<P(w 33XH34-6%<34 a1 "./1 "
S:26L5I('3 *1 &.1 &
αk,s
:C&IXY3
S = Sk + Sk−1 + · · ·+ S0, Sk = SSl ∈ T k−l
γ (Ek,s)1b./3 *./3
l ≤ k − 1.j+AN<Cml%'%./:4./*1b346Fa@O;% "( "
C(S) = αk,sS(C − αk,s
uv:)Sk−l = −N(Sk−l+1) ∀l ∈ 1, · · · , k
Sk−l ∈ T lγ (Ek,s).
,5G1 "#$(>' &R%./:4./-wQ5S(wd 5&T13("XY3S5L11l5 w('FQ
Ek,s
MS) .3 xI(5:3HP4>#$D'QIp5a& &./&(>'#ELm &#E5 "XH34./QXY3*"
Sk−l+1wT:l5
T l−1γ (Ek,s)
m5I(. &
N(Sk−l+1)5L11l5 w('F9
T lγ (Ek,s)
15 TIp5-1 &./1b./"./ 4MFz 1 &;'&'Ym
T lγ (Ek,s)
&V:;%'./#E1b./"c%./#$#$A34 &./#$#EJ: &%<c:N@B'&1l5%'Z1 "./1 &':
Cm+%'./#E#$ It@O(4:(XY3 IZ&=;. &>#$ MT,5 "w &%'&(./ :JIt@O./1b; 5S34
C − αk,s
uv: r%s=5%3 :K%'-&./3fv'&1l5%'?w-3^#i3I&(1I|"%C5I(5( "i./HfY3Io: Iy@B:F&;LmT:@B.1R Iy@BgP("&%'c'ZIy@B34(%';J:
Sk−l
MNo;'k("&./ZI(5XY3l5Fkl%C5L&(./
Q1l5
Q|Ek,s(S) = S.G @B'"G;'6H:#$#E'Fa34-h ["'%'./M
G '"?h ["'%'&(./ &5L"d 5
Q LXh = LXh Q ∀h ∈ sl(m + 1, R).j+ 'm1b./3 K.34
S ∈ Ek,s
mQ(LXhS)
'LXh(Q(S))
1l5 ws58F I1 &.1 &;';'*&36L5F /
G G&./F:R6/%<'3 &1 &./14 &:C:G6L5I3 T1 "./1 "
αk,s
1l5L &%GXY3:@O3A1l5 "m
C%'./#$#i34A5C6/'%
LXh
1.3 $&./34h'E:@B534 "V1l5L "Cm
LXhS5L11l5 w('FG
Ek,s
15 *Ip5i1 "./1b./&(. 4m ,3 *& &#$-:-:8 &;
kwagP45%'&#$F
LXhSm
uvI(G5141l5 "&('FGTγ(Ek,s)
1l5 *I(5!1 &./1b./"./ M,5D1 &'#E> &+15 "&(Q:TI(5D1 &'346/R5"&3 "95I./ &NXH3N@BI(:.6/'F %'. 4%(:4 M
T 1
E)(+ $ # ( ' % $ (-,5c1 &./1b./"./\"36L5FZ" 5cde./Hf:l5#$Fs5LI(1b./3 Ip5i"3&0/
/M@ %# /1254 " 0
[γ(h), C] = 2∑
i
γ(εi)ρ∗([h, ei])
%' 2 > @9/14 % I " -=@2 " -=2 h ∈ g1.
G ./#$#$93K./1b; &5L'3 x: G 5L&(#$( W%.#E#i3D5C6%Ip5i "1 ";&'Fs5L&(./J%./ " &'&1b./:l5F&m ./A5 /
[LXh, C] = 0,[LXh , C] = 0.
G ;XY3l5S(.*de./Fa'A"./ "&-XH34
[LXh , N ] + [γ(h), C] + [γ(h), N ] = 0.
B1b34E6/./ $d 5%'(I#$F$ 34&(I(5FKI(5 1 "./1.&&(./ XY3[γ(h), N ]@B5Y3IMo-1I3ml.3G576./
[LXh , N ] = 2∑
i
(LXhγ(εi)LXei − γ(εi)LXei LXh).
,A& "#E'c:@O./ &: " w &%'&#$Fc"31;' &3 :l5A%<& <P1 &'&"(./:./6/Fo@B5Y3I(' 13&XY3
[γ(h), C]'"2:@O./ ": & 4MuvI &3 Ko:./%i:4i%./I f
I('%'' *I('& "#E'*:@B. &: " Mj+V34&(I(5LYI(5Kde. &#i3IO gml./V1b3a6./( GXY30I?14 &#$(' * "#E?w
:@O./ &: "-&31b; "(34 *./3A;'8F5I Ml,-&'%./4:A' &#$2@O;%' &
−2∑
i
γ(εi)(LXhLXei + LX[ei,h]).
,*' &#$:@O./ &: "K:5%<&-gP414 &"&./V"./F
2∑
i
γ(εi)ρ∗(DX [ei,h]),
:@O.SRZI(2 &;'&3Is5LM
"( ( % ')"% ( a.3!5II(./|5:l5L14' i(%'9I('0./3(I|14 &;"F;'E:5iI(5V&'%'./ cI(5
&3l5L&(./ %'./3 "hLM
T 5E ' (')$ % ' $ ' $ % (-,5-%'./w &34%'./K:It@B5l5Sf
I(./83-%./3 "h|:0I(5EXY3l5F&kl%5L./c5 $|whl5";i"3 oI(5$:4(; &'Fp5S(.(F6L5 &(5FLM jT N<Cmbc3(I(&5Fa%'II(<f %yml./c1b34a 5L"de./ "#E' a34c )H#!fh.I(
S ∈ C∞(P, SkV1,V2
)'Z3Z./1;' 5L&3 9:(; "FI
Qω(S)58/("5F*"3
I('Dde./%'&(./4f ∈ C∞(P, V1)
'A1b./&5F
Qω(S)(f) = 〈S,∇ωk
s f〉. S
jxP414I((%'&#$FCm &I( )H#ihb./I(S'@B;'% "
sA⊗h1∨· · ·∨hk
5C6/'%s ∈ C∞(P )
m
A ∈ V ∗1 ⊗ V2
<h1, · · · , hk ∈ Rm ∼= g−1
ml5I(. &*./J5
Qω(S)f =1
k!
∑
ν
sA Lω−1(hν1 ) · · · Lω−1(hνk)f,
.SRν15 &%'./3 "|./30I(|1b "s5L&(./4|:'!(:4(%'
1, · · · , k'0.SR
swa%'./"(:;' &;-%./#$#$23Z./1;' 5L&3 :-#i34I1I(%C5L&(./NM
]S ∈ C∞(P, Sk
V1,V2)w
H−;XY36L5 &(5FCm7It@O./1b; &5L'3
:(; "FIQω(S)
2& 5wde./ &#$21l5I*de./%<(.H−
;XY36L5 "p5F&de./%<(.
H−;XY36L5 &(5F'M7j+- 'Cm7"
fw
H−;'XY36L5 "p5FCm'Ip5de.%'&(./
∇ωk
s fwi&'3I('#E'F
G0−;'XY36L5 "p5FLMx,5Zde./%'&(./
Qω(S)fE%./ " &'wf
1.:c:./4%01l5o 3|&'%'./c:E2(M)
M$a.325II(./#$./F " GXH34iIt@O./1'34R#E.H:kl +IG*)H#ih.I(
S1l5 T:'& &#$T%'./ & "%'&deR:@O./ &: "R(4de;' &3 "
1.3 * &'#E;':(' K%'-1 &./h4I(>#$M (+ ( ( ')( ( ' # ' $% 54%<&!"%<(.m
S:;"(8/4 5Z3 ;I;#$F?:C∞(P, Sk
V1,V2)G0
'f ∈ C∞(P, V1)G0
"#E5 wfXY3./XY3-%'Ip5!5"&3 "?Id 59XY3
Qω(S)(f)511l5L "FG
C∞(P, V2)G0
gM)+3("XY3+de./:l5#$F5I('#E'Ym../34I(1 &;'& w6/'Y Ip5
G0f ;XY36L5 "p54%m
./3|&./#$#$014 &(4%(15I('#E'Y?Y&; "&";!1l5L 0Ip5g1
fv;'XH346L5 &(5%'MW,r &;gf&3Is5LD&365LYG'"GIp5!%I;2:26/. -:-.& &-#E;<=.H:0/
/M@ %# /1254 " 0
Lh∗Qω(S)(f) − Qω(S)(Lh∗f) = Qω((Lh∗ + γ(h))S)(f)%' 2 >$/ # / .#&% I " - @2 " - 2 f ∈ C∞(P, V1)G0 . h ∈ g1
% 2 S ∈ C∞(P, SkV1,V2
) ,5K &I(5L./rXY3?./3G5C6/.o!1 "./346 G@O;% "
〈S, Lh∗∇ωk
s f −∇ωk
s Lh∗f〉 = 〈γ(h)S,∇ωk−1
s f〉.G ./#$#ETI(' :43P0#E'#ih "W&./F
C∞(P )fvI(;C5L( &'
Sm IH"3 KQ:R6/;' &k
%<&0 &I(5L./J1./34 o3c*)H#ih.I(0%'./ws5FSXY3Q5EIp5$de. &#$
Xk ⊗ Amb.SR
X ∈ g−1
'A ∈ V ∗
1 ⊗ V2
M 54|%'0%'./:./mNI(K#E'#ih "$:4 8F534%s=@O;%' &
A(Lh∗Lω−1(X) . . . Lω−1(X)f − Lω−1(X) . . . Lω−1(X)Lh∗f)
T ''"a;8F5I
A(∑k
j=1 Lω−1(X) . . .(j)
L[h,X]∗ . . . Lω−1(X)f)
= A(∑k−1
j=1
∑
i>j Lω−1(X)(j). . .
(i)
Lω−1([[h,X],X]) . . . Lω−1(X)f)
−A(∑k
i=1 Lω−1(X)(i). . .Lω−1(X)ρ1∗([h, X])f),
:@O.SRcI(| &;'&3Is5L215 oI(5 1 &.1./"&(./ 4Mj+ 'CmNI(a%s=l5#$1ω−1(X)
'
ω−1([[h, X], X])%./#$#i34&F13&XY3
[[h, X], X] = −2〈h, X〉X.
+ ! ' ( ') % ( # ?,1l5 &5I(I;I(&#$'F &GI('&3l5L&(./4Q1Ip5SD<%'./3 "h&348/8/> ":*:;'k( T3$5l5LI(./8/34D:It@O./1b; 5S34
CM
a.3:;'kl4(&"././3:@ 5Lh./ ":A34A5Ll5I(.8/32:N
J1b./5F
Nω = −2∑
i
γ(εi)Lω−1(ei).
j+&3LmF./3R1.346/./4:;<kl RIT./1b; &5L'3 &Cω
'Cω
15 TI(34 &T &'" "(%gf./53Pr"1l5%'
C∞(P, Ik,s)/l1.3 D./3
S ∈ C∞(P, Ik,s)m./J1b./&
Cω(S) = αk,sSCω(S) = Cω(S) + Nω(S),
.SRαk,s
waI(506L5I('3 *1 &.1 &-:C"3
Ek,s = C∞(Rm, Ik,s)M
,[email protected];' 5L&3 Cω
5!Ip5i1 "./1 &;'&;?"36L5F&0/
" - @ 2 " -=2 h ∈ g1 . "0 /
[Lh∗ + γ(h), Cω] = 0 - @ C∞(P, Sk
V1,V2)G0
W./34G:@B5hb./ &:ml4./3G5C6/./
[Lh∗ + γ(h), Cω +Nω] = [Lh∗, Cω] + [Lh∗, Nω] + [γ(h), Cω] + [γ(h), Nω].j+&3
[Lh∗, Cω] = 0134(&XY3
Lh∗
"5hI(("%s=l5XY3?"1l5%'|1 "./1 "
C∞(P, Ik,s):4
CωM oZIp5 #!#$ # 54(> "m ./3E5C6/./
[γ(h), Nω] = 013&XY3 1l5 I(5i1 &./1b./"./ m
[γ(h), γ(εi)] = 0, [γ(h), Lω−1(ei)] = 0
13&XY3γ(h)
58/9&'3I('#E'Y"3 *It@O"1l5%'2:@ 5L wf &6/;'
SkV1,V2
Mnx(5I('#E'Ym4./3./h4&./4
[Lh∗, Nω] = −2∑
i γ(εi)[Lh∗, Lω−1(ei)]= −2
∑
i γ(εi)L[h,ei]∗
= 2∑
i γ(εi)ρ∗([h, ei]),8/ L%-KIp5
G0
fv;'XH346L5 &(5%'MBZ%./4%I(3a15 *Ip5!1 "./1.&&(./ M
Tj+Z%2XY3N%'./%' &[email protected];' 5L&3
Nωm./34*5C6/./Io &;'&3Is5LD&365LY /
- " I3;@9/12 % -=@ NωIJ@'; % @ > %# / G0
H9;9+.-=4?>$/1@ 45/10 7 % %'L" 0 72:4 " 0
G <&01 &./14 &(;<;?w3|%./4&;XY3'%K:|Ip5K1 &.Sf1.&&(./ r'-:3^d 5XY3KIp5r:N;' &'Y&p5L&(./V(F6L5 &(5Fi1 ";" "6EI(5
G0
f;XY365L &p5L%LM B$5&3%'%"&6/#$Fm1b./3 Q&./34
f ∈ C∞(P, SkV1,V2
)mu ∈ P'
g ∈ G0
/
(Nω(f))(ug) =∑
i γ(εi)Lω−1(ei)f(ug)=
∑
i γ(εi)(∇ωf)(ug)(ei)=
∑
i γ(εi)ρ(g−1)((∇ωf)(u)(Ad(g)ei))=
∑
i ρ(g−1)(γ(Ad(g)εi)(∇ωf)(u)(Ad(g)ei))= ρ(g−1)
∑
i γ(εi)Lω−1(ei)f(u),:@O.SRZI(2 &;'&3Is5LM
$ #(% $ ' ( ')$ % ' $,5 %./w "3%'&(./^:!I(5rXH35Fk%C5L&(./ '"?hl5";K&3 -Ii1 &.hI(>'#Ei:4
6L5I('3 &*1 "./1 ":-It@O./1b; 5S34 CωM
q#E5 &XY3./4G&./34a:@ 5Lh./ ":VXY3|Ip5K%./w "3%'&(./A:0I(5$"%'&(./ M w%'./ &26L5I(5hI(o:l5I(2%5%./34 &hbM
4 #&% 7 " - I #&% (V1, V2)0 - %' 2I/ 7@ 4?2:45+.- % . I " -=@ 2 " -=2
S/I I/1@ 2 % 0J/1032
C∞(P, Ik,s) .4 #!% 4 2 % -=0 % -=0345+.- % L" 0 72:4 " 0 S
/10 C∞(P, Tγ(Ik,s))
2 %#?#&% +.- %
S = Sk + · · ·+ S0, Sk = S
Cω(S) = αk,sS.
S
% I # - . 4 S %' 2 G0
H9;9+.-=4?>$/1@ 45/1032 . / #&" @ S %' 2 G0
H9;9+.- 4?>$/1@ 45/1032 ,5|de./4%'./
SgP("&'9wD3XY3MHuvI"3 K*&#Kf
1I('#E'F!: .& iXY3 I!%'./:(.0:l5L S|&./FK%'I(Ii:4 -.SRCw0 "#$1Ip5%';!1l5
Cω'|.SR
Nw0 "#$1Ip5%';K1l5
NωM,$1b./Y?1 "(f
%1l5I*XY3G1' &#$'&5K:4V &;'&./34: & i;<s5EXY3N
'F6/./T l−1
γ (Ek,s):l5T l
γ (Ek,s)M G !:' & Gd 5o"-' &;5I(;034i1 "./1 &;'&;i:
γ'-15
%./4&;XY3'F./34a5C6./
Nω(C∞(P, T l−1γ (Ik,s))) ⊂ C∞(P, T l
γ (Ik,s)).]:014I(3
Sw
G0
fv;'XH346L5 &(5FCm 5I./ &S'"
G0
f ;XY36L5 "p5FMo|d 5CmSw|.h4'H3 J15 "&( 2:
S 511I(XY3l5F-"3%%'&"6#$F!I('-./1;' 5L&3 "
Nω'WI('1 &.S["'%'&3 " :
T lγ (Ik,s)
&3 "W%./#$1.5F'( & ";:34%'hI(''W%'./1b; 5S(.1 &;'&' "6/'FQIp5
G0
fv;'XH346L5 &(5%'+:de./%<./ e6./( 1 "./1b./&(. gM
G G &;"3I5LR1' &#$'R::;<kl( RIt@O(48/ &;':('YQ1 &%1l5I" "6L5FD-:4;'kl Ip5iXY3l5F&kl%5L./M
]311b./&. XY3TIR%'./31I(V1, V2)
T&./1l5% "&(XY3M,+@ 5141I(%C5L&(./
Q : C∞(P, SV1,V2) → C∞(P, SV1,V2)wo5LI(./ "DIt@O<PH'&./JI((4;C5 &o:4?Iy@ 5L&&.H%(5L./S 7→ S
M,+@ 5L11I(%C5L&(./
Q5!Ip5i1 "./1 &;'&;?#E1b./ ws5Fa&36L5F /
0 /
(Lh∗ + γ(h))Q(S) = Q(Lh∗S),
I " - @ 2 " -=2 h ∈ g1% 2 2 " - 2 S ∈ C∞(P, SV1,V2)G0
,5 1 &'346/ "A&#E14I(#$F 34 5L:l5145L./ :
%'I(I(o:3J&=;. &>#$ MuvI"3 KG:26/; "kl' Ip5!1 &.1 &;';21b./3
S ∈ C∞(P, Ik,s)G0
e1./34 &./34k'
s<MC) ./3 W3-'I
SmIp5Dde./%<(.
Q(Lh∗S)w*:;'k(15 /H%@OwIt@O3XH34a6/'%'&3 *1 "./1 ":Cω
5L11l5 w5F
C∞(P, Tγ(Ik,s)):+6L5I('3 1 &.1 &
αk,s
'W:.YWIQ "#E+:4R14I(3=l5L34W:8/ ";w
Lh∗SML,D#$#ih ":4D8F53%s=49:DIy@B;'XY3l5L./ 5
Lh∗T%'./#$#ET' &#$
:!1I3=l532:8 &;!13&XY3γ(h)
5Lhl5("&!Ii:8 &;!:!"-5 "8/3#$F'MuvI511l5L "F?%I(5( "#$F|
C∞(P, Tγ(Ik,s))MWj+4klmW13&XY3
Q(S)w
G0
f;XY365L &p5LYm I(5V1 "./1.&&(./ (#$1I(XY3$XH34
(Lh∗ + γ(h))Q(S)"K34
6/'%''3 G1 &./14 &-:Cω
:265LI(34 *1 &./14 &αk,s
M
G Z ";&34Is5S 1I34 %s=4(XY3'A1b "#E<&'FZ:@B;<s5h4I(( I(V&=;./ ">#$1 &%1l5IyM
4 #&% 7 " - I #&% (V1, V2)0 - %' 2 I/ 7@ 4?2:45+.- % . / #&" @ # / L" @ H - #&%
QM : (∇, S) 7→ QM (∇, S)(f) = (p∗)−1[Qω(Q(p∗S))(p∗f)]=; 6 034?2 - 0 % +.-/1032:4 6879/1254 " 0,0J/125- @ %#?#&% IJ@ "KL% 7254?> %M% 032 4?03>$/1@45/1032 %
W./34 :@ 5Lh./ ":mTIp5 de./ I(J5 34 &' /RIp5 de./%gf./
Qω(Q(p∗S))(p∗f)"
Hf ;XY36L5 "p5F&MF,5
G0
f ;XY36L5 "p54%*1 ". 6HF:3^=;'./ &>'#E '|:$Ip5 &'# 5 "XY3 M,W5
g1f ;XY36L5 "p54%K1 &.76H('F|:4
&'Ip5L&(./
Lh∗[Qω(Q(p∗S))(p∗f)] = Lh∗[Qω(Q(p∗S))](p∗f)= Qω[(Lh∗ + γ(h))(Q(p∗S))](p∗f)= Qω[Q(Lh∗(p∗S))](p∗f).
j+&3Lm7I(+ )H#ihb./I+1 &%1l5I:4QM (∇, S)(f)
'"W<P45%<'#E'YSM7uvIF&3 K
:.L 9XH342I(a& "#E:41I(34D=l534D:8/ ";:Q(p∗S)
(p∗S
<*:@O34I(& I('* &;"3I5L:3J%s=l514 "M
zo&ymQM(∇)
'"a30XY3l5LY&kl%5L./M j+I(I2w1 ".S["%<6#$FF65 f &(5F21l5 *:4;'kl./ :
ωM
j+4klmlI(5il5L&3 5I&;o:'GXY3l5LY&kl%5L./5&:;'k('G'"G3-%'./&;gfXY3%'|:4?I(5il5L&3 5I&;:2./34GI*./hY["'G34&(I(&;'D:l5I(50de./ &#i34I(M
o '+l N
,Ah434 :4V%A%s=l51 "A'"E:@B;<s5h4I(( K34Jde./ "#i3IJgP1I(%&Z1./34 EI(5XY3l5Fkl%C5L&(./ :4./F-(IQ'"-XY3w./ 5L3 %s=51& & 4M G '&&!de./ &#i34I(igP414I(f% w %'./w &34J^1l5 w( 0:@B.1;' 5L&3 "El5S3 "I(K"3
M'$%'./"&&3
Ip5K8/;4; 5LI((&5L./AE3 ./ ": &|XY3I%./XY30:ode. &#i3Ia53PA:'3P(>#$0< "./("(>#$o. &: ":;[&K13h4I((;'*1l5L ./3l5 " &./3: [*:l5L 3'G:l5L yM
,5rde./ &#i34I(!@O./h4&('Y?8/ %KZ:'?.34I(gP41b./";0' :4;'s5L(IQ:l5L!FyMa./34Z5I(I./ I &511bI(' rh "(>'6#$FJ# 5L( 4./3Z(F6H&./rI I(%<'3 Z%./4&3I *%''"? &;'de;' &'%?@O(I:4;& &-:'G4de./ "# 5L&(./49&3411I(;'#E'Fs5 &'M
E) $ ( ' $G ./#$#E|./cI|d 5 &'# 5L &XY3 orIp5$1l58/ : E 3tm3i%'./gP4./
:@BjT=4 &"# 5LΥ
&34 P 1M
511l5L "'l5FZ 3Vw "3%'&3 &c14 &.S["'%'6/P:./4I(3ri34&'%'&(./
GL(m, R)fv;XY36L5 &(5F
σΥ
:P → P 1M
'I(I(XY3
σ∗Υθ0 = Υ
<+'I(I(9XH34σ∗
Υθ−1
'"T;8/5IlIp5ode. &#$D%C54./(XY3*:4P 1M
M]45%s=l5LY0XY3@
Υ%. & &'&1b./:\34 ";:3%<./ :
P 2M
GL(m, R) e6/.(
%s=l51 " SgmFIp5-&'%'./σΥ
5&".%'(auIy@B34(XY3G &'1>' &G:o%''&&a ";:34%'./
XY3&-1 ".S["'"-&3 uM jxP1I(%&#$FCm./A5K:4./% /
σΥ : P 1M → P : (xi, uij) 7→ (xi, δi
j,−Γijk).(u
ij, 0),
.SRiIΓi
jk
&.YQI(x )H#ihb./I('x: G =4 &(w.1I%./ " &'&1b./:l5FTI(5a%'./gP4./ΥMj+ d 5CmFIp5?%./ " &'&1b./:l54%XY3 6H(F9:@Q< "o:;'% &aF "%./4<P(.
'R&'%'&(./R;'5hIT3Gh ["%'&(./K'Y& &GIt@O"#ih4I(G:R%./4<P(.95L11l5 f'l5FR2Ip5"& &3%<3 "1 &.S["'%'&6/
P<TIt@O"#ih4I(*:'R"%<(.
GL(m, R)f
;XY365L &p5LY& :P → P 1M
M o d 5CmD%''"^%./ " &'&1b./:l54% "A;'6H f:#$#$FK(S["'%'&6/LMTj+I(I( '"E:Z1I(34$"3 e["'%'&6/ /+&
σ′ '"E3Z&'%'./
GL(m, R)f ;XY36L5 "p5F&mH5I(. &9IgP("&3de.%'&(./
α|6L5I('3 &9:l5
g1'GL(m, R)
fv;XY36L5 &(5F'I(IXY3
σ′(u) = σΥ(u). exp(α(u)).,5K"%<(.
σ′ 1b3o5LI(./ "*@O;% "( "?:4?I(5i# 5L(>' &2&346L5F0/
σ′ : P 1M → P : (xi, uij) 7→ (xi, δi
j,−Γ′ijk).(u
ij, 0),
&IΓ
′ijk
"./Fa:4./;'G15 %'"- &'Ip5L&(./ /
Γ′ijk = Γi
jk + αjδik + αkδ
ij,.SRZI('
αj
"./FaI*%'./#E1b./&5F:-I(5 f de./ "#Eo &14 &;"F;'i1l5 αM
]σΥ
"I(50&'%'&(./ %'./ & "&1b./:5Fi32%'./gP4./Υ&34
P 1Mm./
1'34:;<kl( 34|5L11I(%C5L&(./τ : P → g1
:-Ip5!#E5(>' &"36L5F&"/
u = σΥ(p(u)). exp(τ(u)).
]8/l5I./DI2 &;"3I5L&346L5F e6/.( QF1l58/ / " - @2 " - 2 % % 72:4 " 0 GL(m, R)
H9;9+.-=4?>$/1@45/1032 % σΥ :P 1M → P . 4 # % 4 2 % -=0 % - 0345+.- % 7 " 030 % 4 " 0 % /1@ 2 /10 ω = θ−1⊕θ0⊕ω1 /12:4 /14 /1032 ω1|(σΥ∗(TP 1M)) = 0.
]σΥ
"EIp5c"%<(. %'./ & "&1b./:5Fr^3J%./4<P(.Υ
&3 P 1M
m./3E511bII(' &./!%''"V%'./gP(./ : G 5L "s5L I(5 7 " 030 % 4 " 0 % /1@2 /104?0 1- 4?2 % I/1@ Υ
'4./3Ip5!.& "./ΥM
,5J%'./gP(./ : G 5L "s5L ./ "# 5LI(ω5&".%'(;EZIp5Z%Ip5L&&K1 ".S["%<6/
:Υ<aI(5!%./4<P(./A: G 5 "5
Υ(:43&21l5
Υ:N>' &FG3XY3#$F
1l5 !I3 "$%'./#$1./&5F'$:5g1
M oZ1I3m+%'./#E#$rI(5^:4(; &'%ω − Υ@B5Y3IJ"3 $I(K%s=l5L#E1$: 6/'%'&3 " 6/' "%C53HPNmTID<P(wJ3 de./%<./
Γ ∈ C∞(P, g∗−1 ⊗ g1)
&I(IXY3
ω = Υ − Γ θ−1.G '"!de./%<(. '"Hfv;'XY36L5 "p5FLM o!d 5Cm
Γw0:;<klT:KIp5r#E5> "
&36L5F0/
Γu(X) = (Υ − ω)u(ω−1u (X))
.SRu ∈ P
mX ∈ g−1
Mxj+\6 "&3 :rIy@Ad
fv(F6L5 "p5%'!:ω<
Υm I+6F
5I(. &D"3%%'&"6#$FCm&g ∈ H
/
Γug(X) = (Υ − ω)ug(ω−1ug (X))
= (Υ − ω)ug(Rg∗ω−1u (Ad(g)X))
= Ad(g−1)(Υ − ω)u(ω−1u (Ad(g)X)).
0@O3V15 "m+&g ∈ G1
mTIp5^ &'"& &(%<./ :4Ad(g)
g1
< I(5 1 ".S["%<./&3
g−1
:?5K &w "(%'&(./Jg−1
"./Fo;'8F5IKIt@O(:'Y&&;M|@ 53 &?1l5 wCml"g ∈ G0
m I(' &w "(%'&(./4 :Ad(g)
g−1
<Qg1
%. %'(:'Y &'&1b%<6#$F5C6/%oI(95%'&(./4%5./XY3:
GL(m, R)"3
Rm<D"3
Rm∗ <D./ 1'345I(. &D%'./%I3 &LM
,5$de./4%'./Γ &'1 &;'&F&i:./%?3 f &"3 a%.76L5 &(5Fa"3
M %L@B'"
1.3 &XY3./9./3KIt@B511bII(' &./0I(,2 % 0 % - @ % =; L" @ /1254 " 0 (6/./ F*1l5L8/
T
mW1l5 &58/ 5L1= M 4M<M G '" de./4%'./\1b./&">: 4.s5#$#$F0I(5V1 "./1 &;'&;&36L5F! e6./( *I#$#E 4M 4M /
(κ0 − κ0)(u)(X, Y ) = [X, Γ(u).Y ] + [Γ(u).X, Y ]
&u ∈ P
mX, Y ∈ g−1
< &κ0
<κ0
&./FZI('Ede./4%'./r:3' &'&1b%'&6#$Fo15 *I('*%./3 "h3 "G:
Υ':4
ωM
' %&( ( $ ( ' $54D%a1l5 5L8/ 514=mH./3%C5I%3I./RI('&'3 D:a:;'de./ "# 5S(.E:l5L
I(+%51 &.S["'%'&d:TIp5*#O'#Ex# 54(> "+XY3@BI/wx%5I(%'3I(;+:5I(+%5%./4de. &#$!Ip5i1l58 K:4OQFtM
nx P4.0.34i:@B5hb./ &: 34 hl5"ei
:g−1
mei
j
:g0
mεi:
g1
M B 55I(. &
Γ(u)(ei) =∑
j
Γ(u)jiεj,
κ0(u)(ei, ej) =∑
k,l
κ0(u)klije
lk
'
κ0(u)(ei, ej) =∑
k,l
κ0(u)klije
lk.
,5!" "3%'&3 &?:@B5I(8>h "2:?,2:g−1 ⊕ g0 ⊕ g1
d 5DJ&./ w-XY3
[ei, εj] = ej
i + δji Id.
o>I./ &'m(IN6HF
[Γ.ej, ei] − [Γ.ei, ej] = [∑
p
Γpjεp, ei] − [
∑
p
Γpiεp, ej]
=∑
p
Γpj(−epi − δp
i Id) −∑
p
Γpi(−epj − δp
j Id)
= (−Γkjδli − Γijδ
lk + Γkiδ
lj + Γjiδ
lk)e
kl .
]TIt@O./ 511bI(Iδκ0
I(5 :(; "%'κ0 − κ0
m./ ./h('F0:.%E' 34&(I(5LYIt@O;8F5LI(&; I* &'Ip5L&(./4D&36L5F /
(δκ0)lklj = Γjk − mΓkj;
(δκ0)kkij = (m + 1)(Γji − Γij).
0@O3G1l5 wCm/IQde./%<./(κ0)
lklj
'(κ0)
kkij
&./FR(:'F(XY3'#E'FY3II('1l5 D4./ &#E5I(;a:
ω e6/.( *1l5 *gP#$1I(E 1l58/ <M0@ 5L34 "?15 "m4I('
T
de./%<(.(κ0)
klij
"./FDI('%'./#$1./&5F':GIp52de./%<./K;XY36L5 "p5F&a"3 P &'1 &;'&F5FoI4&34 a:-%'./3 &h43 &-%. & &'&1b./:l5LYo!Ip5!%.<P./
ΥM
j+c 'mN5L11'I(I./RI(-'&'3 :?%.3 &h34 &i%'./ & ""1./4:l5F2
Υ'
:;"(8/4./015 fI(5Ade./%<./ ;'XY36L5 "p5FE &'1 &;'&F5F
R&3
P 1MM uvI
6H(F?5I./ &'mN"+./^.ΩIp5 ftde./ &#$0:4$%'./3 "h3 &K:
Υm"Q./^34&(I(&iI(5
:;'k&(./ :Ω:4./;'E:l5L!(ET1l58/ Z<0"+It@O./^:;"(8/4$15
πI(5
1 &. ["%'&(./Z:P 1M
&3 M
/
(p∗f)(u)klij = f(p(u))k
lij
= (p(u)−1Rπ(p(u))(p(u)i, p(u)j))kl
= Ωp(u)(X1, X2)kl ,
&p(u)i
mp(u)j
&./FDIR%./#$1b./5F&i'
j:3r "1b> &
p(u)<D"
X1, X2 ∈Tp(u)P
1M"./Fi'I(0XY3
π∗p(u)X1 = p(u)i
'π∗p(u)X2 = p(u)j
MQ]RIt@O./:;"(8/4D1l5
θP 1
I(5ade./ &#$9%5./4(XY3*:P 1M
m/.!5+XY3θP 1 +Υ
%./4" f3A3c%'./gP(./ :4 G 5 ws5 "3
P 1M 6/./ ,QFa1l58 SgM B21'34
5I(. &28/ %EAI(5Z:4;'kl./ :4θP 1
1 &': " 1b./3 X1
'X2
I('26/%<'3 &
(θP 1 + Υ)−1p(u)(ei)
'(θP 1 + Υ)−1
p(u)(ej)M]It@O./A "#E5 &XY3?XY3
Ω = σ∗ΥΩ0
m
Ω:;'&8/l5FIp5!%'./3 &h43 &-:
Υ e6/.( QF3158/ gm(Ib6F
Ωp(u)(X1, X2)kl = (σ∗
ΥΩ0)p(u)((θP 1 + Υ)−1p(u)(ei), (θP 1 + Υ)−1
p(u)(ej))kl
= Ω0σΥ(p(u))(σΥ∗(θP 1 + Υ)−1p(u)(ei), σΥ∗(θP 1 + Υ)−1
p(u)(ej))kl
= Ω0σΥ(p(u))(Υ−1(ei), Υ
−1(ej))kl
= κ0(u)klij.
o2d 5m4./A5!:@O3-1l5 wσΥ∗(θP 1 + Υ)−1(ei) = Υ−1(ei)
M jTJ 'm
Υ(σΥ∗(θP 1 + Υ)−1(ei)) = ei
J6/ w3c:43Ad 5XH34σ∗
ΥΥ'"o;8F5I
θP 1 +
Υ + σ∗ΥΥ1
6/./ <QF01l58/ SJ<^: Ip5:4;'kl./ :Υ
XY3i:cXY3
Υ1|(σΥ∗(TP 1M)) = 0M
0@B534& &V15 "mκ0(u)(X, Y ) = κ0(σΥ(p(u)))(X, Y )
1.3 $&./3X, Y511l5L "'l5Fa
g−1
6/./ F1l58 9'G./Z1b34a5I./ &*%'./%'I(3 "MaJ1b3Z:-%C5I%3IN1' &#$'G5I./ &*:@O./h4& *It@O<P1 "&"(./Z:3Z'&'3 G:
:;'de. &#E5L./ !1l5 w( *:4 &'Ip5L&(./ ' /
Γjk =Rickj
1 − m+
m trRjk
(m + 1)(m − 1),
.SRRic
<trR
&'1 &;'&F&FI('Rde./4%'./R;XY365L &p5LY&&3 PXY3b%. & &'wf
1.:Fx &"1'%'&6/'#E'FT53|&"3 x:Dq(%'%Y'QaIp5* &5%:9I(5G%'./3 "h3 &LM
∇ωl
Divωl 1
!! $ ∇ωl
Divωl
54I*h34R:@O./h4'( +:4Tde./ &#i34I(x<P1I(%+1b./3 TI(5XH35Fk%C5L&(./m./3D576./Dhb"./( :%.l5 &aI('9:;<6/I./11b#$F*:4D./1b; &5L'3 &
∇ωl
'Divωl 'Jde./4%'./Z:@O./1;' 5L&3 "&3
MM
]./Υ34E%'./gP(./ &3
P 1M:.l5F?I(3 3$:; "6L5L./ %.Sf
6L5 &(5F∇
'|5141l5 "&l5LY0 3$"& &3%<3 "E1 ".S["%<6/PMG..
ωI(5
%./4<P(.J:4 G 5L "s5LA"3 P5&".%'(;'M
]./J;8F5I#$F(V, ρ)
34 "1 ";&'Fs5L&(./ :4GL(m, R)
:35F3J &'1 &;'&F5L./
(V, ρ∗):
gl(m, R)MR]./ .
ρ(l)∗
Ip5 &'1 &;'&'Y5Sf./ %C5.(XY3 &3
⊗lg∗−1 ⊗ V
<V"s ∈ C∞(P 1M, V )GL(m,R)
m5LI(./ "
F ls := ∇ωl
(p∗s) − p∗(∇ls)w:./4;01l5L GIy@B:3%<./Z&365LY& 6/./ DQF
1l58/ /
F 0s(u) = 0
F ls(u)(X1, . . . , Xl) = ρ(l−1)∗ ([Xl, τ(u)])(F l−1s(u))(X1, . . . , Xl−1)
+Sτ (Fl−1s(u))(X1, . . . , Xl−1)
+S∇(F l−1s(u))(X1, . . . , Xl−1)
+SΓ(F l−1s(u))(X1, . . . , Xl−1)
+ρ(l−1)∗ ([Xl, τ(u)])(p∗(∇l−1s)(u))(X1, . . . , Xl−1).
G '"?<P1 "&"(./J@O;%' &a%'./#$#E234?"./#$#E:4-& "#E'G:4?I(5ide./ &#$
aρ(t1)∗ (β1) . . . ρ(ti)
∗ (βi)p∗∇js
.SRawT3!&%5Ip5 &LmLI
βl
&./F+:'+% ".%s=<T&; ";xd 55Fx(F' "6/'( x% fs55 &8/34#E'Y&
Xl
mIR:(; "FII(R&; ";∇rΓ
;<6L5I(3;'&3 D:N@ 53 &'5 &83#E'F
Xl
'RIy@ 5141I(%C5L&(./τMY,
tj1 &'#E "T5 &8/34#E'Y&
X1, . . . , Xtj&./Fo;'6L5I(34;5L1 &>'oIt@B5%<(.A:ρ
(tj )∗ (βj)
mIo534 "o5 &8/34#E'Y&5141l5 5L(wf5FD0I('3 9: "./&a"./F*;'6L5I3;95C65LYM4,'9 &5"de. &#E5L./
Sτ
mS∇
'SΓ&./FG:;<kl(':-Ip5i#E5(>' &2"36L5F /
|,T@B5%<(.2:Sτ
&'#E1I(5%x%s=l5XY3+' &#$aρ
(t1)∗ (β1) . . . ρ
(ti)∗ (βi)p
∗∇j
1l5 3-&.#E#$:-' &#$*XY3@O./V./h('FJ&3hw3l5F
−12[τ, [τ, Xl]]
$34τM
S|,W5V& 54"de./ "# 5L&(./S∇
&'#E14Ip5%' %s=l5XY3r& "#Er:5F l−1 1l5L
3&./#$#$*:* "#E'[email protected]./h('FT$&3hw3l5F23Γ./3E
3T:T&' :N;' &'Y&('I(I('5:;' &6;T%.76L5 "p5F∇Xl
M jT?./34& &LmC./5 ["./34&|3J' &#$?"311I;#$Fs5L( &-:l5LaIXY3'I
∇js"a &'#E14Ip5%';
1l5 ∇Xl
(∇js)M
∇ωl
Divωl |,W5\& 5wde./ &#E5L&(./
SΓ
&#$1I(5%c'4kl %s=l5XY3^' &#$c1l5 34&./#$#E:' &#$RXY3@B. ./h4&('YR' &34h"&35F
Γ(u).Xl
-3τM
j+ d 5Cm%<!5I8/./ "=4#Ei1b3 '& &K "6Hp5I#$F-I(;C5L &(";K:EI(5J#E5> "&36L5F0/
% =;> %#&" I I %M% 032 % ∇ωl
(p∗s)(X1, . . . , Xl) - "1 H
2:4 % 0327 "$ M% -=4?2
∇ωl
(p∗s)(X1, . . . , Xl) = ρ(l−1)∗ ([Xl, τ ])(∇ωl−1
(p∗s))(X1, . . . , Xl−1)
+Sτ (∇ωl−1
(p∗s))(X1, . . . , Xl−1)
+S∇(∇ωl−1(p∗s))(X1, . . . , Xl−1)
+SΓ(∇ωl−1(p∗s))(X1, . . . , Xl−1).
4f ∈ C∞(P 1M, ∆λ(Rm))GL(m,R) .
/ #&" @ ∇ωl
(p∗f)(X, . . . , X) %' 2 -=0 % 7 "$! 4?0J/14 " 0 # 4?0 ;9/14?@ % % 2 % @ M%' %N# / L" @ M%
(⊗n−1τ ⊗ p∗(⊗nl−2∇l−2Γ ⊗ . . . ⊗⊗n0Γ ⊗∇qf))(X, . . . , X). 4 # - " 0 0 " 2 % T (n−1, . . . , nl−2, q)
-=0 2 %# 2 % @ M% . T (n−1, . . . , nl−2, q) " 030 %
# 4 % - /10 #&% =;> %#&" I I %M% 032 % ∇ωl+1(p∗f)(X, . . . , X)
(−λ(m + 1)− 2l + n−1)T (n−1 + 1, . . . , nl−2, q) + T (n−1, . . . , nl−2, q + 1)
+
l−2∑
j=−1
njT (n−1, . . . , nj − 1, nj+1 + 1, . . . , nl−2, q).
B 6/./D'JN<*d 5%I('#E'YXY32It@B5114I((%5L./$:Ip5i1 "#$(>' &21l5 w2:-It@B5I(8/. &&=#$a:.
(−λ(m + 1) − 2l)T (n−1 + 1, . . . , nl−2, q).,5K:43P(>'#E-1l5L ":./4-XH35F!II(
n−1T (n−1 + 1, . . . , nl−2, q).,5!& &./&>#$1l5 w(o%./F "(h34|
T (n−1, . . . , nl−2, q + 1) +
l−2∑
j=0
njT (n−1, . . . , nj − 1, nj+1 + 1, . . . , nl−2, q).
,5KXY3l5S &>#$21l5 w(o &4:VXY3l5Fa!I(I
n−1T (n−1 − 1, n0 + 1, . . . , nl−2, q).
BA'A:4;:3G5I(. &5&;#$FI2%./ "./I(I(5( "a"36L5F /
∇ωl
Divωl 4
f ∈ C∞(P 1M, ∆λ(Rm))GL(m,R) . ∇ωl
s (p∗f) %' 2- 0 % 7 "$! 4?0J/14 " 0 # 4?0 ;9/14?@ % % 2 % @ M%' %# / L" @ M%
τn−1 ∨ p∗((∇l−2s r)nl−2 ∨ . . . ∨ rn0 ∨ ∇q
sf)," r
=; 4 10 %# /I/1@254 %$ ;2:@ 45+.- % 1-O2 % 0 % - @ % 4 7'74 14?> 4 ; % I/1@ 1−m 4 # - " 0 0 " 2 % T (n−1, . . . , nl−2, q)
-=0 2 %# 2 % @ M% . T (n−1, . . . , nl−2, q) " 030 %
# 4 % - /10 #&% =;> %#&" I I %M% 032 % ∇ωl+1
s (p∗f)
(−λ(m + 1)− 2l + n−1)T (n−1 + 1, . . . , nl−2, q) + T (n−1, . . . , nl−2, q + 1)
+
l−2∑
j=−1
njT (n−1, . . . , nj − 1, nj+1 + 1, . . . , nl−2, q).
G../4i./34i:@ 5Lh./ ": XY3 Ip5V15 "&( *)H#E;< &XY3:
Γ& &;:43i
r15 i5F&( )H#$;'& &(E:3 &"3
trRM uvI&3 KK5LI(./ "0:
.& TXY3o"∇ωl
(p∗f)(X, . . . , X)'"9;8F5I ?3G%./#0h(l5L(&.$I(;5( "*:
' &#$*:-Ip5ide./ "#E
(⊗n−1τ ⊗ p∗(⊗nl−2∇l−2Γ ⊗ . . . ⊗⊗n0Γ ⊗∇qf))(X, . . . , X),5I(. &
∇ωl
s (p∗f)wE;'8F5I*^Ip5c%'./#ihl5&./ I(;5( " %. & &'&1b./:l5LY&Z:
' &#$*:-Ip5ide./ "#E
τn−1 ∨ p∗((∇l−2s r)nl−2 ∨ . . . ∨ rn0 ∨ ∇q
sf).j+ 'Cm9I( :43P : "(' &r'&'3 &Z"./FJ5I(./ " ;'8F53P 13("XY3@BI(r"./F./34RIT:'3P$ )H#$;' "(XY3'9'RXY3@O(I+&./F;8F53HP$I(./ "&XY3@O(I(+"./FR;'6L5I(34;RX l
M
q#E5 &XY3./4*XH342It@B5%'&(./r:4It@B5I(8./ &=#$&3 *Io' &#$8/;4; &XY32:3:;'6I./11b#$Fi:
∇ωl
s (p∗f)1'34 < " &;'&3#$;LMQj+\N<Cmx%''"r5L%'./
:./4-&./34:@B5hb./ &:
(−λ(m + 1) − 2l + n−1)T (n−1 + 1, . . . , nl−2, q).j+I(I(o:./-'&3
n−1T (n−1 − 1, n0 + 1, . . . , nl−2, q).j+4klml'I(I(ad 5G58( DI(5i:; "6L5L./ %.76L5 "p5F
∇s
&34
(∇l−2s r)nl−2 ∨ . . . ∨ rn0 ∨∇q
sf.,2 &;"3I5LG&36L5F;%'&"&|3JI('#E#$ /
4S ∈ C∞(P 1M, ∆δRm ⊗ SkRm)GL(m,R) .
/ #&" @ ∇ωl
(p∗S)(X1, . . . , Xl)%' 2 - 0 % 7 "$! 4?0J/14 " 0 # 4?0 ;9/14?@ % % 2 % @ M%' 7 " 0 2:@ -=4?2
%D# / /1034 ,@ % -=4?>$/1032 % 0 ;>$/ # - % - / " @ p∗(∇qS) -=@!7 % @ 2 /14?0 Xi% 2
" 0,7 " 032:@ /=72 % #&% @ ; - # 2 /12 - 07 % @ 2 /14?0 0 "$! @ % % L" 4 /1> % 7 τ 07 " 0325@9/=72 %
% 0 - 4?2 % #&% $! "$#&% "1 2 % 03- /1> % 7 %' 2 % 0 % -=@ % % 1@ ; "1 2 % 03- % 0
∇ωl
Divωl T7 " 032:@ /=72 /1032 %' p∗(∇tΓ)
/1> % 7 t + 1 Xi 0 - # 2:4I # 4 %! ;25@ 45+.- %M% 032
#&% @ ; - # 2 /12 I/1@- /1-=2:@ %' Xi N0L6 0 . " 0 - # 2:4I # 4 % #&% 2 " -=2 I/1@ %' 0 "$! @ %'
"1 2 % 03- % 0 ;>$/ # -/1032 τ -=@ %' Xi% 2 %' p∗(∇tΓ) - @ t + 2 Xi
B 6; &klo'r 'D5(";#$FDXY3aIt@B5%'&(./$:aIt@B5If
8/./ "=4#Ea"s5Lh(I(&oIp5ide./ &#$:.;?:l54GIy@B;'./%';M
4S ∈ C∞(P 1M, ∆δRm ⊗ SkRm)GL(m,R) .
/ #&" @ Divωl
(p∗S) %' 2 -=0 % 7 "$! 4?0J/14 " 0 # 4?0 ;9/14?@ % % 2 % @ M%' % # / L" @ M%
〈τn−1 ∨ p∗((∇k−2s r)nk−2 ∨ . . . ∨ rn0), p∗(DivqS)〉.
4 # - " 0 0 " 2 % T (n−1, . . . , nl−2, q)-=0 2 %# 2 % @ M% . T (n−1, . . . , nl−2, q)
" 030 %# 4 % - /10 #&% =;> %#&" I I %M% 032 % Divωl+1
(p∗S)
(γ2(k−l)−1(m+1)+n−1)T (n−1 +1, . . . , nl−2, q)+T (n−1, . . . , nl−2, q +1)
+
l−2∑
j=−1
njT (n−1, . . . , nj − 1, nj+1 + 1, . . . , nl−2, q).
G./3a576./GK%C5I%3I
(∇ωl+1
(p∗S)(ei1 , . . . , eil+1))(εi1 , . . . , εil+1).
G ./#$#EI(5!1 &'#E> &215 "&(:3J:;<6/I./11b#$Fa:
∇ωl+1
(p∗S)(ei1 , . . . , eil+1)
&'I(./ZIt@B5I8/./ &=#$G"
(ρ(l)∗ ([eil+1
, τ(u)])∇ωl
(p∗S)(u))(ei1, . . . , eil),./3G5C6/.:@ 5Lh./ ":V!%5I(%'3I('
[(ρ(l)∗ ([eil+1
, τ(u)])∇ωl
(p∗S)(u))(ei1, . . . , eil)](εi1 , . . . , εil+1).
G '"-: "(>' &-<P1 ""&(.VwG;8F5I-
[ρ∗([eil+1, τ(u)])(∇ωl
(p∗S)(u)(ei1, . . . , eil))](εi1 , . . . , εil+1)
−l
∑
j=1
(∇ωl
(p∗S)(u)(ei1, . . . , [eil+1, τ(u)]eij , . . . , eil))(ε
i1 , . . . , εil+1),
tMOMl
[ρ′∗([eil+1
, τ(u)])(∇ωl
(p∗S)(u)(ei1 , . . . , eil)(εi1 , . . . , εil))](εil+1)
+l
∑
j=1
(∇ωl
(p∗S)(u)(ei1, . . . , eil))(εi1 , . . . , εij [eil+1
, τ(u)], . . . , εil+1)
−
l∑
j=1
(∇ωl
(p∗S)(u)(ei1, . . . , [eil+1, τ(u)]eij , . . . , eil))(ε
i1 , . . . , εil+1),
&ρ′ :;"(8/ Iy@ 5L%'./:4
GL(m, R)&3 Ir*)H#ih.I( : :8 &;
k − lM
, :43P(>'#EA'E &.(&>#$ZI((8K:AIt@O<P1 &'&"(./ 14 &;%';:'F :./4F &'&1b%'&6#$F
2l<
−2l' &#$*:l5LI("XY3I
n−1'"G538#E'Y&;2:@O3
3;MY,'3 &R%./F& &(h434./T@B5=(IFT:./4%M BE%'./"5La5&;#$FRXY3Ip5i1 "#$(>' &2I(8/:4./-XY3l5Fa$'I(I
γ2(k−l)−1(m + 1)T (n−1 + 1, . . . , nl−2, q). %o53rI#$#ELmH. 6/./DXY3I(5|:'3P(>'#E15 "&(G:It@B5I8/./ &=#$*:./4
n−1& "#E'9.SR
n−1:<6F
n−1 + 1MH,oI#$#E1b &#$'D53"&b:#$./F &'
XY3-Ip50 "./("(>'#E15 "&(2:-It@B5I8/./ &=#$G%./F "(h34|
T (n−1, . . . , nl−2, q + 1) +l−2∑
j=0
njT (n−1, . . . , nj − 1, nj+1 + 1, . . . , nl−2, q).
,5KXY3l5S &>#$21l5 w(o &4:VXY3l5Fa!I(I
n−1T (n−1 − 1, n0 + 1, . . . , nl−2, q).
q#E5 &XY3./4*XH342It@B5%'&(./r:4It@B5I(8./ &=#$&3 *Io' &#$8/;4; &XY32:3:;'6I./11b#$FD:
Divωl
(p∗S)1b38'& &a &;"3#$;M4j+r 'm%''&&5%<./
:./4-&./34:@B5hb./ &:
(γ2(k−l)−1(m + 1) + n−1)T (n−1 + 1, . . . , nl−2, q).j+I(I(o:./-'&3
n−1T (n−1 − 1, n0 + 1, . . . , nl−2, q).j+4klml'I(I(ad 5G58( DI(5i:6 &8%'
Div"3
〈(∇k−2s r)nk−2 ∨ . . . ∨ rn0, DivqS〉.
' ( &*+! %& zo3K6H3E:4T1 &.1./"&(./x1 &;'%;:4F'm4I(5XY3l5Fkl%C5L&(./K'@B;'% &T%./#$#$
3-%.#ih(5("./ I(;C5 &a:2 "#E':-Ip50de./ "#E
〈〈τn−1 ∨ p∗((∇k−2s r)nk−2 ∨ . . . ∨ rn0), p∗(DivqS)〉, p∗(∇l
sf)〉.5%''&&gP1 &'&&./m (I&3 K :"T1 &;'.%'%31b W:' &#$1b./3 I('&XY3In−1 = 0
M jTJ 'ml&3141./"./GXY3-It@O<P1 &'&"(./
k∑
j=0
〈aj, τj〉
S
:l5TI(5XY3II(DIQde./%<./aj
&.YHf ;XY36L5 "p5F&Q"./
Hf ;XY365L &p5LY&M
a.&./R./349:@B5h. &:rXY3Lh∗τ = h
1b./3 &./34h ∈ g1
e6./( FN1l58/ <M
,?d 5oXY3|It@B511I(%5L./ :Lh∗
, *:./401b./3 .34
h ∈ g1
.3:*5I(./ "*XY3
k∑
j=1
〈jaj, τj−1〉
wH34ItmC:.%Hfv;'XY36L5 "p5FCM'j+-;' 5FI+1 &.H%'&"3m7./2& &./346T1./34 Nkl
XY3ak = 0
M B2 ' :;:43-5LI(./ "o:i14 &.H%s=! 1 ".%s=4iXY3!I('ode./%<(.aj
&./FGY3I(I*1b./3 j6L5 "p5F*:4
1
kM
, ";&34Is5S&346L5F:.FWI(':;'6I./11b#$F<P1I(%'':∇ωl
s (p∗f)':Divωl
(p∗S)/
% 2 % @ M% % % 1@ ; t % 0 τ /10 #&% =;> %#&" I I %M% 032
% ∇ωl
s (p∗f) %' 28; / # (
lt
) t∏
j=1
(−λ(m + 1) − l + j)p∗(πl−t(
l−t∑
j=0
(∇s + T1)j)f),
4 πl−t
=; 4 10 % # / IJ@ "KL% 72:4 " 0 - @ #&%'O" I3;@ /12 % - @ % % 1@ ; l − t #&% % 1@ ;
% ∇s
;2 /1032 . #&% % 1@ ; % T1
;2 /1032 % 2 4 # / @ %' 25@ 4 72:4 " 0 % T1
/1- 2 % 0 % -=@ j L" 4 7 " >$/1@ 45/1032 M>$/ #&% - @ /10 #&%'λH % 0 4?2 ; %' 2; / #&%
(−λ(m + 1) − j)(j + 1)L" 4 #&% IJ@ " 1-=4?2 $ ;25@45+.- % I/1@ r
8/1@ 7 " 03> % 032:4 " 0 . " 0OI " % @ / +.- %#&% IJ@ " H1- 4?2 ∏t
j=1(−λ(m + 1) − l + j) %' 2; / # 1 4 t = 0 )W./3 T&#E1Ikl' I+.Ls5L&(./'m:;'&8/./T1l5L
βI
./#ih4 &−λ(m+1)
ML,W5Gde./ &#i34I(R"Q;'6H(:'#E#$F 6 &5(R&l'
t"./F+;8/53P
0M]3411.&./oIp5!de./ &#i34I(-6/; "kl;'01b./3 G./34
t["3&XY3@B Iy@B./ ": &
l − 1MN]
l − t ≥ 2'a&
t ≥ 2mb5I(./ "I(- "#E?:?:8 &;
t
τEIy@B. &: "
l'"a;8F5I
J34I(5FDI(5!1 &.H%;':3 &-:- ";%34 & &'%| /
(t + 1)
(
l − 1t + 1
) t+1∏
j=1
(β − l + 1 + j)p∗(r ∨ πl−t−2(
l−t−2∑
j=0
(∇s + T1)j)f)
+
(
l − 1t
) t∏
j=1
(β − l + 1 + j)p∗(∇s(πl−t−1(
l−t−1∑
j=0
(∇s + T1)j))f)
+
(
l − 1t − 1
)
(
t−1∏
j=1
(β − l + 1 + j))(β − 2l + t + 1)
p∗(πl−t(
l−t∑
j=0
(∇s + T1)j)f).
qG'# 5L &XY3./*XY3
(β − l + t + 2)(l − t − 1)r ∨ πl−t−2(
l−t−2∑
j=0
(∇s + T1)j)
wa;'8F5I
πl−t(T1(
l−t−2∑
j=0
(∇s + T1)j)).
o>RI(. &mFI(5?"./#$#E*:'& &./Q "#E'%'f :"&3";8/5I(G-3 #i34I1I(9:
p∗(πl−t(
l−t∑
j=0
(∇s + T1)j)f),
%-#i3I1I(a;'s5LYG;'8F5I
t∏
j=2
(β − l + j)(
(
l − 1t
)
(β − l + t + 1) +
(
l − 1t − 1
)
(β − 2l + t + 1)),
tMOMlt
∏
j=2
(β − l + j)
((β − l +1)(
(
l − 1t
)
+
(
l − 1t − 1
)
)+ t
(
l − 1t
)
+(t− l)
(
l − 1t − 1
)
).
BA%'./%'I(34'A34&(I(5LY9Ip5ide./ "#i3I(o:3J& &(58/I:?)x5"%C5IyM,G%C5
l − t ≥ 2 & t < 2, l − t < 2 & t ≥ 2, l − t < 2 & t < 2&2 &5&F:-#E5(>' &25Ll5I(.8/3M
% 2 % @ M% % % 1@ ; t % 0 τ /10 #&% =;> %#&" I I %M% 032
% Divωl
(p∗S) %' 2; / # (
lt
) t∏
j=1
(γ2k−1(m + 1) − l + j)p∗(πt−l(l−t∑
j=0
(Div + T2)j)S),
4 πt−l
=; 4 10 % # / IJ@ "KL% 72:4 " 0 - @ #&%'O" I3;@ /12 % - @ % % 1@ ; t − l #&% % 1@ ;
% Div;2 /10328H . #&% % 1@ ; % T2
;2 /1032 H % 2 4 # / @ %' 2:@ 4 7254 " 0 % T2
/1- ! "$#&%' % % 1@ ; j %' 2; / #&%
((m + 1)γ2k−1 − k + j)(k − j + 1)L" 4 # / 7 " 032:@ /=7254 " 0 I/1@
r 8/1@ 7 " 03> % 03254 " 0 . " 0 I " % @ / +.- % #&% IJ@ " 1-=4?2
∏tj=1(γ2k−1(m + 1) − l + j) %' 2; / # 1 4 t = 0 ,5-1 &'346/Gw'F(>' &'#E'Y&#EIp5 &R-%II(D:4aI(5
1 &.1./"&(./r14 &;%';:'FM
B '":;"./ &#E5(T' #$&34 &G:@B;'% "( &GIp5de./ "#i3I*<P1I(%:./5F9I(5XY3l5Fkl%C5L&(./ 5L3 "II(21 ".S["%<6#$FG(F6L5 &(5F:43A%s=l5L1& & /
/ +.-/1032:4 6879/1254 " 0 QM
1- 2 3; " @ , M% %' 2 " 030 ; %I/1@ # / L" @ - #&% -=4?>$/1032 %
QM(∇, S)(f) =
k∑
l=0
Ck,l〈πl(
l∑
j=0
(Div + T2)j)S, πk−l(
k−l∑
j=0
(∇s + T1)j)f〉.
' ( B 1b34rd 5L%(I#$Fr#$./F " r8 %' 53P :;'6I(.11gf#E'FR:
Divωl
(p∗S)':
∇ωl
s (p∗f)XY3oI(5de./ &#i34I(:3E&=;. &>#$ '"
Hfv;'XY36L5 "p5FLMbjT^N<CmN"Q./ #E1b./&|I(?d 5oXH34!Ip5E15 "&(i:0:8/ ";
τ"./-;'8F5I($ J:5?%'"$de./%'&(./Nm./ .h4F?&3 ?I('?%'. $%('F
Ck,l
Ip5i &'Ip5L&(./ :4? ";%'3 & "%' !:3Z=4;./ ">#$ M]HIt@O./|511bI(I
QI(5de.%'&(./|&3
P:./l5LYxIp5XY3l5Fkl%C5L&(./NmSIY&3 K
5I(. &:!#$./F " 2XY3Lh∗Q(u0) = 0
1./34 ./3h ∈ g1, u0 ∈ P
M]HQIt@O./&i14Ip5%'i^3 1b./(F
u0 ∈ PmN./ %'./&:> "i3!%'./gP(./
Υ&3
P 1M:./5FoI(3AE340"%'&(./σΥ : P 1M → P
1l5"5F21l5L u0M B 5 5&
τ(u0) = 0M
,5i&./#$#$o:'& "#E'D:2:'8/ &;2w &%'&#$F*&341;' &('3 τ:l5L
Q'@B;'% &
k∑
j=2
〈aj, τj〉,
.SR I(-de./%<./aj
&./FHfv;XY36L5 &(5F'M]TIt@O./ 511I(XY3
Lh∗
A%''&&de./%<(. '[email protected];<65LI(32I2 &;"3I5LG
u0
m./J./h4&(FG5I./ &
k∑
j=2
〈jaj(u0), τj−1(u0) ∨ h〉 = 0.
R $o aN o8 R4o *T a a4 oYbT w
uvIG"r%./Y3 XY3A:l5rIA%5 :4r./1b; &5L'3 & :(; "FI(E58/&&5FF ":4";'m I('XY3l5Fkl%C5L&(./4
sl(m+1, R)fv;'XY36L5 "p5F'&34
Rm"./F
3XH34T$:=4./ &T:'R"&3l5L./Q%' &(XY3'M G <&34(%;*@O(#$1I(XY3915It@O3%&;o:'XH35Fk%C5L&(./9l5L34 &II('D1 &. ["%'&6#$FG(F6L5 &(5F9:./F(I"GXY3'"./A53Z%s=l51 &4M,-h3:-%-%s=l51 "2"a:2#$./F " *XY3%'DXY3l5Fkl%C5L&(./49a&./F*1l5LD3XH34m#O'#Ear:4=./ "D:'*&&3l5L&(./% "XY3'M
BA%'./#$#E'%21l5L Dd 5( "2Ip5i &'# 5L &XY3-&36L5F /
0 % +.-/103254 6879/1254 " 0 0J/125-=@ %#?#&% IJ@ "KL% 7254?> %M% 0324?03H>$/1@ 45/1032 % QM
0 - %' 2JI/ -=0345+.- % 4 % 2 % - #&%M% 032 - 4 # % 4 2 % - 0 % /I I # 4 79/1254 " 00J/12:-=@ %#?#&% IJ@ "KL% 72:4?> %M% 032 4?03>$/1@ 45/1032 % 0 " 003- #?#&% / 14 ' /1032 % 032:@ % Sk
δ (M) % 2Sk−l
δ (M)I " - @-=0 7 % @2 /14?0 k % 2 - 0 7 % @2 /14?0 l > 0
,9XH35Fk%C5L&(./R;'5F9:4Dh ["%<(.mYIp5-./Hf3%&;x:@O3TXY3l5F&kl%5L./2l5L&3 &'I(I( 1 &. ["%'&6#$FW(F6L5 &(5FQ'";XY3 f6L5I('Y&?$Iy@BgP4"&%'0:?:3HPJXH35Fk%C5L&(./Gl5L&3 &'I(I*1 ".S["%<6#$F(F6L5 &(5F'
Q'
Q′ 'i:@B34 5114I((%5L./ l5L34 &II($1 &. ["%'&6#$FK(f6L5 &(5F
T:4
Sδ(M):l5L
Sδ(M):N;' &F& :4 Iy@B:F&;\&II(\XY3
Q′ = Q TMuvIG:./ gP4"& r5L3 #E.($3
k&IGXY3cIp5 &'" "(%<(.:
TSk
δ (M)&.:(; "F:4oIt@O(:4F;M G ./#$#$a34oXY3l5Fkl%C5S(. :./
1 &;'&' "6/' ZI(V*)H#ih.I(c1 "(%'(1l5LItm+Ip5 1 &.S["'%'&(./ :V%'" ""& &%'./ "3
Skδ (M)
:4./< "?;'8F5I2!It@O(:'Y&&;M,'G14 &.S["'%'./*:-I(5i &w "(%'&(./ "3
Sk+lδ (M)
m 576%l > 0
m:./6/F '& &-Y3I(I*<a.J1b3o5LI(./ "*%./%'I(3 "M
a.3+5II(./Whl5" QI(5a%'./"& &3%<./i:'+511I(%5L./l5L&3 &'I(I('x1 ".S["%gf6/#$Fo(F6L5 &(5FGF& &i'&1l5%'2:| )H#ihb./I('o"3 o3 %' ws5A'&'3 XY3-./3G5II(./DK14 &;"FF &.H:3 &LM
E $ (
]ω:;"(8/?Ip5!%'./gP4./V:4 G 5 ws5J./ "# 5LI(-5"&.H%(;'0K3?"& &3%gf
3 " 1 ".S["%<6/Pm I(5Jde./4%'./
κ:3$1l5 0&5V%.3 &h34 &r1b./&">: 34
(#$1. "s5LY&21 &.1 &;';-:@O(F6L5 "p5%'O e6/.( F1l5L8/ /b"h ∈ H
mu ∈ P
T
'X, Y ∈ g−1
mlIp50de./4%'./κ ∈ C∞(P, g∗
−1 ⊗ g∗−1 ⊗ g)
6/;' &kl /
κ(X, Y )(uh) = Ad(h−1)κ(Ad(h)X, Ad(h)Y )(u). S
o2d 5m4IN6H(FG&34%%'&"6/'#E'FCmbJ34I((&5F*It@Ad
f (F6L5 &(5%'o:ω
/
κ(X, Y )(uh) = K(ω−1(X), ω−1(Y ))(uh)
= −ω([ω−1(X), ω−1(Y )](uh))
= −Ad(h−1) ω(Rh−1∗([ω−1(X), ω−1(Y )](uh)))
= −Ad(h−1) ω([ω−1(Ad(h)X), ω−1(Ad(h)Y )](u))
= Ad(h−1)κ(Ad(h)X, Ad(h)Y )(u).
][email protected]%./4&(:4> &!I%'./#$1./&5F'2&I./g0
:2:43Pc#E'#ih &': <m(Ib6F /
κ0(X, Y )(uh) = ρRm⊗Rm∗
(h−1)κ0(ρRm
(h)X, ρRm
(h)Y )(u),.SR
ρRm⊗Rm∗ <ρRm :;"(8/4FR &"1'%'&6/'#E'F9IR5%'&(./4+:
H"3
Rm⊗Rm∗ < Rm
M o0d 5Cm I('a%'./#$1./&5F'&'I(./g−1
:Ad(h)X
'Ad(h)Y%.t%:F? &'&1b%<6#$F0576%
ρRm
(h)X'
ρRm
(h)YM o$1I(3'mI(!d 5
XY3κ−1 = 0
d 5*J&. "-XY3?Ip5i%'./#E1b./&5F-&'I(./g0
:4
Ad(h−1)κ(ρRm
(h)X, ρRm
(h)Y )(u)wa;'8F5I-
ρRm⊗R
m∗
(h−1)κ0(ρR
m
(h)X, ρRm
(h)Y )(u).
,5 de./4%'./κ0
'" :./%Hfv;'XH346L5 &(5FLMTjTII(Z &'1 &;'&'Y&^5I(. &K3
'&'3 :? )H1b (
13
) "3 M
XY3?It@O./Z5141'I(I(2I 2 % 0 % -=@ % !%$#
$ #(% $ ' !! % ' $0$'( 0! % $ $ ' # ')$
]j'"03 l5L&3 &'I9&341;' &('3 0.3\;8F5LI9
2m .\:;'k03 de./4%'./
Hfv;'XY36L5 "p5F
W ∈ C∞(P, S2jRm∗):-Ip5i#E5> "2&365LY&0/
W (u)(ei1, . . . , ei2j) :=
∑
ν
∑
r1,...,rj
κ0(u)r1
iν(1)iν(2)rσ(1). . . κ0(u)
rj
iν(2j−1)iν(2j)rσ(j),
.SRσw341' &#i345L./i:
1, . . . , jM1AT3KIp54./ &#E5I(;R:
ωmσ*:.
Ip5&" *534%3J;I;#$FG(%s=l5L8/;M,VI#$#EZ&346L5F 1b "#E<r:A%C5I%3I $It@O;%5 " Iy@B;'XH346L5 &(5%'J:4
:(; "FII(9(F6L5 &(5F*;' &;':-Ip50de./%<./W
/
T 1
0 / # / L" @ - #&% -=4?>$/1032 %
Lh∗∇ωk
s W = −k(k + 4j − 1)h ∨ (∇ωk−1
s W ),I " - @ 2 " -=2 h ∈ g1
,501 "3462wD&#EIp5 &*|%II(G:I(5?1 "./1b./&(.
:3V%s=l51 & 4M]Hk = 0
mI(5$de./ "#i3I(?w6H 5M B2V1 &.H%>':!5I./ &a1l5 (:43%'&(./M]
X ∈ Rmm
(Lh∗∇ωk
s W )(X, . . . , X)wa;'8F5I
Lω−1(X)Lh∗(∇ωk−1
s W )(X, . . . , X) + (L[h,X]∗(∇ωk−1
s W ))(X, . . . , X).)x5 *= )H1b.=>'&|:- &;%'3 & "%'m I(214 &#$(' D' &#$2"G;'8F5I
−(k − 1)(k + 4j − 2)(h ∨ (∇ωk−1
s W ))(X, . . . , X).j+A%'?XY3%'./%' &4|I2:3P>#$-& "#ELm./J./h4F
(ρ∗((h ⊗ X) + 〈h, X〉Id)(∇ωk−1
s W ))(X, . . . , X).,2 &;"3I5LG1 &.76H('Fo5LI(./ "D:-Ip5i:;<kl(.J:4
ρ∗
M
B!wRo1 &;'&'YR'!#E'&3 "*:D%."& &3 &D:'T511I(%5L./Wl5L34 &II('1 &. ["%'&6#$FG(F6L5 &(5F*'F &-'&1l5%'a:- )H#ihb./I(' /
4S ∈ Sk
δ (M) % 2 4 l ≥ 2j . 2 " - #&%' - # 254I #&%' %# - /I I # 4 79/12:4 " 0
S 7→ p∗−1
(
l−2j∑
r=0
Ck,l,r〈Divωr
p∗S,∇ωl−r−2j
s W 〉)
" 032 0J/12:-=@ %# % 2 IJ@ "KL% 7254?> %M% 032 4?03>$/1@ 45/1032 4
Ck,l,r =(l + 2j − 1)!
(m + 1)r(l + 2j − r − 1)!γ2k−1 · · ·γ2k−r
(
l − 2jr
)
, ∀r ≥ 1, Ck,l,0 = 1.
,5c1 "346J'"!'F(>' &'#E'Yi&(#$(I(5( "Kc%'I(I(E:3=;'./ &>'#E :43A%s=l5L1& &4MuvI"3 Ka:4-6; "kl' GXY3?Ip50de.%'&(./
l−2j∑
l=0
Ck,l,r〈Divωr
p∗S,∇ωl−r−2j
s W 〉
wg1
f ;XY36L5 "p5F&M G 'Ip5 5KI(3J8/ L%|53AI#$#E214 &;%';:'FCm$Ip5!1 "./1.Sf&(. !:3A%s=51& & !'a53Zd 5*XY3-Ip5! &'Ip5L&(./r"36L5F&?'"6/;' &kl;"/
Ck,l,rr(m + 2k − r − (m + 1)δ) = Ck,l,r−1(l − r − 2j + 1)(l − r + 2j).
T ' (
,511I(%5L./xXY3GIt@O./!6H('YR::. T"./F9:'T<P#$1I(':N@ 51Hf1I(%C5S(.a5L3 "II(o1 &.S["'%'&6/'#E'F-(F6L5 &(5FaF "!"1l5%'-: )H#ihb./IM=Go:'&%' &(1(.E%'./#E14I(>'&:aIy@B'&'#ihIG:4o%'95L11I(%C5Sf./*"#ihI?5&"?:;I(%C5SM
BA1b3#E./F& & iIp5i#E5( oXY3?Ip5iXY3l5LY&kl%5L./ l5L&3 &'I(I1 &.Sf["%<6/#$F0(F6L5 &(5F$"03XH34-["3"XH3N@ 53\& &./&>#$!./ &: " :l5LI('Q&3l5S(.Q./Hfv% "&(XY3'MLuvI4&3 KT:*%'./&:; " Q&./34'QI('T511If%C5S(.l5L34 &II('*F "
Sk(M)5:
Sk−l(M) &=
1 ≤ l ≤ 3D'
:K#$./F &' 2XY3@BI+@O<P(wK1l5-:K%.#ih(5("./cI(;C5L( &0:!%'051Hf1I(%C5S(.|XY39&./i1 ".S["%<6#$FK(F6L5 "p5F$:l54!I(0&&3l5L&(././Hfv% "&(XY3'M
j+ 34I(5FRI('9#$;'=4.:'D:;'% &'9:l54DI(a%s=l51 & mH./r1b./3 & &5:;' &6 :'Dde./ &#i34I(9<P1I(%91./34 *I(5L11I(%C5L&(./4:3 &=;./ ">#$ 4Mxzo3P XY3l5S &>#$r'K%'(XY3>#$r./ ": &'m+&9./ :;"(8/ 15
TI(5
de./%<./ ;'XH346L5 &(5FE&3 P 1M
%'./ & ""1./4:l5FKW
5C6%j = 2
mI('G511I(%C5S(.9:3Z=4;./ ">#$ K"./FG;8F5I* "&1b%<6/#$Fo
〈S, T 〉'a
〈S,∇sT 〉 +8
(m + 1)γ2k−1〈DivS, T 〉.
+ a
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