Lasten
G- hie
group
A prinzipal E - bundle ⇐ fiber bundle p : P - M with stand .
Aber G and structure gr .
Glactieg
an itself by left uueltpl .
) .
There is a
right
-
actien
of G an P :
r : Px E - P
• it nesticts to on action on the fisers , r :
TI x G → !
which is free and transitive
=)
UEPxiudacesadiffeau.E-pg.su
- E =P
,
Prop.2-11-suppose.pe.
P → M is a Smooth Suri .
ueophetuoeea ne fahs .
and r : Px G → P a Smooth right actien
of G that preserves the f.has
of p
and aus transitive
ly and freely an edu f.her .
Then p :
P → M is a principe G bundle
⇐ poden.bg
↳ ud Smooth Kai aus .
In particulier ,
if p :P→ M is a the kuehle
wright-
alien
with a zuootu fisher-
pre
wrang
oden
of G an P that is transitive
and free an euch find ,
then p : P→ M is n
ßriucipd E bundle
.
ProofF)
/ Chase an
gen
neuer • U =
{ Ua} , £,
of M
so that there is a Loud decken
Sa:
Ua → P
of p
.
Then di
'
:
µ .
× E →
p
-
'
( Ua )
(× ,
g) '→
sg
(* )
is the inverse a a smooth Loud trivialitäten Ola :p
- "
(4) → ↳ × G .
Note that 4) is a
hijea.ae ,
since ? × E → P,
is free Handtasche
✓ x EM .
For
my x eWas
= Von Up # µ ( for dis EI ) .
7 ! yßalx) EG
5. t .
salx ) =
Sgk ) :
Ysak) . Implizit fd .
Then .
ich
plies
that
Yß,
iv. es
→
E is Smooth ( see also Seetiere 2.5 ) ,
( ×
, g) =
9. (sek) g)
=
d. ls.KYE.gl
= ( ×
, Ysak ) g)
.
=)
{ ( Ua
,
da ) } , #
is u
principe E bundle
oteos her p :P → M .
D .
Def.2.12-supp.se y
: H → C- is a hie
group homomorphie
between hie
gray , It and G and
q
: Q - N and p
:P- M
u
principe H (
resp . E- ) bundle .
① Then u
morphin of principe l bundles over
Y
between
pg
end p
is n fiber bundle morphin f : Q → Ps .
t .
f- ( u .
h) = f- Lu) .
4h ) v ne Q ,
the H -
② Special lose of ① : H is a aozeed sub
group of G and
y = i : H - G is the in andere .
Then a reduction of Structure
group
to It ofa
prinzipal
G- - bundle p : P → M is u
principe H bundle
q : Q IM
|ho
getan wih
auuopu:L of principe unedles f : Q - P
over i : Hanf www.lonerskeerdenlihyen M ( t.e.fm-
id
µ ).
Q
f- P f- Luk ) f-
Lu) -
ilu)
9)
µ
!p
fln
) -
h
Note that f is
iujeaine
Q ↳ P .
③ A morphin of!riucipd G bundle
p :
P - M to it self
loweriug
the
identity :
PIP
r ↳
µ
! p
is called a
gange
transformation .
( Gange
theorie in
physics one forueuhehecd internes of
Principe ) bundles und principe l lonuectiaes ) .
leuuuue2.JP : P )
M
, q
: Q → N Principe kuehles .
① If f : P - Q is a
ueorpuism of pnuci pol hnedleg and UEP
,
then fln) detemines die wehe
off on
Pp," ,
.
② Any morphium f :
P →
Q
of principe G -
bundles
sie .
f- : M → N is uol.ffean.is an
isomorphem
of principe handle .
In particulier , ony gange
transformation is an
isomorphen
Y pniucipd E bundle>
.
PHI
① n
'
E
Ppm,
n
'
= U -
g
for a
unique ge
G .
sur Typ
srr.gr
f- Lu
'
) =
flug)
= fln ) .
ylg) y : E- → Ht 99
= wie
gr . hauen .
% ! µ
%:
" :*
an .
flug) - fln) g)
.
Since
a I is
hijeclive ,
this im
plies f is bijectieu .
Remains to show that f-
^
is
Snake
,
since then it is
ouhoudiudlyamarpn.in of the Indies and E egu.v.wiaecygfimph.esE
-
eguivoieucy of f-1
.
her 9 :p
Yu)
→ Ux G and 4 :
g-Yv) K
G
he priucipd fihehudle vorn her p andy with f-(U ) =
V -
Then
,
( 4. too
"
) 4. g) =
(Ek), µ
E)
g) f-er seine
-
Smooth
luop µ :
U → G .
=/
y - m ( f-
'
(y ) )
-
^
is smooth ,
since I is odiffea.
und innersten in G i ) Smooth -
=) 01 of
to
y
-
1 neuste he equal to
lag) - Hills ) ,
nett
-
4)Fg )
which is also Smooth .
D .
Event
① HE E- closed
Sungr .
of a hie
gr .
G-
Then
p : E →
% is a
principe Hbundle
.
② N = real welcher
Space of din -
n
D= flinker isauarpu.zsu.IR
"
- IV} → M Lpt}
.
priucipd-GLlu.IR) - bundle one a
point
Principe right
even
is
give by :
n EP ,
AEELLn.IR)
u . A = no A : IR
"
→ N E P
Px ELLu.IR ) - P
( u
,
A) - no A
UNEF , then A : = n-tv : IR
"
- R
"
E GLK.IR ).
and U A
= v
.
( i. e. ocheuistneus.hu )
The action - oho
hee ,
since not = u =) A- = U u =
idü
Equiuobutly ,
elements of P con le
identifier wih the
see has is for IV . →
t.nl}
→ baiser
G- In Standard bois .
per
TM
③ M n alien
. vufd .
,
x EM .
gegen
SH
-
x
Ee : -
3=2
FIM) : ↳
MFXIM) This is a
priucipd-ELK.IR) -
bundle
<
=
with the
principe night adrien
gun byP ! } S
u e- FILM) , A C- Ella, IR)
M - O u . A =
no A .
\
"
IM n
C- Llu , IR) - athos :
"zu IM ua :
U
,
→
udk ) EIR
"
42
(U. .ua) chart of M .
"
" '
⑦ IM
Ina :
IU,
I R
"
da : P
-
"
(Ua ) → Ua ×
GLLn.IR)
-
IM
KK) : =
(pl) ,
Tu.
. r
)
v://FI.ae
Ph)
% .
%
-
^
( x. A) = (x
,
Tus . Tui : A) = 4. Yak) .
A)
-
I
Yak)
4g .
:
↳ → GLIn.IR ) smooth .
By Prog . 2.4
,
FLN) → M is a
priucird-CLIu.IR) -
hudle
,
Called the ( linear ) frame handle of M .
Note that any
Loud secieu s :
U → FLN ) of the FLN)
girls
rise do a Local frame of TM defiued an W .
{ and Kennedy .
④ Supreme V → M neckar handle with Ihadod Jw N .
L
FW ) : =
: 4. EnEV) Ew) ) linear isau .
N→ ! }
! Ethnie of K } .
P
M
Riu quellend the treue bundle of v .
is a
priucipd ELIN) bundle
; u EEN) A- EGLIN)
U . A =
hat .
Wecker handle Worts for V
9.ve rise to
priucisd Indie (harter FLV)
↳ cd Kunden) rechtens of FW ) ¥ ↳ coltranes
of V .
( FLTM) = FLM ) ) .
-