Laatzen :
P → M principe C- -
bundle
We can form as sociale bundles : G- ✗ F → F Smooth left-
action
of
E on a mtd .
F
PFF → M
Gfiber
handle wihshomdodf.hu F and Structure
group C- .
For F = N ne
representation
of G
,
V : =
PEN - M
is a wecker bundle .
.
1- ( P ✗
a.
F) =
(P
,
F)
£
t-uuctoridpropertiesofossociahedbundle.suppoze p
: P → M
principe ( G - bundle and lel G- ✗ F→ F
and Ex E F-
Smooth left action
on wfd .
Fand E .
Amy smooth uiop OI : F → E that is G- equivanaet.
i. e .
§ (g. f) =
g. ☒ (f) V-gc-G.tt c- F .
in denes a morphin of fho bundles with Structure
gap
Glover
.mg
the identity on M :
←
to
PEOI] :
_ Pz F →
PEE
Eln , f)Im Eu , Elf)]
Note that is G- equiväaer
f-er (u
,
f) .
g
=
(• u
-9 , g-
'
f) ,
have it iuauce, a well -
defiued
luop PIE] and it is Smooth , Since
g
: Px F FGF
i )
le
surject.ve Sub merken .
H i ) also o
morphin of fiber bundle,
wir structure
group
C- ,
since
locdly Peg] ho > the face
4. f) - K . # H )) and OI is G- equiv .
We how on induced
wop
1- (PIF ) 1- (
PEE )
Is is
GLP ,
F)
↳
→ c- ( P
,
E) £
Ü :p → F - ☒oh :p →
Er
In particular ,
for any morphin € : N →
of G- representation,
we
get
an induced morphin of wecker handle, <
V :
=P ^ W : =P
EINT
→
µ
!
If § is on
isomorph ihn
,
Keen so is PEOI] .
•
p
:P → M and F : F → NT principe bundles with structure
gr .
G- and £ respect -
,
T : C- → Ä hie
group homo mon .
und ☒ : P → F a
morphin ogfpriucird bundles out
love hey E- : M → Ü .
Given a left action G×→F ,
then we
get a left adieu
C- ✗ F → F by g. f =
Tlg ) f .
Then we
get
on induced luop :
☒ ( F) : P ✗
cF
→
FE F
[nit ] =
[☒n ) ,
f ] .
This delius a
www.izneffiserbundlescoueriuy#:M-M~.1fF--NamdGxT-→
F o
rezepte >
erhoben
, ☒ IN] i)
u
morphium of Vector bundles .
Exomptes
① FLM) → M freue bundle of on n-dim.cn/d.
M .
Any representation N
of Gun ,
R) iuduco, a weder heute
✓ = FIM ) × N → M
G-Un.IR)
land
Isomorphie repneseut .
iuduce isomorphe wecker bundles -
*
N =
IR
"
,
F. (M) x
R "
⇐ TM FLM/× R
" "
⇐ Tµ
.G-Llu, /R) G-Un, IR)
•
Tensor bundle, come >
p.to representation s ⑦PIR "
⑦ ① 9
/Ra -
.
Also , if N is und irreducible and NEU,
⑦ . .
⑦ NN
Is the delon position iuho Urne ducible uaeporonl )
,
then
V =
FLM ) ×
N ± # (m ) ✗ Yt - ☒
NN
G- Un , /R ) G-Un .IR )
= FIM /✗ N,
☒ . .
☒ FCM), Na
|, GLLu.IR)
µ Gllu, /R)
=
V
, ⑦ . . ⑦ VN
④ Survive this a
mtd with
a G strache
( P-
→
µ
!
PEE = TM .
If C- = Oln) ( i. e. M is
equ :p> red wir o Riemann - wehe ) ,
then IR"
= IR
" *
es Oln ) reroseut
.
VI. <v. → <
g.v. w > =
< v. g-1. w >
nefkcting the face thai TM = OLM) × IR
"
an ,
and THE 044 Rua
Dre Isomorphie via
g .
an)
R c- 1- ( 5 ( ATM ) ) SYTIR"
B
- elements in
SYNTH
whiusotisfy du Matty
RK.n.eu/-R-Hi,j)
Bianca
identity
-
5ps MIR
" *
= ☒
☐
☒ ¥ ☒ IR
„ „
*
if„
weylcuruteettroce-beeR.ca + Scola
Gruebel .
Curv
.
b) Oln ) reyreseul
. this Kos 3- irre
duckte
Laepouecefg _
-
RE
# FM ⑦ End ( TM ) )
R
" ☒
R "
II.CONNECTIONS-3.tlRecoKfromGlob•lAndym-AffiuewmeH
M h din
.
mfd .
,
TM -1M
Affine Connection on A is what we will hehe well a linear Connection
on the hello bundle TM M
.
What is the idea of this
loucept ?
Au affine is o device
www.idaiifiestangeut Spaces
at near
by point
( on more
generelly a Linear Connection on o hector bundle is a
device
that ident if ie, point, in near
by fihars ) .
Why do we Core about such a
device ?
•
y : I- M Smooth Curve
, gilt ) cTaff
lt t EI .
Acceleration of y con con la le de find w . r .
to on office connection :
j
"
( o ) = / im
&
"H%T ueokes no sense
,
+→ o E since
j (+ ) cTg„µ
and
J
'
lol cTj,%
t.ie
indifferent• Directiond der ivahhe of vecwor
Space)n
neuer
fields : MTTM ( or a sect/ en s : M → V of any lecker hmdk)in direction
of a
vector ( field) µ c- IM ?
We home T} :
TM → TTM ( Ts : TM → Tv )
but this
iguoe ) the wecker huelle strecke .
We would like
to View vector fields ( or action
of any
now bundle) os
Vector-
valued
huop, on M and differentiale than o, such in
direction
of woher fields .
If TM =
Mx IR
"
→ IR "
is trivial ( V Mx
IV → M) ,then g Lou hexen 6 > o
Knop S : M → IR"
( or as
maps M → N)and
we can
olefine tue dirediauddeivetihe (D)4)(y)
( Ds ( x ) ( y)) of } ( resp . s ) in direction
of q c- 1-(TM) os
tuned :
If y
: I → M -
Cerne
, ylo) = ✗
.
y
'
/01--4 ,
then
Ds 4) Ira ) =
dä) s ( ru) ) =
1in
"""+→
1- =
o
and Ds K ) : IM → IR "
is a Linear
map .
luokl) ho
Simi ↳
rly , for s : M → IV .
Kuk formen-
bird
TM .
Au affine Connection (resp .
linear Connection on
any
Vector handle V74)is a
Cho ice of identification of the
long cul
spaces ( of the /ios )for
newby point> . LH „ Connect>
"
the fho) „ t neu bis point ) -
Two ( equiv . ) View point) ou affine Connection we hae
olreody teen ,
① Directv and derivohiepeswct.ie
☐ : TLTM ) - T (1-*
M ①TM ) linear
Sir .
Gfl =
fGg
+ dt (g) s tsiy c- HTM) ,
tttt (m, #
( Leibniz rule )
② Parallel transport .
For
any
G- Curie
f i I -1M
,
ts ,
tz c-I
Me Kos a / in der Donner pur an :
Pt
"
tz
( g
) :
IHM,
→ TM ,
Jltz)what Pt!; (g) = Id and it sah fies Sane oder pronotie .
① → ② : For
any
wecker
9g *f-
T
) ,
7 ! D- parallel
neuer Jidd
solang j sie.
Sltr ) =
Sylt, )
und Pt
# (g) (sg , " ,
) =
Sltz ) .
② →
① Pri ) k ) =
E) (PE )
"
(slm ) ter
1--0
f , I→ M G -
Cone
flo ) = ✗
, J
'
G)
=p;
There is a 3rd
again point of view on un affine connection
which we will di> Cass now in the context
of linear Connections an
vector
hindley .
3-2.lineorconnectionsonuecnorbund-Tp.TE-TMDef.3.jp: E → M is a fiber bundle wir Shand - the F
.
Then the Vertical bundle of p
is the wecker Sub bundle
of TE → E de ): und
hy Ver LE ) : =L SETE :
Tps = 0 }
=
Ker Ep) E TE .
It f- ihn >
Vern LE) : = Ver (E) „
E Tu E are called the
vertical
Subspoce> .
Rennen . Vector Sun bundle D E TE
of TE → E = smooth distribution
D on E =
Sunset D C- TE s.
t .
Ds Tut = :
Du is a know subspoce
and the bestricken
of TE → E ho D- E is a near bundle .
Note that Vern (E) =] Epa) ( Epa
tu
'
C- E :
plu
'
)
-
=
plu )
}
Exercise Check that Vesle) → E i ) a weder subhuudle
of
TE → E
of reuk rk Wer (E) ) = dim (F) .
More over
,
it p : E - M is a never bundle , Epen,
is a wecker
Space and no one
way identification :
Verde ) =
Tu Ep" ,
=
Ep," ,
for
any
u c- E .
F ' " " C- F- pm )
:
☒„
i
Epa ,
→
Von LE) hlu) adf.HU
+ tu '
)
is the naturel Hamer puh ohne -
-