Def-3.2-suppc.se p
: V → M is a vector bundle .
A a linear
Connection on ✓ → M is a linear
uuop :
D :
TV ) → TLTTM ① V
) =
1- (Hau LTMN ) )
s →
9s " TM - V
☒g) In) = :
Ffsit Tits
=
f- Rss +
df (g) s t s c- TV )
, the TEM)( *)
V-f-CULM.IR) .
( Reine . D : 1- (TM ) × TV) → 1-(u)
"
IR - bi linear ( Kis ) -
Is
) ( *) leibuitz.nu/e-←
IM .IR) linear
in
R .
•
and Soli> fies (*) in S .
Exaupte Affine Connections on
More linear Connections
outM-M.Tbm.3-3-supr.aep :
V→ M is a (smooth) vector bundle .
Thou
there ex ist a linear Connection Von ✓→ M and the Space
of all linear connections is on off, ne Space modelled on the
vector
space 1- (TM ⑦ Hau ( v.v ) ) = 1-(TM ☒ ↳④ V )
of 1 fernes
on M with vdues in the vector bundle Hau µ,
V
).
Proof.
Existence :
{ ( Ua , da) }•
⇐
vector bundle atlas for ✓→ M
,
da :
p
-
^
( Ua ) ins Ua × IR
"
.
For
any ✗ c- I and a Sechtem s c- TIV ) wir
Support in ↳
,
olefine 75 (x ) : =
&! ( ( ×
, ( s .
sik )
,
. .
,
s Suk
) ) -4-4
- whee
Da ↳ K )) =
(×
,
Srk ) . .
. Puk ) ) ,
and 5)(
y) = O
Hye Hwa .
Now let { f, } be a
partition a) unity suberiodiude to two love
U =
{ Va}, < ±
. Then
T.gs : =
fas) for s c- TLTM) ,
> c- ITU)
✗
EI
de
feines a linear lonuectiae an ✓ → M .
+
Freeden .
A,
c- 1- ( TM
⑦ f) an / v.v ) ) A : TITM) ✗ Mv)
→ 1-( v)
Üs : =
Tss + A ( s ,
s ) is
ogoih
• liner
/ MIR ) bi
linear
.
p Connection
Üf >
=
Bfs + fs ) =
f- Ist dfls ) s +
ACE )
-
=
Fps +
dfls ) s
Converse↳
,
Ü and Done two Connection ) and Set
Als.is ) : =
Fs Dis
.
Evideutky , Alfs ,
s ) =
f Als .
s ) und
Als As ) =
Ffs Ffs
= f- (Fs Fs)
+ df G) s -
dflsls
-
-
=
f- Als .
> ) .
= , A c- 1- ( TM ☒ Hau ( V.V) ) .
☐
How do things took in locd ↳ordinäres ?
v Suppe ( Ua , na ) is • chart for M
.
(din LM) =
n )
t and ( V2 , da ) is a neuer huelle Choir for ✓ → M
M
✗
( naeklv ) m
) .
Then p ) - ↳ ✗ 1pm
"
<
✗ id
→
↳ (4) ✗
REITER"
is a chart f- er V .
-
h
g =
{ si D-
"
Ö
,
is the Loud
in
Dai
i
S =
{ SJ
e)
ja Sector
of Voeefiud
% =
"
{ [Sitz sie.
=
• ↳ sie .
in in
☒
% ( EK)/ =
(x
, e)
=
§,
si (E) %. + Sisi Ei
"
e) j standard
go.is
⇐
iii. K
=
wecker in
/Rue .
for Smooth function Tj; ( i = 1
, . .
,
n
, jik =
1
,
. .
,
m )
via rederi and
by %;
=
[ Tj? % .
Fei k-1
More
loupoctkg , wo also nie tei. ) hohotieu : 5 C- 1- ( U )
Sa : ↳ → IR "
(10×0 5) (x ) =
(x
.
Sak)) ttx c- Ua .
(g)•
=
dsals.
) + Als /¥)) Az RTL .
gllm.IR))
÷ -
4.
SAALE ) = : Ai
(Ai ); =
Tj
"
;
=
Simulierter ,
os for affine Connections and hector fields we Love :
Fer
y
: I → M CA -
Curve
,
le Seetiere of V
okay g is
a Socken
of ✗
*
V and we home on gesehen
%
: Tljv)
→
TGV) .
Prop.3.4-supp.ae ✓→ M is ne wecker bundle
equipredw.huhinan nenne Chien T .
① Supreme y
: I- M is n G- Curve and v c%.
) ,
↳ EI .
Then I ! Sechtem s
along y sie .
(F) G) = 0 lttct
and s (to ) = v
.
② In the
setting of ① scypoze [ to.tn ] EI .
Then
Pt
"
+ .
( J ) :
↳lt. )
→
% )
v - s ( ter s as in ① )
is a linear
isomorphe .
It is called the parallel transport
doug Jr detiued by D .
/
ProA %,
s is wellde)
. hdeaos os for office Connection .
and also the best
of the proof is as for affine lonnectieus
( cf . Global
Analysis ) .
Reuig There " I on abstract hohlen of pwdlel transport for never
bundles which t )
äquivalent
to a linear Connection .
The lineor Connection is ne Covered from the parallel transporte by
Fsk ) ÷
¥) (PE )
"
(s ( KH) ) ,
1whee
J : I →
M i ) n Smooth Curve wih flo ) - ×
,
840) →×
.
Right hand
siehe just depeuds an
jlo) =3
×
and not on the
Curve j .
Follow> d. riecht
y
freu the Local loordiudeexpessiae
of Pps) tt ) .
Prop.3.5-supp.se ✓ → M is a vector bundle
equippeed
with a linear
Connection T .
① For
onyx
c- M und v. c- ¥ ,
there ex ist se 1-(v) wirst) = v
and
) (x ) = 0 % cIM
.
② For ve k Set Hu : Is
(EM ) c- IV for a clone
of sect, en s os in
① .
Then Hu C- IV is
independent of the more
of social
s o >
in ① and it is
a linear kompliment to Ke! (V):
Tutor) treu .
Moreau ,
H =
↳ „
Hu C- TV is a hector subhuedle
of TV → V
,
Called the horizontal distribution de terminal
by T .
Pr¥
① Follow) from the Local Coordinator express Ieee
of
(Dgs ) (x )
µ hin show, that
(7) (x )
depemdsa.ly
ermahne and the first deiuohhe
of s at ×
.
For s (x ) =
v 7 ! unique 1-Id
j ? s sie.
Es (x ) = O -
② For •
Us ectieus s o ) in ① ; Is : IM →
IV
is the Some and Lance Hu is independent of the
Chord
of s .
pos =
1dm
IHN -
Since
Ip is the inverse
a)
✓
Is : IM →
Hu
,
1- ✗
s is a linear isomeren and TV Hu
⑦ V9 (v).
Check that It c- TV is a Smooth neuer
subhuudle
¥ (Exercise) .
Def .
3.6 V → M wecker bundle wir a linear Connection T .
-
=
=
Then to ✗ c- M ,
v c- Vx
,
we Love a linear
map ,
Called the
horizontal litt ,
Tv -
A.☒
V94)
Hoi :
IM → Hu c- TV
TIP I.{
×
↳ ghor
v ) IM
Whee shor Im HEIM✓
is the
Unique wecker in Hv sie .
Ip [
↳'
=3
, .
For a
never fields c- TLTM) ,
s
""
: v - s !
'
olefiuesu vector fielden V
,
Called the
horizontal lift of s
,
war .
ho P .
Reinartz:
Horn : IM → Hu c IV
v40) ¥
/ +
=P.tt(Hlu ) c- Hr
for any y
wir
v10) - ×
(
Pt-X://gcntterus-flowofeueaard.deSupport y : I → M is a
YQ-iouuehohok.pe9 Tv) .
Smooth Curve
,
then for
any
✓ c-
Yu,
7 ! [ : I
'
→
✓
,
I
'
c- I sie .
v
If"
Comunio and
Elo ) = u
[
'
( t ) cHd+)
KTEI '
.
ju is Called the horizontal lift of y
w . r .
ho T and v
F. ( t) =
Pt! (g) (v
) c- Volt)
For
any y c-
p
-
^
( glr) ) c- Vgl + | ,
let S
}
'
to the horizontal
lift of ( t ) .
,
contre This con Le extended do a
noche
2-ehd den ending on all
of V .
Then
I is the
unique
integral of this rector find wir [ (o ) = ✓ .