Afewwmmeuts.IOConnection :
←
↳ , Rx and their flows
F (B) = Betx
←
Äh
=
↳ V-BEGLln.IR )
FLEX (B) =
e
"
B =) F
↳ =
bis.
FLI"
FH (B) =
↳(ek) -
Bett
• -
2720¥ +2×001=0-
32-2%1 t2y¥ - O
~) X , Y E <X.
43 invdnliue
E is spannend by ⇐ , #z
for me oppropriehe Chart ( Kh) .
② Chonburi In
⑨ ,
ak.y.tl
¥ ↳
y
) = alx ,
y ,
fk,
y) )
=
zcosy
¥ Key) =
ßK , y , fkiy) )
7 sanken ⇐
§#k.y.tk,
g) =
¥
@KqfkyDsmeEGEEojFEkisIEk.y.tkiyDIIR→
Efta::
"
Def.h.21-Ag.ro ded derivaten
of degree r
of CGM )
,
^
)
is a liked ueop D : RIM) → RIM) sah .
¥5:'(M )
D : Sz
"
LM ) → R
" "
LM ) and
Dmg) = DCW ) n
htt-1551dg
µ for WE R
"
( M) .
Lasten
M mfd . of dim - u .
WER
"
IM )
WIE Windeln . .edu "
for a Choir IV, n ) .
w! .us
U → IR
Ian .
. II)
.
If ( V.v ) is onother Chart wie V - U .
Then :
wi.in
"
: ulu) → IR d : nu ) → v ( U )
-
01 = Von
-
1
„ µ )
der ( Dot) wi.
. üv
-
too
-
fürn"
"
=
ufji.ir?dlDetboIequdsE)Rn up to a
sign {ja
! on
"
Heule , integral of 6cal coordiuede
erpressten of W
"
Up to a
sign independent of Ubersee Chart .
5. 1. Orientalen
Suppose
V is an n dim
. Uecker Space
.
Ihr { an . .
.
Ion } ieud 9h .
. .
, hieß one ordnend boris of V
,
then
7 ! linear
luop An V → V S- l. Ala ;) =
bi t i .
The two ↳ „ s Sai } and { bi } have the Save arieuhatieee
,
if
del LA) > 0
,
if del (A) < 0 ; they home oprositeorieuhotieu .
•
„ Moving the sause orientetieu
"
defines an
eqcivdeuce relation
den tue sehr
ef ardvad boris of V
and 7 exackytwoegu.name
dieses '
•
An orieuhotiauouv-istheao.ee of one
of these two
Classes and a weder
space wih Chose
anzuhalten is Called
on oriented wecker
space .
Haring wogen an onaehotieee an
V
,
we well the ordeed basis in this Chose
egu.ir . dass es positiven
oriented and the aus in den aber dass
hegdiedy araehad
-
•
Standard an euhotieu an IR
"
is bleiern: und by the Standard
↳ is { es .
. . sei .
A basis { es .
. .
, on} is
positiveleg orieeted
Wir ho
Kee standard wa orientalen
,
if der ((es , . .
son ) ) > 0 .
• Gieren hwo n dim
.
dreimal veaö
Spaces
V end W ,
then a Linear
isomorph A :
VTW is orientierten
preseruey ,
if A mops a lheuce
any) positive leg an eekeed has to a
positive
G oriented basis .
Other Wise
,
it is called
orientalenneuerung
.
Oh man ifdd , we can Talk about orientatieus andere
taugen Spaces ; we weed nation
of Suu ohne :
Def.5.IM mtd .
① M is well ed oriente ble , if one can aoose an orientalen
° " IM t x EM such that the
fokowiugkdeds
:
For
any loud from { se ,
. .
, gib on an
open
Connectionnet UEM
,
the house { snlyl,
. . ,
snly) } of Ty U IM
ie eitler positiveleg oriented
tty EU or
uegdively
oriented VYEU .
② If M is o neu hohe
,
a Chaco
of oneukatiaea.IM # e- M
as in ① is Called an oriente time on
M .
An arieuhable mfd . wie a
Closer anatolien is Called
an oriented ueouifdd .
. If M is nennedid and aieihedde,
it is
easy
to see
that on dienenden is
alreooly by the done of au einholen
an die
tangente Space .
Heule , on a Connect aieuhablemfd .
7 exoctly two oricuhatieus.
a. An open kebset U of on men Led info .
M is ihelf in
a naturel
way aseeheduefd .
Def.5.2-supp.ae Maud N are anziehend uefds. und f : MTN
a Loud
diffeau .
Then fis called
oneukatieupreseru.bg , if
the hin .
isdn .
If i. IM →
TIN is anatolienpoeseru.tt/EM.0rieulatieuviespecidatloses:Def.5.3-M
mhd .
① An ¥4 an M is an ottos A- {( Ua .ua ) , *I }
for M s.
g. for
any als EI W.tn Von Us #0 the transition cuop
Ußoui
'
:
Ua ( Van ) -1
Uß ( Udp ) hos tue großartig keot
DU det ( D (ußouä ) ) > 0 on Walks) ②
Two oriented atom on
M are Called Übel,
if their union in agoiu
an oiaehudatbs .
Prop.5.TN mfd .
of dim . n
Then the followmy
one
aequivalent :
① M is orieuhable
② Module an asanhed athos
③ 7 n form
WE du (M) s - t .
WK ) # 0 VXEM .
PRI
① ② Support M is orientale and fix on anbeteten .
Cheese athos an M
,
A- =
{ ( 4. u.
) :< EIG such that Wa is
↳nneched Here I Ina
: IU,
I.¥14) is eine
areuhotaepneservy
Epic
Vx E Ua or Orient .
neuerung-
THEUa .
In the first love we
keep tue hart es it is
,
in tuesecaud
linear
leere We
Coupon tue
www.houorieukeetieee-reuersmykuegsIR "
-1 IR "
(
exchange ten first two verwies ) .
In this
way we anton an Oriented alles .
② ③ Lel A- =
{ LK ,
u
. ) : < EI } he an aiahedatlos
for M .
and
9¥ ! ⇐ „
a
partition of cnn.by
that is
subaridiuate to U =
{ Ua } ,
°
For each IE IN devon d ; EI s?.
Supp ( ti ) c Ua
;
and dehne wie SMM )
by W ! -
fidujnr.edu!
( extended
by Zero to M wie
bunptd .
).
Ser W : =
{ wi Since suprllr ) is
locdly heute
,
JEN W is a Smooth n fan
an M .
Fix × e- M .
Then Ef :
k ) =
1 im ph.es Zi s.l.fi/x ) > 0 .
i EIN
By definition
,
wilx )
6¥.
) . .
,
%;) > O -
=
fik )
Since the athos is ariahed and dl Ag
'
s
Leon men -
hegdeve
Vdues an M
, we home wik ) (¥;)
. .
,
¥;) 70 VJ .
wk ) * 0 .
③ → ① WER "
( M ) nowhere
raising .
For XEM we hallo haisis Iss ,
_ .
, sie ) IM
positiven erntend
,
it wlxtlse ,
_
Ku
) > 0 .
This yields orientieren an M .
D
Rein
• Ever y
oriented athos determinen an erahnten on M and two
areuhed etwas an M defeuiuud tue saure erroetete if
they are anzutreten -
eyn.name
.
Simi harley , any nowhere
warming n ton
W de herum an araieotieee
and
any
other sum foiuvdewm.us Leuten einhalten , f
7
positive 0-wopfi.M-IRS.tn .
w =
FT .
*
Giveu an unrentable mhd M
,
a voice of orientalen an M
is
equ N deut ho a voice
of
Üuox, und areeheolotloses (an
egal .
Cross of erahud atloses) und also hob Gore
of nowhere
von idi
wg
n fein
up to multiplenNeee by a
poste seh fcf .
¥ M = IR
"
is orieuhable .
Er Möbiusband is not orrenhedsle .
¥ IRP
"
is
oneuhdde ← n is odd .
5.sn/utegralsSupposeMis an oriented n dim
. mhd .
and let A- 9 ( Ua ,
u.
) : a EI
}
he an aranhed altes
regsreseeday
the eu orientierten .
We Witte
supplw) ÷ fxEM.eu#FO3- for the
Support
of a a
dift .
form w ER
"
( M ) und RI ( M) for ten
Space of K focus
with
Compact Support Suppe
w E RI ( M ) - Since
surplw) is
hanpoa,
7 fiu.by
way Charts ( Ui , u ;
) i =
1
,
. .
,
l of them the areeked
atlas A s. ) .
supplw) c
Uno - U
Ue -
Furnier , lel fj : MT [ b. 1) he suuooh td .
5=1 . .
.
,
e s.
) .
supplfj) c Uj and
Iff;)
=
1 .
Supplw)
(Goose u
partition of cnn.ly suborioinde to { Us ,
- ,
4. Mlkepplußand het f , he the Sun
of all tds wir
Suppen in
Uy ,
fz the Sun
of the
reeuaicug Ids .
war
deppert M
Uz and
be on .
.
)
, u; lfiw) (es .
nen
)
-
Kw
: Is
!¥;)heilig)) Haien.
.
.
. I en )w
=
Ätiw t.io er:( u ;) .
tiwlii
'
b) ) 1¥:
In
y) ) . .
, ¥-4:)
Supp (fiw ) is a con pod Set
of Ui ,
so tue
right
- hand -
siehe
equeds a Luke sum
of integrals of fds .
wie
Conrad Support,
und Lance this
ruwgrd is Luke .
Let us Check that
fuw is well detmed
,
i - e .
Independent of all Koreas we made :
Suppe (Vjivj)
are
fiuhely many Cheers of the maximal altes deleuiad
by A
and
gj
, M → [ 0,1] flhrhedg way sand fds es ohne Conor .
wie
'
respect ho here Charts .
- w =
?GJW und Lance fiw =
{tigjw
-
? ! luiilgllltguies .
.
.
, Tsui 'er )
-
=
.
! !!.jp?wlui'b) ( →
-
)
=
%.
Sfigiwtnt ( →
-
)
suppltigjw ) hilinvj)
(
Uinvj %!!
teils) ) Linien . -
ein
.im/-=fdetlDkioui)tigiwkilsDlTiiij!;)uitvisvs)
=
!! ii. Yi" ) Hüte. .
. .
Türen ) =
iF"
Since
l :
gjw
veiishes outside
Vj ( U: s Vj ) .
Deuce
, fnw is well definiert
.
Prop.5.5-Mmf.ee .
of din u .
Then
JM :
RL (M) → IR is a sur jedem linear
uuop .
PI.
liuaaiy toll aus freu tue de find ie and Lively of Negra
of Kt - an IR" .
Susjechvily :
It remains ho aww that 7 WER ! (M) :L
1µW # 0
.
Lewon u am (Un ) and
a Snack war soo 1a .
f: M - HR
> •
wir Conrad
Support in U .
W: = f- du's . . nduu Lae
we extended
by too to on dauert in
RY (M) .
→
Smw f.für
> 0 .
Special
① M = IR eoenipred wir Hadad verrenken , for a < b
and w - f dt E RTR)
!!
=
ÄH) dt .
② Line
inhcgnds
: VEIR
"
gen Kessel ,
w -
¥,
widxi
C- NN !and g : I → V -
Cure
,
IEIR open
Merced .
→
¥
*
w =
.IE/lwioJ)lg-ilt) ) dt ( I
like integral uf w okay g.
Ü!