5.tn/-egrotionoumanifoldsRecall
that the transformation formula for multiple
integrals ( or wordinahe
charge
formula for
integrals ) :
Suppe UEIR
"
open subset and § : U → ☒LU ) o diffeau .
between open subsots
of IR
"
.
Let f- :
OILU) → IR be a smooth function with uaupoct
Support :
ff =/(f. E) lotet Dot / ( * )
low) u
-
.
↳ oks like the Transformation
of n - form) ou uefds .
of din .
n :
Suppe M is a suooth mfd .
of dim .
n
,
WER
"
( M)
and (U , u) is a chart of M :
Then I ! G- für .
w
"
^ . .
„
: U → IR s .
L .
w
/ u
= w!.
„
du^^ . .
ndu
"
( w !.ru
=
w
# , . . .
.
¥) : U → IR ) .
Support v : U → v ( U) c- IR
"
is on other chart for M
with domain U .
Then w
/ •
=
wir.
.
„
drin . . ndu "
.
Locd werdende expressives
W ! .no
u
"
: hlu ) → IR
of the two function > wir _
} wir. .
„
• ✓
^
: HU) → IR
ho ( Un ) resp .
( v.v )
wir. .
.
In- ^
ly) ) =
wlüly) ) (Eier , . . .
, Jähn )
wir.
.
"
( v
-
"
(z) ) - WK-
'
Iz ) ) ( Eu -
'
es , . . .
. Er%)
.
Now let Ä :
uhu ) →
vlu ) , § : =
von-1
= ) v
-
1. § = u
"
and
Tyu"
=
☒yj
_
?
Ty
$ .
-
-
w! „
hi "
y) ) = w In-
"
y)) Fyn
%
.
. .
,
Tyu
-
'
en ) ÄHH)
: Blog , . . .
, Tag,jj:
qq )
→ " "
"" ""
"
IEEE)
=
detl
Dyp ) wlü '
ly ) ) Fuji'er . . .
.
§ en )
= det ( Dyp) w
„
K"
ldly)) .
¥
y
c,
since D is od://.edu .
If we as since that U is connected
,
heute also u ( U)
,
then der (Byd ) is either always positiv or
always negative an u ( U) .
(*)
soy,
that integral one Local Word mole expressen of
w is well defiueud
up
to a
Sign .
( i. e. independent
of Voice of chart up
to a
Sign ) .
5.10rientatiousupp.aeV i ) an dim
.
Vector Space .
If { an .
. .
, an } and { b
, . . .
,
bis } one two ardened basis
of
V
,
then I ! | , waer uuop A : V →
V sie . Alai ) - b
;
k i
.
und
{ hi
} < „
how the some orientalen ,
The two bei >
{ •
ihn : in
it alt (A ) > 0
; if der /A) < o , they are > aid to home
opposite Orientalions
.
•
„ Having the some oriental-
iae
"
defines an
equivalauce
relation on the seh
efwdoed
bons of V and I
exoctly two
equivabuce
Classes .
• An orientali
en on
V is the do ice of one
of
those two
Classes and a wecker Space with a do ice
of
Oriental"en
is Called an oriented lecker spoce .
Having morgen an oriente tien on
V
,
any
odered boss
in the Chula Cross is called positivelegieren had and
any
that
is not
negativen oriented .
• Standard orienvatieu on R"
is the orientalion
detern in ed
by the standard basis her .
. .
,
en } .
A boris { an .
. .
.
an } is positiven Oriented w .
r .
ho the Jhoedod
anzutreten
, if der / Las ,
.
. , an
) ) > °
.
• Given two n din
.
oriented wecker Spaces
V and W
,
then a linear isauopu.su A : V - W D uelled
orieuhatien
pnesevmg ,
if A
mops a
I have
any)
pontivdy
Oriented basis to a
positiven oriented basis .
Other wie it is called orion retten rewersing
.
On mfds . we von talk about orientatieus on the
tagein Hoch
,
but we need seine nation of Smooth ness :
Def.5.IM mfd .
① M is toll edorient-be.it wie war ckoose on on
en
toten
ou IM THEM s .
t .
the
following hdds :
Fer
any
Local f-raue { sei , . .
in } of TM on air connected
open sunset U c- M
,
the boss { si! .
i. (
y) } of JU -
-
TYM
is either
positive Ly orrcnnad ty c- Vor
negativen oriented } c- 0 .
② If M orientierte o Choi ( e of orienhetieu on IM the M os in ①
is Called on orienhtiai on M A
orientierte uufd .
with a Cho>
en orientotiae is called on oriented
mfd .
•
If M is connected and Orientale
,
it is
easy
to see
that
on orientalen i )
olnhdydeheui, med
by
the eseuhotias on one
on
tangente Space .
Have Ya connected orientabk mtd .
3- two bereute tieus .
• An
open Sunset of on oriented held .
is itself in a no herd
way on oneuhed mfd .
Det-5-2-supp.ae Mond N one oriented mhds . oudf :
M → N
is a lokal
diffoauedpiisvn .
Then f- is Called on
-
enhotiae
preserviug , if the linear
isomorph im If :
IM→
Ty!
i) •
nen toten
preservrng
tx c- M .
Orientalim
vie Special atloses :
Def .
5.3 M mfd
-
.
① An oriented atlas on
M " an athos d- =
{ (Ua .ua) :
✗ e- I }
for M sit .
for any d. ß c- I with ↳ n ↳ + of
the
transition wop ueiuj
'
:
↳ (↳ ntfs ) →
↳ ( Oas )
Kos the
pnopehy that old ( Dlusouj'
) ) ) O on K ( Un ) .
② Two Oriented ehrloses on M one called area hatten egu.ve/euh,
if their union is
again on oriented ethos .
Prop.5.IM mfd .
of din . n .
Then the
following
are
eguivoleut
:
① M is orieukeble
② M admin on oriented otto,
③ I on n form
W ER "
LM) s.ir . WK ) -1-0
K ✗ c- M .
Proof
① ② Suppae M is orieuhableaudfixonon.eu/-otrai .
( Kool on athos an
M
,
lt {
( Ua
,
Ua ) : ✗ c- I} 5. t.lk is Connected
Hat I .
=) Ihn :
I ↳ →
T "
• LK ) = TIR "
=/R "
is either
halt
) 4K ) =
orieuhokieupesev.mg
V-xc-Uaarorieuhotiarhenesmgk-c-Ua.luthe first lore we Keeps the chart Os it is
end in the
second Lease we can
pose it with on orion hatten ne
verging
linear
uuop IRU → IR "
( for in stare
exchange
the first
two variables) .
In this we we abtoin an
oneutedatlos .
② ③ Let d- =
{ (Ua ,
u
.
) : ✗ + I } be an Oriented
atlas on
M and
{ ti }, ⇐
mir
•
partition of unity that
is subordmahe to the war U =
{ Ua }of
of M -
For ie IN choöse ✗
;
c- I sie .
Suppe (Ai ) E ↳ .
.
und dehne wie R
"
(M)
by Wi : = f- ; du
; ^ . . n du
!:
( extended
by zero to all
of M ) .
Set
W : =
Sei .
Since
supplf :) is ↳
colley Janke ,
IEIN
W is a smooth n - fernen M Fix
✗ cM
.
Then { fik) =
1
impu.es 7 ist .
Pik ) > 0
i EIN
ßy definition
, Wik ) (¥ / - - -
/
du
,;
E) =
f. H ) > 0
.
"
i
Since the athos is oriented and all fjl,
home non negative
volumen M
,
we home Wik ) (¥.
, _ .
¥;) 70 ltj .
=) W (x ) # 0
③ ① we R"
LM) nowhere
vanishing .
For ✗ c- M
,
we well a how ) 24 , _ .
in } of FM
rosihudy
oriented , if WK) 141 ,
. .
, s ) > 0 .
This
giuesononcukotrae
On M .
☐
Reiuork Prop . show >
•
Emery Oriented ottos determine, an oneuhotiae and two
Oriented athos on M de Le wine the some omen
la Hat ⇐>
they are ostentative eyuivaleut .
•
Simihorly , any nowhere
vorweg n - tour de termine, an
orient at ion
on M and
any
two such n form
W end E
determine tue sone • nen hohen ⇐ Ja positiv Smooth fcl .
f- : M → IR on M sie . c- =
fw .
• Given oneuhoble mfd .
, a choke of orieuhotion en M
i )
eguivoleut to a
Chord
of a maximal (or on Kriminal / Guidance
( Iag )
on.cn/-edotlos ,
and also to a
choke
of nowhere
vanishing
n fern
w up
to luultpl .
by a smooth pontes function .
E- .
M =
IR"
is Orientale .
E- Möbius hand is not oncutable .
( see Tutorial) .
IRP
"
is orieenhable n is add
.
5.2lntegrosupp.aeM i ) an oriented h - dim . mfd .
and let
d- =\ ( Ua
,
u ) : ✗ c- I} he on oriented athos (
giving rise
to the fix ed oienratieu) .
Then we con olefine on
integral for n form
, on M og
fellows :
we write
supplw) :
={xEM:wH*→ for the
Support of w c- Rh (M) und R! IM ) ter the Space of
h form
) with Leonrod Support .
Suppae w c- RI (M ) .
Since supplw) i)
uaipoct
,
7
finitely Wong Charts ( U; ,
u
; ) i =
1
,
. -
,
e of the maximal
oriented
atw.lt/def;n.uytueon-atatieu)s.l.Mpplw)E4U-.uUeFurther,letfj: M → [ 0,1] he Smooth functions g-=p
. . .
.
e
sie .
supplfj ) c- Uj and
( [ f; ) 1=-1 .
5=1
Supplw )
[ ( hoon
partition of Unity Subordmete to { Us ,
. .
, Ue , MI keppko)
and let f ,
he the Sun
of dt functions with Support in Us
,
fz o,
the
reuuoining
functions whal Support in lonhoined in Uz,
. .
)
( of fiw I u ;
so :-.
iiiiiiii.EE?::;:.;jMi=nuiNiTfiwFdw =
{ f ,
w
fiwi ER
:(Ui ) .
Fi!"
:""
"
.
¥!
:},
Is
=
Since
supplfiw ) is a
uaipod Sunset of U
;
,
the night-
hood
Si de
equal, a finde kein
of Integral,
over Ladies with uaepodn
Support .
Haue ,
this
integral is hnihe .
het us Check that
)µ
w is well -
de/med
,
i. e.
independent
of all the
clones :
Suppe ( Y ,
u
;
) one An
itelyuony
Charts of it and
gj : M → [0,1] functions 6s ohne
Wushu ched w .
r . ho there Cheers .
w =
{ gjw
und heule tiw =
Efigjw
j
i
-
{ ) fiwluily )) / Ty "
:& .
. .
,
Tgqi
'
en )i
uilvil
=
{ Stig>
wlüilyl ) (Ju es .
. .
.EU?en )
ii, uilu ;)
=
E) → supplfigiw)
in u
; / Uinvj )
c-
Uinvj
Jfig;
w Kitty) (Tg leiten .
.
.
.
Tyui
'
en )
uiluinvj )
=
/ detlDG.mil/fig,.wlu.ilyD(Tjj.;ijji-.iE-
'
en )
Hui'
)/
y)
"
ilvinvj )
=
Jfigjwlv; Ly ) (Trier.
. .
. Ty:L ) =/ → ,
P
"
i / Uisv;) µ
IN;)
trauten .
ruhe
tormhegnels
Since
figjw vereistes
and
detlDK.eu;-) ) O but ).de
of 4- ( U
, svj ) .
Heuce , /
„
w i > wdtdetined .