Recall
:
Frobenius.
)
• E ETM Smooth distribution that is iuvduliue .
→ E is
iutegroble .
• E iuvdut.ve dir .
← FE fdiatiou .
Some
applications of Then .
3.38 ( Frobenius Thun. ) to the Steady
of PDES :
Ex Cousin
system of PDES for a fcl . f : IR
'
→ IR
Hey, z )
wordin.ie,
¥
-
ZZ
?
¥0 t 2 ×
¥z = 0 "neu System of first
Ndr PDES
on IR} -
37T
GG t
2g 0¥ = 0
Does 4) has
any
non- Laurent solutions f ?
→ see Tutorial .
¥ lausiger system of PDES for a fol .
f : IR
>
→ IR
¥4,
y
) = a Kis ,
f Kis) )
(**)
0¥ ↳ g) =
ß (x. y,
fley))
< Iß Smooth fcls de find ou an
open
subset.VE/R3 .
¥ When does (** ) has a solution ? Conditions on a
, ß ?
→ see tutorial .
On the opposihe ending of integrale distributions ( among
all dishr ) are the nocalled
bnecket-geueratingdislr.in.
:
Def.3.LI A smooth distribution EETM on a mfd .
M
is called b rocketgenerativeg
, if any
Loud freue { 4 ,
- ich }
of E to
getno with its iheahed hie brocken Eli ,
sj ] ,
Tsi , Esj , sei]
.
.
form
a loud frame of TM Reina
If a Loud freue is brocken generatrug
around some
point ,
then so is ocean french around that point .
¥ Standard unterm au RI
E =
<
¥ ,
¥ +
y ¥) suooku rank 2 distribution ,
[
% ,
¥ ty ¥] =
¥ KE not a seckau
of E .
TIR'
spannend hy % , ¥ ty # ,
¥ H .
¥ Lautet uuanifohds
M odd dim .
uf of dim . 2+1 . A distribution
E ETM ran is a connect distribution ,
if Eko> reuk Zu
s.hr . L
×
: Ex × Ex → IM
#ER is nen -
degenerde
" ( ' '
2) n
% ( LTTE ) k )) the M .
Levi -
brockeh
Äh
I. [ are extension
of g.
g.
to Load uf . Wand ×
and
%
: IM →
IM
#×
is tue Standard hprojedien .
Heute A odd div .
mfd .
eqnippeedw.hu lachend
distribution is called a wartet mhd .
n, war ted
geometry ! topology .
EIN
Driviug a war .
Configuration space ) phone space of a cor : M = IRZX 51×51
↳ '
y ,
d
, ß )
T
×
\
lxtlwszsg.tl sind ) ) ( Hg) Position of
| :
%:: ::*:"
to × -
taxis
b-→
es sheenwgaegteof
y
the freut wheels .
Moving the Cor traversen a Curve clttkltl.gl/-),aHfsG))
In M .
¥ III) is werd to Is!!)
÷ !! !!!!)
) " - ↳ Is:c!:')
j
x
'
lt) sin alt ) -
y
'
(t) Cosa la) = 0
↳
'
lt ) - l sin ldr)) a
'
(r) ) sin lautslr)
) -
(y
'
It) t l los alt) äh)
los (alt) ßlt)
) =D
n ) Solution ←
¥::') - *
„ t.li::: ←
←
"
d
steer vf : X : =
§ Fs
ß
drive uf : Y : =
lwsß .
( Wsa
Gz + Sha
) süߧ
.
] d
The two ,
neutral
'
vector fields X and Y spon a
rank 2 brackel Generalsrang distribution an M .
( X ,
Y
,
EX, 4)
[4
, IX. YD
(TM is
spannend by X ,
Y ,
EX ,
4] ,
EY,
EX ,
4]]) G-
Nen E ETM a Smooth in ✓ du live distribution ,
we
know that
through eau point x c- M we love an
integre
subufd.by FB Thun
.
¥ What about maximal integral subufd .
through a
point ?
There one in
general und subuetd .
but so called initial
Sub mhd .
IR
?
( x.
y
) ( x.
y
) vechor tidd an IR ?
.
{ IT
I - ¥ ta
§ DER
Tz
le
"
,
eig)
( =
Rye)
Integral eures www..tl! )
IR
'
s is IT related
to auf .
a. T '
n )
integre Cervus of it
are
the
image of the
Delegat
Cerues
ofg wo T .
little. . ) ) ) =
( e
"
.in/cT ?
-
If d rational ,
That's a subcufd .
If a is irrational
,
it's not ,
because it wirds
deuselg
or and the Torus .
In
opproprihe dort around a
point
( eit,
eiat ) caesists of haenhably many
line
Segments
Def.3.4.IM mfd .
of din .
u .
① For a
Sunset ACM and x. EA let GIA ) : {
+ EA : 7
6 Smash und
( : TO ,
1) → M
Wireless NA
and do) -
Xo
and ch) }
② N EM is Called on initial submfd .
of M of din -
K
,
if for any
+ EN 7 a
non (Ku ) for M with x e- U
adulx) and u ( ↳ (un N ) ) = u LU) n
"
× Sos ) .
H NEM is on initial subwfd .
Then 7 ! mhd
.
tue in Clustern
Structure bu NS.t.li:1,
M is on in
jede immense
wir tun
pnorohg that for
my ufd .
P und
auuop
f- :P-1N
we home
f n smooth i. f is suook . ¥) .
I.The
home und Loup . one
2nd lautholde hat
huuaenkddy way
of than . ( so N
night
he und 2nd caenhoble ) .
One uns ein on athos D= { ( ( × LUNN ) , a)×
! her
warts es in
② .
-
•
Eqnipp N Für tordogy genesend by ¥N) -
sets
→ this to
pology is in
geued feiner than the
subspoce
topocogy an N indie cad treu M .
In
particulier .
it is still Hausdorff .
Transitives
mops of B one scuooh ,
since resrrctians of
Smooth
Kings .
'
Uniqueness fellows free H) ( el .
sunuf .
)
④×
( Un N ) not open is
sunspocehopobg .
;
If i is
a haue an .
ouko ins
image then it it is and 414N )
=
Vs N
for V E M
open and Un U
,
Ulvsu ) is a
submfd .
hart
)
Converse
ly ,
one
way Show that the
Droge of on in
jede
immerzu i : N ↳ M wir
popeiey 4) Is on initial Iubcuf .
Gaming how ho
integriere dir .
/ fdiatieus :
hingehe
EETM
✓
wir torrey .
hedidie E FE -
For
my
x EM bt FI : =
f y EM : 7 are
c : Eo,
e) → M
SI . do ) = x and CH)
and c
' l t)
feat)
httEEO
,
1)It is called the
ten of NI .
Nahe that it a
plaque in Wien FF it nun ,
the lachend
in that
Knef . Ideen ce ,
kuepcoq es war kein ed in
EE und
reshictiees
of Chart
uueps war he used to
give EE the
Structure of a k dim
.
mf .
• i : FI - M is initidsubmfd .
( Hansa .
t AAZ ) .
.
It is an
integre susmf .
l TYFE =
Ey Ty M tg ) .
.
Any uonnedud
iuhegrd ( mild ) subuf . tudiekrseds EE
ueuhoineud
in EE ( EE =
„
maximal
nkgr.int .
trage x
"
.
)
Fdiatieu FE de vieles Mirko K dir
.
iutidesubmfds -
Rennen We war
equipp
M wih a different uefd .
Ihr .
ME whee athos
gun by pr,
aua :
ni
'
(Wax la? ) → ! ER"
her ( Ua ,
u.
) E FE .
Topdogy on ME fuer teuer an M hut Id :
ME → M
is in
jede knarre .
Neu holouauic warst oiuhs : Lou >Wants on
pontiac and
uedoakg
that can not we
integrated
ho coustroinls an positive only :