Prop.3.30-M.IN mfds ,
f : MAN a local diffeau .
① f-
*
Es ,
y
] = It's ,
ftp.V-s.ge HIN ) .
In
particulier ,
if U ← M is on
open
Sunset and i : U - M indeesiee
,
then Is
) ) Ei
'
Es ,
q
] =
Eis ,
i ) =
Eslu ,
Rlu] theHIM).
- Hexe
.
SIE 0
implies Es ,
27150 t
ME HIM ) .
② For
s.ge HIM ) ,
M :
a)FL
qlx) =
Esigflx) .
to C-
IM
③ Suppen ( V. n) is a and on M and
Size HIM ) wih
SIE ?II and
4; ? ni # . then
[
hin]
!
§
Hindi # , wbve
tun:{ HiFi reif
) .
Prof
① FT = SEHLN) f. MAN
GE IN .IR) .
( Fs lg.tl/lx)--HIf*.g.l)--LIfEIk).g--f;jgi-e.fI.lgof)-(i.
g) of
It's , f) .
(g. f) =
f ; .
(ff.
(g. f ) ) ff
.
( ff (g.
f) )-
④g)
of
-
h
.
( s g)
of
=
(Is
).
g) of =
E):(
g.f) .
③ By ① ,
Thriller =
Klo , Klo ] =
=
!
"
:
÷ ::X *
ii. ¥)
"÷.se?dsiIIiEui-nihIE) , since
[
¥ ,
¥,
] =
0
wie teuer t ! ¥] f
=
¥ f) ¥
-
II. f)
j.tn
deriv. of
fait
j
2nd Pokal
derivdie
of fait .
= 0
by sguuery of 2nd partie
der vehives -
②
74¥ =
Ein] H .
t '→ ITFI, h . F) ) k ) loudlg abfindennein IM .
_
_
- d
Caesi der a : IRZ → IR
g.hn by f c. PIM, R )
altes) : =
g. LFLIK)) .
( f. FI! ) =
= TFL! helft
'
k))>
f) =
ITFI! .
q.FI ! '
f
-
q
-
alt,
o ) =
q ( FIG) ) .
f ←
alois
) =
(TFL
; Kk ) ) .
f =
gk) .
( f. Fl! ) ←
¥alao ) =
¥1+
!
#Ek) ) .
f =
¥ ftp.f) ( Ffn)
-
si
( g.f) .
←
← ( o.o) =
4k) .
¥!!! Es
'
) =
µ
.
(s f)
.
←
ftp..lt.
t)
=
sik f)
-
a. Ist) -
flat)-
Elst!
Ätna) f
.
Ü -
f
D
Grins 4in EH IM )
[
4g] = 0 ⇐
Flieg =
y
wheneuer defuad
F- I. FLY =
FLY .
Elf when euer olefrueed -
*
:*
in"
¥
) Für#
MAIN)
Prof. see Tutorial .
|
HEIKKI nk
)
live =
Erik)+ o
DIf.3.es#M,Nmfds. ,
f : M → N -
map
.
Then s EHLM ) und qz
HIN ) one f- related
,
if
If I K ) =
µ Ifk) ) ttx EM .
Rennen Einen aveaeo find [
EHIN ) (or SE HIM) there
is in
general no veaor Lied ( se HIM) ( resp .
RE HCN))
so that they are f- related .
If f is a Local difteeaueud
YE HIN ) ,
then 7 ! f- related vf ,
uaudy frz .
Prop.3.IN f : M → N U -
map
hehweeu ufels .
Suppen 4,4 E
HIM ) one f - related to
get
HIN)
resp .
GEH IN) .
Then this,
] is f- related to trug.
] .
PI see Tutorial .
n
3.7 Frobenius Theorem
- Existeuce
of flows of vector fields revisited :
- =
{ E HIM )
• for HEM 7 an
integre Curve c : IN M
,
OEI
,
do) - x
( CH) =
FLIK) )
If
41×7=0 ,
then dt) - x neusten Curve .
If
Sk ) # 0
,
then
sly) to Vg EU ,
U
neighbhrofx .
=)
integral Curve,
through x is a 1- dein .
subuefof M
Heule
, s deuanposes V into a Lanier
of 1 dies
.
Subuifds given by
the
images of the integral und>
through YEO.
The
hangeul Space of sum a subutd
through y
EU
equots Rs
ly ) ETGM .
.
.
If we
upon s
by fs,
for a nowhere
vaishiugf-CLM.IR )
then the
integral Curve> of A and 4 one
just teprareuohrizatieeeg
of each other ; Lance the defiue the some
fomily of
1- dim - Sublabels. ( of U ) .
.
Suppose D : x n KEIM is a
map that
ossigus
to
each XEM a line Lx tragen ( ke .
a 1- die .
subspaie of # M)
sit -
7 an
open
neuer { U ; } of M und local redd fields
S ; EHLU;) s.t.si (y) span,
lg ttgt Ui Ui
.
Then for each x EM 7 ! ↳ ad Scuook submfd . Nx EM Srh -
TYN.
=
ey ETGM Hye Nx DIEBE
M mfd of dim .
n .
① A distribution E of rank K an M is
giveu by a k dim
.
Subspoce Ex EIM for each x EM .
② A ( smooth) SEHE ETM is a uecker tiddvf M
s.t.sk) E Ex tx e- M .
A Local sectieu oft defiued
°"
open Sunset U EM is a ↳ adveadtidd SE # LU)
s.IS k) E Ex Fx EU -
③ A distribution EETM of rank K is udleed Smooth
,
if
for any
× e M 7 on
open neighbhd .
U of x
and Loud Seetiers
4 , . .
.sk
EHLU) of E s.hr .
{ ssly ) ,
. .
, S.ly) } is a boss
her Ey Hye U .
Sun collection of Local secti aus i
Called a Local frame of E A
Smooth distribution is also called a (smooth) vector Kebbundle
of TM .
④ A distribution EETM is called invdutive ,
if
for any
Local seehaus s ,
q of E their hie backen Ey]
is also a Load sectien oft .
⑤ A distribution EETM is called Integrale , if for FFM
7 a Smooth subufd N E M wir × e NS.t . hier any ge
N
IN =
Ey ETYM .
S" m
submfds one called integral subcufds of E .
Exrsheuce of flows for vector fields uuplies
Prop.3.35-A.my Smooth distribution
of rank 1 on a mhd is
Integrale .
Distributions of high Main are not
always in
hegroble .
A necessary
Landi time for
iuhugrasilihg of a distribution is
invdutivihy :
het EETM he a
integrale distribution and NEM
is on
integral Subufd ,
v. e. IN =
Ex EIM KEN Assune
s ,
y are Local sectieus of E defined an open nlignbld.
U of x EN in M .
Replouhg N
by Nn U , we
mag ossuu NEU .
Stu
and
R ) ,
one i - related ho weder tredds T.IE HIN )
when i : N ↳ U EM is the Nansen .
( Tyi :
TYN Ey
↳
IM
Mondeodmd
)
=) ES
Iu ,
4) u
] a is i related
to IT.
ÜIE HIN ) .
Prop . 3.34
=) Es ,
q] ( y ) E im ( Tyi ) =
Ey ty EN .
- Frobenius
Thin Shows
theol also the couverte is true
,
i.e .
any luueluhive Smooth distribution is integrale ,