Differentialgeometrie: Kurven - Flachen - Mannigfaltigkeiten

Wolfgang Kühnel Differentialgeometrie

Wolfgang Kühnel

Differentialgeometrie Kurven – Flächen – Mannigfaltigkeiten 5., aktualisierte Auflage STUDIUM

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über abrufbar.

Prof. Dr. Wolfgang Kühnel Fachbereich Mathematik Universität Stuttgart 70550 Stuttgart E-Mail: [email protected]

1. 2., 3., 4., 5.,

Auflage 1999 überarbeitete Auflage 2003 überarbeitete und erweiterte Auflage 2005 überarbeitete Auflage 2008 aktualisierte Auflage 2010

Alle Rechte vorbehalten © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2010 Lektorat: Ulrike Schmickler-Hirzebruch | Nastassja Vanselow Vieweg+Teubner Verlag ist eine Marke von Springer Fachmedien. Springer Fachmedien ist Teil der Fachverlagsgruppe Springer Science+Business Media. www.viewegteubner.de Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlags unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dass solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu betrachten wären und daher von jedermann benutzt werden dürften. Umschlaggestaltung: KünkelLopka Medienentwicklung, Heidelberg Druck und buchbinderische Verarbeitung: MercedesDruck, Berlin Gedruckt auf säurefreiem und chlorfrei gebleichtem Papier. Printed in Germany ISBN 978-3-8348-1233-9

,/8,/3 >-S-S \'< -DXSX"D* "\S FRA-S\D:-D \ +$-R *"S -O) *>P *-R
\XFR ^>-*-R"A:-FB-XR>- I*"S S>D* *>- "H>X-A G09 *-S \'D-R -$-D3"AAS ->DS-B-SXR>:-D FRA-S\D: \ +$-R >-B"DDS'- I "H>X-A 70.JK >- ^-S-DXA>'SS- SFAAX-D >D *-D \ +$A>'\BS IGK0YK -B-SX-RJ $-R->X:-SX-AAX S->D( >D-"R-
A:-$R" \D*
D"A`S>S) ->DS'-;A>'< >4-R-DX>"A= \D* DX-:R"AR-'D B-'< *"S \'< "AS -:A->XA-@X+ \R- a\ ->D-R FRA-S\D: "$ *-B 9K -B-SX-R) \D* a^"R "\S*R+ \'@A>'< "\'< 3+ \R -AX $-= SFD*-RS 3+ \R *"S "H>X-A . 0 "\'< 3+ \R @SX\*-DX-DK "X\R:-B+ "; @"DD *-R
DSHR\'< D>'<X S->D) *"$-> ^>SS-DS'-AB-S'4-R-DX>"A:-FB-XR>-RSX -RB+F:A>'D *-D
D3"D:S0 "H>X-AD ^>R* :RF;-R -RX "\3
DS'-RX ^>R*K >- D"'<
DS>'<X *-S
\XFRS $-SFD*-RS ^>'<X>:-D >D:- S>D* >D + "SX'D:-R"- $-SFD*-RS '<X B+F:A>'< :-^-S-D F- DX-RSX+ \Xa\D: B->D-R X\*-DX-D \D* >X"R$->X-R) *>- a"''-A- R->SSA-R) *>S-DX-DS>] FRR-@X\R :-A-S-D A*-R B>X *-B FBH\X-R"A:-$R"=`SX-B 
 -RSX-AAX) "R'=A>]-R X= XF X-A U $->:-SX-\-RX) A]" "*-R-R HX ]-RX->AX ^\R*-J >D 
 :-S'-$-DK '-;A>'< D ]-RS'-*-D-R >D= S>'<X -RB\X>:X) \D* *-B -D-R:>S'Ra-$R\'< >SX -S a\ ]-R*"D@-D) *"SS -S SF S'D *>- ->-^-:=X\*>\B
\3$"\@\RS "XB \D> GEEE

5  - a^->X-
\8":- ^\R*-D "AA- "H>X-A :R+ \D*A>'< \ +$-R"R$->X-XK D B"X+ ->D>:- a\S+"XaA>'A*\D:-D ->D:-3+ \:XK
\'< SHR"''< ^\R**-R -_X ]-R$-SS-RX) *"S A-XaX-R- \DX-R *-R @\D*>:-D
DA->X\D: ]FD "R-D FRSK ->D "D@ :-<X "D "AA-) *>- "\3 RRX+ \B-R >D *-R -RSX-D
\8":- D:-^>-S-D 'A*-R -RSX-AAX -S-R X-AA- DF'< *>- -D:A>S''"A F'>-X` >D *-R ->B "D\"R [bbY

5  D *-D ]-R:"D:-D-D "X>] \D* -RB\X>:-D* ^"RK !"'@ F*-R -_HA>a>X- -R$-SS-R\D:S]FRS'-S- SX-XS *>- 3FA:-D*-D \D@X-( GK S SFAAX- B-'A*\D:-D Da\:-@FBB-DK >- B->SX-D *"]FD SX"BB-D "\S *-R -*-R I$a^K *-B FBH\X-RJ ]FD "$R>-A- R->SSA-R \D* >'-A- \D:-A+ FSX-) *-R-D +FS\D:-D S>'< ]>-A3"'< >D "D*-R-D + \'SX a\ X *>- *R>XX-
\8":- B-\B ->:D-XK -B >-^-:=-RA": >SX a\ *"D@-D 3+ \R *"S DX:-:-D@FBB-D \D* -RSX+ "D*D>S $-> *-R R^->X-R\D: *-S \'X-A V -R:+ "DaX) \D* >DS:-S"BX + \R *>- ]>-RX-
\8":- ^\R*-D *>- $\D:S"\3:"$-D + ^\R*-D B--DJ a\ "D@ ]-RH8>'<X-XK X\XX:"RX) >B "D\"R [bGb

5  D:\D:-D "D R+ \BB\D: \D* FRS>FD K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K G9 [ >- R-D-X0A->'- "\;0
$$>A*\D: \D* R+ \BB\D:-D ]FD A+ "'"DX-
$A->X\D: K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K E9 9 "R"AA-A]-RS'-$\D: \D* -F*+"X>S'- "\;0A->'\B K K K K K K K K K K K K K K K K K K K K K K K K K K K Gb[ 9 -R "\HXS"Xa *-R AF@"A-D A+"'- "\;0 R+ \BB\D: >D SH-a>-AA-D "R"B-X-RD K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K GGb 9 -R "Xa ]FD "\;0FDD-X K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K GGV 9
\S:-^+"'S

 $'((/# (($"&2$"%$2(

+3)

7
-R "DD>:3"AX>:@->XS$-:R>4 K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K G9b 7 -R "D:-DX>"AR"\B K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K G99 7 >-B"DDS'@-D K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K G9E 7 -R >-B"DDS'*>S'- >D *-D 3FA:-D*-D "H>X-AD [ \D* Y ]FR:-SX-AAX- >4-R-DX>"A:-FB-XR>- I"\'< *>S'R *>- XFHFAF= :>S'D -SX"AX ]FD R-Da^-RX-D) F4-D-D -D:-D) >4-R-DX>"X>FD \D* DX-= :R"X>FDK \R'<
\Sa->'D-S 3-SX-D \D@X-S "AS RSHR\D: >SX -S B+ F:A>'*>S'R >D *>-S-B \'< >B ^->X-R-D -RA"\3 "\'< X\D ^FAA-DK !\ *-D R\D*$-:R>4-D "\S *-R >D-"R-D
A:-$R" ]-R^->S-D ^>R "\3 *"S \'< ]FD 5*8( 7) a\ R\D*$-:R>4-D *-R
D"A`S>S I->DS'-;A>'< :-^+ F= '4-R-DX>"A:A->'S-D ^>R "\3 5178< 7)
D"A`S>S G)[) a\R DX-:R"X>FD \D* a\ >4-R-DX>"A3FRB-D "\3 5178< 7)
D"A`S>S YK +-+- . 0 &/ %2*..5' '$2 $((.' .*5%2 -R 0 >SX -R@A+"RX "AS *>- -D:- "AA-R 7=\H-A R-AA-R !":SX-D >:-DS'X+ "X >D P! R     GK P! R 1 R! P I@// ;9+ J   [K P! P , b 3+ \R "AA- b (1 P  0 I21:+;+=  %0+;) +;J >-S-S @"A"RHRF*\@X -RB+F:A>'<X -S) *>-  0' ]FD -@XFR-D *\R'< *>- 19/ 7  P! P P (1

[

G -a->'A3SB>XX-A "\S *-R
D"A`S>S

a\ B-SS-D) SF^>- *-D +0- . ? a^>S' -@XFR-D P! R (1 b *\R'<
P R  P! R

! 1 'FS ? 1 P R P + R ->Da\3+ \S'S' \D@X-D P! R ^>R* *"$-> -R@A+ "RX "AS FRB *-S -R$>D*\D:S]-@XFRS R$PK "B>X ^>R* *-R 0 a\ ->D-B 019/+ 9; 0  -;199D-RS->XS \D* a\ ->D-B / ;9+:) 0 D@-AB-SS\D: "$:-A->X-X ^-R*-DK >D- 9;)1019/.:+: \R . (1 /K I@\Ra( *:+:J >SX ->D- "S>S $# ! = = = ! $0 B>X $( ! $( 1 G \D* $( ! $) 1 b 3+ +-4- . 0 &/ 2*,*&*"$/#. 5' >- FHFAF:>- *-S D 0 $"S>-RX :"Da -DXS'*-D* "\3 *-B -:R>4 *-R F4-D-D %=B:-$\D: ! IPJ 1 R  0 D P $ R 4 % ->D-S $-A>-$>:-D \D@X-S PK "B>X -R@A+ "RX B"D *-D -:R>4 *-R 10= 9' 0A \D* *-D *-S 9 0A> 9; : =10 AB-D:-D I->DS'-;A>'< *-R A--R-D -D:-JK >D- -D:-
^>R* ' :).1:: 0 :-D"DDX) ^-DD >SXK +-3- $.(2$2$*( $' 0 -R ^>'<X>:SX- -:R>4 3+ \R *-D DF= "RX S>D*) SF^>- I"AA:-B->D-RJ ]FD
$= D-D) *>- "\3 F4-D-D ->AB-D:-D  . 0 -R@A+ $>A*\D:-D ]FD F4-D-D ->AB-D:-D  . 0 >D *-D / K >4-R-Da>-R$"R@->X $-*-\X-X >D-"R>S>-R$"R@->X I$>S "\3 -RB- A*\D:
H ( 0  / :>$X) SF *"SS >D ->D-R :-->:D-X-D B:-$\D: " IPJ :>AX  IP L QJ 1  IPJ L
H IQJ L 9IQJ= "$-> $-*-\X-X *"S `B$FA 9IQJ) *"SS *>-S-R -RB 3+ \R Q  b :-:-D \AA @FD]-R:>-RX) \D* a^"R "\'< D"'< ]FR:-R >]>S>FD *\R'< QK S >SX *"$-> DFX^-D*>:-R^->S-
H *>-?-D>:- A>D-"R-
$$>A*\D:) *>- *\R'< *>- A*\D:  >D P >SX *"DD -R@A+ "RX "AS *-R "D: *-R \D@X>FD"AB"XR>_K + \R \DS-R- !^-'@- >SX *-R ^>'<X>:SX- "AA *-R?-D>:-) $-> *-B  >D ?-*-B \D@X P   *>4-R-Da>-R$"R >SX \D* \ +$-R"AA B"_>B"A-D "D: D *>-S-B "AA- ->D- // 9:+10 I3"AAS 7 ! 5J $a^K ->D- A*\D: I>D-"R>S>-R\D: ]FD  J R-HR+ "S-DX>-RXK >-*-\X\D: *"]FD ^>R* *\R'< *-D 3FA:-D*-D "Xa \ +$-R >BHA>a>X- \D@X>FD-D @A"RK +-!- 2: 5  . $',&$:$2 5(%2$*(( >D- +/2.+A+; R* aKK :-:-$-D *\R'< ->D- A->'D-"R >SX) SF >SX *>-S D\R ->D- R":- *-S "D:-S ]FD  K "AAS  D>'<X A>D-"R) "$-R SX-X>: *>4-= R-Da>-R$"R >SX) SF @"DD *>-S ->D-RS->XS >B "AA:-B->D-D D\R .1-. :-A>D:-D I*"S S>-<X B"D

G -a->'A3SB>XX-A "\S *-R
D"A`S>S

Y

S'D- *>4-R-Da>-R$"R- \D@X>FD ]FD P I$a^K \B:-@--S :>AX >D "AA-D >B-DS>FD-D ^>- 3FA:X) ^F$-> ^>R \DS *-R >D 5178< 7)
D"A`S>S [) 2. :-:-$-D-D FRB\A>-R\D: "DS'-;-D( : : + 0 # . * a>X- A->'A*\D: ]FD B"_>B"A-B "D: SF ]-RA*\D: ]FD B"_>B"A-B "D:K R+"a>S>-RX ^>R* *>-S *\R'< *-D SF:-D"DDX-D 0':;A) *-R ->D- FA:-R\D: "\S *-B "Xa \ +$-R >BHA>a>X- \D@X>FD-D >SXK >D- FDS-M\-Da >SX *-R SF:-D"DDX- ;A <  9 + /- )9+.4-R-D= a>-R$"R-
$$>A*\D:T *>- $->*-D $-X->A>:X-D ->AB-D:-D *-S 0 D-DDX B"D "\'< +# 1* /192) a\->D"D*-R) ^-DD ->D SFA'4-FBFRHSB\S -_>SX>-RXK
\;-R*-B 3+ \:3"AX>:@->X) ^>- "\'< >D 5178< 7)
D"A`S>S Y) 2G9 ->D:-3+ \D:S ^-R*-D DX-RB"DD>:3"AX>:@->X-D >B 3FA:-D*-D "H>X-A [ :"R D>'<X \D* >D "= H>X-A Y D\R "B "D*- $-D+FX>:XK -R -S-R @"DD "ASF *>- 3FA:-D*-D -6D>X>FD-D GK7 0 GK. a\D+"'<SX \ +$-R:-R-@X B>X "H>X-A [ $-:>DD-DK +-- ($2$*( IDX-RB"DD>:3"AX>:@->XJ >D- -*+/ 0:+10. 0; 9/00+'".;+'- +; I*-R A"SS-   J . 0 >SX *"*\R'< *-6= D>-RX) *"SS AF@"A *>- \AASX-AA-DB-D:-  # IbJ ->D-R I=B"AJ SX-X>: *>4-R-Da>-R$"R-D
$$>A*\D:  0 1  $ 0* * D 1 D \R B>X B"_>B"A-B "D: >SXK "S ;X :-D"\-R( "D:I H  J 1 "D: 5 5H H 1 7 $ 0 3+ # ?-*-S P   ) ^F$->  1  IbJ 3+ \R ->D- :-->:D-X- B:-$\D:  ->D-S ?-*-D \D@X-S ]FD K F@"A @"DD B"D "\'< "AS >A* ->D-R // 9:+10 *-R A"SS-   $

* 1  $ . 0

9

G -a->'A3SB>XX-A "\S *-R
D"A`S>S

B>X "D:I( J 1 0 $-S'$-DK >D SFA'SX *"DD >D?-@X>] \D* ;X ->D- .1-. 9/ ;9+:+ 9D- 9; ]FD K B:-@-SX *"S >A* ->D-R B= B-RS>FD ( D>'<X >BB-R ->D- DX-RB"DD>:3"AX>:@->X) "\'< *"DD D>'<X) ^-DD ( >D?-@X>] >SXK >- !"SX *>- +/ 0:+10) \D* 7 $ 0 ;X "\'< *>- 1+/ 0:+10 ]FD K H-a>-AA *"S 3FA:-D*- "H>X-A [J) 0 1 [ \D* 7 1 Y I.) 0 +/ . ) *"S @A"SS>S'-J SF^>- 0 1 7 $ G I@2 9& ) 0 +/ 0 JK +-1- ($2$*( I"D:-DX>"A$+ \D*-A *-S 0 J 0 0 0 SX   (1  &  ;X 0' 0;+. *"$-> H 0 (1 P & 0 *-R 0' 0;+.9B \D@X P I1 "\B "AA-R "D:-DX>"A]-@XFR-D >B \D@X PJK \R'< *>-S- 3FRB"A- -6D>X>FD \DX-RS'*-D ^>R a^>S'XS \D*  -* ;19 0 "D*-R-RS->XSK >-
. +;SS-D 3+ \-
$A->X\D: ( H @"DD *"DD "AS A>D-"R-
$$>A= *\D: a^>S'="XR>_ H K -B+"; GKY :>AX 3+ \R *>-
$A->X\D: D"'< -6D>X>FD ( IP L % + J 1 ( IPJ L % + H ( IJ L 9I%J= +-0- ($2$*( I"D:-DX>"AR"\B "D ->D- DX-RB"DD>:3"AX>:@->XJ S S-> . 0 ->D- -*+/ 0:+10. 0; 9/00+'".;+'- +; \D* -S S-> ;  K -R "D= :-DX>"AR"\B "D >D ; >SX *-R?-D>:- DX-R]-@XFRR"\B 4 . 4 0 ) *-R *-6D>-RX >SX *\R'< 4 (1 (E IM & * J 1 (E IE * J 3+ \R ->D- "R"B-XR>S>-R\D: ( (   B>X ( IMJ 1 ;) ^F$->  0 * ->D- F4-D- -D= "D:X D>'<X ]FD *-R "SXK -R "D:-DX>"AR"\B 4 >SX -*+/ 0:+10. \D* S>-R\D: ( "$ I5178< 7)
D"A`S>S Y) 2G7JK A ,  (1 4 4

;X "\'< *"S 0' 0;+. B \D@X ;K "RD\D:(  B\SS ]FD & * S-'< "AS 19;)1'10. : 1/2. / 0; ]FD 4 -R:>$X( 4 0 1 4  ,4

 

  *6

%0*&6

"$-> $-a->'- FRXD ]-RS'-*-DSX-R ->S- "\3) a\B ->SH>-A "AS RF6A@\R= ]-D ]FD X-'S'SS- *-RS-A$-DK
\3 *-B ^->;-D !->'- H\R) *>- ->D A->SX>3X F*-R ->D "D*-R-S !->':) B"D SHR>'<X *"DD "\'< ]FD ->D-R 9/ ;9+:+ 9 "$SXR"-RX B"D ]FD ?-*-R >'@-) *>- ->D- IR-"A-J \R]>D >R:-D*->D-B >DD- A*-K "$-> SFAA-D SF^F- "R"B-XR>S>-R\D: "AS "\'< *>- >A*B-D:]-RD+ \D3X>:- >:-DS'- ->D- B"XSS ]FD
D3"D:S:R+ \D*-D ->D-R \R]-DX- 6D*-X S>'< $-R->XS >D *-B \'< ]FD 5178< 7)
D"A`S>S [) 29K >R ^-R*-D *>-S -R "$-R D>'<X ]FR"\SS-Xa-DK

5
/+36/7+ '*
7 "XD- :- \R]-D) *>- ->D :"Da-S \"*R"X \RA>'D *-R $-D- \ +$-R*-'@-DK# "SX -S ]FB X"D*H\D@X *-R
D"A`S>S S-'< *>- >4-R-Da>-R$"R@->X F*-R $-SS-R *>- SX-X>:- >4-R-Da>-R$"R@->X a\ 3FR*-RDK
$-R "\'< *>-S XR>4X D>'<X *-D -RD *-R >D:-) *" >4-R-Da>-R$"R@->X ->D-R
$$>A*\D: D\R $-*-\X-X) *"SS *>-
$$>A*\D: *\R'< ->D- A>D-"R-
$$>A*\D: "HHRF_>B>-RX ^-R*-D @"DDK + \R *>- >A*B-D:- $R"\'<X *"S *"DD D>'<X a\ :-AX-DK FB :-FB-XR>S'DD]FAA) *>- AF@"A-
HHRF_>B>-R$"R@->X *-R >A*B-D:- *\R'< ->D- -R"*- a\ 3FR= *-RD) *K- \R]- >D ?-*-B \D@X ->D- "D:-DX- "AS :-FB-XR>S'D-"R>S>-R\D: $-S>XaXK >-S $-*-\X-X "$-R :-R"*-) *"SS *>-
$A->X\D: *-R
$$>A*\D: ]FD D"'< 0 D>'<X ]-RS'D*-D *"R3K "D D-DDX ->D- SFA'A*\D: "\'< ->D- // 9* :+10K "S ;X ->D3"'B"A B+ F:A>'  JA !/4 A!2F/K !/4%-!B !/B8/!2 B/!-! 5*- 7FD ( $ 0 ) *-6D>-RX "\3 ->D-B DX-R]"AA 0 K "S $-*-\X-X) -S :>AX , 1  +$-R"AAK A (1 b \ -R -@XFR II , 3J 1

" DD D "I AA

;X 0' 0; 0= -;19 "D  >D I3 ) \D* *>- ]FD >D *>-S-B \D@XK >- >SX ->D- :-FB-XR>S'B"X>FD ]FD -RSX-R , 3 J L 9IIJK R*D\D: >D *-R B:-$\D: *>-S-S \D@X-S B>X II3 L IJ 1 II3 J L I + II + >D- 9 'D-
M\>]"A-Da@A"SSR-:\A+ "R-R H"R"B-XR>S>-RX-R \R]-D \DX-R R-:\A+"R-D I\D* FR>-DX>-R\D:S-R 1 %! &J ?( %! & $ %! &!

? , b!

? $>?-@X>] \D* SX-X>: *>4-R-Da>-R$"R)

 \D*  ) ? ^-R*-D *"$-> "AS 6SX >D]"R>"DX \DX-R ISFA'FD-DK @"A>S'< @"DD B"D ->D \R]- "DS-D
$- >SX R-:\A+"R) 3"AAS *>- FB-DX"D:-S'D*>:@->X  , D>-B"AS D\AA ^>R*K 4-4- '' -*- R-:\A+"R- \R]- @"DD B"D D"'< >D >DXS]-@XFR >SXJK %  A *8 S S-> *>- \R]- ( %! &  0 :-:-$-D B>X -S"BXA+ "D:- 1    A  "IK >R S-Xa-D *"DD %! & 1 %b! & \D* 3+ \-S *-6D>-RX ->D-
$$>A*\D: A( %! &  %b! &K S ^>R* *"DD : >  # 1 1  ) SF *"SS  ) ? 1  ) A # A A A  (1 b) "ASF :>$X -S ->D- B@-FD ? (1 A D"'< *-R F:-DA+ "D:- H"R"B-XR>S>-RX >SXK >-S- "R"B-XR>S>-R\D: >SX ->D*-\X>: $>S "\3 ->D
-RS'-$\D: E # E L E3 F*-R ->D- >'<X\D:S\B@--$>:- R-:\A+ "R- "R"B-XR>S>-R\D:

IEJ , 1 

 1

$-a->'- "R"B-XR>S>-R\D: D"'< *-R F:-DA+ "D: A  >

S-> *-R "D:-DX-D]-@XFR S-> *-R >DXS="D:-DX-D]-@XFR

DS$-SFD*-R- :>AX *"DD , 1

>  A 

1 ,   \D*   1 G=

[
R-D-X0 \R]-D >B 0

U

4-3- $/,$& GK IIJ 1 II! IJ) ->D-  9 >D X"D*"R*="R"B-XR>S>-R\D:K -:-D , 1 I! J >SX *-R "R"B-X-R *>- F:-DA+"D:- :-D"\ *"DD) ^-DD 0 L 0 1 GK >- "R"B-XR>S>-R\D: "R 3+ \R I 1 bK IIJ 1 II. ! I. J $-S'$X :-D"\ *>-S-A$- -R"*-) >SX "$-R D>'<X R-:\A+ [K IIJ 1 #0 I'FS [I! S>D [IJ) ->D 9 +: B>X "*>\S #0 K -:-D IIJ , 1 I$ S>D [I! 'FS [IJ :>AX  , 1 G) "ASF >SX I *-R F:-DA+"D:-DH"R"B-X-R) *KD @FDSX"DX-S >-A3"'S'< -DXSX-<X  "AS "D-D R->SK

ä°{

ä°Ó

qä°{

qä°Ó

ä°Ó

ä°{

qä°Ó

qä°{

$& 4-+ R->S \D* 'SX IIJ , 1 I[I! YI0 J B>X IbJ @->D- R-:\A+ "R- "R"B-XR>S>-R\D: ]FRK "XS+ "''< - B"'<X *FRX ->D-D D>'@O \B *-D >D@-A @K P >- >4-R-Da>-R$"R@->X ]FD  "AS
$$>A*\D: SX-<X *-B F4-DS>'<XA>'< D>'<X -DX:-:-DK 7K IIJ 1 II!  'FS< A J B>X ->D-R FDSX"DX-D  (1 b) *>-  ;; 0.+0+ K >- -DXSX-<X "AS *>SX"$>A- ":- ->D-R I$-A>-$>: $>-:S"B :-*"'<X-D) "$-R S'S' 3-SX-D \D@X-DK -:-D IIJ , 1 IG! S>D< A J >SX I D>'<X *-R F:-DA+ "D:-DH"R"B-X-RK

[ \R]-D >B 0

.

ä°£

ä°ä£ ä°ää£

ä°£

ä°Ó

ä°äx

ä°ä£ qä°ää£

qä°ä£

qä°£

$& 4-4  *-S'- 9-;9+? IF*-R ). 22-SX *"*\R'< :-@-DDa->'- "D:-DX- D"'< @FDSX"DX-R +"D:-  ->D- 3-SX- -R"*- 8 IaKK *>- R=
'<S-J 1 * %A XR>4XK "D @"DD *"DD ->D-D
SX *"]FD *\R'< IIJ 1  -_HIIJ! 3 G $ -_HI[PJ "P ) * 1 %A8 $->*- a\S"BB-D *\R'< IIJ 1  -_HI$IJ! 3 G $ -_HI$[PJ"P H"R"B-XR>S>-= R-D) SK >A* [KYK £

£ ä

q£

Ý

£

ä°x

£

q£

$& 4-3 -XX-DA>D>- \D* '-R-D( IEJ 1 IbJ L E IbJ L

E0  E.  IbJ L  IbJ L 9IE. J [ V

"BA>'< *>- 0' 0; "D  >D E 1 b >- >D-"R>S>-R\D: IbJLE IbJ $-S'$X ->D- -R"*-) D+   I^-:-D  IbJ (1 bJK >- M\"*R"X>S'B"X>FD IbJLE IbJL >0  IbJ $-S'$X ->D\"R"$-A I3"AAS  IbJ (1 bJ) *>- SF:-D"DDX- I-\@A>*>S'-R-D ^>R *>- 3FA:-D*- -6D>X>FD( "D:- H"R"B-XR>S>-RXJ S>'< "D S":X) *"SS a^-> \R]-D # IEJ \D* 0 IEJ I$->*- D"'< F:-DA+ >D E 1 b =10 -*; 9 90S a\R 0=X-D R*D\D: *FRX \ +$-R->DSX>B= B-DK "D @"DD "\'< S":-D) *>- ->D- \R]- S'-:X S>'< ]FD 0=X-R R*D\D: "D *>- "D*-R"DK !\B ->SH>-A $-R+ \- F$>:- '-:H"R"$-A *>- \R]- >K"K D\R ]FD a^->X-R R*= D\D:) \D* a^"R >D >XX-R R*D\D: $-R+ \X-ASK
D"AF: @"DD B"D D"'< @\$>S' S>D* @\$>S'D-S ->D ^>'<X>:-S -R@a-\: >D *-R 'FBH\X-R:-SX+ \XaX-D -D :-->:D-X-S -a\:SS`SX-B a\R -S'$\D: *-R \R]-) *"S "D *>- \R]- FHX>B"A "D= :-H"SSX >SXK >-R ^>R* B"D -R^"RX-D) *"SS *>- -@XFR-D  !  !  ! = = = *"S AF@"A- -R: S>D*K >-S BFX>]>-RX *>- 3FA:-D*- -6D>X>FD( 4-!- ($2$*( I7 / ->D- 9 '- 7=B"A SX-X>: *>4-R-Da>-R$"R \D* D"'< F= :-DA+"D:- H"R"B-XR>S>-RX >SXK "DD D-DDX B"D  ->D- 9 0 ;!: S>D*K "S $-:A->X-D*R-D  !  ! = = = ! %0#& >D ?-*-B \R]-DH\D@X A>D-"R \D"$SX *"DD ->D*-\X>: $-SX>BBX *\R'< *>- 3FA:-D*-D -*>D= :\D:-D( I>J $# ! = = = ! $0 S>D* FRX-RX \D* HFS>X>] FR>-DX>-RX) I>>J 3+ \R ?-*-S 0 1 G! = = = ! 7 $ G :>AX .7I$# ! = = = ! $* J 1 .7I !  ! = = = ! %*& J! ^F$-> .7 *>- A>D-"R- + \AA- $-a->'D- R-D-X0 \R]- *>- -*>D:\D:  (1 bK "B>X S>D* A-*>:A>'< -D*-H\D@X- "\S:-= DFBB-DK + \R 7 1 [ :>$X -S @->D- -'<X-D >DS'DDX $# ! = = = ! $0# "\S  ! = = = ! %0#& *\R'< *"S 9/!)/+;:) 9;)1'10.+* :+ 9D*\a>-RX *\R'< *"S R-D-X=7=->D ->D*-\X>: ->D- \R]- >D *-R X>-3-A= \R 7  Y "DD>:3"AX>:@->X "AA-R 7=->D- >B 0 K B:-@-AX *>-S >K"K D>'<X) ^->A aKK 3+ ->D @FDSX"DX-S 7=->D @->D-R R-D-X0 \R]- -DXSHR-'D- 9 0 ;!: *>4-R-Da>-R$"R >SXK -R 0' 0; 0= -;19 >SX $# 1  ) *-R 19/. 0= -;19 >SX $0 ) *-R "\S $# *\R'< R-:) "ASF  1 1 + $0
\S  !   1 b 3FA:X  !  1 bK "D*  \D* $0 A>D-"R "$X ->D-R S@"A"R-D \D@X>FD 1K >-S- \D@X>FD 1 ;X *"DD *>- IFR>-DX>-RX-J 9 - \R]- I$a^K > $-*-\X-X 1 , b) *"SS *>- "D:-DX- S>'< D"'< A>D@S *R-<X \D* 1 4 b) *"SS S>- S>'< D"'< R-'<XS *R-<XK D ->D-B  0 2X 1 1 b >SX *>- >'<X\D: *-R "D:-DX- SX"X>FD+ "RK S :-AX-D *"DD *>- 3FA:-D*-D
. +;_) *>- S'-S- A->'- \R]- >D ->D-B "D:-H"SSX-D @"RX-S>S'D"X-DS`SX-B $-= S'$X *\R'< IIJ 1 II! RIIJJ B>X RIbJ 1 RIbJ , 1 b II >SX ?-XaX D>'<X *-R F:-DA+ "D:-DH"= R"B-X-RJ) SF :>AX IbJ 1 Ib! bJ!

IbJ , 1 IG! bJ!

+IbJ 1 Ib! R+IbJJ 1 Ib! 1IbJJ=

1 * 0 "\3) >- R+ \BB\D: 1IbJ 1 R+IbJ XR>XX *"R-@X >D *-R '-:H"R"$-A I # I! J%3& 0 I *>- "AS M\"*R"X>S'FD ]FD  >D I3 1 b -DXSX-<XK
AA:-B->D-R $-R+ \'< ?- a^-> -$-D- \R]-D ]FD 0=X-R R*D\D: :-D"\ *"DD) ^-DD S>- >B 'XXH\D@X *>:A->'- *>- :A->'X @FDSX"DX-R R+ \BB\D:J >D- R-:\A+"R- \R]- >B 0 - ->D ->A # >SX I3"AAS 1 (1 bJ F*-R ->D ->A ->D-R -R"*-D I3"AAS 1 1 bJK ->D-S R->S-S B>X "*>\S + -R  A *8 3FA:X *>R-@X "\S *-D R-D-X0A->' a\D+ "'<SX 1IE3 J (1 b 3+ \R ->D 3-SX-S E3 K 4-D$"R >SX IEJ L +%># & $0 IEJ :-D"\ *"DD @FDSX"DX) ^-DD IEJ ->D-D ->A ->D-S R->S-S B>X "*>\S  +%># &  $-S'$X) ^->A *-R >4-R-Da]-@XFR *>- @FDSX"DX- + "D:# # #    +%> &  ]"A-DX a\R FDSX"Da 1 1 1IE3 J) ^->A ?"  L +%> & $0 1 $# $ +%> & 1$#

[ $-D- \R]-D \D* "\B@\R]-D

GG

:>AXK "SS 1  b D\R 3+ \R ->D X+ \'@ ->D-R -R"*-D :>AX) >SX XR>]>"A ^-:-D $0 1 $1$# K >-*>D:\D: b 1 1$0 IEJ 1 $# IEJ 1  IEJ >BHA>a>-RX *>R-@X  IEJ 1
\D* IEJ 1 E
L 
I^F$->
!   0 @FDSX"DX S>D*JK 4-0- '.%5("( GK + \R ?-*- R-:\A+"R- \R]- >D *-R $-D- B>X D>'<X=]-RS'D*-D*-R

R+ \BB\D: *-6D>-RX *-R R->S \B IE3 J L +%># & $0 IE3 J B>X "*>\S  +%># &  *-D SF:-D"DDX-D )/+ '-9 +: F*-R "\'< 9 D*-\X>: $-SX>BBXK >- "\S "AA *>-S-D >XX-AH\D@X-D :-$>A*-X- \R]# $0 IEJ E # IEJ L +%>& ;X =1.K"K D>'<X R-:\A+ "R S->DK `H>S'SXR-X-D H>Xa-D "\3 ^>- $-> *-R  *-S'A* [K[JK >- ]FA\X- ->D-R AA>HSD-R -$-D-D \R]- *\R'< > *>- R+ \BB\D:S3\D@X>FD 1IEJ ]FR:-:-$-DK "D @"DD *"DD S-Xa-D * 1 $# 1 'FSIIEJJ! S>DIIEJJ B>X ->D-R :-S\'<X-D \D@X>FD IEJK FX^-D*>:-R^->S- :>AX * 1 $0 1 $ S>DIIEJJ! 'FSIIEJJ = >- R-D-X0A->' "D-S "D:-H"SSX-D

FFR*>D"X-DS`SX-BS @"DD B"D "DD-:-D "D:-DX-D]-@XFR $ # %> "ASF IEJ 1 3 1IIJ"I= FA:A>'< ^>R* :-S\'<X- \R]- IEJ 1 IPIEJ! RIEJJ $-S'-$-D *\R'< $ > $ > ($ ? / ($ ? / 'FS 1IIJ"I "G! RIEJ 1 S>D 1IIJ"I "G= PIEJ 1 3

3

3

3

[ \R]-D >B 0

G[

+ \R @FDSX"DX-S 1 3+ \-S DF'< ->DB"A "\3 *>- $-@"DDX-D + FS\D:-D D"'< [KVK "AAS 1 ->D- A>D-"R- \D@X>FD ]FD E >SX)0 -R- SF:-D"DDX- 2+00--A* [K9K 4-- 5'%5.6( + \R 7 1 Y >SX ->D- R-:\A+"R- Y=B"A SX-X>: *>4-R-Da>-R$"R- \R]- ->D9 0 ;!- >SX D"'< FR"\SS-Xa\D: SX-XS HFS>X>]K
AS
. +;- +0_='$^->S- @"DD B"D S>- ^>- 3FA:X a\S"BB-D3"SS-D( 5 . 5 . 5. b 1 b $# $# , $0 3 1 , $1 b J 3 , $0 3 $. $. b $J b  . 7,?/& / GK >D-  0 -RX) *"SS $. @FDSX"DX >SX) \D* *"SS  >D ->D-R a\ $. S-D@R-'<X-D $-D- -DX'<X "D *-D R-D-X0A->'- S>'< >D *>-S-B "AA "\3 *>- R-D-X0 A->'-R-D B>X Da\:-3+ \:X-D \AA-D >D *-R "XR>_K [K "AAS  IEJ 1 b >D D\R ->D-B \D@X :>AX) SF D R-'<XSS->X>:-S \D* ->D A>D@SS->X>= :-S R-D-X=Y=->D B>X ->D-R FRX_O "D *-R 'XXSX-AA-K "B>X >SX P -S B+F:A>' >D ]-RS'-*-D-D $-D-D A>-:-D*- \R]-D *\R'< ->D- :A"XX- \D@X>FD  \:-D) *"SS ->D- R-:\A+ "R- \R]- -DXSX-<X) *>]FB `H ( IPJ 1 $#@H SF a\S"BB-Da\3+ /2!A K64 JAK!4 !/ !4!4 =JAF/B- /4  /BF '4!F 34 O;; /4 5*-- / -! 2BB/'F/64 6% -N8!ABJA%!B 6%  !J2/!4 B8! L/F- 8A22!2 -/+-!A 6A!A %J43!4F2 %6A3 F-; !/FB-A/%F   DG(#D*G 9755P:

[ -*>D:\D:-D "D R+ \BB\D: \D* FRS>FD

GY

D>'<X B-D ->D-R $-D- -DX'D-D :-^>SS-D R-<S>DD F*-R 'B >DD- ->D-R R>-DX>-R\D:) "\'< +04-R-DX>"A:-FB-X-R SHR-'B $-:A->X-D*-D R-D-X=Y=->DJ >- :-^+F'D IEJ 1 IbJ L E IbJ L

IEJ 1 IbJ L IEJ$# IbJ L IEJ$0 IbJ L +IEJ$. IbJ L 9IE. J B>X :-^>SS-D F-5a>-DX-D3\D@X>FD-D ! ! +) *>- ^>- 3FA:X $-SX>BBX ^-R*-D( !\D+"'<SX :>AX D"'< *-D R-D-X0A->'-S >BHA>a>-RX *"DD / E0 E. (  1 $0 $ 10 $# L 1J $. L 9IE. J 1$0 L [ V ( E 0 1 E . 1 / ( E. 1J E . 10 / $# L L $0 L $. L 9IE. J= 1 IbJ L E $ V [ V V >- RF?-@X>FD-D >D *>- ]-RS'-*-D-D I$( ! $) J=$-D-D S>D* *>- 3FA:-D*-D) ]:AK >A* [K7( IEJ 1 IbJ L E$# L

I$# ! $0 J* 0 I)/+ '  0 J( E0 1IbJ $0 IbJ L 9IE0 J [ >- RF?-@X>FD >D *>- '-:-$-D- >SX "ASF ]FB `H ->D-R 9 . I$>S "\3 9IE0 JJK IEJ 1 IbJ L E$# IbJ L

I$0 ! $. J* 0 I19/.  0 J( ( E0 1IbJ E. 1 IbJ / E. 1IbJJ IbJ L $0 IbJ L $. IbJ L 9IE. J IEJ 1 IbJ L [ V V >- RF?-@X>FD >D *>- FRB"A-$-D- >SX ]FB `H ->D-R  +.:) 0 9 .) 3"AAS J IbJ 1 ( b I$>S "\3 9IE. JJK I$# ! $. J* 0 I9 -;+%A+ 9 0  0 F*-R ;9 -  0 J( ( E. 10 IbJ / E. 1IbJJ IbJ $# IbJ L IEJ 1 IbJ L E $ $. IbJ L 9IE. J V V >- RF?-@X>FD >D *>- R-@X>6a>-R-D*- $-D- >SX ]FB `H ->D-R -B 0

G9

œÀ“>iLi˜i

,iŽÌˆvˆâˆiÀi˜`i Li˜i

â

-V…“ˆi}iLi˜i Ý

Þ

Þ

â

Ý

9: -3/!+!!4!

â

Þ

9: 6A32!!4!

Ý

9: !1F/'O; !4!

$& 4- R-> RF?-@X>FD-D ->D-R "\B@\R]- P$# L R$0 L S$.

5 '+#6+#+ + /6  **6+# 6+ ,/0',+ >R D- R-D-X0 \R]- >B . B>X @FDSX"DX-R

R+ \BB\D: \D* FRS>FD J 1 b ->D X+ \'@ ->D-S R->S-S S->D B\SS I^->A >D ->D-R $-D-DX-S-B "AA

R+ \BB\D: \D* FRS>FD @FDSX"DX S->DK B:-@-SX ?-*- R-D-X0 \R]- B>X @FDSX"DX-R

R+ \BB\D: \D* FRS>FD ->D- '-RX) ^F$-> *"DD *>- R+ \BB\D: $a^K FRS>FD "AA->D a\R -= S'$\D: *>-S-R \R]-D "\SR->'<XK B:-@-AX( >- \:-AF$-R8+ "'J S S->  ->D- R-D-X0 \R]- >B . B>X J IE3 J 1 \B *-D \D@X G 1 IE3 J $0 IE3 J $ $. IE3 J! IE3 J L 1IE3 J J IE3 J10 IE3 J \- \R]- >D E3 ]FD *R>XX-R R*D\D:K >- >SX *\R'< *>- *\R'< IE3 J :-<X) $-R+ *>-S- >:-DS'-:X :-D"\ *"DD >D ->D-R H>>J S S->  ->D- D"'< F:-DA+"D:- H"R"B-XR>S>-RX-  . = \R]-) *-R-D >A* >D *-R   >DXS=HSX  ->D- R-D-X0 / -Xa- (1 -XI!  !  JK

\R]- B>X R+ \BB\D: 1 1 G L 0 \D* FRS>FD J 1  HIGL 0 JK >- RF;@R->SS>D* *\R'<  b :-@-DDa->'- A->D@R->S- *\R'< @FDSX"DX-S (1 bK  A *8 + \R ->A I>J :-R ]FD *-B >XX-AH\D@X 5IE3 J ->D-R X :-^>SS-D F-5a>-DX-D ! ! + "\SK !\ *-R-D -SX>BB\D: $-R-'R
$A->X\D:-D  *-R \D@X>FD CIEJ 1 5 $ IEJ! 5 $ IEJ (   C 1 $[ 5 $ IEJ!  IEJ !     C 1 $[ 5 $ IEJ!  IEJ L [  IEJ!  IEJ !   C 1 $[ 5 $ IEJ!  IEJ = "S FHX>B"A-
DS'-:-D *-R \:-A "D *>- \R]- $-*-\X-X ->D3"'AXK S -R:>$X S>'<   ( 1 /  / (  1 J G IEJ $0 IEJ $ $. IEJ 1 $ $. IEJ I5IEJJ 1 IEJ L 1IEJ J IEJ10 IEJ 1 J 10 "ASF >SX 5IEJ :-D"\ *"DD ->D @FDSX"DX-R \D@X I*-R \:-AB>XX-AH\D@XJ) ^-DD *>- >4-= R-DX>"A:A->'D I>>J :>AXK
\S 5 IEJ 1 b 3FA:X C IEJ 1 b) ^"S B"D 3+ \R *>- + \'@R>'<X\D: DF'< $-D+FX>:XK

[ \R]-D >B 0

GV

S ]-R^\D*-RX D>'<X) *"SS *>-S ->D- >4-R-DX>"A:A->'S'-S-R >4-R-DX>"A:A->'-S>:-DS'AX  !  1 $  !  1 $G) "ASF 3FA:X  1 $ L  &  \D* *"B>X 10 1  !  1 G L 0 , b= -RD-R :>AX $0 1 +#  \D* $. 1  & $0 SF^>-  !  1 bK "B>X -R:>$X S>'<  1

G G
G  J 1 $ $. ! $0 1 $ I  &  J !  1 $ 0  &  !  L .  &  !  1 1 1 1
 1 G*

 G 1 $ 0  &  ! $ L  &  1 0 -XI !  ! J 1 0 = 1 1 1  "$-> 3FA:X *>- ]FRA-XaX- A->''< "\'<  &  S-D@R-'<X "\3  & K
 . 7,?/& / GK >- -X-RB>D"DX- >SX S-A$SX ->D- >DX-R-SS"DX- R+ F;- 3+ \R *>- \R]-) \D* a^"R >SX S>- *>- R+ \BB\D: >DD-RSX ?"  &  *-R?-D>:- >DXS= ]-@XFR) *-R S-D@R-'<X "\3 *-R \R]- SX-<X \D* X"D:-DX>"A "D *>- HA S-D@R-'<X a\B RXS]-@XFR JK "DD >SX 1  !  &  *-R
DX->A ]FD  ) *-R X"D:-DX>"A "D *>- H'< "\S *-R A->'>J -R:>$XK -> FR:"$F:A>'X *"3+ \R *\R'< /]FD 1 $>-X-X I>>>J ->D- -_HA>a>X- + >D3+ \FD 1 - 10 $ GK "D $-XR"'<X- *"a\ *"S A->'X 1 1 b "\3XR-X-D @"DD) aKK $-> ->D-B/RF;@R->S F*-R A->D@R->SK -R "AA 1 1 b @"DD D>'<X "\3XR-X-DK
AS -SX S-Xa- B"D 1 1 G L 0 \D* J 1  HIG L 0 J >D *>- A->'D I>>J ->DK 4-++- 2: I+FS'D>-DJ "M\>]"A-DX( + \R ->D- R-D-X0 \R]- >B . S>D* *>- 3FA:-D*-D -*>D:\D:-D + I>J S :>$X ->D N  .  b *-R"RX) *"SS $# ! N @FDSX"DX >SXK I>>J S :>$X ->D N  .  b B>X $0 ! N 1 bK I>>>J S :>$X ->D N  .  b *-R"RX) *"SS $. ! N @FDSX"DX >SXK I>]J -R \FX>-DX

J >SX @FDSX"DXK 1

[ -*>D:\D:-D "D R+ \BB\D: \D* FRS>FD

GU

DS$-SFD*-R- -DX:\D: ->D- + FS'BHA>a>-R-D N 1 $# L$. B>X FDSX"DX-D ! K " N @FDSX"DX >SX) 3FA:X b 1 N  1 $# L $. 1 I1 $ J J$0 ) "ASF +B 1   = B:-@-BHA>a>-RX *>- FDSX"Da ]FD +B *>- FDSX"Da ]FD N (1 +B $# L $. I^-:-D N  1 bJ \D* *"B>X I>J) I>>J) I>>>JK -R >D *>\R :A->''<X\D: a->:-D*- 71?BS'DX-R-SS"DX) ]:AK [KG[K


$& 4-1 +FS'D>- >D ->D-B -:-A I-10+:) )9D ->D-R HB . J + \R :-:-$-D- FDSX"DX-D ! !  >SX *>- 'S>-RX) 3"AAS ->D- R-D-X0 \R]- >B . ) 3"AAS  , b!  (1 bK >- >SX D"'< F:-DA+ G 1  0  0 L 0 = >- :-D *R-> A->'- ->D*-\X>:- +FS\D: 10 L J 0 0 1 

1

1HI10 L J 0 J

0

1

J 0 HI10 L J 0 J=

-*- R-D-X0 \R]- >B . B>X @FDSX"DX-R R+ \BB\D: 1 \D* @FDSX"DX-R FRS>FD J >SX ->D ->A ->D-R 'SX *"$-> "AS R->S B>X -DX_ I]:AK $\D:S"\3:"$GEJ . b 1  1 , $1 b b $J

XR>XX "\3 >D *-R FRB"A3FRB *-R S'-3= 5 b J 3! b

\D* a^"R @"DD B"D $0 "\'< $-R-'-RX-D I\D* *"S'SX $0 1 $I10 L J 0 J ->Da>:-R D>'<X]-RS'D*-D*-R >:-D^-RXK "B>X >SX  $>S "\3S FRa->'BBX) \D* ^>R -RD ; *"SS-A$- R-D-X=Y=->D $-S>Xa-D SF^>- *>-S-A$- R+ \BB\D: \D* FRS>FDK >- $-XR-4-D*- '