if

PROJECTIVE DIFFERENTIAL GEOMETRY or K
CERTAIN. SURFACES m ASPACE
or mun mmmsnons
THESIS FOR THE DEGREE OF M‘ A.
' PaulL; Dressel
.  1934 ‘ -

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LIBRARIES
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stamped below.

 

 

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A Thesis
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of
jICfiIJAI STATE CGLLZJE
of
AJRIC ULT"R€ AID APPLIED SCIZNSE
P3 rtial Fulfi lluent of the

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mequi: ments tor the De fee

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PRCJEC IVE DIFFTRZ-T1AL GECZT RT 0? CTRTAIK SURFACTS

~.. Q ~.r‘<‘r‘\ A 1 (\T‘ - ‘fi;rfi'r.A.v'/_N
Ill A c1 3...); vi“ FucR DI...L..O.L\_.1.Q

4* I 3 n 14"": A" ‘v ‘r‘ v .- "L — Iv /.H‘ -' " av

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co editions for tLe tnIee c111e1euo1al equatlons lelJ-

-: -,-< r-, T, “‘ fir , I" ——. f , . r3 ‘1— ‘ -.. c; ’11 ‘ 1- . ——

1J3 the buIIflCS ale cer1vec. Tue o1f_elential egoat10Ls

co lote systen of invariants is found.
Two COVPriants points in 8*d1tion to x , xm, xv are

found and a covariant local pentahed_'n of rele1cnv-

4. .x‘ 4 , ,3 .4 - . - V,“ - -0 -, .-
etchlleuéo. Cahon1cal exyans1018 lor tse coord1: 1.3t 33
.9 4 4. .. ’1. . (or, ,L‘ ‘ ,,. ° * _ . 4 1-4, 1
01 a p01no on the surtgnc are OetclJlnCO- Certa1n Lats

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on the 81118ce are c1£c~cs:c. FlDullJ general all 8: 1t

'~~~.--—m ‘-.° . ‘1 «n 1- 4,..c'" 14 m
Lgyclauac11cs ale louna and orlcllj are c cterized-

23. T; .1 LITT‘CIZT TIAL E“ TATIOIIS

(l) xi=xi(u,v) 'tzl)2)3)"l)5.

 

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xuuu. : q)(l-ML 1' lngv +3qu +MXV +JX)

quLV 2 a‘qu—L. +LXK+MXV+DX,

xuv 2’ (walfbxv +QX)
/ /

qutv 2"— bxvv-r L’xu+MXv+ DX)

I I I .
xVVV :: xxuu+ JXVV '4' 1 Xu +WXV+JX )

L: 0—b+c.+a.u) LI: Chg-ray ,

M=b1+b¢ , MI?- “b+¢'+b">
I

[Dr-bead“) D=o.c+cv)

.VL _. f n .9 ,1 '1 .4 .44., ,1 .9. 4.1,
o-(3 L-UCOLLLL and. 100.1%“ p'elhfl. Octrllhsa "”2 - ..

rsntistion. TLi

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“v+fly=%+L) Xd+c§D’+w’c. +dL=bJ’+ L'c,+ 0;,
fl; +pv 4"“ =90.) yaL-tc‘L’ +1,“ +o.\~\’+d’=, 51%. LICL+ L’v ’
«Lav +fl1'=L.t+aM+D, xv“ sm’+m;= bu w; + 0',
°‘ M +16+MV+J+flWI=am+bM+Mu) “Yd-Ya +1“: bY,
OLD +c1 +4., +/BJI= ad +cM+ Du ) flY-f-J‘L: bv +Ml.

grow the first and last of (7‘ we find rs°4‘l7 tfist

1 {1.1.

(J/

OLV +1bv :: (fa-+- 310,“

11131109 from (4) a finction 9 exists 511011 1:56:13

eu=d+1b ’ esz-f-QQ‘ .

7—) B ”Asmar-da“
A \
“"’“*“/37\L) u= Qu‘5A“)

 

.. 3 u ‘.

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JSLILg, t- e 11‘1varis1ts flJYJJJJJ a.) £56. to 1.0“. 13100063
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:..1 9.. gives om 0.1 1:117:11 1:111 ts

H ‘ ‘

.' .- ' ._.‘ ‘ - ‘ v-45 . : I: ,.~ ‘ - , , -'. -1.‘ , I _ ' ‘ ‘.._ J
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integrcoility conditions
w = £05) , «4:42), 1’: {3(1) , w’: 49“),
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x: -:Lb + “Lg/ax)“,
cf: ’24- + 3(1’5’68)v .

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V;-VC/ L" {Q ”J4! J' b ‘_- -L b.-.V' Ulfgt J; L1,:

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a.) he, fiJY, J, J’ are :_jvo11 as; ftmctiorzs 2:1: 4.; anc- v

 

 

gub*eot to the intelpnoilitirconfiitions.

 

Let us introduce two symbols of operation
(19) '- b Au ’ — a. 3-9 ) ‘
EviCortly the functions

V(A) ) V(b) ) V01); V‘(C) ,

17(8) , Um , Um, WC),
are also invaritnts- In fact the trans
fanations by (13) or

flag): U3) ) ‘66) 7.14%? )
iiai) : ER?) fiUE> : “EEQQL

 

(20)

u
‘3
4!
g
u
E
x;

1'36")

- __ q-.- .- : .0 1" -\ ‘ ‘ . ' «‘5 “
WTWI; itpetltxon o; tuls process also LJ1me us a“

9-1:.

inruriant.

From two invariants A and H for which
" ' A ~ .1_

A :. —-1 =
another, its Wronskian, can always be formedv*'

We find

Ava «1 ._’Aw4 «1
H "' H )
whence by logarithmic diffs ren ti ation we find that

Mariette; 1e)

Hence the function

 

(21) (99)“) = MHAu‘lA Ha

is an invariant for which

Au
(28) (EMAM) : £§:7:%+' '

We shall speak of (HJAQ) as the Wronskian of H and A

 

with resnect to u. .

By combining the Wronskian process with the operations
{7 and.V7 , it is clearly possible to deduce an infinity
of invariants from the seven original ones. We now
proceed to show that any invariant is a function of those

obtained in this manner.

, , I
We lave alrea .dy snovsn tnat a.) ch,J)d’ gy may

C:
CD

eXpressed in terms of the seven invariants (18). Since

. . . I .
all invariants are functions of 0L, '0, c.) did) fl) y 8.21:;
tgeir Corivatives, any inveriLnt may be e: pres . ed as o

 

* . _- .n- W..- - q ‘5‘
E-J.Wilczynski, Progective Dirforential EoogetiJ bl

 

Curved Surfaces. First Keroir; Transactions of the

 

American Istie.3 001 Society, Vol. 8 (1307), p. 231.

A

Hereafter referred to as Surfeo.

(D
‘5!)
O

function of (13) and the derivatives of these quantities.
Introduce infinitesimal transformations by putting
(35) 4’00: u+Hu)§t , WW) 2 v+ 8(v) <51: .

Hence

¢u 1“” Pu 6t ) Wu. “:— O

¢v:O) WV:‘+8VC¥tJ
¢qu-¢vq’u: '+(‘Fu+8v)<§‘t .

The infinitesimal transformations of (18) become

(34)

60;: Ell—CL: %_ 0,: a.(I—3V)Jt -a = -gvotJt’
ébe—«Cuth, (SA: —3fiAJt

(25) ’
(SJ: ”3‘Fu46‘t) SB: ”33‘186‘17)

SJ": 'Bng’5t) (SC '-‘— O .
The infinitesimal transformation,<§F7, of any function
F of the functions (18) being known, the infinitesimal
transformations §Fg_enui<§FQ of its first derivatives

nay be easily found. W's find that

 

8F“: £-&E: 3_(_____F*JF)_L.;;t-a __E.—. 3§5F)_{upu&_)
(35) i
Jsz- if- 1:“: MR“) 1 -AE: alga- Fat.
dv 3w ”3th o‘v 9" V

Thus we find that
J but : “ (‘uu bf 1““ bu)Jt) Jbv '-'- " (8v 4' (a) bv (St)
61“ “(3" +4“)a“ 6t) cg‘L\/'=‘(gvv‘l'1'3-fivav)cr_t,
61“, Z -(3(ou+q'€u J«)Jt') $dv : -(3fu +8V) &V 6*,

<2?) 84% “(m-«ma; e, 54",: 4331,44 43,4351:
5A“ -(3 411A wages 5A - -(34 +351 5e
38“ “(fuuC- +2; “(05);" J8v ‘ ‘(38vv8+‘48v 8‘96};
SCuz -‘(;U.Cth) SCV~“ “SVCVs-t‘

All absolute invariants n" which involve only (18)

(3. 1'1

d their first deri vs

0.)

ci-
<$

es must, therefore, satisfy
en of partial difier ntial equations obtained

’3 “ .'.‘ '. 1' ‘1-I . 'c . 'r‘ . J.‘ "‘ V . " 'L" W!“- 'N'?/‘ ' ~~-" ‘- . s 1 V
iorgimmg the 1niinuiuesime]. t n31:u1n+u1on 'ncar an

arbitrary function Tr of these arguments and e uating

t

are

"0
fi.‘

dispense with the latte r tno eoua. ions. We omit the

’d

O

l“ 0

zero the coe fficie21ts of ¥uu3gvv3€43c3v . If we

looking only for relative invariants, we may

of of this and also certain other necessary dis—

ussion since it may be found in W’ilcs ynshi's memoir

*

q

in exactly the for n nee so here.

8.1”

021.

Form the infinites i2nal transformation <5" of an

d-
' 4)

rory urction 11 of (1a) and their derivatives
é‘rr=‘-:-%_Ja + fiJb+éfiJd+i3 {er-r %"JA+%%JB
+318“ + g1 Ail.“ + éfiflfluéfi JJ‘+ +3 “dim“
+-%—5Cu+ «9353354 +—31vaefb +33% +9433 SJ

+é—g-JA +43 43 +3315“)

Q1

, A a
(28) = (4&3? 4.35%"; ~3fid 35-33%! 3(if-satuAillA— .. 33v8fi;

*-

_‘F“ 404 bill. 56-14.“ 5“ i o‘éf'wgv -‘uau%fla ’3'Fwdg—g'u-q‘cudufi-‘L
‘- I I ' _ 3

3M- swans? arm/at- was... staa—gu-tagg
.. aficu if ~3w°~5§ ‘13V6‘V3'3‘ 43.15.]? -42“ b‘, ELEV-34m 5:10",
_ a 003’ ,

3“." Si “33"”? 3%}; t‘fava’vfi- -31‘ Avéfl- 3A ~3VAV 9%.,
-33w8 5-9;- 4QVBI? 33V -C3v 31%;) 31— .

 

E'J‘Wilczyfiski+ Surtacas. U9.au4,255.

TT'

(7
.4 U

queting the coefficient

obtei

0’

b

E4

(29)

9/

6
Q

l

0..

9.4

I
v+31? gs;
All

:
l

C!-

ir t d iva ives are isot

U‘,

1

Q)

ff)

7r
.1
v

3')

_ '1
' G- p,

.LL;C,-

endent variables I

cepenoent solutions.

(30)

434%311+ 3A

1elative invariants
tem of two equations (23) w"

Thepe
1, 11,4, 4; A, B, c, we, no, MLWC), 1761),

a‘
‘—

01

cu.“ 3.116.

3.9.
AAu.

+38 3%.,

involv'

O

O

7‘ 1' "I
l '. ‘L
k '.

only (13) and thei

ric solutions of the complete

0
'1 'rvwv
.5...L V

‘ich olve t enty—one

"(a

ADJ»

,1. +
S ‘ V9

t t1 efore nine 1n—

s- may be selected in the for;

mama), (01,141,111), (8, at), (1,1), (A, c.)

r11

(3
V

‘ —,. 2' ' —u - - ':l\“ ~-r
T-LCt-e l 01691'13' 11251.net}

ent-

 

w‘ ‘1 ’3‘. r" ‘L' . . ‘ 70 ‘4 . I ‘ v“. . 1n . \ V r_ e ‘1'“
All ichblVG intaiients urlCu involte 0130 one
3 . .1 'I ‘_C‘ C ' :‘ ' . ‘ -. u 4‘ ’ I
secoio CSIlVothBS i (1:) W111 oe isocwric solutions
N'? - 4 - 4" -- V~-- A— J :1 !‘fi ': .~ . 7‘ ,- ,\ I‘- ‘ '4--
U1 2:. Canslt? u -‘ E- St: '31 1011: 1-16.01) oil-Cicrl t 311;? 1110118
:7 .L- 1, ., H,» _ - y 11,, . 1 ~_ .1 ,, :_ .4-
1n iortJ—two ‘o‘laulBS- dense onero ale tiirtJ-eiflnu
', - 7 .2 , V_ .. _ V, 1.1- -‘. ._J_: ,-V 1, '1' _ .1. - \ f 4-"
4‘... Li'or til-41‘ Li; L! b‘Jl-. d... Jl_|‘:3 . t. L L L- ) k: .L C‘ v \A U. ‘.\,\I... y .
«1“ J— .. »- w. ~, . ‘ . . I z\ ' ‘ . 1“ -v
I; U..J.LS pLOC 4L)??? J8 tll-lls'4M’ .Lt Cr‘ll Lje 8-40‘M'Il LI'J
J-‘ .1 4. - -' .1 .,-'+.“ , .L“ - .1- .1-‘ - 1 1 ._‘
1 U1-L .(L blbal 1111,11» L11- Li. 1 L, 1 1 1-1; 7I<+5
‘ 1.1, ',~,11 . - .' ,z ,...:.1_- ,1. 1.0 ‘
nil-ALT. 5 .L uL‘ - \I L lh- -\V \, L“; _L hL/ U—«Q '(tk Ix. I: Lx \- \J-ln‘V Us C/J- ter
4— a - ' . 1,. -2. .,.1 .7 , ' JR
List-Vi LL61 ilub (1'3) 0 1111.33 t..C to 9k 1 1'1. 111‘: )1 4.11\.I:3..L.1.w-iun3
"‘ "T "' n ‘ ‘ - r N 1" . I '3' ~ . —,c" 1 m
\_, J- C.r.l.( LJ: 1.68:) t .‘C I]. 11‘...‘ (,g‘xALevL to “ .LS
V1 7.
(31) (7k+5) J- (7H +17" +I"/)
a.
K=O
. « ’ ' N. “Tn —- r. (.7
E'U.',-";.L pL:;.1S;'.l, 13.3.1.1:130680 ‘r,p. Lu1,wuuo

b. A wCVARIAhT PSHTAEEERCJ

‘f‘ --\ F- 1“ - --,J-fi '- ”-1 7; 4—~- 1‘“. ~c'~ I i'\
:AaL‘»\ C; Xu (ind Xv CO'VC—irlai.ub0 .‘ue A-C‘V \‘ ‘u'.'.‘t;,'—L €4.1JIC tL-e

+0 v, 1
U0 Lv-L‘

fill-es COVE:I'ia.1;t {mints x) wav' T220 .222: @0111
required to outergine a Covariunt pentahedron of
reference. Tgeee we proceed to determine-

T;e covariant point
(32) e: fixvv +""\xv

J-‘ . + ,,' ' .3" f. J—r‘ .. - .LV .t m -1 ..
es. v-0 peer“ e1; ‘..-..J.L,A- the L.(..:1bc3:;t no ;-e Valve u-c,

Par-l
k4
H.
(H
H
O
(13
6')
d-
: I
(D
O
(A.
O
L
FJ
m
d-
'—

y 4

't;-l'OCLc".:; ".12: to the curve
_. xv .. S3

V:C.at X . A £111 a CLertcterization may be jade for
tge point
I

T use Characterizations mey be easily verified by

~ ‘\

. ‘?\

totlcing that from equations (~x

XML“ -o¢xu,_~l xu—Jx = (3' xw +VV1 Xv
' 00’ - Y)! +,(’

XVVV "Jxvv-w XV- - “u xu"

q

Tie points x’xu’xx’ed(y nerefore, are coveriant

 

 

\

.‘f.. 4— r- .3 do 51 a -‘-,‘ 7 “4-4; f\ o I v. —\ 0 .~ ,
LuJ’ipS (lib. JO...“ tn: V91 wees C; 8 COVRllBTlt pGI‘tc..-‘-C-

{1.1

T7"

 

4-\-- ~-1- . ',.‘ A3
L.... .A.\ - ..- \, AVE].
*

 

~ 7‘11 ~ e - ‘
U. VIOLA-1 0.1.1 .40 LI¥L O1-.N)d- dLINJ

- \ ‘ '-| ~ . 3‘ 0‘ . ’ . r- '5‘? ." -v “ "‘ - N ‘L > \O, 3 f‘ x '. J' P. j“
T.--U CUULLLLMJJJCCS x); 51.1.} EJUllib X ilL4Lil d lJOJ-llb PK \JLL

3"“ S ’5' ' 1“ 17.2“".wctnfin-‘f- :5 ‘~.v~ '1 0-5 \ ~ "_‘."",\v‘r‘; ~
gut, \Afll'. 1.3 L £.'i.\.~,~\.i--~v'.3 1. L4;‘ L1. TLA‘iUI'S 1,..x‘JU4LvDJ—Oli

v

(0

‘2

J

a
f‘.)

(5 ~

of t;e form

X: x+ quuH— vav + (XMLAuz +514“W uuw+ xwavl)/2 +...

h". \1‘ 7.x," -.—~ , 7 .- ' r 4-: ,‘-. ‘-\ l ‘3 ‘,~ . IV ‘I‘ 4- . -‘. ‘R f‘ . ‘ 1 ‘g —.a
13,;iuuss oi equetiods \u), tnxaeagM-tious outeinea.ticie—

Iv

' 7* 4—, 44-2 . ” ' “is: ;J_,_
10;; together hit; the lutehlrluilibw'

conditirus, and equations (38),(35) it is possible on
e;-;£_~1'ess every derivative of x uniouelv as a linesr
eefi-inetion of X,Xu,Xv, 9’0” . Hence X :1113 DC exm"e:sse<i
ugiquely in the form.

X: X,x+X,—LX..L+ X3 Xvi-qufxsa')
wimrein x,~~~x$ are power series in Au,Av - The

it'u;'fctitns x, x5— ere coerdirates of the point X , tie

1-},

..~ ,‘ ‘3 J— (‘4 ‘ ‘ ’1 .0 w. \ I' : ‘ "‘I ' ‘
jugbthuron oi e erence eeini x,xg)mn 9,0' . P€L~

-

-: ~ 4.1. .- .'. .- ' H ., ,_. q- _' -i '~ ,.
iClmlH; sue CulCu et_e;s lgul ate», we iinc tqit

t—n
(J
(3,.

X, 2- 1+ eAuAv + J5(DAu‘Av + D'AuAvL) + ,
"1 : A“ ' 1% M" + “MW +£[H ~‘-‘é’)Au3+ 3(L—3é'pu‘nv +3L'Auovfl+.-. )
(35) X3: AV ‘ fl AVL+bouav+f [0‘4" %%V3+3(M'— 9/5.")Auav'lg 3Mw‘dv +. .. )
X4 = aLxAu1+ 7'.— (537 Au3 + 3%Au‘av “3.3).... .... )
x5: 33 “12% (Au? + 3% Ausz+%Ay3) +

Tx" 4— '3'“ J: 3 fjfi ~7- -‘ f‘" ‘>\ ,, 1- z. -‘ Al,» I (a 'Q I v“ r‘ t A- -, I ‘9' +-:‘«‘ 0 Ir» .A -o . ‘fi 4 J— }. 7‘ "'A O
.LaA v'i ULLV.LJ.1(, n‘JJ-L_Jr$‘3'1lo L‘ts L1.) 10 COOI‘KA—LALQ“ LS 8.).f VA \/ '~‘.Q-¥- 11‘4- LJ .LI,).Li‘.)

(36) x: :3: y: I; 4:: 7‘! .15
V:

V
1‘
x.
:1
\.
2
II

z A -1: A 2‘ A .L -0118, 3
X M. ay LL +¢Au V 4 5(1 FY)A.LL +2-L(L‘ %’_1C)AUIAV

I
4-1:. 2.
2 ALLAV 1...“)

Y : Av— —"‘—"Av"+bmav 4 .g Au‘Av +3E(M'- big—ac. AuAV"

(37) + {~(MI— %—K)AV3+-n-

_, i 2. g 3

Avl-t—(JJAu-s -+

2
ll
\Zit

‘9 AuAv" i 3
a? +GflAV+H .

‘ 4'- W v I" ‘m\ _‘ t - A g" C' F '1‘
A1'L\j L4- ‘— V‘A‘JL -éQJ-’.)o--S (U(/ ul- it.) t.) ‘DK'A-ULC UV, (‘J LIJV V;
n
- .- av fl — x r "P 7‘ ) ~ g f) a 1
\4h‘ \ lev-l .1le ‘Ur E A--\A w S ‘. 1" .1 |_, ,L-nyk.) 11‘ x Q&-KA

izw tlat tie expe;siexs (e7) satisfy tLe equation
0 terms of any desired ord3r.
B? tiis gethofi we obtain

2: fixt+iy(%+—§)x3-§? KN +’cl§‘(3+'”“ ,

5'3Y1'*2L"3* abp Xx *Jp (5/ V3.’?)‘I3“‘ .

P~ V a ': -, "or ', 1“»x "‘ I 4".
T.‘>: Eaélblar-lSJ-OA'}. for % .ijjJIk/bk_nt8 ULAG 83 " v:¢iC:: is LII-Le

 

 

(VJ‘1"..'1- fir t~ A C.)“a4-fi \C‘ ‘WY‘W'CI 3 t) '~Vv “t fi {‘53: \r‘gr +l'11
L‘L} LA. O.‘ C, n.\_,..[.lC‘CtL, ILn/4Vu( L‘s L C “-1.le (A
*A —
l‘ - _ ‘ fi '5 .. 4-
‘ -‘-'- A 10
eqj'tiens £1), n t.e nei “eel! C( on t e seine x' is
—— -‘ .—- _—

 

 

 

1 11.

n — ‘ 4 F5 . "ffi
— "'-' 1x~+ + +~ -- ’- -‘ -.-~-: "w
CC L. '.'k_' LLI L; 0 u--e S L -‘L‘ ,7 (.74 ti-L: ‘ '_'. .L C1) (1-4 f 0
— 4—— —

 

 

 

.- 2.
~Q°+QIX+CL1x +..~

Y
__ 1
‘3‘ co+ 91X“! 31X 4" ' ‘

C‘
L 0
O
C)
H
[G
(L
H
F.1-
('D
U)
F...
:5
cf-
(b

i;e::e-uoe ocaugueerl t;.e ts“) ~\.

(“J
L)
* .'
’4.
f')
G
'-i
3
H
-<
P
D
Ft:
H
C)
(J
c+
*4.
{J
H
5'.
H

powers of ‘x covic be

L‘ J" "\ " f“ ' 4" -:. .' ~*' '\' a ‘Lj‘ ~' 'I — v .D --- .: > ' 'V V' V “,'c
Oubaiuud to iurt_ei MAC studt oi tflis curve. :onexos,
a I " - ‘ + r - J— . L. a J'- +'1 ‘ ‘-

LE) siaxli zzoti Colitizuie t; b Eiuhiflf ;A3re.
n if"! -- *yflrrr“ u rfi I: '1' 'r 'r P \ 1‘1".
7 . C‘JL| \1 _JLK-‘- Li‘J I‘I‘ LLTU '1’ J- 31 .‘ixjgo

~V‘ I o ’3 J. 1. n I _ v
Coesioer two oistinct Certes CA

tively by the oifxere; :31 equitions

(39)

 

 

*--4 x ‘ '“I ‘. fl I-: 1' ' 'r \ - ¢v - my I '__ "v VI ‘
CuiVCS end 3d faces. Tue UAlVb s1 3 oi VthJhO Pi ss

 

Tye osc eel ting; plane to CA is defined tic-:3 points

/_ 1/ 2
(40) X , X -Xu+/\xv , X = xuu-M xw+2athu+ (A’+1bA)xv +:2.c./\X.
Tle local Coordinates of these points, sing the

A. . '1 i
(3 ‘3 r/ JCLiYe‘LBV

r‘f‘

sent: neoron X, X“) XV) qu) 1'

“T'O
4-.1

va

(41) (5,0,0)0}o)) (o, l)/\)ojo)) (ask) 2a))1bk) 1) A1) .

w- ‘ .4. " ..
ii u.€ iy'perplane
Z 5’. w o
\
I“
" ’ ‘- ~ 4’ 'II‘ 'I' ‘~' ‘4" 1 f“ I "" \‘f ‘ ’\ . 4' " * : ‘ J." 7“ "‘
£7?.Li_‘t.;s U-.IUIA{LL$ LLBLC tn. ‘1'?“ 14/.) 5-11 we, lb: €31.1th .-‘Il 'iL-lgu ..«u‘

Xxx—x3 + (A’+ 1b) —aa.,\")x.,:o )
1 _

-_ ’5'; -?,'-\’p .3 -‘»'J.»\ ,4— :u 4'."\ "\ ‘fi “L.- "‘ (‘3‘? C '3 '
Li.'L‘l€.Lp.Le lfltc‘loLCu .Ln U.;L', USleC.blflw planlic t0 A (,1;

it: plane to CH at x has t;;e equations

HXL‘X31' (u’+1bg—1mH’)Xq=O,

H‘xq -x5= o .

Sir-Ge CA 82d CH are distiz-ct it follows the

:1.“ o

.wv ,__;_- -. ..-.'. J.’ A .L.,, no, :A .. 4.“: ..,
Re ......LL;.Li-. tee BCLMtlUllS oi oi.,.€ t .o i)lt..es um; i bulb

Condition they are

Axl -x3 + (Au-I- AX, +1bA~anz>Xq=

z O
) x1.- X§=° ,

(45)

Axa + X3 + (KHAN, +1.5) +21X")Xq :0)
(458) A1

‘ . . . ~ o n a -

16W“ W~ , 1... 1 ~~ - X 4' t‘v‘
T;.‘.\Is.‘\1 p 911ng lam“ beut lll wilt? 1.1-3. 334111.12.) L0 1113
“pf-AJ-

LIU-‘.1;L;

(46) y : » (2.5 +41%)“ + A"(T ~2¢)xv +x1111+A‘xvv .

he SUEfece of the form

 

 

 

 

 

 

4. 1... 4. .4. ' ,1 . w
‘Vw 11 1x 1-3;0 e?1~»£:€*1* diuue line t'toa “ x cur;
.L‘ ‘ 3 \ .. - - - .
v-0 Io1nt u;t3- 1‘nu ‘:“’ (40‘ P“ v Lstltutinr t “te';
\‘L‘

 

 

 

 

 

.9»
9'
0"
03
F
p.
(D
S‘
H
V
H
D
«3
L
2
2*
I!
’rN
O
C
I
(D
O
r U
r A;
(F)
H
t)
.3
FJ
r
O
l‘

Tfie net (47) becomes

(48) of 6L”:- back}: 0

b 7.
b a. “34%”; myfixudtifiw .

TLe net (48) may 3e 0118wr eterized as follows:

3. _‘ : . ' 7-1 , 4. .0 - ..., . fl ' x: J- :1

The pulnt XKV 18 the yOLflb o- lnt615LCthH 01 bfle
4"" ,«1 v14- ‘1 ' on. ‘ “ '14?" "1" ml— 4- - 1 ‘- t‘\ ..-1(-:‘
1111.fL;\r.€11 Ub 10 LLVC. 00. tile Saildce 536112310 UGCL U3" X“ , 14.0w
tie te‘1‘:1:1-.:1t to Vza 011 te 51:15:10»: genszi': . 1‘; by XV -

TLe tangent to the curve wloee differential equatian is

17

pa

U)

ses through the point Xuv . The tangent to the

\

curve defined by

ado-4 Eddy»: O

D)
(‘f'
P
(D
hJ
’11
H

ionic conjugate of the tenrent to the curve

a&— bob.“ :0

with respect to the tengents to the parametric curves.
Consider the pencil of nets

(50) d“"‘-— A24": dual-2 0

Tue curves of this net are the integrel curves 0? the

'§ 0

siffcrential equation
51 I: L. 1 _ do-
() HTH+AKLH1H“ZZ'
The osculating planes to the curves of this pencil
inte:sect in the line joining )< to
_ 1 z
(52) Y“ ‘A t‘ va ‘f'xuu +l‘tkl(%y"2a)Xv~(1b+—AS5) Xu. 0
The local coordinates of the point Y“ are
X|::. O,
(53) X1: '(1b+)fl,
Z. 2.
fl 2. 1-- A) x5 : A I" o
It is possible to eliminate F1 from these local
coordinates in two different ways, thus obtaining
- >1, ..
”3 (T 2“) Y5“ 0,
x,_+(1‘i-‘—+2.b)x.,= o .

Each of these equations represents a hyperplane. The

5

‘

two equations taken tofetner represent tne into section
of the hyperplanes, this intersection being a plane
through the point )( . It follows, tierefore, the he

locus of the axis of the nets of e nencil 's a nlnne

 

Such a systefi of curves is of the type called by
Bonpiani a two—axial system. The plane (54) is the
plane of which th- given system of curves is the two-
anial system-

The envelOpe of the osculeting planes to all the

integral curves of (51) is the cubic cone*
(55) [XS-(AXL“1a)¥5-)axq= [x1 + (3f+zb)xq]2x5 .
The tangent plane
X” .2 )(5 =- O
is a sinple directrix plane of (55), wnile the plane
determined by equatiens (54) is a double directrix

p lain e .
8. A PHOJECTIVELY IRVARIAZT IJTEGRAL

The differential form flYoLqu is easily shov-rn to

q

We absolutely invariant under the transformation

 

* E- Bomnieni, Projective Differential Geometri_of firmer-

 

unce- Lectures given in the University f Chicago,

Sumner 1930. Part IX- Hereafter referred to as Lectures.

19
X: .. ~ —- —- . .
Ax 2 “" ‘ 4“), v: IP(v) . Tne ‘efore the integral
l
(56) $(fiYv’)"cL....
calculated along a curve is invariant. We may call

this integral the projective arc-length of the

 

,urve along which it is calculated. We now make us.

(0

Of Euler's equation for the extremals of an intejrel

of the form

we»; WM»,

H ¢V" C‘uv’ -¢vv' VI)

A
U.
x}

V

s-

<‘

a
<
H

and find the Ci ierential equation of the extremals

of tie projective arc length to be
<58) (if ¥s)v’-<%’+%)v".
This is an equation similar to (51), both of them
belonging to the genera form
(59) V": A+ Bv’+Cv’L+ Dv’3 .
Evidently for (52)

A30, |3=(L’3195’).i , C3“ (10316399 0: 0.
It has been shown by Bonpiani*' that on all surfaces
sustaining a conjugate net, equation (68) represents
a two—axial system of curves. By a little computation
it is found that the plane which is the locus of tee

lines of intersection of the osculating planes to

 

*- . .
E. Bomplanl, Lectures, Part IX.

W
O

tle extremals of ‘$(§Xv§i4u. is the plane deter—
mined. by the points

(60) X , {3‘1 Afipflflmm, [1" + “"358” x“- ”““°
If we call the extremals of f@¥v')’£ol- projective
feodesics, and the plane determined by (80) the
projective normal plane, we 1333’ state the theorem:

T e os ulating_plenes of all the projective

 

 

eoCesics throng; x , intersect the projective

 

 

 

norfiel plane determined “v tne points £60) in lines;
and, moreover, this plane is unique for each point.X .

 

 

 

 

9. ASSOCIA ED HYPERQUADRICS

T? e eqt ration of the most general non-singular

W'reroueiric is

z.
(61) “-.. x +0. ‘1‘“ 3L” fiWX" +a 55"5 +a XML” ,3X. X3+ am xnfq ”Mama;
+%x}3*arzq“1'l + (25‘ 1x5+034 XJX.‘ +Q-35X31‘5 +aqsxq ‘5. :0 , '31 Kl¢o .

:1)

If no dezn nd the t th series (35) satisfy this

equation identically in AuVAV as.far as terms of
the fourth degree we obtain the «J’iyperquadrics

[mam ”(ta-+19% - mm + ems]
(52) ‘9' 0.33 [xslvafifixs +1<§+V~JX3X5 -2bp)l,.x5 +. ”J'fJszxv]
+0‘w "414'0‘55‘5 +31% “4"" = 0 0

These Lyperquaerics have four point contact with

the surface at the point X . The tangent plane

'S
x
v

xq=x3=o intersects this four point qua ic in the

(N
H

We have therefore the theoreh: The four point hyper—

 

 

 

ouaorics of S zrt F; intersect the mnggmt plane

to £3 at P, in lines paired in involution, with the

 

 

conjugate tan;ents as double lines f the involution.

_ —‘

 

 

 

Consider a curve u.=c. on 8 through PX . ..t P,‘
and two neighboringjpoints FZ,F; on.the curve
cons ruct the osculating p anes of the curves v: C.
:Tov: let PUP1 approach PX independently- Does
there exist a hyperquaoric such that these three
os ulating planes are plane generators?

The osculating plane to the curve |4=<L through

If near P

x is determined by the three points

sz'

X: x+ vav 4-va 3+

X

2.
4: X¢+ Xuvbv + x$VV 4—5-1» ... .- .

X4“: X...“ + xuuvAv + xuuvv $311+ -. -.
Any point if in the osculating plane may as defined
by a linear combination of the form
(64) Y: X + KXu—f '1qu .

By use of equations (2),(32),(33),S( can be expresses

in the form

Y: "J 4' xaxu+53xv +qu*x56-

P“
Jr-)

where the local coordinates x; -x3— of the point

I

Sf are given by

X.= I+K¢AV +‘QDAV4' 6—0.,“ 2" +h2'OAVz’ + Lat-Av; + l‘|[)v

14V2+u

J

x1: K- i? .4. Kaav _. hag-(MVPkLAV +KLI1_AV hi} ’szy. haLAV“

-l10. 1’

2.
“5"?" + 5%VAV‘+-~o
X = Av+KbA —."‘..3 ‘- I
(65) 3 v +|1MAV 313°“ ’1‘: “‘3, “Mm/2+ “05M Av"
kLL
+ VL- %‘Av‘+LMvA:L+h_D AVLu-l-
J. ’
X 2 L, -+-Av+»¢ a 1 _“_
5 3’6 2 AV 4- 9. av + . .. .
De2. endin; that the equation of a general hyperquadric
be satisfied by the series (58) for,n---x5 identically

in ng and identically in.‘Av as far as the terms of
third degree we find that such a hyperquadric exists
if and only if
MY=o
Since X10, M must be zero and hence
O) b: 353. ,(bro) (2) bao

For such points the hyperquadric has the equation

(66) 0-,,[x:- -.=m:(x+bx,_+b1&.)]+a— Wren-55x5 —0

-01 espondine to (88), there exists at points for

C)

‘k';:.i011
I
0) °" v-ocz ) (0- 10) L2) as o

the hyperquadric
l. 2.
b 1[x; — 1Yx.‘(x. +043 + Quad] + b“ X1Xq + b.” x., :: o

osculnting

(87)
Consider the hyperquadrics of the fanily of
planes
, and K-1 neighboring points

curve C at x
t approach X

X; (i=l,--'K-I) as these poin s .
We shall say that such

indepen—

 

dently along the curve-
K plane contact with the curve

have
We shall now determine all of the hyper—

1- 1 ’
nvnorcuacrlcs
LL

Ceix°
quadrics which have three plane contact with the

curves of the conjugate net on our surface.
The osculating plane to the curve ll=¢ at If near

X is determined by the points X) Xv ) va whose
" the

coordinates can be represented respectively by

series
X: X+XVAV 1-va [iv-t1- )
Xv: xy+xw 4v+wi £§‘+..... ,
€2?*” " 0

KW: xW +I'vvaV + wa
Any point if in the osculating plane may be defined
linear combination of the form

SK= 3:“? k:Xal‘*‘K}rvv

If this be eXpressed in tne form
Y- x,x+x1xu+ xaxv +xqe+ x50-

the local coordinates x, ----x5 of Y are given by
:I+Kd’4v+(hd’+ k31+ +kf’c.+kd') 93L“
X1: K (YL' ij’i-JIV 6:1." ‘ ° "

+....

)

A
0')
CO

V

X
I b

3 1‘.— EEJ-(l‘{lg-"gréKM)DV+(‘1m-m~law6+KMY‘K€M+KSM’)%+6.’

q Kov+(k+k¢+K_Vi’L+k6‘)Agft+ ,)
X5: %1'(%J)AV+7§‘(I4k5+k81+k<sy+km’)951+ ..

De endinfi that the equation of the ge ne eral hy per-

suadric be satisfied by the series (58) for rm.--x5
ic‘iez‘itically in ‘1,“ and identically in AV as

far as terms of the third deg ee, we obtain the
hyperquadrics Q, , whose equation is

(83) Qt: aazxt+aquszm155xlx +5LIG(%1’ YL-jéo.) 2.5 ‘41:" .
The equation of the corresponding hyperquadrics

for the curve v=<:is obtainable from (69) by the

transformation (lo). There are, therefore, a double

 

infinity of quadrics having three plane contact

 

with the curves gf_a_con1ugate net 9n_g surface i§;‘9 .

 

 

Consider the conic
(70) Q':X|: X1510 .

Evidently the line ecr is ten ent to the conic (70)

 

 

pg: the pointy a‘o

 

If we impose the condition that the tangent plane
to the surface at )< be a plane generator of (69) we

find that oa£:o - Hence the equations of all

to
C ‘.

‘

hyperquadrics havin: three plane contact with LU=Q
at x and having;r he tangent plane at x as a plane

generator are

3 may! XIX“ +‘515 XLXS' + A (Lasxqzz. 0

hear-we).

(71) ’

Such hyperouadrics form a pencil.
‘L A _ =......__._._...i

 

By use of transformation (10) the hyperquadrics

Qt ’ corresponding to (54), for the curve V=c_

are found to be

2.
535 Xaxs + ‘33.; {3X4 + B 0.3., x5 .—. 0

9; 334% \M-flMrmu) .

The intersections of the hyperquadrics (64) and (85)

(72) ’

are the cubic cones

(73‘ XL: ' A “2.5 ‘10’35‘: + by!) , x4 = t (alqt+“as)(bsgt 4' 933“) )
2

X3 ‘1 " B by; (0‘th +015); X5 = (41.415 +m35)(b35t+b3") ‘
wherein “t is a parameter. The vertex of tiese cones
is at the point x .

Tl'e intersection the hyperplane x..-.o with

IP’)

 

 

the cubic cone £Z§L

P.

s §_twisted cubic. Any point

 

 

‘

F’ on this cubic is on CE . Hence the plane

q.

‘eternined by the point F3 and the tangent to Lch.

 

 

 

_

lane generator of G, . These plane generators

[—1
U)
| m
*d

 

 

 

intersect in_a line, namely the line Joining )( to F).

 

 

It follows that the locus 93 the lines of intersection

 

 

 

 

i

f the plane generators of QUQL which intersect

 

 

in 8 line, is a cubic cone with vertex at x

In order to charecterize the single hyperquadric

to vxnich (0d) reouces if a11:a1q= o)

J. Y I __ I 1_
we note that the polar hyperplane, with respect to

A

the ";§'§;erquaacir‘ic Q, , of any point in the tangent

’—

plane Xq=x5 =0 is the hyrmrplane
r _
(7'3) 2‘11)“. +914 X4 +415. x5 " O .

Hence this hyperplane (75) may be called the polar

A

 

LA

 

 

biraerplene with respect to the hyperqundric Q. of

the tangent 919178 Xq=Xg=0

 

(1) It has three plane COUthCt with use, it. X .
(3)

(3)

 

 

hes the tangent plane 8 plane generntor.

 

 

H
c+ c+

hes the plane: xfzo es the ppler hvper—

1

 

plane of the ten;ent plane.

 

A
..-

-orr tZLe

0.)

Similar statetents may be reds ssociated

cuedric which may be obtained fro: the equation (74)

.-

1

c? the transformation (10)°

B BLIOJRAPEY

Bompimni, E. , Projective Differential Gecretry of

Hyperspace. Lectures given in the University

 

of Chicago, Summer 1930.

Grove, V-G- , On Canonical Ferns of Differential

}-

4quet‘ons. Eulletin of the American

l

Tethereticnl Society, August 1930.

L536: E-Po , Prgjective Differential Geometry of
“urves and surfaces. The University of

Chicago Press. (1952)

Tilczynski, E-J. , Projective Differential Geometrg
of Curved Surfsces- First flemoir.
Transactions of the American Kathematical

Society, Vol. 8 (1907).

   

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