2-families of lines in P 3 and pairs of congruent lines in P 3

2 ~t ~ M I I , t E S

t~i

('(~N(ill|'IINT

|,iN}

t,, Z.

|.INES

S IN

tN Pa

Kruglyakov

UDC 5 t : L 0 1 3 . 2 3

Up :<> ~|+e p r e s e n t , m a i n l y t h o s e c l a s s e s of e o n g r u e m p a i r s -,,f,he ~h.r(:(,,dimcn:~i(,naI p r o j e c t i v e s p a c e P~ h a v e b e e n s t u d i e d in d e t a i l b y u s i n g P l i i c k e r i n t e r s e c t | t i n s , to which t w o - p a r a m e l e r f a m i l i e s ( 2 - f a m i l i ~ , s i of l h w s p , , s s e s s i n g s u b f a m i l i e s of d e v e l o p a l f l e s u r f a c e s ( t o r s e s ) c o r r e s p o n d in the f i v c - d i : n e , : s i o n a i p r o )(.(7~iv( ~ s p a c e P:, ( s e c [1, 2, 71~. S o m e ( : l a s s e s of c o n g r u e n t p ' d v s for w h i c h the 2 - f a m i l i e s of " R o x e n f e l ' d L i n e s " [2[ h a v e no t o r s o s , .m}d a.Iso p a i r s A of S. E. K a r a p e t y . ' m [71 a r c c o n s i d e r e d h e r e i n . In S(.c. 1 a ' c o n d i t i ( m is e s t : O ) l i s h e d for w h i c h the f i r s t d i f f e v e n l i a d n c i g h b ( w h o o d of a l i n e of the 2fa:;xilics is f o u r - d i m e n s i o h a l in PS- In t h i s c a s e t h e r e e x i s l s a o n o - t o - o n e c o r r e s p ( m d t , t l ( ' ( , between: p~)ints of t h e i i n e and the r u l e d s u r f a c e it d e s c r i b e s ( s e e [61). Such 2 - f a m i l i e s . : , f ! i n e s a r e e a i t e d pseu(l,)focal. In Se(:.2-i,a(:a:m,~iealframeof 2 - f a r o . | l i e s of l i n e s in P s i s c o n s t r u c t e d will', an a b s o l u t e , a P i i i c k c r - l < i e i n h 5 pcvqua,.h'ic (,)4~, and its , : o r v e s p o ~ d i n g f r a m e of c o n g r u e n t l i n e p a i r s in Pa, the B . A . I l o , : c n , c l d "fo(.:fi qua, h ' i c , " is c e m s i d v r c d . "~hit'h is the , g e o m e t r i c l o c u s of ci',avac{evistics (~f o-' )~a' ~(,s. p o l a r {(, li.qe:; ~)f ..')[`at~ i[h:,s r e l a t i v e t~ Q,"." :\ c,)r;grvent p a i r in Pa w h i c h has a t'",~agon, ; ' l i u c a v ('I a p,u,p.})cv of t)vop~,r~i~>> ~f tit(, c~w.gr~c,~t p a i r A[7] is a l s o exap,~incd. 1,1 cor~ctusion, s o m e clztss[fica~i~ms e~f2-fa~>.ilies of t i - c s i> P5 a r e .~iven. St':CTI()N

1.

PSEIJD()I,'()CAL

L('t a 2-!'a?~{~y ,~i ]i'.~cs :\1.\2 !)e r e f e r r e d

2-t:ASlII.

IES

(i!

I. I N E 5

to s o m e f r a m e {:\~.~.} !<~. ~. 5

IN

i, 2 . . . . .

f) a /;), *.vi~,sc (b.q'i,.azi~,,~

dA,, := ~,)./;.1~,

( t. I

~*~({ ~hr f(~l'~ns a2o~ sa~ is[ 3 ~h(, stru~,itlvc e q u a t i o n s

!.(.~ u • be p a v a m o t e r s {bus

.,.~f the 2 - 1 a m i i y .

T h e n the different|'0_| equatiop, s of this 2 - f a m i l y r

o~d" == a~,,Pdu" .ks is knmv,~ [5I, a t a n g e n t 3 - s u b s p a e e

li,~<~s is wvitt(:n

(i, z =: 1, 2: p := 3, i, 5. i;).

(I.:~)

at the point x l A i of the ra-- of a 2 - f a m i l y has ti~e f o r m

To ~ (AI, A2, a.~!xi.lp, a,zvx'Av), ~}~,.:,re ~):x~-:x ~. A tangent 3 - s u b s p a c e of a o n e - p a r e m e t e r G rassm~ product

sul)family du2:du !

Ta == (At, A=, at.vdu*Av, az~vdu*Av).

(I.'i) ?, is (h'fined by a:~othcv (!.5)

In the general c a s e linear spans of s e t s of these two subspaces coincide with I)s for alI p o s s i b l e p and X Let us elucidate the condition of coincidence of the 3 - s u b s p a c c s T o and T x. "I'o do this, let (is write their equations in tangential coordinates X~ = O, a~pz~Xp = 0,

(t.S)

Xf = O, a~rXvdu~'=O,

(i.7)

where X ~ are tangential coordinates of the hyperplane.

The subspaces T o and T~ coincide if and only if

Translated from Sibirskii Matematicheskii Zhurnal, Vol. 9, No. 3, pp. 554-567, May-June, I968. Original article submitted April 2, 1966.

417

t h e r e exist q~anltties cv j ( j , v : 1,2), such that

a,,~'.,'

, . ~,,. ,h,'.

(1.8)

Elimlnallp.g cVJ f r o m (1,~,) and p-~l)l, P2, I);t. mid f,,r P: ['1. P2, P4, w h o r e the Pk {k ~ 1, 2, 3, I) at'(, f,~nl' different s u p e r s c r i p t s f r o m 3, 4, 5, t;, we obtain dr, ~ (..! ~~~ . . r , & : . i . l s: ~',.r: ,tt, * 4 , t : ~ 'z ~du" -{ A~ ' ,.r ~&~') ...... 0. ( 1.'.0 'h*'~{.lid '.r~'du ~- *{ .! l a t ' . d d u ' + A:~ > . r t d u : + Aaf '.r ::: 0, w h e r e AixP~ and AixPa a r e e o f a c t o r s of the e l e m e n t s aixP~' and ai.,,.Pa of different rows of the d e t e r m i n a n t of the m a t r i x []aixPll. Only two of the four equations in (1.9) a r e independent. T h e y will be equivalent for du x ;" 0 if and only if the c o l u m n s [)~ and Pa or the r e c i p r o c a l m a t r i x to the matrLx [lai~.Pl[ a r e p r o p o r t i o n a l . i, e., the r e c i p r o c a l m a t r i x I[Ai~P[[ is d e g e n e r a t e , and t h e r ( f o r e

deti!a,~i] --= 0.

(1.t0)

For d u X = 0 (~.=1, 2) Eqs.(1.9) a r e equivalent if AIxPiAe~,Pa-A2xP4AIxPa =0 (not s'umnmd o v e r y. !) or a u det[lai~.PH =0, w h e r e / X • is the d e t e r m i n a n t of the two s y s t e m s f r o m (1.S) fox" p =PI. P2, du~"=0, is nonz e r o s i n c e it is a s s u m e d that solutions of these s y s t e m s exist. If condition (1.10) is satisfied, then the single r e l a t i o n to which (1.9) r e d u c e s will e s t a b l i s h a o n e - t o - o n e c o r r e s p o n d e n c e between points of a r a y of the 2 - f a m i l y A t + p A 2 aml t h o s e r u l e d s u r f a c e s du = =),du x, for which the 3 - s u b s p a c e s Tp and "I'}, coincide. We call such points pseudofoci of the c o n g r u e n t ruled s u r f a c e s of the 2 - f a m i l y (in c o n t r a s t to [61 w h e r e they a r e called feel of the r u l e d s u r f a c e s ) . Let us mention a~ottmr g e o m e t r i c c h a r a c t e r i s t i c of condition (1.10). xiAi to the r u l e d s u r f a c e du2:du t is (,'It, A:.

The tangent pt.'me at the t,oint

(i, x -~- l, 2; p :-= 3, r~. 5, ~;) ,

a,,d'z,da~Av)

(I.ll)

By v i r t u e of (1.10), the c o o r d i n a t e s of the iyaint ai• a r e l i n e a r l y independenl, and t h e r e f o r e , the l i n e a r span ()f all t h e s e pIanes, which coincides with the l i n e a r spans of s e t s of ~.he 3 - s u h s p a c e s Tp and "I!~,, are four-dimensional. D('fi1~iti~2n!. A 2 - f a m i i y of lines Ps whose first differential neighborhood of each r a y is f o u r - d i m e n sional will be Called p s c t d o f o v a i . F o r a pscudofoeal 2 - f a m i l y of lines in t)5, to each of its rule(1 s u r f a c e s passing through a given r a y t h e r e c o r r e s p o B d s a single point of this ray, its p s e u d o f o c u s . The local s t r u c t u r e of a pseudofo(:al 2f a m i l y of lines is the s a m e ms for a 2 - f a m i l y of lines in P i (see [G, 12l). In o r d e r for a 2 - f a m i l y to be s e m i f o e ~ [10I, i . e . , to p o s s e s s one s u b f a m i l y of t o r s e s , it is n e c e s s a r y m~d sufficient that the r e l a t i o n to which (1.9) r e d u c e s should d e c o m p o s e into two f a c t o r s one of which depends only on p, and the othec only on ~. T h e y define the focus of a r a y and a t e r s e of a 2 - f a m i l y , i . e . , the focus is a pseudofocus for all ruled s u r faces of the s e m i f o e a l 2 - f a m i l y , and all points for the t e r s e of such a f a m i l y a r e pseudofoci [11l. P,y virtue of (1.10), the condition to which (1.9) r e d u c e s for a focal 2 - f a m i l y of lines [10l, and only for this family. will be s a t i s f i e d identically, and the f i r s t differential n e i g h b o r h o o d of the r a y is t h r e e - d i m e n s i o n a l . R e m a r k . It T h e o r e m 2,1 f r o m false. T h e fact is generally assured

follows f r o m the r e s u l t s of this s e c t i o n that, as has a l r e a d y been r e m a r k e d in [11], [51 on the e x i s t e n c e of pseudofoei for s u b m a n i f o l d s of any 2 - f a m i l y of lines in Pn is that the condition det[lainPXp[~ =0 for s o l v a b i l i t y of (1.6) with r e s p e c t to x i cannot be for an)* c h o i c e of Xp.

SECTION

2.

CANONICAL

FRAME

In the s p a c e P5 with the absolute Q42, let us c o n s i d e r the f r a m e {Ac~} relative to which the hyperquadric Q42 is given by the equation #~a + ~ # + # x ' -----0.

(2.t)

Its s t a t i o n a r i t y conditions result in the following additional relations on the structure Eq. (1.2): O)t 2 ~

0.~ I ~

o ! I + or= =

418

r

~ ~

0}~ s ~

o , 3 + ~4 ~ =

{0.~ ~ ~

0},~ S ~

~5 ~ + o d =

O~

0,

*+ +++~ [ r

b+'+

+,+,~, } ++++t

~,+++.}+ <,+++

|

+,%1

+ +~ ++++

++qt

t,+ + + + +

~.+ p +,@

+ + ~+++

{t

+:+::3

+~.

~ h~:'c ! h,., m , r m a t izal ion ~t

{-tb|.+++l~ i~.'t,.l.)

| , _~ ,

~2.::;~

h a s a h ' e - t d y been c a r r i e d (mr. F r o m t h e p a h ' w i s c c o n j u g a c y of the f r a m e s AIA ~, ,A@4 ,anti As:k. it fi i t ( , ~ s (st,(, ~iI~ tha~ t h e i r c o n g r u e n t t h r e e p a i r s of l i n e s P3 f o r m s o m e t e t r a h e d r o n !.SIk~. Its d e r i v a t i o n formul:u~ a r e

d+tta =: c~j'Lff.~

(k, l, m : : 1, 2, 3, t ) ,

(2,~)

w h e r e the f o r m s Vk m s a t i s f y the s t r u c t u r e equations D~,," =

{_+.,~,,-,

[,.~,r+-q.

L e t the e d g e s c+f'the t e t r a h e d r o n in P3

.~, === [v-'l, .~: = l:+~l, t+ = 1231,

"Li '

,+,)

[+q, ,~

==

l + 3 l .4+ .... l , - I

(2.,;)

c o r r e s p o n d to t h e v e r t i c e s A o, w h e r e [ k m l = ( M k M m ) + T h e r e f o r e . the c o n g r u e n t p a i r (MtM,:+-(.Ma?;!.a} c o l rcsp,.mds to o u r 2 - f a m i l y . W r i t i n g (2.4) in l i n e a r c o o r d i n a t c s ( s e e [2i), we ()blain ttw %il,.)wit~g r e l a t i o n s b e t w e e n the f o r m s w o : / 3 ~u~(l Vkm: UI I :-:: l'j

10i ~,

-~;': hi2;.

U2 ~` =

t,~l t,

U2 3 =.~ (1tt5+

U~2 ="~= --(02 ~,

F32 ~

/.'l ~ =.= ~,11~',

. t02g,

Ui ! : -

0~2 ~,

2.7)

.~vn

O.

t2~)

Sei,.+ctin~ rico., f~ r n : n -zl3 and ~.+'.'~as the b a s i s , we w r i t e (1,3) as to.j~ - - a~,q3 -}- a.:o)z~, oq ~ == bimta -~- bzojz~.

I n s e r t i n g (2.7) h e r e , we o b t a i n d i f f e r e n t i a l e q u a t i o n s of c o n g r u e n t p a i r s in P3. I ) i f f c r e n t i a t i n ~ ( ~ ~ Iy, ~ e o b t a i n t h e f o l l o w i n g d e p e n d e n c e of 6a a m~d 5b a on 7rfl'Y for fixed p r i m a r y p a r a m e t e r s

+ .....

6at = 2aina~ -- a z a ~ ~ ~ - ( a s - - atbs)rc~ ~ - - aza'Jt3 ~ -}- (a.. - - a t b z ) : n i l , 6aa == 2a~.'~ ~ - - a2a~:t~ ~ -t- (a~ - - a ~ b e ) a ~ ~ - - a~a~n~ ~ -}- (a~ - , atb;):~c, 6a~ = a~(2.'~,.~ - - a~t + ~s~) - - ( a t + a~a~).'t~ ~ - - a ~ b ~ , s - - a,~a+n~ ~ - - a~b~n; ~, 6a.~ ---- a+(m. ~ --I-as s) - - ( a z + a~a~)a3 s - - (aa ~ n- | ) . ~ s a~Zzf, aj)aa~, 6an ~--- a.. (2nz" ~ ~ * - - Ztss) - - asasa3 s - - a ~ b ~ 5 - - (a~ -}- a z a s ) n ~ ~ - - a~b~:t,;s, ~a+ = a d a , t - - n++) - ~ a r

6b~ ~

-

a~b+n, ~ - - (a,. + a ~ a d n ?

- - (a~b, +

l)aL

--2btm* + (b~ - - a~bz)a~ ~ - - b~b~'r~ ~ J c (b~ - - azb~)nn ~ - - btbzn~ ~,

fib, = b , ( ~ ~ + ~tnt - - 2gt ~)

acb~z ~ -- (bz +

8 b s ---- - - b ~ ( g ~ +' nt" liar) ~ asbo+zs _ - - (b~ +

b~P+rl~s _

babs)nt~--a~b'~za ~ - -

(2+I0)

bJ~;rt~ a,

(anbs nt- t)n~ ~ - - (b~ -+- b z b ~ ) n ; ~,

b~b~) n~ s.

D e m a n d i n g that t h e p l a n e A1A~Ar b e t a n g e n t to the r u l e d s u r f a c e w l ~ = 0 at the point Al, and the p l a n e AIAzA 5 b e t a n g e n t to the r u l e d s u r f a c e w~ ~ = 0 at the point A~, w e o b t a i n

419

T r~ d - l h i s tl is n p(,es~itry t~l put

b,(~,,m ~

z~~) .-:. 0,

bd,,zd :

0,

f r o m which for (212)

a~acbd,~ r 0

we obtain ~I~$ ~

Again n o r m a l i z i n g

/I~, ~ ~

aa --= /,~. ai t -- 2.%~ -- .'t~t :

~3~ ~

(2. t3)

.'It ~ : : : 0 .

0; a; = --G. a,; + z~~ = 0,

as = --ba. al t -- . ~ = 0,

(2. t4)

we t e r m i n a t e fixing the f r a m e of 2 - f m n i l i e s of lines in P5 and its congruent f r a m e of eongrtu,nt line p a i r s in Pa- Equation (2.q) b e c o m e s o~,a =: a.~,)=l

SECTION

3.

~l=s --- _ bat@,

GEOMETRIC

(,@ = #<(,h: -- h:,(@.

CItARACTEIlISTIC

OF

(2.15)

TIlE

FRAMES

The points of i n t e r s e c t i o n A 1 and A 2 of a r a y o f 2-families in Ps uitt~ the h y p e r q u a d r i e Q 2 d e s c r i b e s u r f a c e s (A t) and /A~) on Q 2 As is known [-~1, the tin,gent 2-piane to the s u r f a c e (Al~ at th e point Al int e r s e e t s Q.ff on two g e n e r a t i n g lines which a r e tangents to c u r v e s on the s u r f a c e (Ap congr,~em to the t o r s o s of the c o n g r u e n c e (.MI.'M2). T h e s e g e n e r a t o r s (..l~A~)and(A,, by.|a-,~ b~ba.1. _L bae.l~,

i,~.1~)

(3.1)

t o g e t h e r with the r a y AiA 2 define tangent pi.'mes to the ru!ed s u r f a c e s o:ha :

0.md(bah5 @- bt)(,'ta @ ,b.~!~e(,@ : : O.

(3.2)

at the point A 1. A n a l o g o u s l y , the tmngent plane to the s u r f a c e (A a) at the point A~ i n t e r s e c t s Q e along the g e n e r at Ors (AzA~),mJ(A2. v:ba.,la q- b~,l~ -b a~.i~ ba:Ad. (3.3) which d e t e r m i n e the o s e i l l a t i n g planes at the point A 2 to the linear s u r f a c e s . ~n~ = 0and(bab5 -}- aa)~z~ -- babr ~ = O.

(3.4)

The r a y s A t & anti AtA 5 define a 3 - p l a n e to which the r a y AaA ~ i n t e r s e c t i n g the h y p e r q u a d r i e Q.,~ at the points A 3 and A4, is c o n j u g a t e r e l a t i v e to Q42, By using the conjugacy r e l a t i v e to Q42, all the r e m a i n i n g e l e m e n t s of the f r a m e of 2 - f a m i l y lines in P5 a r e e a s i l y c h a r a c t e r i z e d . F o r the c h a r a c t e r i s t i c s of the congruent f r m n e of the congruent pairs (MIM a) -(M3Ma), we write, r e s p e c t i v e l y , the feel, the t e r s e ,and focal plane equations 1) Fl ~ b~llt + M,, bzb6va' -= (bi + Nbs)v~ a, dFl =-- (M, lh.lll), 2) Fz ~- Mz, vt 3 : O, dFz = (MtJl2, baJla-{- btM4), 3) & ~---M~, v.~' = 0, d & = (Ma,14, 3I, q- baJh), (3.5) 4) Fi ~ az.lh + baMi, (az + bab~)va~ = --bab6v=~, dF~ = (3Iyl[iMl).

It is h e n c e s e e n that M 2 and M 3 a r e p l a c e d at one pair of foci by the r a y s /I=MiM2 and /2 =I~IaM4, and M 1 and l~I4 at the points of i n t e r s e c t i o n of the focal planes at the o t h e r feel with the c o n g r u e n t r a y . F r o m (3.5) w e a l s o o b t a i n t h e g e o m e t r i c c h a r a c t e r i s t i c of the i n v a r i a n t s b 1, b 3 and a 2 in P3. L e t us e l u c i d a t e w h a t h a s b e e n e l i m i n a t e d f r o m the a n a l y s i s by c o n d i t i o n ( 2 . 1 2 ) . 1) It f o l l o w s f r o m (3.1) and (3.3) that t h o s e 2 - f a m i l i e s of l i n e s for w h i c h the t a n g e n t p l a n e to t h e s u r f a c e (Ai) at t h e p o i n t A i i n t e r s e c t s the h y p e r q u a d r i e Q42 a l o n g one l i n e h a v e b e e n e l i m i n a t e d by the c o n d i t i o n b a t - 0 , and the c o r r e s p o n d i n g p a i r s o f p a r a b o l i c c o n g r u e n c e s h a v e t h e r e b y b e e n e ' . i m i n a t e d .

420

x~Ai ,~i

I, : l . :hen lh~ h~,,t :u:::ior,~,~ o [ t l w '2,Ia*:~i', :,I l}i. - ~:',l.,i a ~ , d : e ~

A

i~, Iht,~hqurmi~iant n-tiM in {l,lUt.

t,2-' ( , & q

i,: ,1 I: m :

1)

CI ~;I

and b : I l l U uqua~ions -q (t'~',u ~ + f'~,,: ~) ' .~/,.~,,~" ........O.

"I>~ ca>es are p o s s i b l e here. a} a2h I = 1.

T h e n the 2 - f a m i l y is p s e u d o f i m a l ,

h~s no torses,

:rod the c o r r c s p ~ m d e n c e

(a.7) h:~'~ b e e n (:sta}llished b e t w e e ~ p : , i n t s of a r a y a n d lhe r u l e d s u r f a c e s . p~eu:lofoci of the r u l e d surfaces

The p o i n t s :\! :m:l A 2 a r c . ru~pectivciy,

(:rs} For ( ha -V b~ b:,) ( b~ -,4-tq ha) -~- b~ r

:= 0

(:~.9)

:;:' o b t a i n the s i n g l e t e r s e (b~ . . bJ.,) ,,..a == b~,',~,,: :

(;;.I,i)

(',q-re:pending to the focus b~.l~

(ha "i b,i,:,).I>

(3.1 }}

b) t _ n T h e n the ~.-,a~,,l{~ :' e ,.," .. has two t o F : a e s . ~13=0 a2).(l ;z2~ :-0. RP:.! :}'tt. c , w t c s l } , ~ : : ~ ! [ : ' . g .,~ ~: . \ , :t~i:} .\.. ~',:,~q'u :}w if:\ i dei)en ~s ~mlv ~m (me p a r a m e t e r , i . e . . the focM s m ' f a c c s ~iAi~ de~en~,wa~e, im'~;" cm-v,.,~ ,.,~.. Q~?. '.u',.~: ! I t , i t c~,rrespo:Miu:4 c o ' : , ~ r u , . , , w e s , into r i f l e d s u r f a c e s ~

Thurui(we,

tr,.)t"m'[I }!, 2-[ami[ius h: P5 and their co.m'csp,,ndin:..~c~m~r~:c:-t, p:d:'~ -r, "~..,in :':~}a~" !~c,t,~:

::1i:::inat ed.

In uo::clusbm,

let us note another interesting fact.

TIiEt)I~E:~.! I. in order for points of intersection of rays of a 2 - f a m U y v,Ph flit.,hyl)c,rqu:v:L,'i.rQI:: It) b:' ice': of :his 2-family,. it [s ::ecessarv. and sufficient that their covreqn:.. ,::lit,,~,c':m,.zr::enr h: Ps :]e~:t'ncr:ite. i~.~to a l'tiled.suFface. I n d e e d , if IX2 is a f o c u s of tile 2 - f a a n i i y , t h e n tile f o e M i t y c o n d i t i o n s in the n o n c a n o n i c a : f r a m e ~ A a ~ : a 3 = a S = 0 for 0213>40. T h e n d [:H] d e p e n d s on one b a s i s f:>rm, and the co,ngr::cnce (.MaMa) d v g e n e r a t e s into a r u l e d s u r f a c e . If Ms:xI 1 d e s c r i b e s a r u l e d s t : v f a c e , th(,n the d i f f e r e n t i a l dA 2::-~:22A2 q ?-'~aA 3 . + c~2tA4 + cc25A5 § 0222A~ s h o u l d d e p e n d on o n e btus is for p,: a~2l, fFOW., W[iictl a 1 : a a a 5 :~!, and t h e point A 2 is a f o c u s of a r a y of t h e 2 - f a m i l y . A n a l o g o u s l y for the point A I, the c . r r e s p o n d i v . . g

}~c('ot::o ~.3. = ..2,.~= ~ ' 2 5 = ~ .c. =0, or a I

conditions is b~ =b 4-- b~ -:0. The a r b i t r a r i n e s s of t h e e x i s t e n c e o f t h e c l a s s a i = a a =as=0 (b e =:b~ =be =01 is three [unctions of two a n d of the class a l = a a = a ~ = b 2 = b , ~ = b ~ =0, sLx f u n c t i o n s of one a r g u m e n t .

arguments,

S E C T I O N 4.

R O Z E N F E L ' D FOCAL QUADRIC

Let us consider the focal manifold of congruences (in t h e s e n s e of [ID of 3-planes A3A:As?~ conjugate to the line AIA2 relative to the hyperquadric Q42. From the condition (dNAaA4A5z%) =0, where N =XPAp

(p=3, 4, 5, 6), we obtain xp(~p~ = 0

( i = I , 2).

(4A)

Hence, for each displacement w24=~wl3we obtain a characteristic line, which is the intersection of the given 3-plane and its neighbor for this displacement: x' = a~ = 0, ~3 + ~..

b,~ nt- ~ nu (b5 -I- Xb~)aa ~- b ~ == 0, _ ~.b~ +

(4.2)

(b, - - ~.b~)~ = 0.

421

(,t~ + a:,r~

ha"

h-.rD{%~

i .*+ i ,~'>+~i b~d~

I'~ ~ z'+": |L

(;:I~

v, htch+ f<~lI~)v:ing B.A. l { o : e n f e l ' d [ 1 ], ;~t' call the "focal qtl,+tdr'ic, " L~'t ~!s e•162 lilws colljuga!~, to the ~'tt~:,o|lt 3-~uh~p~l('e~ Tp and "l'~ ~,ith t'r to Q~. Takin~ acre',mr of the du:dity nf the pnlar eonjt,~acy of the points, wv ub~ain that troy family ~f s,~rai~:h~ linv t:enel'ators ,.,f t h e q u a d r t c (.I.3) c o n s i s t s of p o l a r s t o M l of 71'A. A p o l a r to the 3-subspaer T#:is defined by t}v.. equa~ ions .r~ =. :

=

O,

bi~" +f .r ~ -f- b : . r " - k

r,.r-' -i v,~.a-+ + (l,+

O/,:,).r:'

-

( t ~ -,+ td,,): "+ -

O,

!,1,:.: := O,

{~'~

The lines (4.4) a r e g e n e r a t o r s of the q u a d r i c (-t.'3) and ahvays i n t e r s e c t the lines (.1.2). Thus the s e c o n d f a m i l y of s t r a i g h t line g e n e r a t o r s of the q u a d r i c (-1.3) c o n s i s t s of p o l a r s to all of Tp. As is known [1], a pMr of lines d i ( i : l , 2), which a r e the lim.q position of lines i n t e r s e c t i n g the pair of r a y s ~.'2"~*:~,a:t3 mtd then neighbors for a d i s p l a c e m e n t c o r r e s p o n d i n g to t 1-- 12, will c o r r e s p o n d in Pa to a polar" to tt:e 3 - s u b s p a e e T x in Ps. T h e s e lines a r e the d i r e c t r i c e s of a bundle of tal/gt, nt linear c o m p l e x e s to the pair of ruled s u r f a c e s ,,'at+~,vl a = 0 . Points of i n t e r s e c t i o n of t h e s e d i r e e t r i c e s with the r a y s l t a~d I~ a r e quasifleenodal points [-t]. A bundle of linear c o m p l e x e s in P3 c o r r e s p o n d s to the polar to T p . Its taxes a r e a pair of lines : i ( i = 1 , 2~ i n t e r s e c t i n g the pair of r a y s l t, 12. The ~.3 linear c o m p l e x e s c o r r e s p o n d i n g to all p,.~ints of the 3s u b s p a e e Tp in P~ a r e involute to the bundle of l i n e a r c o m p l e x e s defined by the pair gi" The pillars to ~he 3 - s u b s p a c e s Tp:= 0 m~d Tp:.m. at the points A 1 and A 2 are, r e s p e c t i v e l y (l,:A:: -- ~,,.-l,,~,.:t, -- A+),

+i.,-,)

(ha.% + .1~, /,a-l~ i- ":.!:,)

{ ;:;}

and i n t e r s e c t Q{~ at points by which they at'{.' given.

(baM~ -}- .112, .I1.)

T~vo congruent pairs -

-

(o2:13 -[- b:+.!l+a. M r ) - .

(M'~, b>lla --;- :q.II,~),

{1.7)

(,1/,~..1I, -~- b;.Ifa).

(~.S)

c o r r e s p o n d to these points in Pa. It follows from (3.5) that eacit of these r a y s p a s s e s through the focus 1,'k {k = I , 2, 3. 1) of the pair (1~) -(le) , and the point of i n t e r s e c t i o n of the focal plane dF k wifl~ the congma,nt r a y (a p a r t i c u l a r e a s e of G.X. ?,Ia+keev focal p a i r s [131}. t ' o l a r s of the 3 - s u b s p a c e s T ~ = 0 and TA_-.-~ i n t e r s e c t Q, 2 at the points b>.~a - A > b s A a - b l . \ ~ and b/,:\ a + :~., bsA ~,~ a2:~. tlence, g e o m e t r i c c h a r a c t e r i s t i c of the invariants a2bl, b 3 and b 5 in P5 indeed restflt f r o m ~ 1.5), (.1.6). F o r a pseudofocal 2 - f a m i l y of lines and only for it, the focal, q u a d r i c (4.3) degenerates: into a cone K ~ with v e r t e x Aa-blA+,, which is the polar h y p e r p l a n e (AIA2As.-~, A a +blA~) containing the whole first d i f f e r entia] neighborhood of the r a y AIA 2. By v i r t u e of the c o r r e s p o n d e n c e (3.7) the f a m i l i e s of 3 - s u b s p a c e s Tp and T x attd their p o l a r s r e l a t i v e to the h y p e r q u a d r i e Q.t~ coincide in this c a s e . This yields a g e o m e t r i c i n t e r p r e t a t i o n of the c o r r e s p o n d e n c e (3.7) in P3. Since the l i n e a r c o m p l e x c o r r e s p o n d i n g to the v e r t e x of the cone K = belongs to all bundles of tangent l i n e a r c o m p l e x e s for any displacements, then it is tangent to the congruent pair (/1) -(12), f r o m which follows: T t t E O R E M 2. A pair of congruent lines has a c o m m o n tangent l i n e a r c o m p l e x (tlc) for each p a i r of c o r r e s p o n d i n g r a y s if and only if its c o r r e s p o n d i n g 2 - f a m i l y of lines in Ps is pseudofocal. The lines of this tangential line c o m p l e x correspond to the points of intersection of the hyperplane (A1AaAsA~, A 3 + btA~) vdth Q~2.

Such pairs will be examined in the next section. Let us just note that the plane H, defined by the lines (4.5) and (4.6) for a2b 1 = 1 will intersect the hyperquadrie Q(~ in a line to which the d e m i - q u a d r i c containing two pairs of rays (4.7) and (4.8) c o m p l e t e l y defining it will c o r r e s p o n d in Pa. By virtue of (a.9) the foeal quadrie for a sere!focal 2 - f a m i l y will d e c o m p o s e into two i n t e r s e c t i n g planes x ~=

422

O, btx* + x ~ - - b~b~x ~

+ b# "~-----0,

(4.9)

': ~*

. , ~ ' ,,I ~' i>r ~ h'n{~

t:

,[th;ina~ , t Lt,cgul'-(' b e " ( h ,

In thi> t ; c - ~ a[i ~[i,, :,-:,il~c.},,.,

i >, v,.d; { , : a ~ n .~ ,h 'm I ' 1 ~ ' at ,'he p~,~!1 ( : L I I ~ , i o ' . ~ l : i c h ~h,, pIan~, r La:- a {,~,'~{!, ~,f [i*~,'s c,q'jtt~VV, e t . ' r ~ ,

i u ~':',::,i>

x~ili h . ,.,,;, ',:;~ar

*,',,":gv~;{

t ' } : V : ~ g ' ' , ' V {h{,

w i t h t h e c ~ , : n : n m ('~,u{er .\:~.~h|.,\,,

i,~'~, ~h,* D a i r A [71 (,,rr,~sp.~mdini., ti) t h i s 2 - f a n H [ y , v,(, ~,btain tlm! { h v r v is a :-i:~:l~ l i n t . a t c , ; v : i , I c x c ~ , r * ~~ -;, ,,'din;* *;, lht, fi..-t~:~, ( : L I l'i in {ht, h u n ( l i e o f l i n e a r c o m p l ( , x c s w i t h ~c,:es l 1 a i m l.,. 9 ' s u c h ~hat a!l l i m , : t r ~' ,: t~i,.>.~,s in *'. ",~hich c ( , r r , , s p n n d tn w d n t s ,if the r o t ' a t p l a n e a r c i n v o l u t e ,'(, t':l('h {)f ~.h(' ~'? [;il}b~('l~{ t i r , a r ( , : : ~ [ ~ x ( , s ~17~t:,, ,,.I p a h ' s - ~ ) { r~iled s u r f ; l ( ' e s ~ff t h e c o n g r u e n t [),~drs { ~ 0 - ~ L ~ * ,M~,,"v(,',cv. all !it~.('ar '.:t,m~ , ; , x~.5 ~d a i ,.t, , ( X~oi'h th(, ~Lxt.s I! a n d 12 hi i'3. w h i c h c o r r e s i . , o n r | t,,~ t~:e t a n g ( , n t p)~an(. ,ff t h e .',-rst, in t>2., a r e g: v~d:~:~ ~v, ~ ' a ( h ~f ~ h t ~2 [ i q e a r q(:,lllpI(,xl.,s c o n t a i n i n g t h e (l('nliqULtdri(: (]('fi ,,,,(l bY ~h(? l'Ltys (4-)7'), { [,~), }:~)r.'h~' c ~ , . > p ~ e ; , ' i y f* 'a; 2 - f a ) : : i t y ~ f f l i n e s e x c l u d e d !{~':. c ,

here,

tt~e f < , c : d q t ' : v h b : , ! v : ~ , n t ' r a . ' . , . ~

i: ~.~': :e~,~ti:~t,,{', ,~ *l v,> t h a t t h e c < m g r u ( m t p a i r T [2 t a n d , , n l v it h a s

: u * , , : t !ira,.

m ~ !i~,s ,q' * t~ . . . . ~, '"

,,~., >

i':~'>,(.~ ,:~.~' *i.i: "}',,Y ('a('}'~ i);li1" <)f ( ' ( ) r F e s p o i > l i i ~ g r a y s , SFCTI()N

5.

('(}MM()N

C()N(;IIV}.:NT

T:\NGI-:NT

P.\Ii~S

1.IN},:AII

}{.\VING

:\

C()MPi;FX

1., ~ , > v: : :< < : <= ,:,,t, r.,.2~v,.1 s i n - f a c e t i (i , 1, 2* b e l o n g i n g i,, t h e c , , r 't~s ~,, , i.>: :.,,:~:;v'a,ncv t,, v:tvi~ ,,i lh,, :: ~5;:

: ~!'.!.:.2 l ' : : \ S

X:~I =:c)'i~.'~

.1:

t,f L1 (',)}~!'tl,t'I3t

p~.t[l',

["',~1" (Ji ;"cr2 [}'('..
.... :{ ;-a:: .

l, - '~-~ i= :~.. ':!:,., i,.,,,!~i(-~ ,,f fi;Mi}'g su('!i a i)L~i!" ~|' ,..,I

:*~1 . ', ;:k!:c{CT"

r

[{~?~.'.:1.~." U , , r ' n i ~ l t ' x c ' s

,ap]a

. . . . :k:~. . . . ~ : i ,

I

" ~.

~ :-;Vil'i:}A'C5 k .l : ~ r ~

L'<>;~-

,

~L.,:~(.,.,:t,)-:!'~,

,.r~.2r .

w}~Cl'('

_

I~:}}

-,

9

,.i

{ /,i.=',' f;.4 !t,[gV~i, :IS: tl ~ :

--

k:

1~

~)It* L "-~" L~2;t

"~" II::"

:-,.2) ,: ; .:: (). ,r . :~

. ( h , -t- h ~ G : ) t~;~ .4: ~ . ~ . , ~

b,,~:,,: ~

H.

i !:i- >v~:~-~':: ~ .'fi,;~.s a .-:i~eaf ,.~f t:mgent lineal" c o m p l e x e s if

,~ (a:b, . t) = O, b , ( b ~ + b~o2) + bathe ---= O, a:~T:(/,:,

!,;.~) ! f,:(~: :~: ~~.

(5.;r

t : : : i : ~ i n : r i n < ~hv . ' . ~ r s c s 2) anti 3) in (3.5) f r o m the a n a l y s i s , s i n c e t h e r e a r c m, s u c h d ( q n i , luadri(':~ f~n" t h e m I..,. a u s v <,f ~.he f i x i n g ( 2 . 1 1 ) , w e obtain the s i n g l e condition a 2 b l = l for w h i c h t h e d e s i r e d p a i r ~f r'.0.,.'d s,'.~rfa.,:s

is f l m n d : a, =

(btba + b~)/b6, or2 ~- - - b i b 6 / ( b , b5 + ba).

(5.1)

Definition 2. We shall s a y that the pair of ruled s u r f a c e s % , a 2 belonging to a congruent pair and t:)~c~sing through the c o r r e s p o n d i n g r a y s , p o s s e s s the R - p r o p e r t y r e l a t i v e to t h e s e rays if P. has a tang(.nt d c m i q u a d r i c for t h e m . If a I =~2, i . e . , the s u r f a c e s of the pair c o r r e s p o n d to a congruent pair and if they p o s s e s s the t l p r o p e r t y for the g i v e n pair of r a y s . then the pair of ruled s u r f a c e s is s t r a t i f i a b l e , and t h e r e f o r e , p o s s e s s e s the R - p r o p e r t y r e l a t i v e to any c o r r e s p o n d i n g pair of r a y s [11. F r o m d e f i n i t i o n 2 and T h e o r e m 2 t h e r e f o l l o w s : T t t E O R E M 3. A c o n g r u e n t pair having a tangent l i n e a r c o m p l e x , and only that one, has at l e a s t one pair of r u l e d s u r f a c e s p o s s e s s i n g the R - p r o p e r t y for each pair of c o r r e s p o n d i n g r a y s .

40.2.3

The nv~v~.'-ar> an~l sufficien? ('~,nditi~m~. f,,r this i s . : i , | h3 ~' I, II: thi~ va~,, o, '.h,' ,,,nr l~ia!w ~,,~}~. ('u~p ~i ~,m" t,.~r,~.e :d the p,,int I"I pas.'-,':~ thr~,u!.,',h tlw |,wu,-- FI, a,ld v(m~,,r.,.ely, th(' v,,i~ti~:,.~,m> ida~w I,* !h,' v u s p a t the p,,it~! I-"I t',mtains F t. This rata, differs h',,m the ~'u~t,,inary i2} ~;ralifiahility,,ft~,r.,.~.;, m t h a t the vorr~,sp(,nd(,nce is (,stabli.qmd only I,(.twevn two ril)> l I an 4 12. Th,, cla,~s . f c~mgl'm,n~ pair8 - : h I }h~ I e M s t s with an arbitrariness of four ftlmqions of t~.to argttnl(.x~ts, awl has it simp[e~v(m,ctric('haractvristic. The focal plane dF 2 i n t e r s e c t s the r a y l 2 at the f(,cus I"~. m~d tiu, focal plan(' dF 3 intersects the r a y l| a,' the focus F I. It foli~w.~ from definition 2 and T h e o r e m 3 that the qtmsifleew~dat p,;ints l-i! flu' p a i r s of ('~rresp<mding r a y s of pairs of ruled s u r f a c e s p o s s e s s i n g the R - p r o p e r t y a r e undefined. The congru(mt pair A~ is characterized hy this p r o p e r t y in [71, anti it t h e r e f o r e coincides with the pair of E . T . lvlev [a], x~hich has a tangent l i n e a r c o m p l e x . By v i r t u e of (:;.'~/. (.I.5), (-t.6), for the congruent pair A 0 the lines (t.7t (or (-t.'qJ pass through the qua.~iflecnodal points ill of the [)air of ruled s u r f a c e s v3 ! =~-rvt'~. ~ h c r e cr = (Y2 ( ~" ' ~rt) from (5. l). T h e r e f o r e , as has a l r e a d y been r e m a r k e d in [31, the first and see
d i s p t a c e m e n t s in Ps congruent to the fundam(,n~al pairs of (3.s) that for the t)seudofoc:fl 2 - f a m i l y v ~ f i i n e s tim t)oiqt At to the first fundamental pair of r u i c ( l s u r f a c e s in P3, and :\2 to the third fundamentaI pair of ruled s u r f a c e s .

On the other hm~d, we find the focal d i s p l a c e m e n t s of the plane I1 d e s c r i b e d by the point B = ba-la -- b~el~ + x(ba.l~

..l~;) + y(ba:la + ;Is),

i . e . , sttch d i s p l a c e m e n t s wal:a:l 3 for which the p l u m II and that infinitely c i o s e to it i n t e r s e c t . condition that dB belong to the tflane (5.5) we obtain

(~,, + .~) [ (b~ + b,b~)(,,:, t,,~,~,,:q = a Yl(b~ + btba)(,q a + bs(,,2'] = 0, tit 4 a'0: + V0a = 0,

(5.5) F r o m the

(5.*;)

where O, -~- (bab~ + btbj:)oJa:'-{- (b,~; + ba~,)~,~ + (t,t~:i + b.~.,_)(o~;,

(fla is obtained f r o m e x t e r n a l d i f f e r e n t i a t i o n and solution of (2.15) by m e a n s of C a r t a n ' s l c m m a ) , we have t h r e e focal d i s p l a c e m e n t s

t) (b~ + b,ba) c o , ~ + b6o,z ~ -~ O, 2) (ba + b,b~)~o24 - - b,b~(~, ~ ~ O,

tlencc,

(5.7)

3) b,02-- O! = O, Since dB ~ (1 + xo~) [(ba + b,b~)~2 ~ -- b,b6o~,~JA, -- - y [ (b5 + btba) ~, 3 + b~r + -QPAv,

w h e r e f~P is a c o m b i n a t i o n of the f o r m s oaqP, db 1 and db a (p, q = 3 , 4, 5, 6), then t h o s e two focal d i s p l a c e m e n t s 1) and 2) of the plane (ll)-(12) c o r r e s p o n d to two fundamental pairs of ruled s u r f a c e s of the pair A 0 of c o n g r u e n c e s II, for which the h y p e r p l a n e s AiA3A4AsA r are focal ( s e e [10]) for the i-th d i s p l a c e m e n t

( i = l . 2). 424

l+, ~++...... + t h , ++!:!~'~+ "~+r+;+c+ + u,l +~ ~+ }++ ",.irtuv +++ i Z . ; ~ . \natt+~U+~+ +3"+ +h+ +d;~,:r \ | \ \+ i+ :+[ !a,+i++c:+++++ CP+++ ++; ~,,+ >t+++f;~t + +hq}m-d b 3 | l ~d ~:+~.7+ at t h c |+~+it+t F;~ i}% + ++I}'~ +~m~[ + }tl++r ~'~}+i~ }i ~ ~J I>~ :;':~++'+t ~+:" +{ +}t + t't:i{:+ :;;+;+ V ++, t +++ [f+,}~c ~2+fa:~+i:y i-'- +(,tt+if++cal +>(++' +:+.!+D. th, + p+,i+:ts }:i :t:~+: +++, ~,:+<~+}:+ v,+~:: t+~ f~++,'+:++~,ft~+c :+a)+ L e t u s e x : u u i n v s t i l l at+t)th(q" p r + q ~ ' r t y of Th(' pail" .-~;.:.

'l~}'~t !['~L~*:)'{..__.!+. [I~ t h e pair" A+, and o n l y t h i s p a i r ,

f(+r e a c h dispia.c(+:+t('nt v~ ! = 0 v l + in t h e bun~Iic ++f iii~d e f i n ( ' d by t i w p a i r of l i n e s It ;uM l:, t h o r ( + is a l i n e a r c(+mi)lex ",~i:ich is t:~++g~ i~,t t(+ tile p a i r ,,f rub.,+t s u r f a c o s d c s c r i l ) c d |)y l i n e s p a . s s i n g t h r o u g h t+~o foot a+td t w o p o i n t s of i n t e r s e c t i o n of t h e foc:d p:.a+~c... at <+ther f+}ci w i t h t h e c<+p,g r u c n t rab +"

~ +:lL ~

{ ' ( } ~ ~ ~ l+~i ( ' X {~ S

Pr,+of. "F-~mr d i f f ( , r o n t c o m b i n a t i o n s of l o c i a r c p o s s i t i l e . : l , e t us c x a , u i t : c t h e p a i r +,f tit:e,~ 3 ! . M i a+:,.t M I M : . tP, o r d e r f o r a : i n c a v c o m p t ( , x f r o m t h e hun~ttc a l ? p '~:`,a:~`tp 1 2 : 0 d c f i n ( ' ( l b y t h e r a y s 11 a,ud l. it> co:+: a i n t w , ~ n(,igh}~(,rs ~,ft}t(, r a y [2:+ l§ and [l;}+d[I4l, if (2.15i is t : r i~i:o acc,um~ t h e r e shc+ut~i !+c f,~v ~}t(~ disp|a('c~ll(,~l! v3 ! =ffVl"t:" a':

~{fq1{'C,

--

ill t~:.~.+ :.~t+'1(+!:3.1 c ' a s o {I~.,:'FC at'( + ~v
a=~a u

=

O,

b,a t: - +

a,:'" =.- O.

(~:~y ~,v,'t>so[utioi:.q

(r:f!

i~:c:-.

a;hl <7 + :' ~!cfi':i::~ :: s:~':'i:t: :i+:~.,:.~.',(~+,~:'+-

',:ith a>.cs l~ a,M l~. (~:~}3+ it: t h e c.::sc a~b I : 1 is thor'(, :t ta~:~crt : i n c a r ,:ispla:'o.,t:~'la:. .\i! ti~r+ vc~aai::i:~g cases arc pr:+v(,d a:udog:~usly, SECTI(~N T}'v f(/.~,',:i:~g c l a ~ s i f i c a : i , q ,

+;.

c,>~,~:picx h i p ?~ + :;:>q~ = :: f , q a_:tv

('(~NCI+USI(~N

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T i u ~':.~.~: K<>(rgd ::+,,qf(,cai 2 - f a : : : i l / , ) f i i ; : c s {,d P5 !:::~ +:,,',>~r.'.cs, :}:c l i:'::i , i : ~ . , ' r ~ : : i a ! "ci:: } .: ~:, ,, ! ,.,: /-: ~a:+ c, ir<'i+ics ,,vii!, T}'u v,i:,,,iu s ~ , : . , - . . T!ics(, 2 - f a t : | l i e s }:av(. !~u,.".: :,,~.'-i,{~ d :i't.i'~" t'>,i~.~(-~.v~+ is s l x {vl:c:i(u:s {d ~.v,L~ :.iI+~.~(:.t,~(?l:.'.s. _\ (+:a-.- , t ~,s(_'t.~!<,',+,~:*~ 2 - J a ~ i!ic.-; ~,~hic}: h a v e ,,,u :,>rsc.-: };,u~ lta~c :~ f,~v:~---i,<:x: fir_.: ~i~,~.:-c::i:E ::~ :g}?{ +,I }:,, .l ~ , i : h c [i~:u is , , x : r a c : c d witi: an ar!.dtrari~:os.< ~! i{\u f'.m(qi at< , t ~v,,, ur:v:{~::,.q:+.. ,\ ',, :.;r :~ ::: i u : ; ' ,!<,sv'ri}~(+i i~% E . ' I ' . I v l c ' , :tr:~i S+ E. K,:~.l'apc'.3a!'a :p.:(] (']'aF:.tc:('v[zcd b~ 'hi. :.xismc: ~ ,, ~d a {:~;?..;+ ::, ii , :~.. c~'~}/.~.... !(,r :u~3 i<:i..'" ;,f :~,I:gruc;#, r a y s c o r r c s p ( u M s :o s u c h a 2 - f a ~ i : y ~,i t h i s v i a s s it. P?,. 5e::~il~.>u:d 2-fap.:i:i'c5 :,vc i s ~ d a t e d wi, h t h e a r b i t r a l ' i ~ c s s t,I' f o u r fu!:cti,,I:s ,.,ft\v,~ ar~gu::c+:::.+ S,.~b~ a 2-',a:~:ily h a s ~mc -~u+ffamily :)ft,+q-ses :rod a fi.,tar-dim.('nsi(.:a.[ f i r s t d i f f ( , v c n ! i a l uci~g!d.>rh,~,+d , t : } : c ra G . F i : c S . E . l'.:arapc',yal} i)aiv A c , r r r c s p o q d s to it it: Pa. C()mt)i(-'.ciy f~cai 2 - f m > . i l i e s a r e e x t r a c t e d w i t h an a r b i t r a r i n e s s Suuh a 2 - f a t } : i l y ha..~ t w o s u b f a m i l i e s of t o r s e s a n d a t h r e e - d i m e n s i o n : d ray. T h e S. P. I : i n i k o v p a i r T c o r r e s p o n d s to it in Pa. LITERATURE

2. ,2.

5. 6.

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CITED

B.A. Rozenfel'd, M e t r i c M e t h o d in P r o j e c t i v e D i f f e r e n t i a l G e o m e t r y and Its ('~mfo;-:~mt and C , r e ! a c t A n a l o g s , M a t e m . Sb,, 22, 4 5 7 - 4 9 2 ( 1 9 4 8 ) . S . C . F i n i k o v , T h e o r y o f Congruent Pairs [in Russian], G I T T L , M o s c o w - L e n i n g r a d ( l ! ~ 3 C ) . E . T . I v l e v , C a n o n i c a l F r a m e o f Congruent Pairs in P3, T r u d y T o m s k U n i v . , G e ( , m c t r i c a t (+oliec-

lion, 160, 1, 15-24 (1962). E . T . Ivl----ev, A Pair of Ruled Surfaces in Pa, Trudy Tomsk Univ., Geometrical Collection, 1(;1, 2, 3-10 (1962). S . E . KarapetymL Projective Differential Geometry of Families of Multidimensional Planes, II, Izv. Akad. Nauk ArmSSR, Se.r. Fiz.-matem., 16, No.5, 2-22 (1963). S . E . Karapetyan, Projective Differentia2 Geometry of Two-Parameter Families of Lines and Planes in Fottr-Dimensional Projective Space, I, Izv. Akad. Nauk ArmSSR, Ser. Fiz.-matem., 15, No. 2, 25-43 (1962). S . E . Karapetyan, The Pair A and Some Properties of the Pair T, Izv. Akad. Nauk ArmSSq2. Ser. Fiz.-rnatem., !2_-- No. 4, 27-34 (1959).

425

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S, I'2. t,i:,r II)~'~y;m. ~lhv Thv,,~", ,~f ('-~,'rm'r~t i'aiv~, | , d , ALa,1. ,",'ai~k .'~rmfi,ql~, f-vr, Vig.~m;,J~,~,,, l,i,

1o. II. 1'2. I:L

I.I.

426

N,,. i .

37-)7

(l')C,l).

R . M . G~,id(,l'man, lh'<>hlvms ~)f Difh,rvntia| (;e~m~etl'y ()f ll,~w,~)~;('m'~,u,~ Si)a(-e.~, Trudy 3I~)s~',)~ It~t. l~,zh. T)'aaSl)~,)'t:~, Qm,sti())~s of Diff(,r(.n),ial G(,om(,~ry, I!;~), 5-2() ('1!~;5). I.. Z. Krt)glvak()v, ('aa(,ni('al I-'ram(, of a N()~ff,,c:fl Tw()-i)a~--~'am(,h,r l:':,miiy ~)f l.i)~(,s i~ | . ' i ~ ( , - I ) i ) ~ w n sic,hal l)r~~)ective Si)a('t', T r u d v. T o m s k Univ.. G(,ometri('aI C'~)I[(,cti()n, I ) I. ( ; . :~(;-17 ,~1 '.~(;7). G . P . B()chill(), On the T w o - l ) a r ~ c t e r Manifold V. ~)f S(,mi-No~euclidczt)~ Si)a('('s i)~ I'~, T~'u(ly T o m s k I'niv.. Geome,~rical Cr~llecti(m, 1!)1. 6, 2:1-31 (I~)(W). G . N . Makccv. Foctd Congruenl P a i r s T ~ d ~ [in Russia)~), Transactiot~s of the i.'irst Scientific C~mference of the Ma~hema~ic."d I'k.pa)-tmcnts <)f the l)<)voIzh'ya lh, d : ~ o g i e h~stiiu~(,s)'l(,~(;l). [)p. 122127. R . M . G e i d e l ' m a n , "I'hc Ttu~gent L i n e a r Complex of a Congruent Dai), of [.i)~es. Izv. Vyssh. t'chebn. ~ ( 1',)66). Zaved.. Matem ., No. 3. .' i .9- . , '.~,