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
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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 ('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.~,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" v: : :< < : 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 seet+++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:~+: +++, ~,:+so[utioi:.q
(r:f!
i~:c:-.
a;hl ~,~:picx h i p ?~ + :;:>q~ = :: f , q a_:tv
('(~NCI+USI(~N
of 2-fa,p~ili(,.< , + f ' . i ~ o s it: I):, "vt~ults fro::: ::'c a},,,~,,. .,
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, irrsc.-: };,u~ lta~c :~ f,~v:~---i,