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22 2013 : .., .., .., .., .., .., .., .., .., .., .., .., .., : .., .., .., .., .., ... : ... CR- - , (. ). .., .., .., ... . 1907- . 3- . : ( ) . 1932- ., , 3- CR- (. A). 1974 . . , ( 1) . 2004- .. , , . ( ) , . , 2011- .. .. ( ) 4- CR- (J.Math. Anal. Appl.374 (2011), 655­672), , (. B). , . : . .. 4- , 80- . .. .... , .. , , ..


. 3- (., 1932): M - , aut M - . C2 - (z = x + iy , w = u + iv ) , (r, ) - (y , v ). 1. dim aut M = (1.1) R3 C3 ( ) (1.2) Im w = 0 (, C в R) 2. dim aut M = 8 (2) Im w = |z |2 |z |2 + |w|2 = 1 ()

3. dim aut M = 3 (3.1) v = y s , s [-1, 1), s = (3.2) v = y ln y (3.3) r = ea (3.4) 1 + |z |2 + |w|2 = a|1 + z 2 + w2 |, a > 1 (3.5) 1 + |z |2 - |w|2 = a|1 + z 2 - w2 |, a > 1 (3.6) -1 + |z |2 + |w|2 = a| - 1 + z 2 + w2 |, 0 < |a| < 1
1 2

B. 4- (., ., 2011): C3 - (z = x+iy , w2 = u2 +iv2 , w3 = u3 +iv3 ), (r, ) - (y , v2 ), (R, ) - (y , v3 ). . 1. - dim aut M = : (1.1) R4 C4 ( 4- ) (1.2) C2 ( 2- ) (1.3) v2 = 0, v3 = 0 (4- , C в R2 ) (1.4) |z |2 + |w2 |2 = 1, v3 = 0 ( C2 ) (1.5) v2 = y s , s [-1, 1), s = 1 , v3 = 0 2 (1.6) v2 = y ln y , v3 = 0 (1.7) r = ea , v3 = 0 2 (1.8) 1 + |z |2 + |w2 |2 = a|1 + z 2 + w2 |, v3 = 0, a > 1 2 2 2 2 (1.9) 1 + |z | - |w2 | = a|1 + z - w2 |, v3 = 0, a > 1 2 2 2 (1.10) -1 + |z | + |w2 | = a| - 1 + z 2 + w2 |, v3 = 0, 0 < |a| < 1 (1.5)-(1.10) - .


2. - dim aut M = 5: (2.1) {v2 = |z |2 , v3 = 2|z |2 Re z } {v2 = y 2 , v3 = y 3 } ( - ). 3. - dim aut M = 4 (3.1) v2 = xey + y ey , v3 = ey , R (3.2) v2 =
x y 1 + ln y , v3 = y , R +1

(3.3) v2 = xy + y

, v3 = y , || > 1, = 2, R

(3.4) v2 = xy ln y + y 2 , v3 = y ln y , R (3.5) v2 = x 1 - y 2 + arcsin y , v3 = 1 - y 2 , R (3.6) v2 = exp(q )(x sin() + ), R = exp(q ), q > 0, R (3.7) v2 = ey , v3 = e (3.8) v2 = e
x+y x+ y

,R
+y

cos y , v3 = ex

sin y , > 0, (, ) = (0, 1)

(3.9) v2 = ex y cos y , v3 = ex y sin y (3.10) v2 = y , v3 = y , 1 < < , (, ) = (2, 3) (3.11) v2 = eay , v3 = ey , 0 < |a| < 1 (3.12) v2 = ch y , v3 = sh y (3.13) v2 = y ln y , v3 = y , = {0; 1} (3.14) v2 = y ey , v3 = ey (3.15) v2 = y 2 , v3 = ey (3.16) v2 = y ln2 y , v3 = y ln y (3.17) v2 = ey cos y , v3 = ey sin y , > 0 (3.18) v2 = y cos( ln y ), v3 = y sin( ln y ), > 0 (3.19) v2 = cos y , v3 = sin y

: 1. .. : , , , 2002, . 57, 1. . 3­44. 2. .. , . . 2004. . 75, 4. 3. V. K. Beloshapka, Moduli Space of Model Real Submanifolds, Russian Journal of Mathematical Physics, Vol. 13, No. 3, 2006, pp. 245­252. 4. .., . . ( 100- ), . . . . ., 2008, 1, 1, 63-67. 5. . . , . . , . , 4- C3 , , ., 72, 3, 2008. 6. V.K. Beloshapka, I.G. Kossovskiy, Classification of homogeneous CRmanifolds in dimension 4, J. Math. Anal. Appl. 374 (2011) 655­672. 7. . . , : , , 2012, . 279, . 20­30.