0

514.765
0 *
.. 1, .. 2, .. 1
1 (, )
2 (-, )
Harmonicity of Weyl Tensor of Left-Invariant Riemannian Metrics on Four-Dimensional Nonunimodular Nondecomposable Lie Groups
E.D. Rodionov1, V.V. Slavskii2, O.P. Khromova1
1 Altai State University (Barnaul, Russia)
2 Yugra State University (Khanty-Mansiysk, Russia)
. . , - . 3 . - .
- . , . , . , , .. .
: , - , .
10.14258/^8(2014)1.2-10
In this paper, Riemannian manifolds with a harmonic Weyl's tensor are investigated. The problem of Riemannian manifolds classification with a harmonic Weyl's tensor is considered to be complicated. Therefore, it is natural to study it in a class of homogeneous Riemannian spaces and, in particular, in a class of Lie groups with a left invariant Riemannian metrics. When the dimension equals to three the Weyl's tensor is trivial. Therefore, there is a question of the Weyl's tensor being harmonic in metric Lie groups with dimension greater than three. Four-dimensional unimodular Lie algebras of Lie groups with a left invariant Riemannian metrics and a harmonic Weyl's tensor were studied by the authors of the paper.
In the paper we study four-dimensional non-unimodular nondecomposable Lie groups with a left invariant Riemannian metrics and a harmonic Weyl's tensor. Some methods with possible reduction of this problem to solution of the system of polynomial equations in Lie algebras are obtained. As a result of this classification, the Lie algebras with metric Lie groups that are not conformally flat, i.e. have non trivial Weyl's tensor, are distinguished.
Key words: Lie algebras and Lie groups, left-
invariant Riemannian metrics, harmonic Weyl tensor.
* ( 13-01-90716-__), ( 921.2012.1),
- » 2009-2013 . (. 02.740.11.0457) ( 8206, 2012-1.1-12-0001003-014).
1. . 4- - , [2, 3], .
4- , .. [5]. .. .. [4], , - , . , .
2. . (,) > 4; , , Z, V . V -, (, )Z = VYV*Z - V* VYZ + V[XIY] Z , = ^^ > (, V)) = ^() . -, = + , , ()(,,^) = (^)(, V) + (, V)(, Z) - (, V)(, Z) - (, Z)(, V).
- , {, [.]} . (. [1]). (, } {, (, }} .
{1, 2,..., } . [, , ] = ?,-, V^, = %, (, ,} = ,, {} , {, } .

-2
(, + , - , - ,). (1)
.
, , -
^ = + ), = ,,, |||| ||8||.
, ^, ^, , , (. [2]). ,

^
1] -
(9 - ) ( 1)( 2)
[8]

(2)
,, = 1, + , + ^ + 1,1 .
1. -, (. [3]), () .
[1,6], .
1. (, ) > 4 - , = 0.
2. , .
3. , , .. ^() = 0, , () = [, ], , .
. [4]. (, } (, }- , 1.
3. . 4- , - .
. - . = 0 , 2.
. 1.
2, = 0, = -2. - . : '1212 = '3434, '1213 = -'2434, ^^1223 = '1434, '1313 = '2424, '1323 = -'1424,

1
1


Aa 4,2 c\ 4 = aL, c\ 4 = A(a 1 )L, 4 = C3 4 = L, C3 4 = CL, 4,a = ( (a - 1) - AC)L , L > 0, a ^ 0, -2
A4j3 cj 4 ^ L, C2 4 ^ AL, c^ 4 ^ BL, c^ 4 ^ CL C,L> 0
a4j4 ci 4 = c2,4 = ,4 = , C2 4 = AL,c\ ^ = BL, Cg ^ = CL, A,C,L> 0
4,5 c\ 4 = L, c\ 4 = A(a 1 )L, Cg 4 = C{a )L, 4 = aL, = {AC{a - 1) + B( - 1 ))L, 4,4 = B, -1 < a < < 1 L > 0, a ^ 0, a + ^-1
4,6 4,4 = °B1 4,4 = AB, 4,4 = c3,4 = B, 4,4 = c3,4 = BL, Cg 4 ^ CL , L > 0, a ^ 0, >
A4,7 Ct 4 - 2A, C<2 g - , C<2 4 - C^', C<2 4 - -A, Cg 4 - Cg 4 - , Cg 4 A, , F > 0
A'3 4,9 c} 4 = A( + 1), 4 g = B, 4 4 = C, c2 4 = A, 4 4 = 44 = ^(1-/3), C3j4 = A/3 A, > 0, -1 < < 1
Aa 4,11 c| 4 = 2Aa, c^g = B, 4,4 = C, 4,4 = Aa, 4,4 = -AD, 4,4 = F, c2' A r3 Aa c3,4 D ' 3,4 A,B,D,a> 0
A4,12 4,3 = c2,3 = A, 4,4 = c2,4 = B, 4,4 = C, 4,4 = B, 4,4 = B, c3,4 = ^ A,D > 0, < 0
W1414 = W2323, (1) :
W1212 = (1/6)(A2L2a2 - 4L2B2A - 2L2B2a2
- 2A2L2a + 4L2B2a + A2L2 - 2L2B2 A2 + a2L2
- aL2 + 4L2B2aA - 2L2B2 - 2C2L2), W1213 = (1/4)(AL2Ba2 - 4AL2Ba - 2A2L2aB + 2AL2B + 2A2L2B - aL2C + 2CL2),
W1223 = (1/4)(3L2Ba - L2B - AL2B - AL2Ca + AL2C - 2L2Ba2 + 2AL2Ba), W1313 = (1/6)(-A2L2a2 + 2L2B 2A + L2B2a2 + 4A2L2a - 2L2B2a - 2A2L2 + L2B2A2 + a2L2 - aL2 - 2L2B2aA + L2B2 + C2L2), W1323 = (1/4)(-AL2a + AL2 + 2AL2a2), W1414 = (1 /6)(A2L2a2 + 2L2B2A + L2B2a2
- 2A2L2a - 2L2B2a + A2L2 + L2B2A2
- 2a2L2 + 2aL2 - 2L2B2aA + L2B2 + C2L2)
(2) :
divW114 = (1/8)L3(2aC2 + 4aB2 + 4aA2 - 11a2B2 - 11a2A2 + 6a3 B2 + 6a3A2 + A2B2 - 12a2 B2 A + 6aB2A2 + 10aB2A + 2AB2 - BACa2 + BA2Ca + B2 - BA2C + 8a - BAC + A2 + 2BACa - 8a2), divW124 = (1/8)L3(-4A3B2 + 12A3a - 8A2B2
- 12A3a2 + 4A3a3 - 12a2B2A + 16aB2 A2 - 4AB2 + 12aB2A + 16aA - 8a2A - 2Aa3 - 8A2B2a2
+ 4A3B2a + 4AB2a3 - 4BC - 4A3 + BCa2 - 6A + 3BCa - 4BAC - BACa), divWi34 = -(1/8)L3(6B + 6 AB + 4BA2 + 12B 3A + 4B3 - 16aB + 8a2B - 12B3A2a + 12B3a2 A
- 12B3a + 12B3A2 + 12B3a2 - 4B3a3 + 3Aa2C + 4A3B + 4B3A3 + 4BC2 - 3AaC - 4A2Ba3
+ 12A2Ba2 - 12A2Ba + 4A3Ba2 - 8A3aB + 2Ba3
- 2BAa2 - 4BC2a - 24B3aA + 4BC2A - 10BAa), divW2i4 = (L3/8)(6A3a - 2A3B2 - 4A2B2 - 6A3a2 + 2A3a3 - 6a2B2A + 8aB2A2 + 6aB2A - 2AB2
- AC2 + 10aA - 2a2A - 4Aa3 + AC2 a - 4A2B2a2 + 2A3B2a + 2AB2a3 - BC - 2A3 + 4BCa2 - 4A
- 3BCa - BAC - 4BACa),
divW224 = -(L3/8)(4aB2 - aC2 - 2a2B2 - 9a2A2 + 6a3A2 - 2A2B2 + 4aB2A - 4AB2 - 4BACa2
- 6C2 + 4BA2Ca - 2B2 - 4BA2C + 4a - 4BAC + 3A2 + 8BACa - 4a2),
divW234 = (L3/8)(4C3 - A2C - 5AB - 6ABa3 + 8CB2A + 2Ca2 - 4Ca + 4CB2 + 4CB2a2
- 8CB2a + 4CB2A2 - 8CB2aA - A2Ca2
+ 2A2Ca + 6A2Ba2 - A2Ba + 4BAa + 7BAa2
- 4C - 5BA2),
2


4,5 _ 3 1,4 2,4 3,4 ^ L>0
4,6 1,4 = olL, 34 = 3 4 = L
4,6 1,4 = 2,4 = 3,4 = PL, \ = Cg 4 = L 3 > 0,L > 0
'3 4,9 1,4 = 2,3 = 2, 2 4 = 4 = > 0
-^4,11 ci,4 = 2, = 2, <31 = | 4 = , \ = % 4 = > 0, > 0
A4,12 1,3 = 2,3 = \ = 2 4 = , cfA = \ = D > 0,D > 0
= -(3/8)(4 - + 4 + 22 + 63 - 10 + 22 - 632 + 632 + 23 - 63 + 632 + 632 - 233 + 22 + 2 3 + 2 33 + 22 + 43
- - 223 + 622 - 62 + 232
- 43 - 42 - 22 - 123 + 22 - 6),
divW324 = (3/8)(23 - 5 - 63 + 42 + 22 - 4 + 22 + 222 - 42 + 222 - 42 - 4 + 622
- 2 + 4 + 72 - 52),
divW334 = (3/8)(922 - 32 - 42 + 222
- 632 - 322 + 1222 - 622 - 62
- 62 - 62 - 32 + 32 - 32
- 32 - 4 - 3 + 22 + 6 + 42), divW412 = 3(1 - )(22( + 2 + 3) + 3
- 4(3 + 2 + + 22) - 3 + 2(2 + 1+ 22 + 2) + 3 + 422 - 2)/8, divW4l = (3/8)( + 2(1 + + 2) + 23 + 63 - 6 + 62 - 632 + 632
- 63 + 632 + 632 - 233 + 2 + 23 + 233 + 22 - 23 - 2
- 223 + 62(2 - ) + 232 + 22
- 43 + 2(2 - 2) - 123 - 4), divW423 = 3(-22 + 2 - 22 (1 + 2 + 2
- 2 + 2 - 2) + 22 - 22)/8.
divW = 0 , , , , :
1. , , , = 0, .
2. = = = 0, , = 0.
3. = = 0,, , =1.
4. , , = = 0, =1.
, {4 2, (, }} .
4 3. - . , ^^1212 = ^^3434, ^^1213 = -W2434, Wl223 = Wl434, Wl3l3 = ^^2424, ^^1323 = -Wl424, Wl4l4 = ^^2323, , (1)
^^1212 = (1/6)2(2 + 1) - (1/3)2(2 + 2), ^^1213 = (1/2)2 - (1/4)2, ^^1223 = -(1/4)2 - (1/2)2, W1313 = (1/6)2(1 + 2 + 2) - (1/3)22, ^^1323 = (1/2)2,
W14
14
(1/3)L2 + (1/6)L2(1 + B2 + C2),
(2)
divW114 = (1 /8)L3(6A2 + 2C2 BAC + 6B2), divW124 = (1/8)L3(4AB2 + 4A3 + BC 2A), divW134 = (L3/8)(4B(A2 + B2 + C2) 3AC 2B), divW224 = (1/8)L3(6A2 C2 4BAC ),
L3
divW2M = (A(C2 - 4 + 2A2 + 2B2) + 4BC), 8
3
divW234 = (C(2 - A2 + 42 + 4C2) - 6AB), 8
divW314 = (L3/4)(B3 + B (C2 + A2 2) AC ),
divW324 = -(1/4)L3(3AB - C - CB2 - C3), divW334 = -(3/8)L3(2B2 + C2 + BAC ),
L3
divW4i2 = + 2 " " 2A2 - 2)),
divW4i3 = (L3/8)(AC - 2B(1 + A2 + B2 + C2)), divW423 = (1/8)CL3(A2 - 2B2 - 2C2).
divW = 0 A, B, C, L, :
1. A, B, C G R,L = 0.
2. A = B = C = 0, L G R.
. , {A4 3, (, }} .
A44. - . , ^1212 = W3434, W1213 = -W2434, W1223 = W1434, Wi3i3 = W2424, W1323 = -W1424, W1414 = W2323. (1)
W1212 = (1/6)A2L2 - (1/3)B2L2 - (1/3)C2L2, W1213 = (1/2)AL2B + (1/4)L2C,
Wi223 = (1/4)L2B - (1/4)AL2C,
W1313 = (1/6)B2L2 - (1/3)A2L2 + (1/6)C2L2, W1323 = (1/4)L2A,
W1414 = (1/6)A2L2 + (1/6)B2L2 + (1/6)C2L2,
(2)
divW114 = (1/8)L3(7A2 + 7B2 + 2C2 - BAC), divW124 = (1 /8)L3(4AB2 + 4A3 - 6A + 5BC), divW134 = (L3/8)(4B(A2 + B2 + C2) - 3AC - 6B),
1. , , , = 0. 2. = = = 0, .
. , {4,4, (, }} .
^', = 0, -1 < < < 1, + = -1. , : W1212 = W3434, W1213 = -W2434, ^^1223 = ^^1434, Wl3l3 = ^^2424, Wl323 = -Wl424, ^^1414 = ^^2323. , , (1),
W1212 = (1/6)(2 - 2222 - 2222 + 2 + 422 + 22 - 22 + 22 + 222 + 2
- 22(2 + 22 - 22 + 2[ + 2 + 22]) + 4222 - 42 - 222 - 42 + 42 + 42),
W1213 = (1/4)(22(22 + - + 2)
- 22 - 22 + 22 - 2 + 2 + 222 - 22),
W1223 = (2/4)( - 2 - + -
- 2 + 2 + 2),
W1313 = (2/6)(22 + 22 + - 2 - 22 + - 22 - 222 + 42 + 22 - 22 + 22
+ A2C2 2 2A2C2 + 2BAC + B2 + 2 + 2
2ACaB 2ACBe 22 + 1),
W1323 = (1/4)(AL2ea + AL2 + 2AL2a 2AL2), W1414 = (L2/6)(C2e2 + C2 2 2 + 22 + A2 + A22 + 2 + B2 2 2B2e + A2 C2 2 + B2 + A2C2 2 2A2C2 2A2 + 2BAC + 2ACB
2ACB 2ACBe + 2 + ).
divW2i4 = (L3/8)(A(C2 6 + 2B2) + 5BC + 2A3), divWii4 = (1/8)L3(12BAC + 6A2 + 6B2 + 42
divW224 = (1/8)L3(2B2 + 9A2 - 7C2 - 4BAC), divW234 = (L3/8)(4C3 11 AB + C (4B2 6 A2)), divW314 = (L3/8)(2B(A2 3 + C2) 3AC + 2B3), divW324 = (1/8)L3(11AB + 6C 2CB2 2C3), divW334 = (1/8)L3(9B2 + 2A2 9C2 3BAC), divW4i2 = (1/8)AL3(C2 + 2B2 + 2A2), divW4i3 = (1/4)BL3(A2 + B2 + C2), divW423 = (1/8)CL3(A2 2B2 2C2).
divW = 0 A, B, C, L, :
+ 2 - 4 - 4 + 622 + 222 + 2
- 2 - 11 - 11 + 10
(2)
+ 2 - 2 + 222 - 222
- 22 + 22 + 22 - 122 + 22 + 222 + 42 + 622 - 122 - 1222
+ 622 + 6222 - 42 + 2 - 22), divW124 = (3/8)(2 - 42 - 82 - 2 + 1232 + + 822 + 82
- 43 + 2 + 162 + 42 + 4323
+ 8AB2 + 4A(C2 + 1)3 - 12A3C22 - 3AC2a2 + 4AB2a + 4A33 + 12A3a - 12A3a2 - 4Aa + 3AC2a - 4A3C2 - 4AC2a2 - 4Ca2B + Ca.B - 8AB2a + 4AB22a - 16A2Ca.B
- 8A2Ca2B + 4Aa - 4AB22 - CB2 - 4AB2
- CaB - 10Aa2 + 2Aa2 - 2A2 + 4A), divW134 = (1/8)L3(2B - 4A2B + 2AC + ACa
- 2Ba2 + 4B33 - 12B32 - 11ACa2 + 3AC + 4A3Ca3 - 12A3Ca2 + 12A3Ca - 4A2a2B
+ 8A2aB + 4ACa3 - 10B2 + 4Ba - 4Ba
- 4A3C + 4B - 24A2C2aB - 12A2C2a2B
- 4A3C3 + 24A2C2aB + ACa + 4A2a2B
- 8A2aB - 4ACa2 - 62AC + 12A3C3a + 4C3a3A - 4C3a2A - 4C32A + 4C2a2B
+ 4C32Aa + 12(B2ACa + A2C2a2B) - 4B3 + 62(ACa + 2B2ACa - 2B2AC) + 8C2 aB
- 24B2ACa + 24B2AC - 8C3a2A + 8C3aA
- 8C22aB + 12A2C2B + 4BA2 + 4C23B
- 12A3C3a2 - 12B2AC + 4A3C3a3 - 12A2C2B
- 4C2a2B - 4C22B),
divW214 = (L3/8)(4A - Ca2B - 4A2BC + 6A3C2a + 4C(CA - CAa - B2 + B + A2a2B + 2A2aB + A2B ) + Ca2B + 2A3C2 a3 + 2A3a3 + 2A(2B2 + C2a3 - 3A2C2a2 + C2a2 + B2a + a3 + 3A2a - 3A2a2 - A2) - AC2a - 3AC2a2 a^CB - CB + 2A(B2 - 2B2 - 4ACB - 2))
- 4A2Ca2B - 2AB22 - 2AB2 - 4CaB - A(4a2 + 2(C2 - C2a - 2a + 2) - 4 - 2Aa + 2A2C2)), divW224 =)(1/8)L3(4a - 4a2 - 6A2 - 4a2 - 3A2
- 6A2a2 + C2a2 - 4ABC(a2 - 2) - 4ACB + 8ACaB - 4ACaB + 8a2ACB - 8a2ACB
- 4A2C2a2 + 8A2C2a + 6A2C2a3 - 4B2a
+ 2B22a - 4A2C2 - 12C2a2 + 42a + 6C2a3 + C22 + 12A2a + 6A2C2a - 12A2C2a2 - 2C2a. + 6aC22 + 2B2a - 3A2a2 + 6A2a), divW234 = (L3/8)(4C3(A2a + a3) - 4Ca2 - 2Ca3 + 8C2Aa2B - 3A2C - 2A2a3C + 10A2Ca + 2AB - 8C3A2a2 + 4A2B + Aa2B + AaB + 4C3A2a3 - 12C3(a2 - 2a) + 4CB2(a - 3) + 8CB22 - 4CB2 - 4C3A2e + 6A2Ca
- Aa2B - AaB - 4A2aB - 8CB2a. - 2C + 4CB22 a - 8C2Aa2B - 4C3A2 a2 + 10Ca^2
+ 2Ca + 4Ca. - 6A2C - 6AB - 8C2Aa2B + 8C3A2a + 8C2A2 B + 8C2AaB - 8C2AB
- 3A2a2C - 2A2Ca2 - 43(C3 + C) - 4a2C^), divW314 = (1/8)L3(4B - 2A2B + 4AC + 2ACa
- 2Ba2 + 2B33 - 6B3(e2 - ) + 23B + 2Ba2
- 8ACa2 + 2A3Ca3 - 6A3Ca2 + 6A3Ca - 2B3
- 2A2a2B + 4A2aB + 2ACa3 - 4B2 + 4Ba
- 4Ba - 2A3C3 - 2A3C - 2B - 12A2C2aB
- 6A2C2a2B + 12A2C2aB + ACa + 2A2a2B
- 4A2a.B - ACa2 - 32AC + 6A3C3a
+ 2C3a3A - 2C3a2A - 2C3^2A + 2C2a2B + 2C3 2 Aa + 6B 2ACa + C2B (6A2 a2 + 4a) + 32 ACa + 6B22AC(a - 1) - 12B2^ACa + 12B2eAC - 4C3a2A + 4C3aA - 4C2^2aB + 6A2C2B + 2BA2 + 2C23B - 6B2AC + 2A3C3(a3 - 3a2) - 2C2B(3A2 + 2 + 2)), divW324 = (L3/8)(2C3a3 - 4Ca3 + 4C2Aa2B + 2C 3A2a - A2 C - 4A2a3C + 8A2Ca + AB
- 4C3A2a2 + A2B + 4Aa2B + 2AaB - 4Ca2 + 2C3A2a3 - 6C3(a2 - 2a) + 2CB2a - 2C3
- 2CB23 + 4CB22 - 2CB2 - 2C3A2
+ 2A2Ca - AB(4a2 + + a) - 4CB2a + 2CB22 - 4C2Aa2B - 2C3A2a2 + 4C3A2a + 4Ca2 - 2C + 2Ca + 4Ca - 6A2C - 6AB
- 4C2Aa2B + 4C2A2B + 4C2AaB - 4C2AB
- A2a2C + 2A2Ca2 - 2C33 + 2a2C), divW334 = -(L3/8)(12BAC - 4a2 + 6B2 - 4
- 2A2 + 3C2a2 - 3ACa2B + 3AC2B
- 1 ACB - 3ACaB + 6 AC aB + 9a2 AC B
- 9a2 ACB - 3A2C2a2 + 6A2C2 a + 6A2C2a3
- 6B2a + 3B22a - 3A2C2 - 12(B2 + C2a2) + 42a + 6A2C2 + 6C2 a3 + 3C22 + 42 + 6B 22
- 6C2(A2a + A2a2 + a - a2) - 2A2a2 + 4A2a + 3B 2a),
divW412 = (L3/8)(1 - a)(2A + 4A2BC - 4A3C2a
- 3C2Aa + 3C2A - 3CB2 + 3CB - 4A2CaB
- 4A2CB - 4AB2 + 2A3C2a2 + 2AC2a2 + 2A3
- 4A3a + 2A3a2 - AC2a + 3CaB + 4A2CaB + 2AB22 + 2AB2 - 3CaB - AC22 + 2Aa2
- 4Aa + 2A3C2 ),
divW413 = -(L3/8)(3ACa - 2B - 2A2B - 2AC + 2B33 - 6B32 + 6B3 + 23B - 3AC(a2 - )
+ 2A3Ca3 - 6A3C(2 - ) - 2AVB + 42 + 23 - 62 - 2A3(C3 + C) - 12A2C2
(2)
divWn4 = (1 /8)L3(BAC + 2 + A2C 2 + 6A2C2
+ 6 - 6A2C22 + 12A2C2 + 222 - 42 - ^^ - 3e2AC + 6A3C^
+ 2C33 - 2C32A - 2C^2A + 2C22B
32
32
22
+ 2C2 - BC3A + 6B2C2 - 8C22 - 4C + 8C 2 2 + 2C4 )/C2,
divW124 = (1/8)L3(4A3C2 - 10AC^ - 2AC2
+ 2C - + 6^ + + 4^ + 6AC22 + BC3 + 3CB + 4AB2C2 + 4BC3
- 2AC22 +4A)/C2,
divW134 = (1/8)L3(4A2BC - 3C^ - A - 4A
+ 2AC + 6ACB2^2 - 2 - 2) - 23 + 12^AC - 4C3^ + 4C3A - 4C22 + 6A2C2 + 2BA2 + 2C23 - 62AC + 2A3C33 - 6A3C32 - 2C2B(2 + 2 + 3A2)), divW423 = (3/8)( - )(2^2 + 2C2 + 2A^2 + 2C3A22 - 4C3 - 4C3A2 + 3AB + 3AB - 4C^ - 4A2C + 4AC2B - 4C - 3AB + 2C3A2 + 2C32 + 2C(A2 + 2 + 2 - 22) + 4C2AB - 4C2AB - 3AB + 2CB22).
divW4i2 =0 divW423 = 0 , = 1 = 1 divW = 0 . , divW = 0 :
A, , C, L R, = 1, = 1.
L > 0 , {A^jf, (, }} , 1,4 = 2,4 = 3,4 = L, L > 0.
A^'g, = 0, > 0. - . , W1212 = W3434, W1213 = -W2434, Wi223 = W1434, W1313 = W2424, W1323 = -W1424, W1414 = W2323, , (1)
L2 1 W1212 = -(2 - /3 - 22 + 2 + 1 - 22 +
6 C2
W1213 = (L2/(4C))(2BAC + (2 - )(C2 - 1)),
W1223 = -(1/4)L2(- + AC + 2),
L2 2
Wi3i3 = -j(B2 -/- 22 + 2 + 2 + 1 - ^),
W1323 = (L2/(4C))(- - AC + 2AC),
L2 1
Wi4i4 = -(2 + 2 + 2/ - 22 + 2 - 2 +
6 C2
+ 2(32 - 5 + 23 + 22 - - 2 ))/, divW2l4 = (1 /8)3(2 + - 422 - 62 + 4 + 2 + 422 - 2 + 232 + 3 + 2 2 2 + 43)/2,
divW224 = -3(3 - 3 + 6 + 322 - 22
- 422 + 622 - 43 - 222 + 422
- 4 - 64)/(82),
divW234 = -3(442 - 242 + 2 + 222 + 44 + 222 + 24 - 422 + 53 + 63 - 46
- 424 + 24 - 42)/(83),
divW314 = (1/8)3( - 2 - 4 + 232 + 24
- - 3 - 23 - 62 + 222
- 422 + 422)/2,
divW324 = -3(63 - 242 + 222 + 44 - 42 - 22 + 53 + 442 + 4 - 22(-22 + 22 - 4 + 22))/(83), divW334 = (1 /8)3( - 4 + 6 + 222 + 422 + 22 - 622 - 33 - 34
- 422 - 322 - 64)/2,
divW412 = -(1/8)3(222 - - 42 + 2 - 4 + + 222 - 2 + 232 + 222 + 33 - 33)/2, divW413 = -3(232 - 2 + 3 + 24
- 3 + 3 - 3 - 42 + 222
- + 222 + 222)/(82),
divW423 = (1/8)3(22 + 22 - 222 + 24
- 2 - 26 - 224 + 24)/3.
divW = 0. divW42 = 0
26 + (22 - 2 - 2)4 + (22 - 2 - 2)2 + 2 = 0.
, =1 = = 0.
2
divW = 0 ( ) =0. , = 0, > 0 = > 0 = 0, = 0.
= 1, ( > 0, > 0, = 0, > 0) divW42 =0 .
, {^ ', (, }} , : ] 4 = , 2 4 = 3 4 = , = 0, > 0,
1 4 = 2 4 = 3 4 = , 2 4 = 2 4 = , > 0, > 0.
2. , '6 1 4 = , ;] 4 = 3 4 = , = 0, > 0 - = 0 , . (., , [1]) . , ^ ' ]; 4 = 2 4 = | 4 = , 3 4 = ] 4 = , > 0, > 0 = 0.
4 , 7. - , : W1212 = W44, ^^1213 = W2434, ^^1214 = W2334, Wl223 = Wl434, ^^1224 = Wl334, ^^1234 = Wl324, Wl3l3 = ^^2424, ^^1314 = W2324, Wl323 = Wl424, ^^1414 = ^^2323. , (1)
^^1212 = (1/6)2 + (1/6)2 + (1/3)(2 2 2),
^^1213 = /2,
^^1214 = /2,
^^1223 = (3 + )/4,
^^1224 = /4,
^^1234 = ./2,
Wl3l = (1/6)(2 + 2 + 2) + (1/3)2 (1/3)2, ^^1314 = /2, ^^1323 = 3/4,
^^1414 = (1/6)(2 + 2 + 2) (2/3)2 (1/3)2,
(2)
divWll2 = (9/8),
divWll = (1/8) (9 ),
divWll4 = (1/8)[4(2 + 2) + 13(2 + 2)]
23 (1/8),
divW123 = (1/2)(42 2 2 2), divWl24 = (1/2) (2 + 2 + 2) (11/4)2 + (3/4),
divWl34 = (/2)(2 + 2 + 2 + 2)
(3/4) (11/4)2, divW2l2 = (1/2),
divW213 = (1/8)(2 82 + 22 + 22 + 22),
divW2l4 = 32 + (1/8)2 + (1/4)2
+ (9/8) + (1/4)(3 + 2),
divW223 = (1/8) (13 4),
divW224 = (1/4)(2 2) (15/8)2 + 2
+ (1/2) + 3,
divW234 = (17/8) (1/8)( 2 + 2 ) + (1/2)(2 + 3 2),
divW312 = (1/8)(2 + 82 22 22 22), divWl = (1/2),
divWl4 = (1/4)(2 + 2 + 2 + 2) 32
(5/8) ,
divW323 = (1/8) (13 ),
divW24 = (17/8) () (1/8) 2 (1/2)2
+ (1/4)(2 + 3),
divW4 = (1/4)(2 2) (15/8)2 + 3
(3/2)2 (3/8) ,
divW4l2 = (1/8)[3 2 (2 (1/2)2 + 2 + 2 + 2)],
divW4l = (1/4)(2 + 2 + 2 + 2 + 2) (1/8),
divW42 = (1/8) (2 + 22 + 22), divW424 = (1/8)(4 ), divW434 = (1/8) (4 3).
divW = 0 , , , , , :
1. , = 2, = = = 0.
2. , = 2, = = = 0.
, {4 , 7, (, }} .
4 9, 1 < < 1. - . ^^1212 ^^3434, ^^1213 = W2434, ^^1214 = W2334, Wl223 = Wl434, ^^1224 = Wl334, Wl3l3 = ^^2424, Wl3l4 =
W2324, ^^1323 = Wl424, Wl4l4 = W2323. (1), ,
W1212 = (1/6)(2 + 2 + 2 2 22 22
22 2 +42),
W1213 = (1/4)(2 + 2 + 2),
W1214 = -(1/2)BD,
W1223 = -(1/4)(AD + CF - CFe) - (1/2)ASA W1224 = -(1/4)BF (-1+ ), W1234 = (1/2),
W1313 = (1/6)(B2 + D2 + 2A2e - 2C2 + F22
+ f2 - 2F2),
W1314 = (1/2)BC,
W1323 = (1/4) + (1/2)AC,
W1324 = -(1/2)BA,
W1414 = -(2/3)2 + (1/6)(C2 + D2 - 2B2 + F2 + F22 - 2F2e).
(2)
divWU2 = -(1/8)ABD(5 + 4), divW113 = (1 /8)(5 + 4AC - FD + FDe), divW114 = (1 /8)(6AC2 - 83 + 2B2A + 7AD2 + 2AF2 + 72 - 2AF2 + 2AF23 - 2AF22 + 6AD^ + DCF - 32 - DCF + 22, divW123 = (B/2)(C2 + D2 + B2 - A2(1 + )2), divW124 = (1/8)(-4DAF - DAF2 - 142 + 4C(D2 - 2 - 22) + 5DAF + 4C3 + 4B2C), divW134 = (1/8)(4DF2 - 14DA^ + 4DF22 + 4D3
- 8DF2 - FAC - 1/2A22D + 4DB2 - 4A2(D) + 4DC2 - 4FCA + 5FCA2),
divW212 = (1/8)BAF( + 3)(1 - ), divW213 = -(1/8)B(4A2 - F2 - F22 + 2F2
- 2C2 + 4A2 - 2D2 - 2B2),
divW214 = (1/8)(CF22 - DAF - 4DAF2
- 14CA2 - 6CA22 + 5DAF + 2C3 + 2B2C + 2CD2 - 4A2C - 2CF2 + CF2),
divW223 = -(B/8)(9CA + 4AC - 4FD + 4FD), divW224 = (1/8)(2AD2 - 6AC2 - 2B2A - 9C2A + 7AF2 - 13AF2 + AF23 + 5AF22 - 4DCF + 8A^2 +4DCF),
divW234 = (1/8)^(B2 + C2) - 7DAC - 10DCA + 12FA^2 - 2FA2 - 4FD^ - 6A23F - 4F33
- 12F3( - 2) - F(C2 - 4D2 + B2 - 4F2 + 4A2)), divW312 = (1 /8)B(F2 + F22 - 2F2 + 4A22
+ 4A2 - 2C2 - 2D2 - 2B2), divW313 = (1/8)BAF ( + 3)(-1 + ), divW314 = (1/8)(3FCA2 - 2FAC - 14DA^
- 6A2D - 4A^2D - FCA + 2DB2 + 2DC2
+ 23 + 2^2 + 2F22 - 4^2), divW323 = -(1/8)(4 + 9 - CF + CF), divW324 = (1/8)^2 - 7 - 10 + 6FA22 + 4FA2 - 2FD2 - 423F - 2F33
- 6F3 + 6F32 - 6A2F + 2F2 - FB2 + 2F3), divW334 = (1/8)(83 - 92 - 9AF2 + 22 + 15AF2 - 3AF2(3 + 2) - 62 + 3CF
- 3CF - 22),
divW412 = (1/8)(3DAF - 3AF2 - 2CF2 + (F22 - 222 - 22 - 22 + F2 - 22)), divW413 = (1/8)(^ - 2FCA2 - 22 + 3FCA - 22 - 22 - 23 - 2
- 2F22 + 4F2),
divW424 = (1/8)(4 - CF + CF), divW434 = -(1/8)(4 - 3F + 3F), divW42 = (1/8)( - 1)(2FA22 + 2F32 - 4FA2
- 4F3 + 2F2 - FC2 + 2^ + 3 + 2F3).
divW = 0 , , , , F , :
1. , = = = F = 0, = 0.
2., F , = -2, = = 0, =1.
3., F , = 2, = = 0, = 1.
4. = = = = 0, F , =1.
5. = = = = F = 0, .
> 0 , { 9, (, }} , : 1 4 = 2 3 = , 2 4 = 3 4 = , = 2> 0, =1.
4 11, > 0. - , Wl2l2 = W434, ^^1213 = -W2434, ^^1214 = W2334, Wl223 = Wl434, ^^1224 = -Wl334, ^^1234 = -Wl324, Wl3l3 = ^^2424, ^^1314 = -W2324, Wl323 = -Wl424, ^^1414 = ^^2323. (1)
W1212 = (22 + 22 + 2222 - 2F22 - 22 + 22 + 24)/(62), ^^1213 = (1/2)CF,
^^1214 = (1/2), ^^1223 = (1/4)(3 + )/, ^^1224 = (2 1)/(4), ^^1234 = (1/2),
W1313 = (22 + 22 + 222 2 222 + 2
+ 22 224)/(62),
^^1314 = (1/2),
^^1323 = (1/4)( + 3),
W1414 = (22 + 22 222 + 2 222
+ 24 422 2)/(62).
(2)
divW112 = (1/8)(9 + ),
divW113 = (1/8) (9 )/,
divW114 = (42 2 + 3 + 1322
1622 + 13 22
+ 4 2 822 + 42 4)/(82), divWl2 = (1/2) (2 + 2 + 2 422), divW124 = (1/4)(2 2 + 2 2 1122 + 23 2 ( 23) + 32 (2 + 1))/, divW134 = (1/4)(2(22 + 22 1122
2 + 23) 32( + 3) + 22)/2, divW2l2 = (1/2)2(2 1)/, divW213 = (1/8) (2(1 4 822)
+ 22(2 + 2 + 2))/2,
divW214 = (52 + 2 + (22 + 22 + 22
2422 2 + 222)2 + 92)/(82), divW223 = (1/8)(13 + 2 4 )/, divW224 = (1522 2 22 + 1224
8 2 3 4 + 222 822
422)/(82),
divW234 = (2 4 17 22 22
422 2 222 + 42 + 4 22 + 422 4
2 24 226)/(83),
divW312 = (2 24 + 8222 222
2 22 222)/(82), divWl = (1/2) (2(2 1))/, divW314 = (24222 52 22 + 24 + 2 22 92 + 2 22 + 2 32 + 22)/(82),
divW323 = (1/8)(13 + 42)/, divW324 = ( 17 22 + 24 + 42
422 2 + 224 424 426 + 22
+ 2 22 + 4224)/(83),
divW334 = (15 22 222 + 122
824 + 3 + 222 822
422 + 43 )/(82),
divW412 = (2 + 2222 2 + (22 + 22 + 2 3 2)2 + 22 4
32)/(82),
divW413 = (222 2 2 22
2 4 + 222 + 32 + 2 22 + 2 32 +22)/(82),
divW423 = (2 + 22 + 22 + 222 222
223 22 22)/(83),
divW424 = (1 /8)(4 + 32)/,
divW434 = (1/8) (4 + 2 3 )/.
divWl =0 ( > 0, > 0, > 0, > 0) , =1. divW = 0 ( = 1) , , , , , :
1. = = = = 0, = 1, .
2. , = = = 0, = 1, = 0.
3. , = 2, = = 0, = 1, .
4. , = 2, = = 0, = 1, .
> 0 > 0 . , {^ 11, (, }} , : 11 4 = 2, 12 3 = , 2 4 = 3 4 = , 3 4 = , 4 = /, > 0, = 2, =1, > 0.
4 , . - . , : Wl2l2 = W434, Wl2l = W2434, ^^1214 = ^^2334, ^^1223 = Wl434, Wl224 = Wl334, Wl3l3 = ^^2424, ^^1323 = Wl424, Wl4l4 = W2323 (1),
W1212 = (1/6)(2 + 2 + 2 22 22, ^^1213 = (1/2) + (1/4)( + ), ^^1214 = 91/4), ^^1223 = (1/4)( + ) (1/2), ^^1224 = 1/4,
W1313 = (1/6)(2 22 + 2 + 2 ,
^^1323 = (1/2)( + ),
W1414 = (1/6)(2 + 2 + 2 22).
(2)
divWll2 = (1/8^ (3 + 5),
divW113 = (1/8)(-22 + 22 + 7F2 + 22,
divWll4 = (1/4)(2 +32 - 32)
- (1/8)^ - 7BF2 - 4FC), divWl2 = (5/8) ^
divW124 = (4F2 + F 2 + 43 + 22 + 22
- 4(2 + 2)( + ) + 23 + 5FB)/8, divW134 = ^ - - 7 - FC2 + 4(F2 - ^ - B2F + F3 + F2))/8, divW2l2 = -(1/8)(5 +3), divW2l = (5/8) ^
divW214 = (1/2)(2 - 2 - 2 - 2 + 3
- 2) + (1/8)(2 + 5FB) + (1/4)(3 + F2 + 2),
divW223 = (1/8)(22 - 22 + 2F2 + 72), divW224 = (1/4)^2 - 32 +32) + (1/8)(72 + 4F - FC), divW234 = (1/8)(4(2 - 2 + 2 + F2 - 2)
- 7BF - BFC - 2 + ), divWl = -(1/2)A(2BF + ),
divW314 = (1/4)^2 - 5 - + 2 F
+ F3 - 32F + F2),
divW2 = -(1/2)(2 + F),
divW324 = (22 - 10BF - 62 + 22
- 2BFC + 2F2 + 23)/8,
divW334 = -(3/8)(3BF2 + 32 + FC + F), divW412 = (22 - 23 + 23 - F2C - 2F2 + 22 + 2 - 22)/8, divW413 = (1/8)(FC2 - 3 - - 2^
- FC + 6A2F - 2F2 - 2F3 - 2FG2), divW414 = (1/8)A(8BF + 3 + 9), divW423 = (1/8)(62 - - 22 + 2
- 3BF - BFC - 22 - 2GF2 - 23, divW424 = (1 /8)(8 + 3FC + 9F), divW434 = (9/8) ^2 + 2).
divW2l = 0 ( > 0, < 0, > 0), F = 0 = 0. F = 0, divW313 = 0 < 0 , = 0. = 0, divW2 = 0 > 0 , F = 0. , divW = 0 , F . divW = 0 ( F = 0 = 0) , , , , F, , :
1. = , = F = = 0, = , .
2. = -, = F = = 0, = , .
3. , , , = -, F = = 0.
4. = V7>2 -52, , , = , = = 0.
5. = -7!)2 - 2, , , = , = = 0.
> 0 > 0 , {4 ,12, (, }} , ]; = 2 3 = , ]; 4 = 2 4 = , 2 ,4 = -; ,4 = -, > 0, > 0. .
3. ^ , , , > 0, = = 1; ^' = = 0, =1, > 0, = > 0, 4 ,12 > 0, = -, > 0, F = = 0.
4. , .

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