0 ()

() 514.764.2
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1 (, )
2 (-, )
On the Curvature Operator Spectrum of (Half)Conformally Flat Riemannian Metrics
E.D. Rodionov1, V.V. Slavskii2, O.P. Khromova1
1 Altai State University (Barnaul, Russia)
2 Yugra State University (Khanty-Mansiysk, Russia))
- . . , . , , .. .. , . . . , - . . -, . . , . , . , . . , .
: ,
() .
10.14258/izvasu(2015)1.1-19
An establishment of communication between various types of curvature and topology of a Riemannian space is important for the research of Riemannian manifolds. The sectional curvature is one of the special types of curvatures. Some of the most known examples are Hadamard Cartan's theorem, M. Gromov's theorem, the sphere theorem, the A.D. Alexandrov V.A. Toponogov's theorem of comparison of a triangle corners, the equations of A. Einstein's theory of relativity, and some other results. Generally, a study of Riemannian manifolds with restrictions on the sectional curvature is assumed to be complicated. Therefore, it would appear reasonable to consider the study in a class of homogeneous Riemannian spaces, and, in particular, in a class of Lie groups with left invariant Riemannian metrics. In this direction there are well-known M. Berger's, S. Alloff N. Wallach's research results and research results of some other mathematicians. Another natural restriction is a study of the sectional curvature and its operator in a class of conformally flat Riemannian metrics. This class of metrics allows convenient analytical representation, and the spectrum of the sectional curvature operator of such metrics is closely connected with the sectional curvature. In this paper, the sectional curvature operator spectrum of conformally flat Riemannian manifolds is investigated. Besides, the spectrum of the sectional curvature operator is investigated for the case of half conformally flat metric Lie groups.
Key words: spectrum of the curvature operator,
(half)conformally flat metrics.
* ( 2263.2014.1), ( 14.B25.31.0029), -
( : 1148), » ( 2014.312.1.4).
. [1]. , .
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() . () . , 4- () , () [5; 6].
1. .
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= + ,
()(, , ^ V) = (, Z)(, V)+ +(, V)(, Z) - (, V)(, Z) (, Z)(, V),
=
1

(, } 2 (1 2,1 2} =
= det(gx()), .
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(, , , V) = ((, ), V).
{1, 2,..., } , - . , (2). .
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(, Z) (, V) (, Z) (, V)
(, Z) (, V) (, Z) (, V)
+
+
(4)
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+ (/, )2 - /,/>-2/ , (5)

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2. 4- . = , {, [.]} . (.: [7]). (, } {, (, }} .
, , dim G = 4 [10].
4. g 4- G . 1 R 1.
. 4- G, W = 0, [10], , :
1) 4Ai
2) , Ai
3) ,9 Ai
4) ,0 Ai
5) Ag,7 0 Ai
6) A%i
7)A4a;6e
8) A4,i2
> 0, , > 0, > 0, > 0, ; > 0, (. : [10; 11]).
{1, 2, , 4} . , (2), . , 1. = ($ ), 4- . .
, , .. = -. -, .. = (., : [5]).
. . , [55]. , [7].
ck = 0 Vi, = 1, 2, 3;
ci c2,3 = C c2,3 = C;
c2 ci ;3 ci c2,4 = c3 = - ci ;2 = c2 = - ci ;4 = "c2,3 = AVl + M2, = /1 + M2;
ci ci ;3 = c2 = = c2,3 = A;
ci ci ;3 = c2 = = c2,3 = aL c2,3 = ci,3 = L;
ci ci ;4 = c2 = = c2,4 = c3 4 = L, a = =1;
ci ci ;4 = c2 = = c2,4 = c3 4 = aL,
c3,4 = c3,4 = L a = ;
li3 = 4 =
AD
ci = c2
c2,4 = ci,4
ci = c2 = c2,3 = ci,3 =
VA2 + B2 ' BB
-J A2 + B2'
c
1
4-
spec (72.) = { 12, 13, if , 2, 2, 34}
1 4Ai {0,0,0,0,0,0}
2 A3,6 Ai {0,0,0,0,0,0}
3 A3,9 Ai ( 2(1+2) 2(1+2) 0 2(1+2) 0 01 ^ Q
4 , Ai {-2, -2,0, -2,0,0}, > 0
5 ;7 Ai {a2L2, -a2L2, 0, -a2L2, 0,0}, a, L > 0
6 a = /3= 1 {L2, -L2, -L2, -L2, -L2, -L2}, L> 0
7 ?;6, a, L > 0 {-a2L2, a2L2, -a2L2, -a2L2, -a2L2, -a2L2}
8 A4,i2, > 0 {-(A2 + 2), - (A2 + B2), 0, - (A2 + B2), 0,0}
2
4-

fl spec (72.)
A'3 4,9 | (23+^45)2; (23-^145)^ ^ 2 fj = H > 0
Aa -^4,11 | (23+^45)22; (23-^45)22; ^^ ^^ 322 ^ 22 j, > Q, ff > 0.
. 4 . , , , -, 6 7 - .
. 4 - ( ) :
1) 4Ai , Ai;
2) A3 9 Ai, A3,3 0 Ai, Afj70 Ai A4,i2;
3) A^'f.
3. 4- . , , . , dim M = 4.
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Z
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(6)
+ - .
2. (4,) , .
4 = 4- {0, [.]}.
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1 , 4
2,
1
2 , 3
2 , 4
3
= 2, 2 = 2 4 = 3 4 =
4 = > 0 1 4 =
3 = 2 = ' 2 4 4 I
4
> 0, > 0.
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3 , 4
= ,
2 4
3 4
= , > 0, > 0
11 4 = 12 3 =
3 4
= ,
= 2, 22 4 = > 0, > 0 (.: [10]).
2 4
= 3 4 =
,
, -
11 4
2,
2 3
= 2 4 =
= 3 , 4 = > 0, = 1 {1, 2, 3, 4}.

, 1
2 = ^ - -'2, 3 = ^ - ^, 4 = = \ 2 + 3 + - 4, , 6, , <1 ( {' 2, 3, 4},
11 4 = 1.
2 3
2,
2 4
3 4
> 0,
4,11 11 4 = 2, 12 3 = 22 4 = 33 4 = = , > 0, > 0 -
= ,
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3 4
{1, 2, 3, 4}. ,
1 =
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+ ^ | | -/44, ,,,(1 ( -
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2 4
3 4

{1, 2, 3, 4}
1 2 ± 3 4 1 3 ± 4 2
/2 ' /2
1 4 ± 2 3
72
(7)
:

(111 0 0 ^14 0 0
0 ^22 0 0 ^25 0
0 0 ^33 0 0 ^36
^14 0 0 ^-44 0 0
0 ^25 0 0 ^-55 0
0 0 ^36 0 0 ^66^
12 + 21234 + 34
11
2

12 34
14
13 21324 +
24
-
13
24
25 =
22 14 + 21423 + 23 14 23
---, 36 =---,
\2 2^1234 + ^34 _
2 ; /-55
14 21423 + 23
13 + 21324 + 24 2 '

^22 = ^33 = ^-44 =
66 =---.
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5. 4- , . ^ 2.
. 4- - , . , 2, , :
, > 0, > 0. . 1
± (., : [7, . 614]).
2. 4- , . (7) -
2
3
4 =
5 =
6 =
14 23 + ^ 1
0,0, -,1,0,0
2^1423
14 23 ^ 1 0,0, ---, 1,0,0
21423
13 24 + ^ 2
0, ---,0,0,1,0
21324
13 24 ^ 2
0, -,0,0,1,0
2^1324
12 34 + ^ 3
21234 12 34 ^ 3
2
0, 0, 0, 0, 1
0, 0, 0, 0, 1
1234
1 = ~(\4 + 23 + ), 2 = 1(^14 + 23 - ), 3 = ^(13 + 24 + ^2), 4 = 1(^13 + 24 - ^2),
1
1
2
3
2
As = \{Kl2 + 34 + F3), 6 = X-{Kl2 + if34 - F3),
= /{23 ~ )2 + 4(142^ = ^ 13)2 + 4(1321)2,
^3= ^12 -^34)2+4(124)2.
1 0, , , , . . . 5 - ( ). , -.
, , . :
vi = ei e2, V2 = ei , V3 = ei e4, V4 = e2 , V5 = e2 e4, V6 = e4.
. g 4- G . jej ey}j<y , K (ej ey).
. g 4- G , W + = 0. [10] W = 0, 1 . W- =0. W = 0, 1 , 2. (7) (8) ,

( Ki2 0 0 0 0
0 K13 0 0 Rl324
0 0 Ki4 RI423 0
0 0 RI423 K23 0
0 Ri324 0 0 K24
\Rl234 0 0 0 0
Rl234\ 0 0
0 0
K34 /
(8)
, (7). .

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