N. P. Mozhey Belarusian State University of Informatics and Radioelectronics NORMAL CONNECTIONS ON SYMMETRIC MANIFOLDS

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1 Òðóäû ÁÃÒÓ 07 ñåðèÿ ñ : -. N. P. Mozhey Belarusian State University of Inforatics and Radioelectronics NORMAL CONNECTIONS ON SYMMETRIC MANIFOLDS In this article we present a local classification of three-diensional syetric hoogeneous spaces allowing a noral connection. We have considered the case of the unsolvable Lie group of transforations. The local classification of hoogeneous spaces is equivalent to the description of the effective pairs of Lie algebras. We describe invariant affine connections together with their curvature and torsion tensors. We have studied the holonoy algebras of hoogeneous spaces and have found when the invariant connection is noral. Studies are based on the use of properties of the Lie algebras Lie groups and hoogeneous spaces and they ainly have local character. The peculiarity of techniques presented in the work is the application of purely algebraic approach as well as cobination of ethods of differential geoetry the theory of Lie groups and algebras and the theory of hoogeneous spaces. Key words: noral connection transforation group syetric space holonoy algebra... (. [] [] (. [] (. []. - - [4] [5] - ; M G G = G M. - ( M G ( G G G G M - G / G. g G g - G. ( gg g g. ( gg - Òðóäû ÁÃÒÓ Ñåðèÿ 07

2 4 Íîðìàëüíûå ñâÿçíîñòè íà ñèììåòðè åñêèõ ìíîãîîáðàçèÿõ g. - - G G G / G. ( g g - Λ : g gl( g - g g g - - = g / g. - ( G G G G G - G (g = s o gs o g G s o o. ( g g. - g.. - g = g + - ( g g. g = g + - ( g g [gg] g [g ] [ ] g. T InvT ( R InvT ( T ( R( y y = Λ = ( y Λ( y [ y] [ Λ( Λ( y ] Λ( [ y] * y g. h - Λ : g gl( ( g g gl ( V + [ Λ( g V ] + [ Λ( g[ Λ( g V ]] + + V {[ Λ( Λ( y] Λ([ y] y g}. a gl ( { Λ( g}. * h = a.. -. ( g g g. { e... e n } g (n = dig. - g e... en { u = en u = en u = en}. - d. n d. n. d n gl ( ( g g. Λ ( u Λ ( u Λ ( u R - R( u u R( u u R( u u - T T ( u u T ( u u T ( u u.. - g - g ( g {0} - :..5 e u u u e 0 u u 0 u u 0 e 0 u u e 0 0 u 0..5 e u u u e 0 u u 0 u u 0 e 0 u u e 0 0 u 0..6 e u u u e 0 u u 0 u u 0 e 0 u u e 0 0 u 0.9. e e u u u e 0 e u u u e e u u u 0 0 e 0 u u 0 e 0 e u u u 0 e e e e u u u e 0 e e 0 u u e e u 0 e e 0 u u 0 e e u u u 0 e 0 e u u 0 u e e 0..6 e e e u u u e 0 e e 0 u u e e u 0 e e 0 u u 0 e e u u u 0 e 0 e u u 0 u e e 0..7 e e e u u u e 0 e e 0 u u e e u 0 e e 0 u. u 0 e e u u u 0 e 0 e u u 0 u e e 0 Òðóäû ÁÃÒÓ Ñåðèÿ 07

3 Í. Ï. Ìîæåé g gl ( ( ( g g. g g -.. g = g + g = 0 [g g ] g [g ] [] g. g gl ( ( gg g g {0} g - :. ;. ;.9 y ;.9 ;.. y y z y z y [ u u ] = ae + α u [ u u] = β u [ u u] = γ u. β + γ 0 a = α = 0 ( gg -. β + γ = 0 a 0 α β + γ = 0 a = 0 α... [ e u] = u [ e u ] = u [ e u] = pe p. [ u u ] = ae + α u [ u u] = βu [ u u] = β u. α = 0 a 0 ( gg π 5(6 : g g π( e = e π ( u = a u π( u = = a u π ( u = u ( g6 g 6 su ( ( g6 g 6 sl (. ( gg -... [ e u] = 0 [ e u ] = pe [ e u ] = pe [ e u ] = qe u [ e u ] = u [ e u ] = pe [ e u ] = u + re [ e u ] = pe [ e u ] = u p 0 -. p = 0 : r 0 - a = = 0 r a > 0 r = 0 ( g g..6 ( - a < 0 = 0 r..7. di Dg di Dg6 di Dg di Dg7 g 6 - sl ( g 7 - su (... ( gg a g g a = g g g g a = g g ( - gl ( y y y. z ;.4 z y ;.5 y z. z z - - g g - : Òðóäû ÁÃÒÓ Ñåðèÿ 07

4 6 Íîðìàëüíûå ñâÿçíîñòè íà ñèììåòðè åñêèõ ìíîãîîáðàçèÿõ.4. e e e u u u e 0 e e u 0 u e e 0 e 0 u u e e e 0 u u 0 u u 0 u 0 e e u 0 u u e 0 e u u u 0 e e 0.4. e e e u u u e 0 e e u 0 u e e 0 e 0 u u e e e 0 u u 0 u u 0 u 0 e e u 0 u u e 0 e u u u 0 e e 0.5. e e e u u u e 0 e e u 0 u e e 0 e u u 0 e e e 0 0 u u u u u 0 0 e e u 0 u u e 0 e u u 0 u e e 0.5. e e e u u u e 0 e e u 0 u e e 0 e u u 0 e e e 0 0 u u. u u u 0 0 e e u 0 u u e 0 e u u 0 u e e gl (. - g gl ( ( gg - g g ( g g.4. h ( g - (0 e g ( h = e u ( = e u g ( h = e u (0 ( g ( ( h = ( [ u u] g ( h [ u u ] g ( h [ u u] g ( h - [ u u ] = = a e + α u [u u ] = a e + α u [u u ] = a e + + α u. p = a + α / 4. p = 0 ( gg π : g g π ( ei i = π( u = u ( α / e π ( u = u + ( α / e π ( u = u + ( α / e. p > 0 ( gg -.4. p < 0 ( gg di rg ( di rg ( di rg ( di rg ( ( g g ( g g ( g g. g - ( g sl( g ( g sl( sl( ( g g ( g g..5. ( g - [ u u ] = ae + α u [ u u] = ae α u [ u u] = ae + α u. p = / = a α / 4 a α / a = α. ( gg - π : g g π ( ei i = π ( u = u + ( α / e π( u = u ( α / e π ( u = u + ( α / e.. 4 > a α. ( gg.5. π : g g π ( ei i = π ( u = p( u + ( α / e π( u = p( u ( α / e π ( u = p( u + ( α / e.. 4 <. a α ( gg.5. π : g g π ( ei i = π ( u = p( u + ( α / e π( u = p( u ( α / e π ( u = p( u + ( α / e. rg ( {0} rg ( = rg ( = {0} g ( g sl( g ( g su( su ( g..9.. Λ( u = Λ( u = Λ ( u = R( u u.. V = = {[ Λ( Λ( y] Λ([ y] y g } ( -. Λ( g = Λ( g [ Λg ( V ] = i j Òðóäû ÁÃÒÓ Ñåðèÿ 07

5 Í. Ï. Ìîæåé 7 = [ Λg ( V ] = V Λg ( V. - a = Λ( g g h * = a g : q 0 r q q..7 0 q q p 0 0 q 0 p 0 q r r r 0 0 p 0 0 p 0 0 p 0 0 p p p 0 p p 0 r r 0 r r r.9. ( : ± 0..5 pq pq pq pq pr r p 0 pr r p qr r q qr r q 0 0 A H 0 H± A A 0 0 pr r p r p 0 0 pr + r p r p B C pr r p+ r p 0 0 pr r p r p D F 0 A = p p + p p H = p p + p p B = p r p r r p C = p r + p r r p D = p r p r + r p F = p r p r r p (000(000 ( q r (000(000 ( q ( 00 p q ( p r 00 ( 0 q r ( p ( p r p + r ( p r p r : p 0 p 0 p p 0 p p 0 p p Òðóäû ÁÃÒÓ Ñåðèÿ 07

6 8 Íîðìàëüíûå ñâÿçíîñòè íà ñèììåòðè åñêèõ ìíîãîîáðàçèÿõ p p 0 0 p 0 p g p 0 p p 0 0 p p p p 0 0 p 0 p 0 p p 0 p p ; : Òðóäû ÁÃÒÓ Ñåðèÿ 07 p p p p ± p p ± p ± p 0 p ± p p ± p 0 p ± p 00 0 p 0 ( ( ( 00 p ( 00 p ( 0 p0 ( p00.4. p p 0.4. p p 0 p 0 p p.4. p =.5. 0 p p.5. p p 0 p p p 0.5. p = : Chen B. Y. Geoetry of subanifoids // Pure and Appl. Math. New York: Marcel Dekker. 97. Vol. 0 no.. 08 p... // //

7 Í. Ï. Ìîæåé //. 07. : References. Kartan E. Rianova geoetriya v ortogonal no repere [Rieannian geoetry in an orthogonal frae]. Moscow Moskovskiy universitet Publ p. (In Russian.. Chen B. Y. Geoetry of subanifoids. Pure and Appl. Math. New York Marcel Dekker 97 vol. 0 no.. 08 p.. Kartan E. Geoetriya grupp Li i sietricheskie prostranstva. Sbornik rabot [The geoetry of Lie groups and syetric spaces. Collected works]. Moscow p. 4. Mozhey N. P. Noral connections on reductive hoogeneous spaces with an unsolvable transforation group. Doklady Natsional noy akadeii nauk Belarusi [Reports of the National Acadey of Sciences of Belarus] 06 vol. 60 no. 6 pp. 8 6 (In Russian. 5. Mozhey N. P. Canonical connections on three-diensional syetric spaces solvable Lie groups. Trudy BGTU [Proceedings of BSTU] 07 no. : Physical-atheatical sciences and inforatics pp. 8 (In Russian ( E-ail: ozheynatalya@ail.ru Inforation about the author Mozhey Natalya Pavlovna PhD (Physics and Matheatics Associate Professor Assistant Professor Software for Inforation Technologies Departent. Belarusian State University of Inforatics and Radioelectronics (6 P. Brovki str. 00 Minsk Republic of Belarus. E-ail: ozheynatalya@ail.ru Òðóäû ÁÃÒÓ Ñåðèÿ 07

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