ON COMPOSITIONS IN EQUIAFFINE SPACE

Transcription

1 O COMPOSITIOS I EQUIAFFIE SPACE Iv Bdev Abstrct I euiffie spce projective tesors d E usig the coectio defie with the coectios, d For the spces A, A d A, with coefficiet of coectio, d respectively, we proved tht the ffior of copositio d the projective ffiors hve eul covrit derivtives It follows tht the coectio is euffie s well, d the coectios cse where E d A d re projective to ech other I the hve eul Ricci tesors, we fid the fudetl - vector ε I [4] copositios with structurl ffior re studied Spce cotiig copositios with syetric coectio d Weyl coectio re studied i [6] d [7] respectively Keywords: euiffie spce, copositio, Crtesi, Chebichevi, geodesic 00 Mthetics Subject Clssifictio: 5Bxx, 5B05, 5B0, 5B5 I Let i differetible ifold а the -vector а δ i i lε, Preliiries cosider field of ffior ε defies euiffie coectio by stisfyig where ε ε,,, is the fudetl -vector of the spce ε i i [, p50] Deote by E the spce i which is give The ffior, for which d [ ] [ ] 0 hold, defies the copositio,,,, i E [], []

2 Aiversry Itertiol Coferece REMIA00 Through y poit of the spce of copositios E there re two positios - P d P of the bse ifolds d Assue tht E hs itegrble structure 4 The projective tesors d re defied by []: d stisfy 5 δ,,,, δ,, δ, 0 The projective tesors trsfor the vectors of their positios ito theselves, d the vectors of trsversl positios ito zero vectors Ay vector v E hs the followig represettio where v v P v v v v v, d v P we c fid the projectios oto P d P [5] Fro 4 d 5, for v d For y covrit vector,, it follows tht 6 I [] the ivrit chrcteristics of specil copositio i ulti-diesiol spces re obtied fro ffior d projective tesors The copositio is Crtesi C C, if the positios P d P trslte prllely log y lie of the spce The copositio is Chebichevi Ch Ch, if the positios P d P trslte prllely log P d P respectively The copositio is geodesic G G, if the positios P d P trslte prllely log P d P respectively These coditios re chrcterized by: 7 0 C C 8 [ 0 Ch Ch ] 9 0 G G

3 0- Deceber 00, Plovdiv, Bulgri Copositio is C Ch Ch C, if the positios P d P trslte prllely log P P The copositio is C C -, if the positio P P trsltes prllely log y lie of the spce The copositio is Ch - - Ch, if the positio trsltes prllely log P P The copositio is G - G, if the positios P P trslte prllely log These coditios re chrcterized by: P P - P P C Ch Ch C 0 0 C C - ρ ρ 0 0 Ch - ρ ρ 0 0 G - -Ch -G 4 5 EQUIAFFIE SPACES OF COMPOSITIOS Cosider the followig coectios :,, 6 Coectio is clled the verge coectio of d [, p64] Suppose E, A, A d A,, d respectively Let derivtives i the spces A A, d A re spces with coefficiets of coectio E, A, A d A re spces of copositios Theore The covrit derivtives of the ffior E, A, A d A re eul Proof Accordig to 4 we hve,,, be the covrit, respectively By E, [], [] of copositio i

4 4 Aiversry Itertiol Coferece REMIA00 - Fro 5 d 6 it follows tht 7 Usig 5 we obti - Siilrly fro 5 d 6 we estblish, ,ie, 0, ie Fro 7 d 8 it follows tht eutios d ccordig to 6 we hve, ie 9 Corollry If oe of the spces Fro the lst E, A, A or A hs itegrbility of the structure, the the others lso hve itegrbility of the structure Corollry follows fro d 9 Corollry The projective tesors derivtive i G, E, A, A d A Corollry follows fro 4 d 9 Corollry If the copositio G Ch, Ch G, i oe of the spces C, C E, A A d is soe of C C,, A,, Ch, Ch hve eul covrit, Ch Ch, G G, or G, the it is of the se kid i the rest of these spces Corollry follows fro the ivrit chrcteristics, corollries d

5 0- Deceber 00, Plovdiv, Bulgri 5 Theore Coectios d re projective betwee ech other Proof Fro 4 d 5, tkig ito ccout 4 we hve δ δ, δ δ Fro 6 we obti δ δ, 4 4 Ad tkig ito ccout 6 we estblish 0 δ δ 4 Thus betwee d there exists projective correspodece The vector of the projective trsfortio is p Theore The spce A δ 4 δ 4, with coefficiet of coectio is euiffie Proof For the tesor of ffie trsfortio, fro 6 d 6 we hve: T δ δ 4 Deote by R d R the tesors of the curvture of E d A respectively The followig eutio holds [, p] ρ R R [ T T ρ [ T ] For the Ricci tesors ] R d R of E d A cotrctig the bove eulity log the idices d we obti R T R R [ T ] ρ ρ [ T ] T T T T T T ρ ρ ρ Thus tkig ito ccout d we estblish respectively, fter R R 4 4 The tesor R of the euiffie spce E is syetric [, p50], ie the right hd side of is syetric Thus the Ricci tesor R is syetric s

6 6 Aiversry Itertiol Coferece REMIA00 well Fro 9 it follows tht the coefficiets of coectio of A re syetric A spce with syetric coectio, hvig syetric Ricci tesor, the the spce is euiffie ie A is euiffie Exple Give coordite syste u,,, i E, we wt to fid is the fudetl -vector of the spce E, wheever Ricci tesors of E d A re eul Fro, for the coefficiets of coectio we hve: 4 0 Fro where we obti l u u u 4 Thus, tkig ito ccout, for the fudetl -vector of the spce estblish 4 u u u ε E we Refereces [] A orde, Affiely Coected Spces GRFML, Moskow, 976 i Russi [] A orde, Spces of Crtesi copositios Mthetics, 496, 7-8 i Russi [] A orde, G Tioffev Ivrit tests of the Specil copositios i ultivrite spces Mthetics, 897, 8-89 i Russi [4] K Yo, Affie Coectios i Alost Product Spce, Kodi Mth, Seir Reports, v,,959, -4 [5] Tiofeev G, Ivrit tests of the specil copositios i Weyl s spces Mthestics 976, [6] G Zltov, Oe Trsfortios of Coectios of Spces of Copositios vol54, 000, 9-4 [7] G Zltov, BTcrev, Trsfortio of spces of copositiosplovdiv Uiv Sci Works Mth 5007, 05- Iv Bdev Techicl Uiversity-Sofi, Brch Plovdiv 5 Tzko Djustbov Street 4000 Plovdiv e-il: ivbdev@bvbg