{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Vessiot_Text" -1 256 "Times" 1 14 0 0 255 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 " Helvetica" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Vess_Ti tle2" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 128 0 64 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 4 0 1 0 1 0 2 2 0 1 }{PSTYLE "Vess_IO" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 3 30 1 0 2 2 0 3 }{PSTYLE "Vess_Title1" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 128 0 64 1 2 2 2 2 2 2 3 1 1 }2 1 0 0 10 10 3 6 3 30 2 2 0 1 }{PSTYLE "Example" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 258 "" 0 "" {TEXT -1 94 " \+ Vessiot Tutorial: Ricci tensors for invariant metrics " }} {PARA 256 "" 0 "" {TEXT 257 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 145 "In this tutorial we compute the Ricci/Einsteint tensor for a v ariety of invariant metrics on homogeneous spaces and cohomogeneity \+ 1 spaces. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 258 22 "Procedures Illustrated" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {HYPERLNK 17 "homogeneousRicci" 2 "homogeneousRicci" "" }{TEXT -1 3 " \+ , " }{HYPERLNK 17 "cohom1Ricci" 2 "cohom1Ricci" "" }{TEXT -1 3 " , " } {HYPERLNK 17 "spacetimeVectToAlgVect" 2 "spacetimeVectToAlgVect" "" } {TEXT -1 4 " , " }{HYPERLNK 17 "spacetimeMetricToAlgMetric" 2 "spacet imeMetricToAlgMetric" "" }{TEXT -1 1 " " }}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">" 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(Vessiot):with(Koszul): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "with(tensors):with(inva riant_metrics):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "with(Ves siot_library):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 " " {TEXT -1 1 " " }{TEXT 259 38 "A check on the homogenousRicci program " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 99 "We \+ take an example of a group action from Petrov which we treat as defi ning a homogeneous space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 256 83 "First we compute the Ricci tensor and scalar curv ature directly on the 3 manifold. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 86 "Then we compute use the homogeneousRicc i program and check that the 2 answers agree." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "DIRECT COMPUTATION" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "coord_init([x1,x2,x3,x4],[u],euc3):" }}}{EXCHG {PARA 0 "euc3>" 0 "" {MPLTEXT 1 0 45 "Gamma:=Lie_lib(Petrov,[32,10],[x1,x2,x 3,x4]):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%ds*This~group~action~is~id entical~to~Petrov~32.25~(epsilon=1),~however~the~transfomation~from~32 .10~to~32.25~(epsilon=1)~has~not~been~foundG" }}}{EXCHG {PARA 0 "euc3 \+ > " 0 "" {MPLTEXT 1 0 31 "coord_init([x1,x2,x3],[],euc3):" }}}{EXCHG {PARA 0 "euc3>" 0 "" {MPLTEXT 1 0 6 "Gamma;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&&%%D_x2G6#%!G,(*&-%$cosG6#%#x2G\"\"\"&%%D_x1G6#%!GF.F .*&&,$*&*&-F+6#%#x1GF.-%$sinGF,F.\"\"\"-FF?FAF.&%%D_x3G6#%!GF.F.,(*&F;F=&F06#%!GF.F@*&&,$*&*&F8F= F*F=F=F>F?F@FAF.&F%6#%!GF.F.*&&*&F*F=F>F?FAF.&FJ6#%!GF.F.,(*&-F<6#%#x3 GF.&F06#%!GF.F.*&&,$*&-F+F]oF=F>F?F@FAF.&F%6#%!GF.F.*&&*&*&F8F=FfoF.F= F>F?FAF.&FJ6#%!GF.F." }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 35 "basept:=[x1=Pi/2,x2=Pi/2, x3=Pi/2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'baseptG7%/%#x1G,$%#PiG#\"\"\"\"\"#/%#x2GF(/%#x3GF(" }}} {EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 43 "Y:=isotropy_subalgebra( Gamma,basept)[1][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG7&\"\"!F &\"\"\"F'" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 28 "D efine a convenient frame. " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc3>" 0 "" {MPLTEXT 1 0 62 "F1:=v_zip([-si n(x3),cos(x3)/sin(x1),-cos(x1)*cos(x3)/sin(x1)]," }}{PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 22 "[x1,x2,x3],plus,vect):" }}{PARA 0 "euc3 > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 75 " F2:=v_zip([cos(x3),sin(x3)/(sin(x1)),-(sin(x1)+cos(x1)*sin(x3))/(sin(x 1))]," }}{PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 22 "[x1,x2,x3],plus,vect) :" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 74 "F3:=v_zip([cos(x3),sin(x3)/(sin(x1)) ,(sin(x1)-cos(x1)*sin(x3))/(sin(x1))]," }}{PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 22 "[x1,x2,x3],plus,vect):" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 34 "dual_Fr:=dual_coframe([F1,F2,F3]):" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 19 "omega1:=dual_Fr[1]:" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 19 "omega2:=dual_Fr[2]:" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 19 "omega3:=dual_Fr[3]:" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "g1 and g2 are the inva riant metrics" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 26 "g1:=o mega1 &tensor omega1:" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 58 "g2:=(omega2 &tensor omega2) &plus (omega3 &tensor omega3):" }}} {EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc 3 > " 0 "" {MPLTEXT 1 0 29 "g:=v_zip([a,b],[g1,g2],plus):" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C):" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 23 "Ricci:=Ricci_tensor(R):" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 41 " DirectRicci:=linear_combo(Ricci,[g1,g2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,DirectRicciG7$,$*&*$)%\"aG\"\"#\"\"\"F,*$)%\"bG\"\"# F,!\"\"F+,$*&,&F*\"\"\"F/!\"\"F,F/F1!\"#" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 60 "DirectS:=coeff_list(Ricci_scalar(inverse_metric( g),R),[[]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(DirectSG,$*&,&%\"aG \"\"\"%\"bG!\"#\"\"\"*$)F*\"\"#F,!\"\"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 91 "Convert everything to L ie algebra data: the algebra, the isotropy, the frame, the metrics." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 33 "L:=vect_to_Lie_alg(Gamma,p32_10):" }}}{EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Lie~algebra:~p32_10G" }}}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 33 "h:=[v_zip(Y,[e1,e2,e3,e4],plus)];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"hG7#,&&%#e3G6#%!G\"\"\"&%#e4G6#%!GF+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 45 "M is a provisional basis for the complement. " }}{PARA 0 "" 0 "" {TEXT 256 51 "Use it to convert F1, F2, F3 to algebra vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 14 "M:=[e1,e2,e3]:" }}}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 77 "newM:=map(spacetimeVectToAlgVect, [F1,F2,F3],Gamma[1..3],M,[Pi/2,P i/2,Pi/2]):" }}}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 57 "gamma 1:=spacetimeMetricToAlgMetric(g1,[F1,F2,F3],basept);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gamma1G-%'matrixG6#7%7%\"\"\"\"\"!F+7%F+F+F+F," } }}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 57 "gamma2:=spacetimeMe tricToAlgMetric(g2,[F1,F2,F3],basept);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gamma2G-%'matrixG6#7%7%\"\"!F*F*7%F*\"\"\"F*7%F*F*F," }}} {EXCHG {PARA 0 "euc3 > " 0 "" {MPLTEXT 1 0 60 "LieAlgRicci:=homogeneou sRicci(newM,h,[gamma1,gamma2],[a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,LieAlgRicciG7$7$,$*&*$)%\"aG\"\"#\"\"\"F-*$)%\"bG\"\"#F-!\"\"F,, $*&,&F+\"\"\"F0!\"\"F-F0F2!\"#,$*&,&F+F6F0F8F-*$)F0\"\"#F-F2F8" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 6 "Ch eck:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 29 "DirectRicci - LieAlgRicci[1];" }}{PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 24 "DirectS -LieAlgRicci[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 256 "" 0 "" {TEXT 260 38 "A check on the cohom1Einstein program." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 50 "We cons ider invariant metrics from Petrov. " }}{PARA 0 "" 0 "" {TEXT 256 35 "First we do the direct computation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "p32_10 > " 0 "" {MPLTEXT 1 0 34 "coord_init([t],[q1,q2,q3,q4] ,RST):" }}}{EXCHG {PARA 0 "RST>" 0 "" {MPLTEXT 1 0 34 "coord_init([x1, x2, x3,x4],[],ST):" }}}{EXCHG {PARA 0 "ST>" 0 "" {MPLTEXT 1 0 44 "Gam ma:=Lie_lib(Petrov,[32,5],[x1,x2,x3,x4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%gq*This~group~action~is~identical~to~Petrov~32.21,~via ~the~transformation~x1=-y1~~~~~x2=y2~~~~~x3=y3~~~~~x4=y4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG7&&%%D_x2G6#%!G&%%D_x3G6#%!G,&&%%D_x1G 6#%!G!\"\"*&%#x2G\"\"\"&F'6#%!GF6F6,&&F06#%!GF3*&%#x3GF6&F+6#%!GF6F6" }}}{EXCHG {PARA 0 "ST>" 0 "" {MPLTEXT 1 0 30 "basept:=[x1=0,x2=0,x3=0, x4=0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'baseptG7&/%#x1G\"\"!/%#x2 GF(/%#x3GF(/%#x4GF(" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 43 "Y :=isotropy_subalgebra(Gamma,basept)[1][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG7&\"\"!F&!\"\"\"\"\"" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 13 "F1:=vect(x1):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 31 "F2:=v_zip([1],[x3],plus,vect): " }}}{EXCHG {PARA 0 "S T > " 0 "" {MPLTEXT 1 0 37 "F3:=v_zip([exp(-x1)],[x2],plus,vect):" }}} {EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 13 "F4:=vect(x4):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 18 "Fr:=[F1,F2,F3,F4];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#FrG7&&%%D_x1G6#%!G&%%D_x3G6#%!G*&-%$expG6#,$% #x1G!\"\"\"\"\"&%%D_x2G6#%!GF5&%%D_x4G6#%!G" }}}{EXCHG {PARA 0 "ST>" 0 "" {MPLTEXT 1 0 37 "dual_Fr:=dual_coframe([F1,F2,F3,F4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dual_FrG7&&%$dx1G6#%!G&%$dx3G6#%!G*&-%$ex pG6#%#x1G\"\"\"&%$dx2G6#%!GF3&%$dx4G6#%!G" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%$dx1G6#%!G&%$dx3G6#F'*&-%$expG6#%#x1G\"\"\"&%$dx2G6 #F'F0&%$dx4G6#F'" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 39 "omeg a1:=dual_Fr[1]: omega2:=dual_Fr[2]:" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 39 "omega3:=dual_Fr[3]: omega4:=dual_Fr[4]:" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 45 "g1:=symmetrize(omega1 &tensor o mega1,[1,2]):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 48 "g2:=sym metrize((omega2 &tensor omega3) ,[1,2]):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 46 "g3:= symmetrize(omega1 &tensor omega4, [1,2]) :" }}{PARA 0 "ST > " 0 "" {MPLTEXT 1 0 45 "g4:=symmetrize(omega4 &tens or omega4, [1,2]):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 14 "[g 1,g2,g3,g4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*&&%$dx1G6#%!G\"\"\" &F&6#%!GF),&*(&,$-%$expG6#%#x1G#F)\"\"#6#%!GF)&%$dx2G6#%!GF)&%$dx3G6#% !GF)F)*(F/\"\"\"&F>6#%!GF)&F:6#%!GF)F),&*(&F5F7F)&F&6#%!GF)&%$dx4G6#%! GF)F)*(FKFB&FP6#%!GF)&F&6#%!GF)F)*&&FP6#%!GF)&FP6#%!GF)" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 65 "g:= v_zip( [q4(x4), q3(x4), q2(x 4), q1(x4)], [g4,g3,g2,g1],plus):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 256 64 "Redefine F4 so that is it normal to the orbit wit h respect to g." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 42 "V:=v_zip([a1,a2,a3,1],[F1,F2,F3,F4],plus): " }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 245 "eq1:=contract_indic es2(g, F1&tensor V,[[1,1],[2,2]]):\neq2:=contract_indices2(g, F2&tenso r V,[[1,1],[2,2]]):\neq3:=contract_indices2(g, F3&tensor V,[[1,1],[2,2 ]]):\n\nS:=solve( coeff_set(eq1) union coeff_set(eq2) union coeff_set( eq3), \{a1,a2,a3\}):\n\n" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 27 "newF4:=helmsimp(subs(S,V)):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 44 "new_dual_Fr:=dual_coframe([F1,F2,F3,newF4]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,new_dual_FrG7&,&&%$dx1G6#%!G\"\"\"* &*&-%#q3G6#%#x4GF+&%$dx4G6#%!GF+\"\"\"-%#q1GF0!\"\"#F+\"\"#&%$dx3G6#%! G*&-%$expG6#%#x1GF+&%$dx2G6#%!GF+&F36#%!G" }}}{EXCHG {PARA 0 "ST>" 0 " " {MPLTEXT 1 0 55 "new_omega1:=new_dual_Fr[1]: new_omega2:=new_dual_Fr [2]:" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 55 "new_omega3:=new_ dual_Fr[3]: new_omega4:=new_dual_Fr[4]:" }}}{EXCHG {PARA 0 "ST > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 57 "ne w_g1:=symmetrize(new_omega1 &tensor new_omega1,[1,2]):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 60 "new_g2:=symmetrize((new_omega2 & tensor new_omega3) ,[1,2]):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 58 "new_g3:= symmetrize(new_omega1 &tensor new_omega4, [1 ,2]):" }}{PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 57 "new_g4:=symmetrize( new_omega4 &tensor new_omega4, [1,2]):" }}{PARA 0 "ST > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 23 "Ricci:=Ricci_tensor(R):" }}} {EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 59 "Ricci1:=linear_combo(Ricc i, [new_g1,new_g2,new_g3,new_g4]):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 55 "S:=coeff_list(Ricci_scalar(inverse_metric(g),R), [[]] ):" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 21 "change_frame_to(RST):" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 56 "DirectRicci:=simplify(diff_to_j et(subs(x4=t,Ricci1),2)):" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 47 "DirectS:=simplify(diff_to_jet(subs(x4=t,S),2)):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 86 "Do \+ everything to Lie algebra data: the algebra, the isotropy, the frame, the metrics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 32 "L:=vect_to_Lie_alg(Gamma,p32_5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7$7%%(Lie_algG%&p32_5G7#\"\"%7$7$7%\"\"\"\"\"$F.F .7$7%\"\"#F*F2F." }}}{EXCHG {PARA 0 "ST > " 0 "" {MPLTEXT 1 0 16 "Lie_ alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3Lie~algebra:~p32_5G " }}}{EXCHG {PARA 0 "p32_5 > " 0 "" {MPLTEXT 1 0 33 "h:=[v_zip(Y,[e1,e 2,e3,e4],plus)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG7#,&&%#e3G6# %!G!\"\"&%#e4G6#%!G\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 96 "M is a provisional basis for the complem ent. Use it to comvert F1, F2, F3 to algebra vectors." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "p32_5>" 0 "" {MPLTEXT 1 0 14 " M:=[e1,e2,e3]:" }}}{EXCHG {PARA 0 "p32_5 > " 0 "" {MPLTEXT 1 0 70 "new M:=map(spacetimeVectToAlgVect, [F1,F2,F3],Gamma[1..3],M,[0,0,0,0]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%newMG7%,$&%#e3G6#%!G!\"\"&%#e2G6#% !G&%#e1G6#%!G" }}}{EXCHG {PARA 0 "p32_5>" 0 "" {MPLTEXT 1 0 25 "restri cted_ad(h[1],newM);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7 %\"\"!F(F(7%F(!\"\"F(7%F(F(\"\"\"" }}}{EXCHG {PARA 0 "p32_5 > " 0 "" {MPLTEXT 1 0 57 "gamma1:=spacetimeMetricToAlgMetric(g1,[F1,F2,F3],base pt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gamma1G-%'matrixG6#7%7%\"\" \"\"\"!F+7%F+F+F+F," }}}{EXCHG {PARA 0 "p32_5 > " 0 "" {MPLTEXT 1 0 57 "gamma2:=spacetimeMetricToAlgMetric(g2,[F1,F2,F3],basept);" }} {PARA 0 "p32_5 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gamma2G-%'matrixG6#7%7%\"\"!F*F*7%F*F*#\"\"\"\"\"#7%F*F,F*" } }}{EXCHG {PARA 0 "p32_5 > " 0 "" {MPLTEXT 1 0 27 "beta1:=matrix(1,3,[1 ,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&beta1G-%'matrixG6#7#7%\" \"\"\"\"!F+" }}}{EXCHG {PARA 0 "p32_5 > " 0 "" {MPLTEXT 1 0 21 "change _frame_to(RST);" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 86 "LieA lgRicci:=cohom1Ricci(newM,h,[beta1],[gamma1,gamma2],q4[0],[q3[0]],[q1[ 0],q2[0]] ):" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 45 "simplify(DirectRicci[1] - LieAl gRicci[1][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 45 "simplify(DirectRicci[2] - LieAl gRicci[1][2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 46 "simplify(DirectRicci[3] -2*LieA lgRicci[2][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 42 "simplify(DirectRicci[4] - LieAl gRicci[3]);" }}{PARA 0 "RST > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 23 "DirectS-LieAlgRicci[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT 261 42 "The reduced equations of Wang and Dancer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 95 "We check the reduced \+ equations of Dancer and Wang, CMP(208) 225-243(1999). Eq 3.1, 3.2, 3.3 ." }}{PARA 0 "" 0 "" {TEXT 256 60 "We take d1 =3 and d2=6 which is one of the integrable cases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 15 "g:='g'; S:='S';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG F$" }}}{EXCHG {PARA 0 "RST > " 0 "" {MPLTEXT 1 0 27 "coord_init([t],[f ,g],euc2):" }}}{EXCHG {PARA 0 "euc2>" 0 "" {MPLTEXT 1 0 40 "gamma1:=li nalg[diag](1,1,1,0,0,0,0,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g amma1G-%'matrixG6#7+7+\"\"\"\"\"!F+F+F+F+F+F+F+7+F+F*F+F+F+F+F+F+F+7+F +F+F*F+F+F+F+F+F+7+F+F+F+F+F+F+F+F+F+F.F.F.F.F." }}}{EXCHG {PARA 0 "eu c2>" 0 "" {MPLTEXT 1 0 40 "gamma2:=linalg[diag](0,0,0,1,1,1,1,1,1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gamma2G-%'matrixG6#7+7+\"\"!F*F*F* F*F*F*F*F*F)F)7+F*F*F*\"\"\"F*F*F*F*F*7+F*F*F*F*F,F*F*F*F*7+F*F*F*F*F* F,F*F*F*7+F*F*F*F*F*F*F,F*F*7+F*F*F*F*F*F*F*F,F*7+F*F*F*F*F*F*F*F*F," }}}{EXCHG {PARA 0 "euc2>" 0 "" {MPLTEXT 1 0 21 "beta1:=matrix(1,9,0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&beta1G-%'matrixG6#7#7+\"\"!F*F* F*F*F*F*F*F*" }}}{EXCHG {PARA 0 "euc2>" 0 "" {MPLTEXT 1 0 86 "Ricci:=c ohom1Ricci([],[],[beta1],[gamma1,gamma2],1,[0],[f[0]^2,g[0]^2],[Av/3,A w/6],S):" }}}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 30 "expand(Ric ci[1][1]/(-f[0]^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#AvG\"\" \"*$)&%\"fG6#\"\"!\"\"#F&!\"\"#!\"\"\"\"$*&&F*6#\"\"#F&F)F.\"\"\"*&*$) &F*6#F6F5F&F&*$)F)\"\"#F&F.F5*&*&F:F6&%\"gGF;F6F&*&&FBF+\"\"\"F)\"\"\" F.\"\"'" }}}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 30 "expand(Ricc i[1][2]/(-g[0]^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#AwG\"\"\" *$)&%\"gG6#\"\"!\"\"#F&!\"\"#!\"\"\"\"'*&*$)&F*6#\"\"\"\"\"#F&F&*$)F) \"\"#F&F.\"\"&*&&F*6#F8F&F)F.F7*&*&&%\"fGF6F7F5F7F&*&F)\"\"\"&FCF+\"\" \"F.\"\"$" }}}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 18 "expand(-R icci[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&&%\"fG6#\"\"#\"\"\"&F &6#\"\"!!\"\"\"\"$*&&%\"gGF'F)&F1F+F-\"\"'" }}}{EXCHG {PARA 0 "euc2 > \+ " 0 "" {MPLTEXT 1 0 17 "expand(Ricci[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&*$)&%\"fG6#\"\"\"\"\"#\"\"\"F,*$)&F(6#\"\"!\"\"#F,! \"\"!\"'*&*$)&%\"gGF)F+F,F,*$)&F9F0\"\"#F,F3!#I*&*&F'F*F8F*F,*&F<\"\" \"F/\"\"\"F3!#O*&&F(6#F+F,F/F3F4*&&F9FGF,F " 0 "" {MPLTEXT 1 0 7 "R:='R';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RGF$" }}}{EXCHG {PARA 0 "euc2 > " 0 "" {MPLTEXT 1 0 27 "Lie_alg_init(create_gl(6)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#ijG% " 0 "" {MPLTEXT 1 0 40 "coord_frame([x1,x 2,x3,y1,y2,y3],[u],E6):" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 34 "g:=canonical_flat_metric(6,0,bas):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "J:=canonical_complex_structure(bas):" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz1:=v_zip([1,I],[x1,y1],plus,form):" }} {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 36 "dz2:=v_zip([1,I],[x2,y2],plus,fo rm):" }}{PARA 0 "E6>" 0 "" {MPLTEXT 1 0 36 "dz3:=v_zip([1,I],[x3,y3],p lus,form):" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 29 "nu:=dz1&wedge dz2 &wedge \+ dz3:" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 57 "nuR:= (1/2) &mult ( nu \+ &plus Vessiot_map(conjugate,nu));" }}{PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 57 "nuI:=(I/2) &mult ( nu &minus Vessiot_map(conjugate,nu));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nuRG,**,&%$dx1G6#%!G\"\"\"%#^~GF+&% $dx2G6#%!GF+%#^~GF+&%$dx3G6#%!GF+F+*,&F(6#%!GF+%#^~GF+&%$dy2G6#%!GF+%# ^~GF+&%$dy3G6#%!GF+!\"\"*,&F.6#%!GF+%#^~GF+&%$dy1G6#%!GF+%#^~GF+&FA6#% !GF+F+*,&F36#%!GF+%#^~GF+&FK6#%!GF+%#^~GF+&F<6#%!GF+FD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nuIG,**,&%$dx1G6#%!G\"\"\"%#^~GF+&%$dx2G6#%!GF+ %#^~GF+&%$dy3G6#%!GF+!\"\"*,&F(6#%!GF+%#^~GF+&%$dx3G6#%!GF+%#^~GF+&%$d y2G6#%!GF+F+*,&F.6#%!GF+%#^~GF+&F=6#%!GF+%#^~GF+&%$dy1G6#%!GF+F6*,&FO6 #%!GF+%#^~GF+&FB6#%!GF+%#^~GF+&F36#%!GF+F+" }}}{EXCHG {PARA 0 "E6 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 51 "su3_data:=create_gl_subalgebra(gl6R,[g,J,nuI,nuR]);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%)su3_dataG7*,*&%$e12G6#%!G\"\"\"&%$e21G6#%!G! \"\"&%$e45G6#%!GF+&%$e54G6#%!GF0,*&%$e13G6#%!GF+&%$e31G6#%!GF0&%$e46G6 #%!GF+&%$e64G6#%!GF0,*&%$e14G6#%!GF+&%$e36G6#%!GF0&%$e41G6#%!GF0&%$e63 G6#%!GF+,*&%$e15G6#%!GF+&%$e24G6#%!GF+&%$e42G6#%!GF0&%$e51G6#%!GF0,*&% $e16G6#%!GF+&%$e34G6#%!GF+&%$e43G6#%!GF0&%$e61G6#%!GF0,*&%$e23G6#%!GF+ &%$e32G6#%!GF0&%$e56G6#%!GF+&%$e65G6#%!GF0,*&%$e25G6#%!GF+&FP6#%!GF0&% $e52G6#%!GF0&FX6#%!GF+,*&%$e26G6#%!GF+&%$e35G6#%!GF+&%$e53G6#%!GF0&%$e 62G6#%!GF0" }}}{EXCHG {PARA 12 "" 1 "" {XPPMATH 20 "6#7*,*&%$e12G6#%!G \"\"\"&%$e21G6#F(!\"\"&%$e45G6#F(F)&%$e54G6#F(F-,*&%$e13G6#F(F)&%$e31G 6#F(F-&%$e46G6#F(F)&%$e64G6#F(F-,*&%$e14G6#F(F)&%$e36G6#F(F-&%$e41G6#F (F-&%$e63G6#F(F),*&%$e15G6#F(F)&%$e24G6#F(F)&%$e42G6#F(F-&%$e51G6#F(F- ,*&%$e16G6#F(F)&%$e34G6#F(F)&%$e43G6#F(F-&%$e61G6#F(F-,*&%$e23G6#F(F)& %$e32G6#F(F-&%$e56G6#F(F)&%$e65G6#F(F-,*&%$e25G6#F(F)&FF6#F(F-&%$e52G6 #F(F-&FL6#F(F),*&%$e26G6#F(F)&%$e35G6#F(F)&%$e53G6#F(F-&%$e62G6#F(F-" }}}{EXCHG {PARA 0 "gl6R > " 0 "" {MPLTEXT 1 0 62 "so2_data:=[(k&mult s u3_data[3]) &plus (l &mult su3_data[7])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)so2_dataG7#,.*&%\"kG\"\"\"&%$e14G6#%!GF)F)*&%\"lGF)& %$e25G6#%!GF)F)*&,&F/!\"\"F(F6F)&%$e36G6#%!GF)F)*&F(\"\"\"&%$e41G6#%!G F)F6*&F/F<&%$e52G6#%!GF)F6*&,&F/F)F(F)F)&%$e63G6#%!GF)F)" }}}{EXCHG {PARA 0 "gl6R>" 0 "" {MPLTEXT 1 0 69 "L:=subalgebra_pair_to_Lie_algebr a_data_pair(su3_data,[so2_data],su3):" }}}{EXCHG {PARA 0 "su3 > " 0 " " {MPLTEXT 1 0 8 "h:=L[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG7# ,&*&%\"kG\"\"\"&%#e3G6#%!GF)F)*&%\"lGF)&%#e7G6#%!GF)F)" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 32 "M:=reductive_complement(h, [0]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7)&%#e1G6#%!G&%#e2G6#%!G,&& %#e3G6#%!G\"\"\"*&%#0~GF3&%#e7G6#%!GF3F3&%#e4G6#%!G&%#e5G6#%!G&%#e6G6# %!G&%#e8G6#%!G" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 19 "M:=ma p(helmsimp,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7)&%#e1G6#%!G& %#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G&%#e5G6#%!G&%#e6G6#%!G&%#e8G6#%!G" }}} {EXCHG {PARA 0 "su3>" 0 "" {MPLTEXT 1 0 13 "B:=Killing();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7*7*!#7\"\"!F+F+F+F+F+F+7*F+F *F+F+F+F+F+F+7*F+F+F*F+F+F+!\"'F+7*F+F+F+F*F+F+F+F+7*F+F+F+F+F*F+F+F+7 *F+F+F+F+F+F*F+F+7*F+F+F.F+F+F+F*F+7*F+F+F+F+F+F+F+F*" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 25 "A:=restricted_ad(h[1],M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7)7)\"\"!F*F*,&%\"lG\"\"\"%\"kG!\"\"F *F*F*7)F*F*F*F*,&F,F/F.!\"#F*F*7)F*F*F*F*F*F*F*7),&F,F/F.F-F*F*F*F*F*F *7)F*,&F,F-F.\"\"#F*F*F*F*F*7)F*F*F*F*F*F*,&F,F2F.F/7)F*F*F*F*F*,&F,F8 F.F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 60 "We find the invariant subspaces for the ad(h) action o n M." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 24 "linalg[eigenvectors](A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6)7%*&%\"IG\"\"\",&%\"lGF&%\"kG\"\"#F&F&<#-%'vectorG6#7) \"\"!F&F0F0,$F%!\"\"F0F07%,$F$F2F&<#-F-6#7)F0F&F0F0F%F0F07%*&F%\"\"\", &F(F*F)F&F&F&<#-F-6#7)F0F0F0F0F0F&F17%,$F:F2F&<#-F-6#7)F0F0F0F0F0F&F%7 %*&F%F;,&F(F2F)F&F&F&<#-F-6#7)F%F0F0F&F0F0F07%,$FHF2F&<#-F-6#7)F1F0F0F &F0F0F07%F0F&<#-F-6#7)F0F0F&F0F0F0F0" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 37 "gamma0:=linalg[diag](0,0,12,0,0,0,0):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 38 "gamma1:=linalg[diag](12,0,0,12, 0,0,0):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 38 "gamma2:=lina lg[diag](0,12,0,0,12,0,0):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 38 "gamma3:=linalg[diag](0,0,0,0,0,12,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gamma3G-%'matrixG6#7)7)\"\"!F*F*F*F*F*F*F)F)F)F)7)F* F*F*F*F*\"#7F*7)F*F*F*F*F*F*F," }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 49 "g0: =array_to_tens(gamma0,[[cov_jet,cov_jet],[]]):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 49 "g1:=array_to_tens(gamma1,[[cov_jet,cov_jet] ,[]]):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 49 "g2:=array_to_ tens(gamma2,[[cov_jet,cov_jet],[]]):" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 49 "g3:=array_to_tens(gamma3,[[cov_jet,cov_jet],[]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g3G,&*&&%'theta6G6#%!G\"\"\"&F(6#%! GF+\"#7*&&%'theta7G6#%!GF+&F26#%!GF+F/" }}}{EXCHG {PARA 0 "su3 > " 0 " " {MPLTEXT 1 0 88 "g3 := _VESSIOT([[tensor, su3, [[cov_jet, cov_jet], \+ []]], [[[6, 6], 12], [[8, 8], 12]]]);" }}{PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g3G,&*&&%'theta 6G6#%!G\"\"\"&F(6#%!GF+\"#7*&&%'theta8G6#%!GF+&F26#%!GF+F/" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 41 "map2(Lie_derivative, h[1],[g0,g 1,g2,g3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*(%#0~G\"\"\"&%'theta1 G6#%!GF&&F(6#%!GF&F$F$F$" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 80 "Ricci:=homogeneousRicci(M,h,[gamma0,gamma1,gamma2,gamma3],[lambd a,1/x,1/y,1/z]):" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 8 "" 1 "" {TEXT -1 33 "Error, invalid subscript selector" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 66 "eq1:=collect(subs(lambda= l^2/2,12*(Ricci[1][1]+Ricci [1][3])),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G,,*()%\"lG\"\" #\"\"\",(*&F(\"\"\"%\"kGF-F)*$)F.F)F*F-*$F'F*F-F-)%\"xGF)F*#F-F)*&*&,& *$)%\"zGF)F*!\"\"*$)%\"yGF)F*F-F-F3F-F**&)F>\"\"#F*F:\"\"\"!\"\"F;*&F' F*,&*&F0F*F9F*F-*&F=F*F'F*F-F-F4*&,&*&F=F*F:F-!\"'*()F>\"\"$F*F:F*F'F* F-F**&)F>\"\"#F*F:\"\"\"FCF;*&F:F*F3FCF;" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 46 "eq1:=collect(subs(lambda= l^2/2,12*(a+b)),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G,&%\"aG\"#7%\"bGF'" }}} {EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 44 "eq3:=collect(subs(lambda =l^2/2,12*(a+c)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G,,*&,(* $)%\"lG\"\"#\"\"\"!\"$*&F*\"\"\"%\"kGF/!\"'*$)F0F+F,F-F/)%\"xGF+F,F/*& ,&*&%\"zGF,%\"yG!\"\"!\"#\"\"'F/F/F5F/F/*&)F:F+F,F)F,F<*&)F9F+F,F3F,F- F9F=" }}}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 45 "eq2:=collect(su bs(lambda=l^2/2, 12*(b+c)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e q2G,.*&,(*&%\"lG\"\"\"%\"kGF*!\"%*$)F+\"\"#\"\"\"!\"#*$)F)F/F0F1F*)%\" xGF/F0F*%\"yG\"\"'*&)F6F/F0F3F0!\"$%\"zGF7*&)F;F/F0F.F0F:*&*&F6F*F;F*F 0F5!\"\"F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT 263 40 "The reduced equations of Berard Bergery" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT 264 38 "An example from Coquereaux and Jadcz yk" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "su3 > " 0 "" {MPLTEXT 1 0 46 "coord_frame([x1,y1,u1,v1,x2,y2,u2,v2],[ u],E8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/frame~name:~E8G" }}} {EXCHG {PARA 0 "E8>" 0 "" {MPLTEXT 1 0 36 "J:=canonical_complex_struct ure(bas):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 35 "K1:=canonic al_complex_structure(2):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 99 "K2:=[[tensor, E8, [cov_bas, con_bas, []]], [[[5, 7], -1], [[6, 8], -1], [[7, 5], 1], [[8, 6], 1]]]:" }}{PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 17 "K:= K1 &minus K2:" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 30 "g:=canonical_flat_metric(8,0):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 12 "V:=vect(v2):" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "E8 > " 0 "" {MPLTEXT 1 0 27 "Lie_ alg_init(create_gl(8)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&&%\"eG6#%#i jG% " 0 "" {MPLTEXT 1 0 47 "sp2_subalg:=create_gl_subalgebra(gl8R,[J,K,g]): " }}}{EXCHG {PARA 0 "gl8R > " 0 "" {MPLTEXT 1 0 49 "sp1_subalg:=create _gl_subalgebra(gl8R,[J,K,g,V]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "gl8R > " 0 "" {MPLTEXT 1 0 84 "sp2_sp1_data:=sub algebra_pair_to_Lie_algebra_data_pair(sp2_subalg,[sp1_subalg],sp2):" } }}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 21 "sp1:=sp2_sp1_data[2]: " }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 19 "N:=normalizer(sp1): " }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 8 "nops(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 27 "L:=reductive_complement(N):" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 8 "Show(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7& &%#e1G6#%!G&%#e3G6#F'&%#e5G6#F'&%#e7G6#F'" }}}{EXCHG {PARA 0 "sp2>" 0 "" {MPLTEXT 1 0 10 "Show(sp1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&% #e2G6#%!G&%#e4G6#F'&%#e6G6#F'" }}}{EXCHG {PARA 0 "sp2>" 0 "" {MPLTEXT 1 0 8 "Show(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(&%$e10G6#%!G&%#e9 G6#F'&%#e6G6#F'&%#e4G6#F'&%#e8G6#F'&%#e2G6#F'" }}}{EXCHG {PARA 0 "sp2> " 0 "" {MPLTEXT 1 0 15 "K:=[e8,e9,e10]:" }}}{EXCHG {PARA 0 "sp2>" 0 " " {MPLTEXT 1 0 25 "map(restricted_ad,sp1,K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%'matrixG6#7%7%\"\"!F)F)F(F(F$F$" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 17 "M:=[op(L),op(K)]:" }}}{EXCHG {PARA 0 "s p2 > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7)&%#e1G6#%!G&%#e3G6#F'&%#e5G6#F'&%#e7G6#F'&%#e8G6#F'&%#e9G6#F'&%$e1 0G6#F'" }}}{EXCHG {PARA 0 "sp2>" 0 "" {MPLTEXT 1 0 13 "B:=Killing();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7,7,!#C\"\"!F+F+F+ F+F+F+F+F+7,F+!#7F+F+F+F+F+F+F+F+7,F+F+F*F+F+F+F+F+F+F+7,F+F+F+F-F+F+F +F+F+F+7,F+F+F+F+F*F+F+F+F+F+7,F+F+F+F+F+F-F+F+F+F+7,F+F+F+F+F+F+F*F+F +F+7,F+F+F+F+F+F+F+F-F+F+7,F+F+F+F+F+F+F+F+F-F+7,F+F+F+F+F+F+F+F+F+F- " }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 60 "B_M:=linalg[submatr ix](B,[1,3,5,7,8,9,10],[1,3,5,7,8,9,10]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$B_MG-%'matrixG6#7)7)!#C\"\"!F+F+F+F+F+7)F+F*F+F+F+F+ F+7)F+F+F*F+F+F+F+7)F+F+F+F*F+F+F+7)F+F+F+F+!#7F+F+7)F+F+F+F+F+F0F+7)F +F+F+F+F+F+F0" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 36 "gamma1 :=linalg[diag](1,1,1,1,0,0,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g amma1G-%'matrixG6#7)7)\"\"\"\"\"!F+F+F+F+F+7)F+F*F+F+F+F+F+7)F+F+F*F+F +F+F+7)F+F+F+F*F+F+F+7)F+F+F+F+F+F+F+F/F/" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 36 "gamma2:=linalg[diag](0,0,0,0,1,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'gamma2G-%'matrixG6#7)7)\"\"!F*F*F*F*F*F*F )F)F)7)F*F*F*F*\"\"\"F*F*7)F*F*F*F*F*F,F*7)F*F*F*F*F*F*F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Check Invariance of g under N" }}} {EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 198 "g:=v_zip( [1,1,1,1,t,t, t],[theta1 &tensor theta1, theta3 &tensor theta3, theta5 &tensor theta 5, theta7 &tensor theta7, theta8 &tensor theta8, theta9 &tensor theta9 , theta10 &tensor theta10] ,plus); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"gG7$7%%'tensorG%$sp2G7%%(cov_basGF*7\"7)7$7$\"\"\"F/F/7$7$\"\"$F2 F/7$7$\"\"&F5F/7$7$\"\"(F8F/7$7$\"\")F;%\"tG7$7$\"\"*F?F<7$7$\"#5FBF< " }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 17 "map(Lie_der,N,g);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7(7$7%%'tensorG%$sp2G7%%(cov_basGF)7 \"7#7$7$\"\"\"F.\"\"!F$F$F$F$F$" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 53 "Ricci:=homogeneousRicci(M,sp1,[gamma1,gamma2],[1,t]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RicciG7$7$,&\"#7\"\"\"*&\"\"'F) %\"tGF)!\"\",&\"\"#F)*&\"\"%F))F,F/F)F),$*&,(F,!\")*&F/F)F2F)F)F)F-F)F ,F-!\"'" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 25 "Ricci[1][1] \+ -Ricci[1][2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"#5\"\"\"*&\"\"'F% %\"tGF%!\"\"*&\"\"%F%)F(\"\"#F%F)" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 11 "solve(%,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"& \"\"#\"\"\"" }}}{EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "sp2 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "sp2 \+ > " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "18 4 0 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }