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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "restart: with(linal
g):with(plots):with(liesymm):with(difforms): setup(ct,phi,theta,r):\nd
efform(ct=0,Z=0,phi=0,theta=0,r=0,p=const,q=const,a=const,Y=0,X=0,E=0)
;" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for close" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for `&^`" }}
{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for d" }}{PARA 7 
"" 1 "" {TEXT -1 34 "Warning, new definition for mixpar" }}{PARA 7 "" 
1 "" {TEXT -1 35 "Warning, new definition for wdegree" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 256 75 "Cartan Connection coefficients  from the
 Jacobian Matrix as the Basis Frame" }}{PARA 0 "" 0 "" {TEXT -1 19 "(C
) R.M.Kiehn 1999 " }}{PARA 0 "" 0 "" {TEXT -1 12 "a4sphjac.mws" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "dim:=4;coord := [ct,r, theta
, phi];LGUN:=array([[-1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dimG\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&coordG7&%#ctG%\"rG%&thetaG%$phiG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%LGUNG-%'matrixG6#7&7&!\"\"\"\"!F+F+7&F+\"\"\"F+F+7
&F+F+F-F+7&F+F+F+F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Position v
ector to a point in terms of spherical coordinates" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 66 "MR:=[ct,r*sin(theta)*cos(phi),r*sin(theta)*
sin(phi),r*cos(theta)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MRG7&%#c
tG*(%\"rG\"\"\"-%$sinG6#%&thetaGF)-%$cosG6#%$phiGF)*(F(\"\"\"F*F3-F+F0
F)*&F(F3-F/F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "JAC:=jac
obian(MR,coord);DET:=simplify(det(JAC));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$JACG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+*&-%$sinG6#%&thetaGF*-
%$cosG6#%$phiGF**(%\"rGF*-F3F0F*F2\"\"\",$*(F7F9F.F9-F/F4F*!\"\"7&F+*&
F.F9F<F9*(F7F9F8F9F<F9*(F7F9F.F9F2F97&F+F8,$*&F7F9F.F9F=F+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$DETG*&-%$sinG6#%&thetaG\"\"\")%\"rG\"\"#
\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Specify the Frame Matrt
ix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "FF:=evalm(JAC);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FFG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7
&F+*&-%$sinG6#%&thetaGF*-%$cosG6#%$phiGF**(%\"rGF*-F3F0F*F2\"\"\",$*(F
7F9F.F9-F/F4F*!\"\"7&F+*&F.F9F<F9*(F7F9F8F9F<F9*(F7F9F.F9F2F97&F+F8,$*
&F7F9F.F9F=F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "GG:=evalm(
simplify(inverse(FF)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GGG-%'ma
trixG6#7&7&\"\"\"\"\"!F+F+7&F+*&-%$sinG6#%&thetaGF*-%$cosG6#%$phiGF**&
F.\"\"\"-F/F4F*-F3F07&F+*&*&F2F7F9F*F7%\"rG!\"\"*&*&F8F7F9F7F7F=F>,$*&
F.F7F=F>!\"\"7&F+,$*&F8F7*&F=\"\"\"F.\"\"\"F>FC*&F2F7*&F=\"\"\"F.\"\"
\"F>F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "DGG:=evalm(d(GG))
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$DGGG-%'matrixG6#7&7&\"\"!F*F*F
*7&F*,&*(-%$cosG6#%$phiG\"\"\"-F/6#%&thetaGF2-%\"dGF4F2F2*(-%$sinGF4F2
-F:F0F2-F7F0F2!\"\",&*(F;\"\"\"F3F@F6F@F2*(F9F@F.F@F<F@F2,$*&F9F@F6F@F
=7&F*,(*&*(F3F@F;F@F<F@F@%\"rG!\"\"F=*&*(F.F@F9F@F6F@F@FHFIF=*&*(F.F@F
3F@-F76#FHF2F@*$)FH\"\"#F@FIF=,(*&*(F3F@F.F@F<F@F@FHFIF2*&*(F;F@F9F@F6
F@F@FHFIF=*&*(F;F@F3F@FNF@F@*$)FH\"\"#F@FIF=,&*&*&F3F@F6F@F@FHFIF=*&*&
F9F@FNF@F@*$)FH\"\"#F@FIF27&F*,(*&*&F.F@F<F@F@*&FH\"\"\"F9\"\"\"FIF=*&
*&F;F@FNF@F@*&)FH\"\"#F@F9\"\"\"FIF2*&*(F;F@F3F@F6F@F@*&FH\"\"\")F9\"
\"#F@FIF2,(*&*&F;F@F<F@F@*&FH\"\"\"F9\"\"\"FIF=*&*&F.F@FNF@F@*&)FH\"\"
#F@F9\"\"\"FIF=*&*(F.F@F3F@F6F@F@*&FH\"\"\")F9\"\"#F@FIF=F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CARTANRIGHT:=simplify(innerp
rod(-DGG,FF));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,CARTANRIGHTG-%'ma
trixG6#7&7&\"\"!F*F*F*7&F*F*,$*&%\"rG\"\"\"-%\"dG6#%&thetaGF/!\"\",&*&
F.\"\"\"-F16#%$phiGF/F4*(F.F7F8F7)-%$cosGF2\"\"#F7F/7&F**&F0F7F.!\"\"*
&-F16#F.F7F.FB,$*(F=F/F8F7-%$sinGF2F/F47&F**&F8F7F.FB*&*&F=F7F8F7F7FHF
B*&,&*&FHF7FDF/F/*(F=F7F0F7F.F7F/F7*&F.\"\"\"FH\"\"\"FB" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "CONJ:=simplify(innerprod(FF,CARTANR
IGHT,-GG)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "The Left Cartan ma
trix is the Shipov Connection (IMO)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "CARTANLEFT:=simplify(innerprod(FF,DGG));" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%+CARTANLEFTG-%'matrixG6#7&7&\"\"!F*F*F*7&F*,$
*&,,**)-%$cosG6#%$phiG\"\"#\"\"\")-F26#%&thetaGF5F6-%$sinGF9\"\"\"-%\"
dG6#%\"rGF=F=*&F;F6F>F6F=*(F0F6F>F6F;F6!\"\"*(F8F=-F?F9F=FAF=F=**F0F6F
8F6FFF6FAF6FDF6*&FA\"\"\"F;\"\"\"!\"\"FD*&,**(FAF6-F?F3F=F;F6F=*,-F<F3
F=F7F6F;F6F1F=F>F6FD**F1F6FQF6F>F6F;F6F=*,F1F6FQF6F8F6FFF6FAF6F=F6*&FA
\"\"\"F;\"\"\"FK*&*&F1F6,&*&FAF6FFF6FD*(F;F6F8F6F>F6F=F=F6FAFK7&F*,$*&
,*FPF=FNF=FRFDFSFDF6*&FA\"\"\"F;\"\"\"FKFD*&,**(F7F6F;F6F>F6FDF/F=FCFD
FGFDF6*&FA\"\"\"F;\"\"\"FK*&*&FQF6FYF6F6FAFK7&F**&*&,&FenF=FZF=F=F1F6F
6FAFK*&*&FhoF6FQF6F6FAFK*&*&F>F6,&FDF=*$F7F6F=F=F6FAFK" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "simplify(evalm(CARTANLEFT-CONJ));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"!F(F(F(F'F'F'" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "CARTANLEFT IS THE negative CONJU
GATE OF CARTANRIGHT" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "NOW use te
nsor methods" }}{PARA 0 "" 0 "" {TEXT -1 53 "First compute the differe
ntials of the inverse matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 119 "for i from 1 to dim do for j from 1 to dim do for k from 1 to d
im do d1GG[i,j,k] := (diff(GG[i,j],coord[k])) od od od: " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 56 "Compute the elements of the matrix produc
t of - d[G][F] " }}{PARA 0 "" 0 "" {TEXT 259 32 "which is the right Ca
rtan matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "for b from \+
1 to dim do for a from 1 to dim do for k from 1 to dim do  s:=0;for m \+
from 1 to dim do s := s+(d1GG[a,m,k]*FF[m,b]); CC[a,b,k]:=simplify(-s)
 od od od od ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 144 "for b from 1 to dim do for a from 1 to d
im do for k from 1 to dim do if CC[a,b,k]=0 then else print(`Cabk`(a-1
,b-1,k-1)=CC[a,b,k]) fi od od od ;" }}{PARA 0 "" 0 "" {TEXT 257 44 "TH
E non zero CARTAN CONNECTION coefficients." }}{PARA 0 "" 0 "" {TEXT 
258 50 "                            C(abk) index (1,-1,-1)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"#\"\"\"F'*&\"\"\"F*%\"rG!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$\"\"\"F'*&\"\"\"F*
%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"\"\"\"#F(
,$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"#F'\"\"
\"*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%
\"\"$\"\"#F'*&-%$cosG6#%&thetaG\"\"\"-%$sinGF,!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%%CabkG6%\"\"\"\"\"$F(,&%\"rG!\"\"*&)-%$cosG6#%&the
taG\"\"#\"\"\"F*F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"
\"#\"\"$F(,$*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-F/!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%CabkG6%\"\"$F'\"\"#*&-%$cosG6#%&th
etaG\"\"\"-%$sinGF,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "These
 results agree with matrix method above." }}{PARA 0 "" 0 "" {TEXT -1 
72 "Now compute the Anti symmetric [bk] components of the Cartan conne
ction:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "for j from 1 to \+
dim do for i from 1 to dim do for k from 1 to dim do  s := (CC[i,j,k]-
CC[i,k,j])/2; TTCCS[i,j,k]:=s od od od  ;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 165 "for i from 1 to dim do for j from 1 to dim do for \+
k from 1 to dim do if TTCCS[i,j,k]=0 then else print(`CartanaffineTors
ion`(i-1,k-1,j-1)=TTCCS[i,k,j]) fi od od od ;" }{TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 53 "If no entries appear here, there is no affine t
orsion" }}{PARA 0 "" 0 "" {TEXT -1 71 "Next construct the induced metr
ic on the initial state (ct,r,theta,phi)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 52 "Christoffel Connection coefficients from \+
the metric " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "metric:=simp
lify(innerprod(transpose(FF),-LGUN,FF));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'metricG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+!\"\"F+F+7&F+F+,$*$
)%\"rG\"\"#\"\"\"F-F+7&F+F+F+,&F0F-*&F1F4)-%$cosG6#%&thetaGF3F4F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "metricinverse:=inverse(metri
c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "for i from 1 to dim
 do for j from 1 to dim do for k from 1 to dim do d1gun[i,j,k] := (dif
f(metric[i,j],coord[k])) od od od: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 241 "for i from 1 to dim do for j from i to dim do for k \+
from 1 to dim do C1S[i,j,k] := 0  od od od; for i from 1 to dim do for
 j from 1 to dim do for k from 1 to dim do C1S[i,j,k] := 1/2*d1gun[i,k
,j]+1/2*d1gun[j,k,i]-1/2*d1gun[i,j,k]  od od od; " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 184 " for k from 1 to dim do for i from 1 to dim
 do for j from 1 to dim do s := 0; for m to dim do s := s+metricinvers
e[k,m]*C1S[i,j,m] od; C2S[k,i,j] := simplify(factor(s),trig) od od od;
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "for i from 1 to dim d
o for j from 1 to dim do for k from 1 to dim do if C2S[i,j,k]=0 then  \+
else print(`Gamma2`(i-1,j-1,k-1)=C2S[i,j,k]) fi od od od;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"\"\"#F(,$%\"rG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"\"\"\"$F(,&%\"rG!\"\"
*&)-%$cosG6#%&thetaG\"\"#\"\"\"F*F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%'Gamma2G6%\"\"#\"\"\"F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"#F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"#\"\"$F(,$*&-%$cosG6#
%&thetaG\"\"\"-%$sinGF-F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'
Gamma2G6%\"\"$\"\"\"F'*&\"\"\"F*%\"rG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"$\"\"#F',$*&*&-%$cosG6#%&thetaG\"\"\"-
%$sinGF.F0\"\"\",&!\"\"F0*$)F,F(F3F0!\"\"F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'Gamma2G6%\"\"$F'\"\"\"*&\"\"\"F*%\"rG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'Gamma2G6%\"\"$F'\"\"#,$*&*&-%$cosG
6#%&thetaG\"\"\"-%$sinGF.F0\"\"\",&!\"\"F0*$)F,F(F3F0!\"\"F5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 61 "The non zero Christoffel Connecti
on coefficients 2nd kind    " }}{PARA 0 "" 0 "" {TEXT 257 50 "        \+
             Gamma2(a,b,k) index (1,-1,-1)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 171 "NOTE THA
T for the Jacobian basis frame, the Cartan Right matrix is EQUAL to th
e Christoffel CONNECTION matrix.  This conjecture works, but I have no
t proved it abstractly." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "NOw co
mpute the Tabk as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 44 "Tright(abk) = Cartanright(abk
) - Gamma(abk) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "for i f
rom 1 to dim do for j from 1 to dim do for k from 1 to dim do s:=0; s \+
:= (CC[i,j,k]-C2S[i,j,k]); CCTR[i,j,k]:=simplify(s)  od od od   ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "for i from 1 to dim do for \+
j from 1 to dim do for k from 1 to dim do if (C2S[i,j,k]=0 and CC[i,j,
k]=0) then else print(`T`(i-1,j-1,k-1)=simplify(CCTR[i,j,k])) " }
{TEXT -1 0 "" }{MPLTEXT 1 0 13 "fi od od od ;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"#F(\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"$F(\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"\"F'\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"#F'\"\"\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"$F(\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"\"F'\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"#F'\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 258 178 "The \"Right Rotation Coefficients\" vanish if the frame
 is an integrable Jacobian matrix.  There is no AFFINE torsion in the \+
sense of an anti-symmetric component of the connection." }}{PARA 0 "" 
0 "" {TEXT -1 118 "The Anti-symmetric part of the RIGHT CARTAN MATRIX \+
vanishes for a frame constructed from the Jacobian matrix of a map." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 261 
73 "Now compute the Shipov Delta connection, which is the Left Cartan \+
 matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 127 "for a from 1 to dim do for j from 1 to dim do for k \+
from 1 to dim do d1GG[a,j,k] := simplify(diff(GG[a,j],coord[k])) od od
 od: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Compute the elements of \+
the matrixx product of  [F]d[G]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 157 "for i from 1 to dim do for j from 1 to dim do for k from 1 to
 dim do  s:=0;for m to dim do s := s+FF[i,m]*(d1GG[m,j,k]); DD[i,j,k]:
=simplify(s) od od od od ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
145 "for i from 1 to dim do for j from 1 to dim do for k from 1 to dim
 do if DD[i,j,k]=0 then else print(`Delta`(i-1,j-1,k-1)=DD[i,j,k]) fi \+
od od od ;" }}{PARA 0 "" 0 "" {TEXT 256 39 "NON-ZERO SHIPOV CONNECTION
 coefficients" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 46 "Cartan
 left matrix =Delta(ijk) index (1,-1,-1)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%&DeltaG6%\"\"\"F'F',$*&,(*&)-%$cosG6#%$phiG\"\"#\"\"\")-F.6#%
&thetaGF1F2F'F'F'*$F,F2!\"\"F2%\"rG!\"\"F8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&DeltaG6%\"\"\"F'\"\"#,$*&*(,&!\"\"F'*$)-%$cosG6#%$p
hiGF(\"\"\"F'F'-F16#%&thetaGF'-%$sinGF6F'F4,&F-F'*$)F5F(F4F'!\"\"F-" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#F',$*&*(-%$cos
G6#%$phiGF'-%$sinGF.F',&!\"\"F'*$)-F-6#%&thetaGF(\"\"\"F'F'F9%\"rG!\"
\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"#F(,$*&*
*-%$sinG6#%&thetaGF'-F-6#%$phiGF'-%$cosGF1F'-F4F.F'\"\"\",&!\"\"F'*$)F
5F(F6F'!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"
\"#\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"$F
'*&*(-%$cosG6#%&thetaGF'-F,6#%$phiGF'-%$sinGF-F'\"\"\"%\"rG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"\"\"\"$\"\"#,$-%$cosG6
#%$phiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"#\"\"
\"F(,$*&*(-%$cosG6#%$phiGF(-%$sinGF.F(,&!\"\"F(*$)-F-6#%&thetaGF'\"\"
\"F(F(F9%\"rG!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"
\"#\"\"\"F',$*&**-%$sinG6#%&thetaGF(-F-6#%$phiGF(-%$cosGF1F(-F4F.F(\"
\"\",&!\"\"F(*$)F5F'F6F(!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
&DeltaG6%\"\"#\"\"\"\"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&
DeltaG6%\"\"#F'\"\"\"*&,(*$)-%$cosG6#%&thetaGF'\"\"\"!\"\"*&)-F.6#%$ph
iGF'F1F,F1F(*$F4F1F2F1%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%&DeltaG6%\"\"#F'F'*&*(-%$sinG6#%&thetaG\"\"\")-%$cosG6#%$phiGF'\"\"\"
-F1F,F.F4,&!\"\"F.*$)F5F'F4F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%&DeltaG6%\"\"#\"\"$\"\"\"*&*(-%$cosG6#%&thetaGF)-%$sinG6#%$phiGF)-F
1F.F)\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%
\"\"#\"\"$F',$-%$sinG6#%$phiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%&DeltaG6%\"\"$\"\"\"F(*&*(-%$cosG6#%&thetaGF(-F,6#%$phiGF(-%$sinGF-
F(\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"
$\"\"\"\"\"#-%$cosG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&Del
taG6%\"\"$\"\"#\"\"\"*&*(-%$cosG6#%&thetaGF)-%$sinG6#%$phiGF)-F1F.F)\"
\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"\"$\"
\"#F(-%$sinG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&DeltaG6%\"
\"$F'\"\"\"*&,&!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F3%\"rG!\"\"" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "These values agree with the matr
ix methods." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The anti-symmetric
 part of the Shipov (Left) CARTAN Connection" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 135 "for j from 1 to dim do for i from 1 to dim do f
or k from 1 to dim do  s := (DD[i,j,k]-DD[i,k,j])/2; TTS[i,j,k]:=simpl
ify(s) od od od  ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "for \+
i from 1 to dim do for j from 1 to dim do for k from 1 to dim do if TT
S[i,j,k]=0 then else print(`ShipovTorsion`(i-1,k-1,j-1)=TTS[i,k,j]) fi
 od od od ;" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%.Ship
ovTorsionG6%\"\"\"\"\"#F',$*&,,*(-%$cosG6#%&thetaGF'-%$sinGF/F'%\"rGF'
!\"\"**F-\"\"\"F1F6F3F6)-F.6#%$phiGF(F6F'*&F8F'-F2F9F'F4*(F8F6F<F6)F-F
(F6F(*(F8F6F<F6)F-\"\"%F6F4F6*&,&F4F'*$F>F6F'\"\"\"F3\"\"\"!\"\"#F'F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"\"\"\"$F',
$*&*(-%$cosG6#%&thetaGF'-F-6#%$phiGF'-%$sinGF.F'\"\"\"%\"rG!\"\"#F'\"
\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"\"F'\"
\"#,$*&,,*(-%$cosG6#%&thetaGF'-%$sinGF/F'%\"rGF'!\"\"**F-\"\"\"F1F6F3F
6)-F.6#%$phiGF(F6F'*&F8F'-F2F9F'F4*(F8F6F<F6)F-F(F6F(*(F8F6F<F6)F-\"\"
%F6F4F6*&,&F4F'*$F>F6F'\"\"\"F3\"\"\"!\"\"#F4F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"\"\"\"$\"\"#,&-%$cosG6#%$phiG#!
\"\"F)F/F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"
\"F'\"\"$,$*&*(-%$cosG6#%&thetaGF'-F-6#%$phiGF'-%$sinGF.F'\"\"\"%\"rG!
\"\"#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6
%\"\"\"\"\"#\"\"$,&#F'F(F'-%$cosG6#%$phiGF+" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"#F'\"\"\",$*&,.*,-%$sinG6#%&the
taGF(-F.6#%$phiGF(-%$cosGF2F(-F5F/F(%\"rGF(F(*$)F6F'\"\"\"F(*$)F6\"\"%
F:!\"\"*&)F4F'F:F9F:!\"#*&F@F:F<F:F(*$F@F:F(F:*&,&F>F(F8F(\"\"\"F7\"\"
\"!\"\"#F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"
\"#\"\"$\"\"\",$*&,&%\"rGF)*(-%$sinG6#%$phiGF)-%$cosG6#%&thetaGF)-F0F5
F)F)\"\"\"F-!\"\"#F)F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTo
rsionG6%\"\"#\"\"\"F',$*&,.*,-%$sinG6#%&thetaGF(-F.6#%$phiGF(-%$cosGF2
F(-F5F/F(%\"rGF(F(*$)F6F'\"\"\"F(*$)F6\"\"%F:!\"\"*&)F4F'F:F9F:!\"#*&F
@F:F<F:F(*$F@F:F(F:*&,&F>F(F8F(\"\"\"F7\"\"\"!\"\"#F>F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"#\"\"$F',$-%$sinG6#%$phi
G#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"#
\"\"\"\"\"$,$*&,&%\"rGF(*(-%$sinG6#%$phiGF(-%$cosG6#%&thetaGF(-F0F5F(F
(\"\"\"F-!\"\"#!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTo
rsionG6%\"\"#F'\"\"$,$-%$sinG6#%$phiG#\"\"\"F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"$\"\"#\"\"\",$*&,&*&-%$cosG6#%$
phiGF)%\"rGF)!\"\"*(-%$sinGF0F)-F/6#%&thetaGF)-F6F8F)F)\"\"\"F2!\"\"#F
)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"$F'\"\"
\",$*&,&!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F4%\"rG!\"\"#F(F3" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTorsionG6%\"\"$\"\"\"\"\"#,$
*&,&*&-%$cosG6#%$phiGF(%\"rGF(!\"\"*(-%$sinGF0F(-F/6#%&thetaGF(-F6F8F(
F(\"\"\"F2!\"\"#F3F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%.ShipovTors
ionG6%\"\"$\"\"\"F',$*&,&!\"\"F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F4%\"
rG!\"\"#F,F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now compute the T
ijk assuming the formula" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 38 "T(ijk) = Cartanleft(ijk) - Gamma(ijk) " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 144 "for i from 1 to dim do for j from 1 to d
im do for k from 1 to dim do s:=0; s := (DD[i,j,k]-C2S[i,j,k]); SHIPTR
[i,j,k]:=simplify(s)  od od od   ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 161 "for i from 1 to dim do for j from 1 to dim do for \+
k from 1 to dim do if (C2S[i,j,k]=0 and DD[i,j,k]=0) then else print(`
T`(i-1,j-1,k-1)=simplify(SHIPTR[i,j,k])) " }{TEXT -1 0 "" }{MPLTEXT 1 
0 13 "fi od od od ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"
F'F',$*&,(*&)-%$cosG6#%$phiG\"\"#\"\"\")-F.6#%&thetaGF1F2F'F'F'*$F,F2!
\"\"F2%\"rG!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"F
'\"\"#,$*&*(,&!\"\"F'*$)-%$cosG6#%$phiGF(\"\"\"F'F'-F16#%&thetaGF'-%$s
inGF6F'F4,&F-F'*$)F5F(F4F'!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%\"TG6%\"\"\"\"\"#F',$*&*(-%$cosG6#%$phiGF'-%$sinGF.F',&!\"\"F'*$)-F-
6#%&thetaGF(\"\"\"F'F'F9%\"rG!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%\"TG6%\"\"\"\"\"#F(,$*&,(**-%$sinG6#%&thetaGF'-F.6#%$phiGF'-%$cos
GF2F'-F5F/F'F'%\"rGF'*&)F6F(\"\"\"F7F'!\"\"F:,&F;F'*$F9F:F'!\"\"F;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"#\"\"$F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"$F'*&*(-%$cosG6#%&thetaGF
'-F,6#%$phiGF'-%$sinGF-F'\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"$\"\"#,$-%$cosG6#%$phiG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"\"\"\"$F(,&%\"rGF'*&)-%$c
osG6#%&thetaG\"\"#\"\"\"F*F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
-%\"TG6%\"\"#\"\"\"F(,$*&*(-%$cosG6#%$phiGF(-%$sinGF.F(,&!\"\"F(*$)-F-
6#%&thetaGF'\"\"\"F(F(F9%\"rG!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%\"TG6%\"\"#\"\"\"F',$*&,(*,-%$sinG6#%&thetaGF(-F.6#%$phiGF(-%$cos
GF2F(-F5F/F(%\"rGF(F(!\"\"F(*$)F6F'\"\"\"F(F;*&F7\"\"\",&F8F(F9F(\"\"
\"!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"\"\"\"$
!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#F'\"\"\"*&,**$)
-%$cosG6#%&thetaGF'\"\"\"!\"\"*&)-F.6#%$phiGF'F1F,F1F(*$F4F1F2F2F(F1%
\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#F'F'*&*(-%$
sinG6#%&thetaG\"\"\")-%$cosG6#%$phiGF'\"\"\"-F1F,F.F4,&!\"\"F.*$)F5F'F
4F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"$\"\"\"*
&*(-%$cosG6#%&thetaGF)-%$sinG6#%$phiGF)-F1F.F)\"\"\"%\"rG!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"$F',$-%$sinG6#%$phiG
!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"#\"\"$F(*&-%$cos
G6#%&thetaG\"\"\"-%$sinGF,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"T
G6%\"\"$\"\"\"F(*&*(-%$cosG6#%&thetaGF(-F,6#%$phiGF(-%$sinGF-F(\"\"\"%
\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"\"\"\"#
-%$cosG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"
\"F',$*&\"\"\"F+%\"rG!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"TG6%\"\"$\"\"#\"\"\"*&*(-%$cosG6#%&thetaGF)-%$sinG6#%$phiGF)-F1F.F)
\"\"\"%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"\"
#F(-%$sinG6#%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$\"
\"#F'*&*&-%$cosG6#%&thetaG\"\"\"-%$sinGF-F/\"\"\",&!\"\"F/*$)F+F(F2F/!
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"\"*&,&!\"#
F(*$)-%$cosG6#%&thetaG\"\"#\"\"\"F(F3%\"rG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"TG6%\"\"$F'\"\"#*&*&-%$cosG6#%&thetaG\"\"\"-%$sinG
F-F/\"\"\",&!\"\"F/*$)F+F(F2F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 109 "The \"LEFT \+
Rotation Coefficients\"  It does not seem to make sense to subtract th
e Christoffel part from Delta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "CONJ:=simplify(innerprod(FF,
CARTANRIGHT,-GG));NET:=evalm(CONJ-CARTANLEFT);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%CONJG-%'matrixG6#7&7&\"\"!F*F*F*7&F*,$*&,,**)-%$cosG
6#%$phiG\"\"#\"\"\")-F26#%&thetaGF5F6-%$sinGF9\"\"\"-%\"dG6#%\"rGF=F=*
&F;F6F>F6F=*(F0F6F>F6F;F6!\"\"*(F8F=-F?F9F=FAF=F=**F0F6F8F6FFF6FAF6FDF
6*&FA\"\"\"F;\"\"\"!\"\"FD*&,**(FAF6-F?F3F=F;F6F=*,-F<F3F=F7F6F;F6F1F=
F>F6FD**F1F6FQF6F>F6F;F6F=*,F1F6FQF6F8F6FFF6FAF6F=F6*&FA\"\"\"F;\"\"\"
FK*&*&F1F6,&*&FAF6FFF6FD*(F;F6F8F6F>F6F=F=F6FAFK7&F*,$*&,*FPF=FNF=FRFD
FSFDF6*&FA\"\"\"F;\"\"\"FKFD*&,**(F7F6F;F6F>F6FDF/F=FCFDFGFDF6*&FA\"\"
\"F;\"\"\"FK*&*&FQF6FYF6F6FAFK7&F**&*&,&FenF=FZF=F=F1F6F6FAFK*&*&FhoF6
FQF6F6FAFK*&*&F>F6,&FDF=*$F7F6F=F=F6FAFK" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$NETG-%'matrixG6#7&7&\"\"!F*F*F*F)F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "43 6 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }