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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart: with(linalg
):with(difforms):with(liesymm):with(plots):" }}{PARA 0 "" 0 "" {TEXT 
256 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 
32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "W
arning, new definition for trace" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warni
ng, new definition for `&^`" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, n
ew definition for close" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new d
efinition for d" }}{PARA 7 "" 1 "" {TEXT -1 34 "Warning, new definitio
n for mixpar" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition f
or wdegree" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT 299 26 "THE HOPF MAP and CHIRALITY" }}{PARA 257 "" 0 "" 
{TEXT -1 37 "R. M. Kiehn July 12,1998 - May 5,1999" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "See http://www22.pair.com
/csdc/pdf/defects2.pdf" }}{PARA 0 "" 0 "" {TEXT -1 50 "for a tutorial \+
on the Cartan structural equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 73 "The Hopf  Map from the sphere S3 to the \+
sphere S2 gives a representation " }}{PARA 0 "" 0 "" {TEXT -1 94 "for \+
the canonical 1-form of Pfaff dimension 4.  It has two different \"chi
ral\" representations." }}{PARA 0 "" 0 "" {TEXT -1 62 "Recall that par
ity violation does not occur until dimension 4." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The canonical 1-form that
 generates the Hopf Map is of the classic Zhitomirski form." }}{PARA 
0 "" 0 "" {TEXT -1 33 "A = -YdX + XdY + (or -) (SdZ-ZdS)" }}{PARA 0 "
" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 85 "This  form admits
 three associated gradient fields, which are transversal.to the form" 
}}{PARA 0 "" 0 "" {TEXT -1 92 "in the sense that each gradient contrac
ted with the direction field of the form yields zero." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "These three gradient f
ields have a non-zero intersection." }}{PARA 0 "" 0 "" {TEXT -1 89 "Th
e triple of functions that are used form the three gradient fields def
ine the Hopf map." }}{PARA 0 "" 0 "" {TEXT -1 64 "The Hopf map has two
 representations with different \"chirality\"." }}{PARA 0 "" 0 "" 
{TEXT -1 72 "These two forms are give below explicitly.  Each chiral H
opf Map can be " }}{PARA 0 "" 0 "" {TEXT -1 144 "used to compute a bas
is frame over the 4 dimensional domain mod the origin.\nThe two chiral
ly different basis frames yield different topological " }}{PARA 0 "" 
0 "" {TEXT -1 71 "and geometric structures.  The two basis frames are \+
used to compute the" }}{PARA 0 "" 0 "" {TEXT -1 82 "Cartan matrix and \+
the chirally different Cartan equations of structure (see below)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "There exi
st an infinite number of integrating factors to A such that" }}{PARA 
0 "" 0 "" {TEXT -1 77 "d(A^dA) = 0.  Hence these closed 3-forms of Top
ological Torsion will produce " }}{PARA 0 "" 0 "" {TEXT -1 72 "contact
 manifolds of dimension 3.   (If d(A^dA) is not zero the manifold" }}
{PARA 0 "" 0 "" {TEXT -1 36 "is of dimension 4 and is sympletic.)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Remark:" }
}{PARA 0 "" 0 "" {TEXT -1 90 "Weinstein has claimed that the canonical
 Hopf map is tight. (No Limit cycles).  Additional" }}{PARA 0 "" 0 "" 
{TEXT -1 90 "components have to be added to the Hopf 1-form to produce
 \"overtwisted\" structures, which " }}{PARA 0 "" 0 "" {TEXT -1 132 "a
re supposed to support limit cycles.  (An unfinished objective is to s
how how the Van der Pohl oscillator fits within this scheme.)" }}
{PARA 0 "" 0 "" {TEXT -1 8 "********" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}{TEXT 308 21 "In that which follows" }}{PARA 0 "" 0 "" {TEXT 257 77 "
Cartan's Repere Mobile will be computed for a representation of the Ho
pf Map " }}{PARA 0 "" 0 "" {TEXT 302 28 "from S3 in 4D to S2 in 3D.  \+
" }}{PARA 0 "" 0 "" {TEXT 303 84 "The Position Vector in 4 space will \+
be presumed to have 4 components R4 = \{X,Y,Z,S\}." }}{PARA 0 "" 0 "" 
{TEXT 258 82 "The position vector in 3 space will be presumed to have \+
3 components r3 = \{x,y,z\}." }}{PARA 0 "" 0 "" {TEXT 259 42 "The orig
ins are presumed to be coincident;" }}{PARA 0 "" 0 "" {TEXT 304 78 "in
 the sense that the four sphere goes to zero when the 3 sphere goes to
 zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "setup(X,Y,Z,S):" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 24 "R4 is the radius of S3, " }}
{PARA 0 "" 0 "" {TEXT 305 96 "and beta is an arbitrary scaling functio
n on the Holder type with constant coefficients a,b,c,e." }}{PARA 0 "
" 0 "" {TEXT -1 61 "beta will be used to scale the Toplogical  Torsion
 current.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 94 "R4:=((S)^2+X^2+Y^2+Z^2)^(1/2);beta:=((e*S^p+a*X^p+
b*Z^p+c*Y^p)^(n/p));r3:=(x^2+y^2+z^2)^(1/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#R4G*$-%%sqrtG6#,**$)%\"SG\"\"#\"\"\"\"\"\"*$)%\"XGF-
F.F/*$)%\"YGF-F.F/*$)%\"ZGF-F.F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%betaG),**&%\"eG\"\"\")%\"SG%\"pGF)F)*&%\"aGF))%\"XGF,F)F)*&%\"bGF))
%\"ZGF,F)F)*&%\"cGF))%\"YGF,F)F)*&%\"nG\"\"\"F,!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#r3G*$-%%sqrtG6#,(*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"y
GF-F.F/*$)%\"zGF-F.F/F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "The T
orsion current, A^dA, (in 4D) will have zero 4 divergence when the exp
onent of the beta divisor is n=2, for any anisotropy ( a,b,c,e) and an
y exponent p." }}{PARA 0 "" 0 "" {TEXT 261 50 "There are two distinct \+
versions of the Hopf map:  " }}{PARA 0 "" 0 "" {TEXT 306 70 "Each vers
ion is distinguished by the chirality factor, defined as rot:" }}
{PARA 0 "" 0 "" {TEXT 300 32 " rot is either plus or minus 1, " }}
{PARA 0 "" 0 "" {TEXT 307 42 "for right or left handed enantiomorphism
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Con
sider the map from X,Y,Z,S  to x,y,z given by the functions:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "x:=2*(S*Y+rot*X*Z);e1:=grad(
x,[X,Y,Z,S]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,&*&%\"SG\"\"\"
%\"YGF(\"\"#*(%$rotGF(%\"XGF(%\"ZGF(F*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#e1G-%'vectorG6#7&,$*&%$rotG\"\"\"%\"ZGF,\"\"#,$%\"SGF.,$*&F+
\"\"\"%\"XGF,F.,$%\"YGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"y:=2*(Y*Z-rot*S*X);e2:=grad(y,[X,Y,Z,S]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yG,&*&%\"YG\"\"\"%\"ZGF(\"\"#*(%$rotGF(%\"SGF(%\"XG
F(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G-%'vectorG6#7&,$*&%$ro
tG\"\"\"%\"SGF,!\"#,$%\"ZG\"\"#,$%\"YGF1,$*&F+\"\"\"%\"XGF,F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "z:=(X^2+Y^2-(S^2+Z^2));e3:=g
rad(z,[X,Y,Z,S]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG,**$)%\"XG
\"\"#\"\"\"\"\"\"*$)%\"YGF)F*F+*$)%\"SGF)F*!\"\"*$)%\"ZGF)F*F2" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G-%'vectorG6#7&,$%\"XG\"\"#,$%\"Y
GF+,$%\"ZG!\"#,$%\"SGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Note \+
that the square of the spherical radius in r3 is indepedent of the rot
 factor -- a double cover." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
80 "r3RH=factor(subs(rot=+1,(x^2+y^2+z^2)));r3LH=factor(subs(rot=-1,(x
^2+y^2+z^2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%r3RHG*$),**$)%\"S
G\"\"#\"\"\"\"\"\"*$)%\"XGF+F,F-*$)%\"YGF+F,F-*$)%\"ZGF+F,F-F+F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%r3LHG*$),**$)%\"SG\"\"#\"\"\"\"\"\"
*$)%\"XGF+F,F-*$)%\"YGF+F,F-*$)%\"ZGF+F,F-F+F," }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 548 "The v
ector field adjoint to the three gradients, e1,e2,e3 form the componen
ts of a  Hopf  \"current\", HTJ.  THe Hopf current HTJ is the orthogon
al compliment to the \"normal\" field, constructed from the three func
tions which define a N-3 = 4-3 = 1 dimensional immersed manifold in R4
.  This current is proportional to the \"tangent\" vector to the 1D im
mersed manifold.  (This idea of implicit functions producing a normal \+
field - instead of a tangent field -  is (in a sense) dual to that con
cept of a tangent manifold produced by parametrizations).  " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Note that ther
e are two distinct Hopf currents (A Left rot = -1 and a Right rot=+1) \+
 in R4 that map to the sphere in r3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "HTJRH:=factor(subs(rot=1
,d(x)&^d(y)&^d(z)));HTJLH:=factor(subs(rot=-1,d(x)&^d(y)&^d(z)));VOL:=
d(X)&^d(Y)&^d(Z)&^d(S);HOPFDIFF:=hook(VOL,[Y,X,S,Z]);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%&HTJRHG,$*&,**$)%\"SG\"\"#\"\"\"\"\"\"*$)%\"XGF+
F,F-*$)%\"YGF+F,F-*$)%\"ZGF+F,F-F-,**&-%#&^G6%-%\"dG6#F6-F=6#F0-F=6#F3
F-F6F-F-*&-F:6%F<-F=6#F*FAF-F3F-F-*&-F:6%F<FFF?F-F0F-F-*&-F:6%FFF?FAF-
F*F-F-F-!\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&HTJLHG,$*&,**$)%\"S
G\"\"#\"\"\"\"\"\"*$)%\"XGF+F,F-*$)%\"YGF+F,F-*$)%\"ZGF+F,F-F-,**&-%#&
^G6%-%\"dG6#F6-F=6#F0-F=6#F3F-F6F-F-*&-F:6%F<-F=6#F*FAF-F3F-!\"\"*&-F:
6%F<FFF?F-F0F-FH*&-F:6%FFF?FAF-F*F-F-F-\"\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$VOLG-%#&^G6&-%\"dG6#%\"ZG-F)6#%\"SG-F)6#%\"XG-F)6#%
\"YG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)HOPFDIFFG,**&-%\"SG6&%\"XG%
\"YG%\"ZGF(\"\"\"-%#&^G6%-%\"dG6#F(-F26#F*-F26#F+F-F-*&-F,F)F--F/6%-F2
6#F,F4F6F-!\"\"*&-F+F)F--F/6%F<F1F6F-F-*&-F*F)F--F/6%F<F1F4F-F>" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "De
fine the 1-form of Hopf Action  AA as" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "AA:=subs([(-Y)/beta,X/b
eta,(-rot*S)/beta,rot*Z/beta]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#
AAG7&,$*&%\"YG\"\"\"),**&%\"eG\"\"\")%\"SG%\"pGF.F.*&%\"aGF.)%\"XGF1F.
F.*&%\"bGF.)%\"ZGF1F.F.*&%\"cGF.)F(F1F.F.*&%\"nGF)F1!\"\"F?!\"\"*&F5F)
F*F?,$*&*&%$rotGF.F0F.F)F*F?F@*&*&FEF)F9F.F)F*F?" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 63 "Prove that the three gradient fields are associate
d directions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "factor(inn
erprod(e1,AA));factor(innerprod(e2,AA));factor(innerprod(e3,AA));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,$*&**%\"SG\"\"\"%\"XGF',&%$rotGF'!\"\"F'F',&F*F'F'F'F'\"\"\"),**&%\"e
GF')F&%\"pGF'F'*&%\"aGF')F(F3F'F'*&%\"bGF')%\"ZGF3F'F'*&%\"cGF')%\"YGF
3F'F'*&%\"nGF-F3!\"\"FA!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**%
\"XG\"\"\"%\"ZGF',&%$rotGF'!\"\"F'F',&F*F'F'F'F'\"\"\"),**&%\"eGF')%\"
SG%\"pGF'F'*&%\"aGF')F&F4F'F'*&%\"bGF')F(F4F'F'*&%\"cGF')%\"YGF4F'F'*&
%\"nGF-F4!\"\"FA!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 262 365 "So it follows that the for the t
wo chiral choices, rot = plus or minus 1, the three 4D gradients are o
rthogonal to the 4D Adjoint 1-form, AA.  The three gradients are compo
nents of exact 1-forms that are dual to the current AA.  From the dual
 point of view, the three vectors e1,e2,e3 are associated fields with \+
respect to the 1-form with components formed from AA." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "TO summarize \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 93 "The mapping functions
 of Hopf define three exact differentials dx,dy,dz with values on R4. \+
  " }}{PARA 0 "" 0 "" {TEXT -1 68 "There are two distinct Hopf maps de
pending on the a \"chiral\" choice." }}{PARA 0 "" 0 "" {TEXT 264 445 "
The object is to define a fourth non-exact differential on R4 to form \+
a basis frame on R4.  That special 1-form will be defined has been def
ined as the Action 1-form :  A = \{-YdX+XdY +rot(- SdZ+ZdS)\}/factor(X
,Y,Z,T).  The arbitrary factor will scale the 1-form of Action on R4. \+
   The components of the exact differentials are orthogonal to the Act
ion. Moreover the position (spherical expansion) vector in R4 is ortho
gonal to the Action 1-form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 265 217 "The Hopf map is a quartic equation in the sen
se that the fourth power of the \"length\" of the position vector R4 i
s equal to the square of the length of the position vector, r3.  The z
eros of the two spheres  coincide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "AreaS2:=4*pi*factor(r3^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
'AreaS2G,$*&%#piG\"\"\",:*&)%\"SG\"\"#\"\"\")%\"YGF-F.F-*()%$rotGF-F.)
%\"XGF-F.)%\"ZGF-F.\"\"%*&F/F.F6F.F-*(F2F.F+F.F4F.F8*$)F5F8F.F(*&F4F.F
/F.F-*&F+F.F4F.!\"#*&F4F.F6F.F?*$)F0F8F.F(*$)F,F8F.F(*&F+F.F6F.F-*$)F7
F8F.F(F(F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 53 "Redefine non-homeneous c
oordinates x=X/S, y=Y/S,z=Z/S" }}{PARA 0 "" 0 "" {TEXT 267 31 "AREA:=S
^4(4pi)\{1+x^2+y^2+z^2)^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 268 200 "The exterior product of the three perfect differ
entials yields a non-zero volume element  except at the origin of R4. \+
 and note that \{ i(A)dX^dY^dZ^dS \}factor = dx^dy^dz .   Similar to i
(Z)dA=d(theta)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 460 "Now use th
e (non-integrable) 1-form Action, and compute its Pfaff sequence, note
 that the arbitrary factor 1/beta can be adjusted such that the Pfaff \+
dimension of the 1-form is 3, for any anisotropy and any exponent p, a
s long as n = 2.  This means that there exists a hierarchy of conserve
d Torsion (topological) invariants, depending on p and the anisotropy \+
coefficients..  Note that the Pfaff dimension of the 1-form of Action \+
is 4, if n is not equal to 2. " }}{PARA 0 "" 0 "" {TEXT 270 111 "Hence
 the 1-form usually defines a symplectic manifold, but when n = 2 , th
e 1-form defines a contact manifold." }}{PARA 0 "" 0 "" {TEXT -1 140 "
There are two distinct action 1-forms depending upon the chirality.  T
he sum and differences produce 1-forms that are of rank 2, not rank 4.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "Action:=evalm(((X*d(Y)-Y*d(X))+rot*(Z*d(S)-S*d(Z)))/b
eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ActionG*&,(*&%\"XG\"\"\"-%
\"dG6#%\"YGF)F)*&F-F)-F+6#F(F)!\"\"*&%$rotGF),&*&%\"ZGF)-F+6#%\"SGF)F)
*&F9F)-F+6#F6F)F1F)F)\"\"\"),**&%\"eGF))F9%\"pGF)F)*&%\"aGF))F(FCF)F)*
&%\"bGF))F6FCF)F)*&%\"cGF))F-FCF)F)*&%\"nGF=FC!\"\"FO" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 77 "Next compute the 2-form f
or the field intensities (Maxwell-Faraday equations)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "F:=wcollect(factor(subs(wcollect(d(Action
)))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FG,.*&*&,&*0%\"nG\"\"\"%
\"YGF+)%\"ZG\"\"#\"\"\"%$rotGF+%\"aGF+)%\"XG%\"pGF+%\"SGF+F+*.F*F0)F,F
/F0F.F+%\"eGF+)F6F5F+F4F+F+F+-%#&^G6$-%\"dG6#F6-F?6#F4F+F0*.),**&F9F0F
:F0F+*&F2F0F3F0F+*&%\"bGF+)F.F5F+F+*&%\"cGF+)F,F5F+F+*&F*F0F5!\"\"\"\"
\"F4\"\"\"FE\"\"\"F6\"\"\"F,\"\"\"F.\"\"\"FOF+*&*&,&*0F*F0F4F0F-F0F1F0
FLF0FMF0F6F0F+*.F*F0)F4F/F0F.F0F,F0F9F0F:F0!\"\"F+-F<6$F>-F?6#F,F+F0*.
FD\"\"\"F4\"\"\"FE\"\"\"F6\"\"\"F,\"\"\"F.\"\"\"FOF+*&*&,.*0F1F0F4F0F6
F0F,F0F.F0F9F0F:F0F/*0F1F0F4F0F6F0F,F0F.F0F2F0F3F0F/*2F1F0F4F0F6F0F,F0
F.F0F*F0F9F0F:F0Ffn*2F1F0F4F0F6F0F,F0F.F0F*F0FIF0FJF0Ffn*0F1F0F4F0F6F0
F,F0F.F0FIF0FJF0F/*0F1F0F4F0F6F0F,F0F.F0FLF0FMF0F/F+-F<6$-F?6#F.F>F+F0
*.FD\"\"\"F4\"\"\"FE\"\"\"F6\"\"\"F,\"\"\"F.\"\"\"FOF+*&*&,.*0F4F0F6F0
F,F0F.F0F*F0F2F0F3F0Ffn*.F4F0F6F0F,F0F.F0FLF0FMF0F/*.F4F0F6F0F,F0F.F0F
9F0F:F0F/*.F4F0F6F0F,F0F.F0F2F0F3F0F/*.F4F0F6F0F,F0F.F0FIF0FJF0F/*0F4F
0F6F0F,F0F.F0F*F0FLF0FMF0FfnF+-F<6$FAFinF+F0*.FD\"\"\"F4\"\"\"FE\"\"\"
F6\"\"\"F,\"\"\"F.\"\"\"FOF+*&*&,&*0F*F0)F6F/F0F,F0F1F0F.F0F2F0F3F0Ffn
*.F*F0F6F0F8F0FIF0FJF0F4F0F+F+-F<6$F]pFAF+F0*.FD\"\"\"F4\"\"\"FE\"\"\"
F6\"\"\"F,\"\"\"F.\"\"\"FOF+*&*&,&*0F*F0F4F0F\\rF0F1F0F.F0FLF0FMF0Ffn*
.F*F0FenF0F6F0F,F0FIF0FJF0FfnF+-F<6$F]pFinF+F0*.FD\"\"\"F4\"\"\"FE\"\"
\"F6\"\"\"F,\"\"\"F.\"\"\"FOF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 
180 "Next compute the Topological Torsion 3-form.  Note that the compo
nents of this 3-form are proportional to the position vector R4 contra
cted with the 4D volume element, dX^dY^dZ^dS." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "H:=factor(Action&^d(Action));" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%\"HG,$*&*&,**&%\"ZG\"\"\"-%#&^G6%-%\"dG6#%\"SG-F06#
%\"XG-F06#%\"YGF+F+*&F8F+-F-6%-F06#F*F/F3F+!\"\"*&F2F+-F-6%F<F3F6F+F>*
&F5F+-F-6%F<F/F6F+F+F+%$rotGF+\"\"\"*$)),**&%\"eGF+)F2%\"pGF+F+*&%\"aG
F+)F5FNF+F+*&%\"bGF+)F*FNF+F+*&%\"cGF+)F8FNF+F+*&%\"nGFFFN!\"\"\"\"#FF
FZ\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 220 "The components of th
e 3-form H are orthogonal to the components of the 1-form of Action (t
ransversal).  The components of the three form H are a current in the \+
direction of the \"radial expansion\" position vector in R4.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 46 " Finally
 compute the Topological Parity 4-form" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "K:=factor(d(Action)&^d(Action));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"KG,$*&*(-%#&^G6&-%\"dG6#%\"ZG-F,6#%\"SG-F,6#%\"XG-F
,6#%\"YG\"\"\",&!\"#F8%\"nGF8F8%$rotGF8\"\"\"*$)),**&%\"eGF8)F1%\"pGF8
F8*&%\"aGF8)F4FEF8F8*&%\"bGF8)F.FEF8F8*&%\"cGF8)F7FEF8F8*&F;F=FE!\"\"
\"\"#F=FP!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 
0 "" {TEXT -1 576 "This result is extraordinary, for it shows that the
 sign of K depends on the chirality factor, rot, which can be plus or \+
minus one.   The integer n determines the homogeneity of the form, and
 the coefficients a,b,c,e, determine the isotropy - or lack thereof.  \+
For arbitrary evolution, K is like a Liapunov function.  When K is pos
itive, then the evolution is divergent exponentially. When K is negati
ve, the evolution is convergent exponentially.  The remarkable feature
 is that the sign is determined by the homogeneity conditions (the cho
ice of n) and the chirality (rot). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 104 "Typically n=1 and p = 2, with a,b,c,e a
ll equal to 1.  Then rot = -1 is stable and rot = +1 is unstable!" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 101 "Note th
at when n = 2, the 4-form K vanishes, such that the Pffaf dimension of
 the Action is 3, not 4." }}{PARA 0 "" 0 "" {TEXT 276 309 "The 1-form \+
then defines a contact manifold, for which there exists a unique extre
mal vector, such that the closed integrals of the Action 1-form and th
e Torsion 3 -form are deformation invariants.  The result is valid for
 any ianisotropy denominator and any value of a,b,c,e and any value of
 p, subject to n=2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 277 246 "The effect of 4D radial expansions is nu
ll for n=2.  The Lie derivative of the forms of the Pfaff sequence gen
erated by A,dA,A^dA,F^F are zero in the direction of H (the torsion ve
ctor is the radial expansion vector in 4D when n=2 any p ,etc.  ) " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" 
{TEXT 301 25 "THE HOPF MAP BASIS FRAME." }}{PARA 0 "" 0 "" {TEXT -1 
86 "Cartan's methods of the repere mobile are applied to the two chira
l Hopf basis frames." }}{PARA 0 "" 0 "" {TEXT -1 52 "The different chi
ral maps produce different results!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 278 55 "For algebraic clarity the constants wil
l be chosen as: " }}{PARA 0 "" 0 "" {TEXT -1 86 "(If you have Maple, y
ou can change these values and Maple will recompute the formulas)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "n:=0;p:=2;a:=1;b:=1;c:=1;e:=1;Adjoint:=AA;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"nG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"bG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"c
G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(AdjointG7&,$%\"YG!\"\"%\"XG,$*&%$rotG\"\"\"%
\"SGF-F(*&F,\"\"\"%\"ZGF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The \+
Basis Frame on R4 then is constructed from the three gradients and the
 adjoint field." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "E1:=[rot
*Z,S,rot*X,Y];INNERE1A:=innerprod(E1,AA);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#E1G7&*&%$rotG\"\"\"%\"ZGF(%\"SG*&F'\"\"\"%\"XGF(%\"Y
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)INNERE1AG,&*&%\"SG\"\"\"%\"XGF
(F(*()%$rotG\"\"#\"\"\"F)F.F'F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "E2:=[-rot*S,Z,Y,-rot*X];INNERE2A:=innerprod(E2,AA);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G7&,$*&%$rotG\"\"\"%\"SGF)!\"\"
%\"ZG%\"YG,$*&F(\"\"\"%\"XGF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)
INNERE2AG,&*&%\"ZG\"\"\"%\"XGF(F(*()%$rotG\"\"#\"\"\"F'F.F)F.!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "E3:=[-X,-Y,Z,S];INNERE3A:=i
nnerprod(E3,AA);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G7&,$%\"XG!\"
\",$%\"YGF(%\"ZG%\"SG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)INNERE3AG
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 202 "Form at the point R4 t
he matrix array of E1, E2, E3, Action, to create Cartan's  Repere Mobi
le.  Orthogonality is preserved for either choice of the chirality.  T
here are two Frame matrices FTR and FTL," }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 259 "" 0 "" {TEXT -1 456 "The extraordinary result is tha
t the different chiralities produce very different structures.  The mi
nimum polynomial for the LH case (rot=-1) is a quartic as a product of
 squares. The trace of the frame matrix vanishes!   On the otherhand, \+
the RH (rot=+1) case yields a cubic term (as the trace is nonzero)  Th
e eigenvalues of the LH case are equal and opposite pairs with only tw
o distinct magnitudes.  The eigenvalues of the RH case can all be dist
inct." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Note as the determinant
 of the Frame is not zero (except at the excluded origin), the Frame m
atrix is global." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "FT:=ev
alm(subs((array([E1,E2,E3,AA]))));DET:=factor(subs(rot^2=1,det(FT)));M
INPOLY:=subs(rot^2=1,minpoly(FT,lambda)):trace(FT):MINPOLYL:=subs(rot=
-1,MINPOLY);MINPOLYR:=subs(rot=+1,MINPOLY);TRACEL:=subs(rot=-1,trace(F
T));TRACER:=subs(rot=1,trace(FT));POLYDIFF:=factor(MINPOLYR-MINPOLYL);
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FTG
-%'matrixG6#7&7&*&%$rotG\"\"\"%\"ZGF,%\"SG*&F+\"\"\"%\"XGF,%\"YG7&,$*&
F+F0F.F,!\"\"F-F2,$F/F67&,$F1F6,$F2F6F-F.7&F:F1F4F*" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$DETG*$),**$)%\"SG\"\"#\"\"\"\"\"\"*$)%\"XGF+F,F-*$
)%\"YGF+F,F-*$)%\"ZGF+F,F-F+F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)M
INPOLYLG,:*$)%\"ZG\"\"%\"\"\"\"\"\"*&)%\"SG\"\"#F*)F(F/F*F/*&)%\"YGF/F
*F0F*F/*&)%\"XGF/F*F0F*F/*$)F.F)F*F+*&F-F*F2F*F/*&F-F*F5F*F/*$)F6F)F*F
+*&F5F*F2F*F/*$)F3F)F*F+*&,**$F5F*!\"#*$F0F*FC*$F-F*FC*$F2F*F/F+)%'lam
bdaGF/F*F+*$)FHF)F*F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)MINPOLYRG,
>*$)%\"ZG\"\"%\"\"\"\"\"\"*&)%\"SG\"\"#F*)F(F/F*F/*&)%\"YGF/F*F0F*F/*&
)%\"XGF/F*F0F*F/*$)F.F)F*F+*&F-F*F2F*F/*&F-F*F5F*F/*$)F6F)F*F+*&F5F*F2
F*F/*$)F3F)F*F+*(,**$F5F*F/*$F0F*F/*$F-F*F/*$F2F*F/F+F(F+%'lambdaGF+!
\"#*&,*FBF/FC\"\"'FDF/FEF/F+)FFF/F*F+*&F(F*)FF\"\"$F*!\"%*$)FFF)F*F+" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'TRACELG\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'TRACERG,$%\"ZG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%)POLYDIFFG,$*&%'lambdaG\"\"\",2*&F'\"\"\")%\"XG\"\"#F(F+*&F'F()%
\"ZGF.F(F.*&F'F()%\"SGF.F(F+*&)F'F.F(F1F+!\"\"*&F,F(F1F(F7*$)F1\"\"$F(
F7*&F1F(F3F(F7*&)%\"YGF.F(F1F(F7F+\"\"%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 280 59 "The determinant never vanishes except at the origin of R
4. " }}{PARA 0 "" 0 "" {TEXT 281 65 "Next create the induced (from the
 basis frame) metric defined as:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "GUN:=su
bs(rot^2=1,innerprod(FT,transpose(FT)));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$GUNG-%'matrixG6#7&7&,**$)%\"SG\"\"#\"\"\"\"\"\"*$)%\"XGF.F/F0
*$)%\"YGF.F/F0*$)%\"ZGF.F/F0\"\"!F:F:7&F:F*F:F:7&F:F:F*F:7&F:F:F:F*" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The induce Metric is conformal a
nd independent of chiralilty." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "FF:=transpose(FT):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "FFINV:=subs(rot^2=1,inverse(FF));" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%&FFINVG-%'matrixG6#7&7&*&,**&%$rotG\"\"\")%\"ZG\"
\"$\"\"\"F.*(F-F2F0F.)%\"YG\"\"#F2F.*(F0F2)%\"SGF6F2F-F2F.*(F-F2)%\"XG
F6F2F0F2F.F2,6*$)F0\"\"%F2F.*&F8F2)F0F6F2F6*&F4F2FBF2F6*&F;F2FBF2F6*$)
F9F@F2F.*&F8F2F4F2F6*&F8F2F;F2F6*$)F<F@F2F.*&F;F2F4F2F6*$)F5F@F2F.!\"
\"*&F9F2,**$F8F2F.*$F;F2F.*$F4F2F.*$FBF2F.FN*&,**(F-F2F<F.FBF2F.*(F<F2
F8F2F-F2F.*&F-F2)F<F1F2F.*(F-F2F<F2F4F2F.F2F=FN*&*&F5F.FPF.F2F=FN7&,$*
&,**(F9F.F-F2FBF2F.*&)F9F1F2F-F2F.*(F-F2F9F2F;F2F.*(F4F2F-F2F9F2F.F2F=
FN!\"\"*&F0F2FPFNFfn,$FUFao7&,$*&F<F2FPFNFao,$*&F5F2FPFNFaoFboFO7&,$Ff
nFaoFfoFinF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "innerprod(F
F,FFINV):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "DR:=subs(rot^
2=1,innerprod(FFINV,[d(X),d(Y),d(Z),d(S)])):sigma1:=wcollect(factor(wc
ollect((DR[1]))));sigma2:=wcollect(factor(wcollect((DR[2]))));sigma3:=
factor(wcollect((DR[3])));omega:=factor(wcollect((DR[4])));dsigma1:=wc
ollect(factor(simpform(d(sigma1))));dsigma2:=d(sigma2);dsigma3:=d(sigm
a3);domega:=wcollect(factor(simpform(d(omega))));" }}{PARA 0 "" 0 "" 
{TEXT -1 87 "Now compute the component of the position vector in 4D wi
th respect to the basis frame." }}{PARA 0 "" 0 "" {TEXT -1 75 "Note th
at the small omega term is not zero, hence affine torsion can exist." 
}}{PARA 0 "" 0 "" {TEXT -1 81 "In fact note that small omega is precis
ely the adjoint field, to within a factor." }}{PARA 0 "" 0 "" {TEXT 
-1 42 "In Gauss Weingarten theory, omega is zero." }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%'sigma1G,**&*&%\"YG\"\"\"-%\"dG6#%\"SGF)\"\"\",**$)F-
\"\"#F.F)*$)%\"XGF2F.F)*$)F(F2F.F)*$)%\"ZGF2F.F)!\"\"F)*&*&F-F)-F+6#F(
F)F.F/F;F)*&*(%$rotGF)F5F)-F+6#F:F)F.F/F;F)*&*(FBF.F:F)-F+6#F5F)F.F/F;
F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'sigma2G,**&*(%$rotG\"\"\"%\"X
GF)-%\"dG6#%\"SGF)\"\"\",**$)F.\"\"#F/F)*$)F*F3F/F)*$)%\"YGF3F/F)*$)%
\"ZGF3F/F)!\"\"!\"\"*&*&F;F)-F,6#F8F)F/F0F<F)*&*&F8F)-F,6#F;F)F/F0F<F)
*&*(F(F/F.F)-F,6#F*F)F/F0F<F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'si
gma3G,$*&,**&%\"XG\"\"\"-%\"dG6#F)F*F**&%\"YGF*-F,6#F/F*F**&%\"ZGF*-F,
6#F3F*!\"\"*&%\"SGF*-F,6#F8F*F6\"\"\",**$)F8\"\"#F;F**$)F)F?F;F**$)F/F
?F;F**$)F3F?F;F*!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&
,**&%\"XG\"\"\"-%\"dG6#%\"YGF)F)*(%$rotGF)%\"ZGF)-F+6#%\"SGF)F)*&F-F)-
F+6#F(F)!\"\"*(F/\"\"\"F3F)-F+6#F0F)F7F9,**$)F3\"\"#F9F)*$)F(F?F9F)*$)
F-F?F9F)*$)F0F?F9F)!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(dsigma1
G,.*&*&,&*$)%\"SG\"\"#\"\"\"!\"\"*$)%\"YGF,F-\"\"\"F2-%#&^G6$-%\"dG6#F
+-F76#F1F2F-*$),*F)F2*$)%\"XGF,F-F2F/F2*$)%\"ZGF,F-F2\"\"#F-!\"\"F,*&*
&,&*&F+F2FCF2F2*(%$rotGF2F@F2F1F2F.F2-F46$F9-F76#FCF2F-*$)F=\"\"#F-FEF
,*&*&,&*&F+F-F@F-F2*(FKF-FCF-F1F-F.F2-F46$F9-F76#F@F2F-*$)F=\"\"#F-FEF
,*&*&,&*(FKF-F+F-F@F-F.*&F1F-FCF-F2F2-F46$F6FNF2F-*$)F=\"\"#F-FEF,*&*&
,&*&FKF-F?F-F2*&FKF-FBF-F.F2-F46$FNFZF2F-*$)F=\"\"#F-FEF,*&*&,&*(FCF-F
KF-F+F-F.*&F@F-F1F-F2F2-F46$F6FZF2F-*$)F=\"\"#F-FEF," }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%(dsigma2G,:*&**%$rotG\"\"\"%\"XGF)%\"YGF)-%#&^G6$-
%\"dG6#%\"SG-F06#F+F)\"\"\"*$),**$)F2\"\"#F5F)*$)F*F;F5F)*$)F+F;F5F)*$
)%\"ZGF;F5F)\"\"#F5!\"\"!\"#*&**F(F5F*F5FBF)-F-6$F/-F06#FBF)F5*$)F8\"
\"#F5FDFE*&*(F(F5,*F9!\"\"F<F)F>FRF@FRF)-F-6$F/-F06#F*F)F5*$)F8\"\"#F5
FDFR*&*(F2F)FBF5F,F5F5*$)F8\"\"#F5FDFE*&*&,*F9F)F<F)F>F)F@FRF)-F-6$F3F
JF)F5*$)F8\"\"#F5FDFR*&*(FBF5F*F5-F-6$F3FUF)F5*$)F8\"\"#F5FDF;*&*(F+F5
F2F5FHF5F5*$)F8\"\"#F5FDFE*&*&,*F9F)F<F)F>FRF@F)F)F\\oF5F5*$)F8\"\"#F5
FDF)*&*(F+F5F*F5-F-6$FJFUF)F5*$)F8\"\"#F5FDF;*&*(F(F5,*F9FRF<F)F>F)F@F
)F)FSF5F5*$)F8\"\"#F5FDFR*&**F(F5F2F5F+F5FcoF5F5*$)F8\"\"#F5FDF;*&**F(
F5FBF5F2F5FepF5F5*$)F8\"\"#F5FDF;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%(dsigma3G,**&*(%\"SG\"\"\"%\"XGF)-%#&^G6$-%\"dG6#F(-F/6#F*F)\"\"\"*$)
,**$)F(\"\"#F3F)*$)F*F9F3F)*$)%\"YGF9F3F)*$)%\"ZGF9F3F)\"\"#F3!\"\"\"
\"%*&*(FAF)F*F3-F,6$-F/6#FAF1F)F3*$)F6\"\"#F3FCFD*&*(F>F)F(F3-F,6$F.-F
/6#F>F)F3*$)F6\"\"#F3FCFD*&*(F>F3FAF3-F,6$FRFIF)F3*$)F6\"\"#F3FC!\"%" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'domegaG,.*&*&,&*&%\"SG\"\"\"%\"XG
F+F+*(%$rotGF+%\"ZGF+%\"YGF+!\"\"F+-%#&^G6$-%\"dG6#F*-F66#F0F+\"\"\"*$
),**$)F*\"\"#F:F+*$)F,F@F:F+*$)F0F@F:F+*$)F/F@F:F+\"\"#F:!\"\"!\"#*&*&
,&*&F/F:F,F:F1*(F0F:F.F:F*F:F1F+-F36$F8-F66#F/F+F:*$)F=\"\"#F:FHFI*&*&
,&FEF+F>F+F+-F36$F8-F66#F,F+F:*$)F=\"\"#F:FHFI*&*&,&*&F.F:FBF:F+*&FDF:
F.F:F+F+-F36$F5FQF+F:*$)F=\"\"#F:FHFI*&*&,&*(F.F:F*F:F,F:F+*&F0F:F/F:F
1F+-F36$FQFenF+F:*$)F=\"\"#F:FHFI*&*&,&*(F.F:F,F:F/F:F1*&F*F:F0F:F1F+-
F36$F5FenF+F:*$)F=\"\"#F:FHFI" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 
"The forms sigma1 sigma2 and sigma3 are of Pfaff dimension 2.  The for
m omega is of Pfaff dimension 4 unless the exponent n = 2, any p, any \+
signature." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 72 "The forms sigma an
d omega are the 1-forms relative to the basis frame FF" }}{PARA 0 "" 
0 "" {TEXT 283 175 "In the example, the 1-forms sigma1 sigma2 and sigm
a3 are proportional to exact differentials of the  mapping functions d
efined by the Hopf map giving x,y,z in terms of X,Y,Z,T" }}{PARA 0 "" 
0 "" {TEXT 284 73 "The in-exact 1-form of Action is  proportional to t
he 1-form small omega." }}{PARA 0 "" 0 "" {TEXT 285 231 "For a paramet
rized surface, small omega vanishes, and such subspaces have Zero Tors
ion of the Affine type.  Such is not the case for the implicit manifol
ds.  (Recall the three functions define a 1 D manifold , a curve, in 4
D space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 
104 "From the Frame matrix,  now use the standard methods to compute t
he Cartan Matrix of connection 1-forms." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 287 46 "See http://www.uh.edu/~rkiehn/pdf/d
efects2.pdf" }}{PARA 0 "" 0 "" {TEXT 288 64 "for details of the Cartan
 method for an arbitrary Repere Mobile." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "dFF:=array([[d(FF[1,1
]),d(FF[1,2]),d(FF[1,3]),d(FF[1,4])],[d(FF[2,1]),d(FF[2,2]),d(FF[2,3])
,d(FF[2,4])],[d(FF[3,1]),d(FF[3,2]),d(FF[3,3]),d(FF[3,4])],[d(FF[4,1])
,d(FF[4,2]),d(FF[4,3]),d(FF[4,4])]]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "cartan:=(evalm(FFINV&*dFF)):" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 289 91 "Evaluate each component of the connection coefficient
s on transverse subspace  of E1,E2,E3." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "Gamma11:=subs(rot^2=1,factor(wcollect(cartan[1,1])));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma11G*&,**&%\"XG\"\"\"-%\"dG
6#F(F)F)*&%\"ZGF)-F+6#F.F)F)*&%\"SGF)-F+6#F2F)F)*&%\"YGF)-F+6#F6F)F)\"
\"\",**$)F2\"\"#F9F)*$)F(F=F9F)*$)F6F=F9F)*$)F.F=F9F)!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Gamma12:=subs(rot^2=1,factor(wcolle
ct(cartan[1,2])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma12G*&,**
(%$rotG\"\"\"%\"XGF)-%\"dG6#%\"YGF)F)*&%\"SGF)-F,6#%\"ZGF)F)*(F.F)F(\"
\"\"-F,6#F*F)!\"\"*&F3F)-F,6#F0F)F8F5,**$)F0\"\"#F5F)*$)F*F?F5F)*$)F.F
?F5F)*$)F3F?F5F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Ga
mma13:=subs(rot^2=1,factor(wcollect(cartan[1,3])));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%(Gamma13G*&,**(%$rotG\"\"\"%\"XGF)-%\"dG6#%\"ZGF)F)
*&-F,6#%\"SGF)%\"YGF)F)*(F(\"\"\"-F,6#F*F)F.F)!\"\"*&F2F)-F,6#F3F)F8F5
,**$)F2\"\"#F5F)*$)F*F?F5F)*$)F3F?F5F)*$)F.F?F5F)!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Gamma21:=factor(wcollect(cartan[2,1
]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma21G*&,**&%\"ZG\"\"\"-%
\"dG6#%\"SGF)F)*(%$rotG\"\"#F-F)-F+6#F(F)!\"\"*(F/F)%\"XGF)-F+6#%\"YGF
)F3*(F8F)F/F)-F+6#F5F)F)F),**$F-F0F)*$F5F0F)*$F8F0F)*$F(F0F)F3" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Gamma22:=subs(rot^2=1,factor
(wcollect(cartan[2,2])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma2
2G*&,**&%\"XG\"\"\"-%\"dG6#F(F)F)*&%\"ZGF)-F+6#F.F)F)*&%\"SGF)-F+6#F2F
)F)*&%\"YGF)-F+6#F6F)F)\"\"\",**$)F2\"\"#F9F)*$)F(F=F9F)*$)F6F=F9F)*$)
F.F=F9F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Gamma23:=s
ubs(rot^2=1,factor(wcollect(cartan[2,3])));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(Gamma23G,$*&,**(%$rotG\"\"\"-%\"dG6#%\"SGF*%\"XGF*F*
*(F)\"\"\"F.F*-F,6#F/F*!\"\"*&-F,6#%\"YGF*%\"ZGF*F**&F8F*-F,6#F9F*F4F1
,**$)F.\"\"#F1F**$)F/F@F1F**$)F8F@F1F**$)F9F@F1F*!\"\"F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Gamma31:=factor(wcollect(cartan[3,1
]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma31G*&,**(%$rotG\"\"\"-
%\"dG6#%\"XGF)%\"ZGF)F)*&-F+6#%\"SGF)%\"YGF)!\"\"*(F(F)F-F)-F+6#F.F)F4
*&F2F)-F+6#F3F)F)F),**$F2\"\"#F)*$F-F=F)*$F3F=F)*$F.F=F)F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Gamma32:=subs(rot^2=1,factor(wcolle
ct(cartan[3,2])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Gamma32G*&,**
(%$rotG\"\"\"-%\"dG6#%\"SGF)%\"XGF)F)*(F(\"\"\"F-F)-F+6#F.F)!\"\"*&-F+
6#%\"YGF)%\"ZGF)F)*&F7F)-F+6#F8F)F3F0,**$)F-\"\"#F0F)*$)F.F?F0F)*$)F7F
?F0F)*$)F8F?F0F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Ga
mma33:=subs(rot^2=1,factor(wcollect(cartan[3,3])));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%(Gamma33G*&,**&%\"XG\"\"\"-%\"dG6#F(F)F)*&%\"ZGF)-F
+6#F.F)F)*&%\"SGF)-F+6#F2F)F)*&%\"YGF)-F+6#F6F)F)\"\"\",**$)F2\"\"#F9F
)*$)F(F=F9F)*$)F6F=F9F)*$)F.F=F9F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 290 29 "The \"Space-S\" components are:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 49 "hh1:=subs(rot^2=1,factor(wcollect(cartan[4,1])))
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$hh1G*&,**&%\"XG\"\"\"-%\"dG6#%\"SGF)F)*(%\"YGF)%$rotGF)-F+6#%
\"ZGF)!\"\"*(F0\"\"\"F3F)-F+6#F/F)F)*&F-F)-F+6#F(F)F4F6,**$)F-\"\"#F6F
)*$)F(F?F6F)*$)F/F?F6F)*$)F3F?F6F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "gg1:=subs(rot^2=1,factor(wcollect(factor(wcollect(car
tan[1,4])))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg1G,$*&,**&%\"XG
\"\"\"-%\"dG6#%\"SGF*F**(%\"YGF*%$rotGF*-F,6#%\"ZGF*!\"\"*(F1\"\"\"F4F
*-F,6#F0F*F**&F.F*-F,6#F)F*F5F7,**$)F.\"\"#F7F**$)F)F@F7F**$)F0F@F7F**
$)F4F@F7F*!\"\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "hh2:=s
ubs(rot^2=1,factor(wcollect(cartan[4,2])));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$hh2G*&,**&%\"XG\"\"\"-%\"dG6#%\"ZGF)F)*(%\"YGF)%$rot
GF)-F+6#%\"SGF)F)*(F0\"\"\"F3F)-F+6#F/F)!\"\"*&F-F)-F+6#F(F)F8F5,**$)F
3\"\"#F5F)*$)F(F?F5F)*$)F/F?F5F)*$)F-F?F5F)!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 49 "gg2:=subs(rot^2=1,factor(wcollect(cartan[2,4]
)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gg2G,$*&,**&%\"XG\"\"\"-%\"
dG6#%\"ZGF*F**(%\"YGF*%$rotGF*-F,6#%\"SGF*F**(F1\"\"\"F4F*-F,6#F0F*!\"
\"*&F.F*-F,6#F)F*F9F6,**$)F4\"\"#F6F**$)F)F@F6F**$)F0F@F6F**$)F.F@F6F*
!\"\"F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "hh3:=subs(rot^2=
1,factor(wcollect(cartan[4,3])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$hh3G,$*&,**&%\"XG\"\"\"-%\"dG6#%\"YGF*F**&F.F*-F,6#F)F*!\"\"*(%$rotG
F*%\"ZGF*-F,6#%\"SGF*F2*(F4\"\"\"F8F*-F,6#F5F*F*F:,**$)F8\"\"#F:F**$)F
)F@F:F**$)F.F@F:F**$)F5F@F:F*!\"\"F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "gg3:=factor(wcollect(cartan[3,4]));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$gg3G,$*&,**&%\"XG\"\"\"-%\"dG6#%\"YGF*!\"\"*&F.F*
-F,6#F)F*F**(%$rotGF*%\"ZGF*-F,6#%\"SGF*F**(F4\"\"\"F8F*-F,6#F5F*F/F:,
**$)F8\"\"#F:F**$)F)F@F:F**$)F.F@F:F**$)F5F@F:F*!\"\"F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Omega:=subs(rot^2=1,simplify(wcolle
ct(cartan[4,4])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OmegaG*&,**&%
\"XG\"\"\"-%\"dG6#F(F)F)*&%\"ZGF)-F+6#F.F)F)*&%\"SGF)-F+6#F2F)F)*&%\"Y
GF)-F+6#F6F)F)\"\"\",**$)F2\"\"#F9F)*$)F(F=F9F)*$)F6F=F9F)*$)F.F=F9F)!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 308 "Note that the Big Omega
 term is a perfect differential, and is zero only when the arguement R
4 is a constant.  That is off the sphere R4 = constant the Omega term \+
does not vanish.  Hence if the radius of the two sphere is expanding, \+
then R4 is not constant and one has a dilatation. (The source of dilat
ons?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 
176 "There are in general two sets of torsion two forms.  The affine t
wo forms, big Sigma, which depend upon the product of little omega and
 the connection components, little gamma." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 293 102 "The second set of torsion 2-form
s is related to Big Omega and the connection components, little gamma.
" }}{PARA 0 "" 0 "" {TEXT 294 46 "See http://www.uh.edu/~rkiehn/pdf/de
fects2.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
295 53 "AFFINE TORSION 2-forms   due to translations  |Sigma>" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 137 "Sigma1:=subs(rot^2=1,(wcollect(factor(omega&^gg1))));Sigma2:=su
bs(rot^2=1,wcollect(factor(omega&^gg2)));Sigma3:=subs(rot^2=1,omega&^g
g3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'Sigma1G,.*&*&,&*$)%\"XG\"\"
#\"\"\"\"\"\"*$)%\"ZGF,F-!\"\"F.-%#&^G6$-%\"dG6#%\"SG-F76#%\"YGF.F-*$)
,**$)F9F,F-F.F)F.*$)F<F,F-F.F/F.\"\"#F-!\"\"F.*&*&,&*(%$rotGF.F+F.F<F.
F.*&F9F.F1F.F2F.-F46$F:-F76#F1F.F-*$)F?\"\"#F-FEF.*&*&,&*&F9F-F+F-F.*(
FJF-F1F-F<F-F2F.-F46$F:-F76#F+F.F-*$)F?\"\"#F-FEF.*&*&,&*(FJF-F9F-F+F-
F2*&F<F-F1F-F.F.-F46$F6FNF.F-*$)F?\"\"#F-FEF.*&*&,&*&FCF-FJF-F.*&FAF-F
JF-F2F.-F46$FNFZF.F-*$)F?\"\"#F-FEF.*&*&,&*&F+F-F<F-F2*(F1F-FJF-F9F-F.
F.-F46$F6FZF.F-*$)F?\"\"#F-FEF." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'
Sigma2G,.*&*&,&*(%$rotG\"\"\"%\"XGF+%\"YGF+!\"\"*&%\"SGF+%\"ZGF+F.F+-%
#&^G6$-%\"dG6#F0-F66#F-F+\"\"\"*$),**$)F0\"\"#F:F+*$)F,F@F:F+*$)F-F@F:
F+*$)F1F@F:F+\"\"#F:!\"\"F.*&*&,&FAF+F>F.F+-F36$F8-F66#F1F+F:*$)F=\"\"
#F:FHF.*&*&,&*&F1F:F,F:F.*(F-F:F*F:F0F:F.F+-F36$F8-F66#F,F+F:*$)F=\"\"
#F:FHF.*&*&,&*&F0F:F-F:F+*(F*F:F,F:F1F:F+F+-F36$F5FNF+F:*$)F=\"\"#F:FH
F.*&*&,&*&F,F:F-F:F+*(F1F:F*F:F0F:F+F+-F36$FNFZF+F:*$)F=\"\"#F:FHF.*&*
&,&*&FDF:F*F:F+*&F*F:FFF:F.F+-F36$F5FZF+F:*$)F=\"\"#F:FHF." }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%'Sigma3G,**&**%$rotG\"\"\"%\"ZGF)%\"XGF)-%
#&^G6$-%\"dG6#%\"SG-F06#%\"YGF)\"\"\"*$),**$)F2\"\"#F6F)*$)F+F<F6F)*$)
F5F<F6F)*$)F*F<F6F)\"\"#F6!\"\"F<*&**F(F6F+F6F2F)-F-6$F3-F06#F*F)F6*$)
F9\"\"#F6FDF<*&**F(F6F*F6F5F)-F-6$F/-F06#F+F)F6*$)F9\"\"#F6FD!\"#*&**F
(F6F2F6F5F6-F-6$FIFRF)F6*$)F9\"\"#F6FDF<" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 140 "GSIGMA1:=subs(rot^2=1,wcollect(Gamma11&^sigma1+Gamma
12&^sigma2+Gamma13&^sigma3));diff2forms:=subs(rot^2=1,wcollect(factor(
GSIGMA1-Sigma1)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(GSIGMA1G,.*&,
**&*$)%\"YG\"\"#\"\"\"F-*$),**$)%\"SGF,F-\"\"\"*$)%\"XGF,F-F4*$F*F-F4*
$)%\"ZGF,F-F4\"\"#F-!\"\"!\"#*&*$F2F-F-*$)F0\"\"#F-F=F,*&*$F6F-F-*$)F0
\"\"#F-F=F4*&*$F:F-F-*$)F0\"\"#F-F=!\"\"F4-%#&^G6$-%\"dG6#F3-FS6#F+F4F
4*&,&*&*(%$rotGF4F7F4F+F4F-*$)F0\"\"#F-F=\"\"$*&*&F3F4F;F4F-*$)F0\"\"#
F-F=!\"$F4-FP6$FU-FS6#F;F4F4*&,&*&*(FenF-F;F-F+F-F-*$)F0\"\"#F-F=F4*&*
&F3F-F7F-F-*$)F0\"\"#F-F=FNF4-FP6$FU-FS6#F7F4F4*&,&*&*&F+F-F;F-F-*$)F0
\"\"#F-F=FN*&*(FenF-F7F-F3F-F-*$)F0\"\"#F-F=F4F4-FP6$FRFboF4F4*&,**&*&
FenF-F6F-F-*$)F0\"\"#F-F=F>*&*&FenF-F:F-F-*$)F0\"\"#F-F=F,*&*&FenF-F*F
-F-*$)F0\"\"#F-F=F4*&*&FenF-F2F-F-*$)F0\"\"#F-F=FNF4-FP6$FboFbpF4F4*&,
&*&*&F+F-F7F-F-*$)F0\"\"#F-F=F_o*&*(FenF-F;F-F3F-F-*$)F0\"\"#F-F=FinF4
-FP6$FRFbpF4F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+diff2formsG,.*&*&
,&*$)%\"SG\"\"#\"\"\"!\"\"*$)%\"YGF,F-\"\"\"F2-%#&^G6$-%\"dG6#F+-F76#F
1F2F-*$),*F)F2*$)%\"XGF,F-F2F/F2*$)%\"ZGF,F-F2\"\"#F-!\"\"!\"#*&*&,&*&
F+F2FCF2F2*(%$rotGF2F@F2F1F2F.F2-F46$F9-F76#FCF2F-*$)F=\"\"#F-FEFF*&*&
,&*&F+F-F@F-F2*(FLF-FCF-F1F-F.F2-F46$F9-F76#F@F2F-*$)F=\"\"#F-FEFF*&*&
,&*(FLF-F+F-F@F-F.*&F1F-FCF-F2F2-F46$F6FOF2F-*$)F=\"\"#F-FEFF*&*&,&*&F
LF-F?F-F2*&FLF-FBF-F.F2-F46$FOFenF2F-*$)F=\"\"#F-FEFF*&*&,&*(FCF-FLF-F
+F-F.*&F@F-F1F-F2F2-F46$F6FenF2F-*$)F=\"\"#F-FEFF" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 296 51 "The two methods of computation give the same resu
lt" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E11:=innerprod(AA,[d(X),d(Y
),d(Z),d(S)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E11G,**&%\"XG\"\"
\"-%\"dG6#%\"YGF(F(*(%$rotGF(%\"ZGF(-F*6#%\"SGF(F(*&F,F(-F*6#F'F(!\"\"
*(F.\"\"\"F2F(-F*6#F/F(F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "d(E11)&^E11;" }}{PARA 0 "" 0 "" {TEXT 309 78 "Now compute the seco
nd set of torsion 2-forms associated with rotations. |Phi>" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#,**(%$rotG\"\"\"%\"XGF&-%#&^G6%-%\"dG6#%\"SG
-F,6#%\"YG-F,6#%\"ZGF&\"\"#*(F%\"\"\"F4F&-F)6%F+F/-F,6#F'F&!\"#*(F%F7F
1F&-F)6%F+F2F:F&F5*(F%F7F.F&-F)6%F/F2F:F&F<" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 105 "Phi1:=wcollect(factor(Omega&^gg1));Phi2:=wcollect
(factor(Omega&^gg2));Phi3:=wcollect(factor(Omega&^gg3));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%%Phi1G,.*&*&,&*(%\"ZG\"\"\"%$rotGF+%\"SGF+!\"
\"*&%\"XGF+%\"YGF+F+F+-%#&^G6$-%\"dG6#F--F66#F1F+\"\"\"*$),**$)F-\"\"#
F:F+*$)F0F@F:F+*$)F1F@F:F+*$)F*F@F:F+\"\"#F:!\"\"F+*&*&,&*&FDF:F,F:F+*
&F,F:FFF:F+F+-F36$F8-F66#F*F+F:*$)F=\"\"#F:FHF+*&*&,&*&F-F:F1F:F+*(F,F
:F0F:F*F:F+F+-F36$F8-F66#F0F+F:*$)F=\"\"#F:FHF+*&*&,&*(F1F:F,F:F-F:F+*
&F*F:F0F:F+F+-F36$F5FPF+F:*$)F=\"\"#F:FHF+*&*&,&*&F-F:F*F:F+*(F,F:F0F:
F1F:F.F+-F36$FPFfnF+F:*$)F=\"\"#F:FHF+*&*&,&FAF+F>F+F+-F36$F5FfnF+F:*$
)F=\"\"#F:FHF+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%Phi2G,.*&*&,&*&)%
\"YG\"\"#\"\"\"%$rotG\"\"\"F/*&)%\"SGF,F-F.F-F/F/-%#&^G6$-%\"dG6#F2-F7
6#F+F/F-*$),**$F1F-F/*$)%\"XGF,F-F/*$F*F-F/*$)%\"ZGF,F-F/\"\"#F-!\"\"F
/*&*&,&*&FAF/F+F/!\"\"*(FEF/F.F-F2F/FLF/-F46$F9-F76#FEF/F-*$)F=\"\"#F-
FGF/*&*&,&*(F.F-F2F-FAF-FL*&F+F-FEF-F/F/-F46$F9-F76#FAF/F-*$)F=\"\"#F-
FGF/*&*&,&*&F2F-FAF-FL*(F.F-FEF-F+F-F/F/-F46$F6FPF/F-*$)F=\"\"#F-FGF/*
&*&,&F?F/FCF/F/-F46$FPFfnF/F-*$)F=\"\"#F-FGF/*&*&,&*(F.F-FAF-F+F-F/*&F
2F-FEF-F/F/-F46$F6FfnF/F-*$)F=\"\"#F-FGF/" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%Phi3G,.*&*&,&*&%\"SG\"\"\"%\"XGF+!\"\"*(%$rotGF+%\"Z
GF+%\"YGF+F-F+-%#&^G6$-%\"dG6#F*-F66#F1F+\"\"\"*$),**$)F*\"\"#F:F+*$)F
,F@F:F+*$)F1F@F:F+*$)F0F@F:F+\"\"#F:!\"\"F-*&*&,&*(F1F:F/F:F*F:F-*&F0F
:F,F:F+F+-F36$F8-F66#F0F+F:*$)F=\"\"#F:FHF-*&*&,&FAF+FCF+F+-F36$F8-F66
#F,F+F:*$)F=\"\"#F:FHF-*&*&,&*&F/F:FFF:F-*&F?F:F/F:F-F+-F36$F5FPF+F:*$
)F=\"\"#F:FHF-*&*&,&*&F1F:F0F:F+*(F/F:F*F:F,F:F+F+-F36$FPFZF+F:*$)F=\"
\"#F:FHF-*&*&,&*(F/F:F,F:F0F:F-*&F*F:F1F:F+F+-F36$F5FZF+F:*$)F=\"\"#F:
FHF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "factor(Phi1&^Sigma1
+Phi2&^Sigma2+Phi3&^Sigma3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 45 "Both of the two forms are ch
iral dependent.  " }}{PARA 0 "" 0 "" {TEXT 310 71 "The two vector two \+
forms of different torsion types are not independent" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "SS3:=factor(Sigma3*(S^2+X^2+Y^2+Z^2));" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$SS3G,$*&*&%$rotG\"\"\",**(%\"ZGF)%
\"XGF)-%#&^G6$-%\"dG6#%\"SG-F26#%\"YGF)F)*(F-\"\"\"F4F)-F/6$F5-F26#F,F
)F)*(F,F9F7F)-F/6$F1-F26#F-F)!\"\"*(F4F9F7F9-F/6$F<FAF)F)F)F9,**$)F4\"
\"#F9F)*$)F-FJF9F)*$)F7FJF9F)*$)F,FJF9F)!\"\"FJ" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 48 "PP3:=wcollect(factor(Phi3*(S^2+X^2+Y^2+Z^2)^2)
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$PP3G,.*&,&*&%\"SG\"\"\"%\"XGF
*F**(%$rotGF*%\"ZGF*%\"YGF*F*F*-%#&^G6$-%\"dG6#F)-F46#F/F*F**&,&*(F/\"
\"\"F-F;F)F;F**&F.F;F+F;!\"\"F*-F16$F6-F46#F.F*F**&,&*$)F+\"\"#F;F=*$)
F/FFF;F=F*-F16$F6-F46#F+F*F**&,&*&F-F;)F.FFF;F**&)F)FFF;F-F;F*F*-F16$F
3F@F*F**&,&*&F/F;F.F;F=*(F-F;F)F;F+F;F=F*-F16$F@FKF*F**&,&*(F-F;F+F;F.
F;F**&F)F;F/F;F=F*-F16$F3FKF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "PP3&^SS3;factor(PP3&^SS3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
*&,**&**,&*&%\"YG\"\"\"%\"ZGF*F**(%$rotGF*%\"SGF*%\"XGF*F*F*F-\"\"\"F+
F0F/F0F0,**$)F.\"\"#F0F**$)F/F4F0F**$)F)F4F0F**$)F+F4F0F*!\"\"!\"#*&**
,&*(F-F0F/F0F+F0F**&F.F0F)F0!\"\"F*F-F0F.F0F/F0F0F1F;F4*&**,&*&F.F0F/F
0F**(F-F0F+F0F)F0F*F*F)F0F-F0F.F0F0F1F;F4*&**,&*(F)F0F-F0F.F0FB*&F+F0F
/F0F*F*F-F0F+F0F)F0F0F1F;F4F*-%#&^G6&-%\"dG6#F.-FQ6#F)-FQ6#F+-FQ6#F/F*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 298 47 "The two torsion two forms have no intersection." }}
{PARA 0 "" 0 "" {TEXT -1 4 "The " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 
"" 1 "" {TEXT -1 0 "" }}}}{MARK "33 1 0" 280 }{VIEWOPTS 1 1 0 1 1 
1803 }