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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "**************************
***********************************" }}{PARA 0 "" 0 "" {TEXT -1 69 "Th
is program calculates the different curvatures of an m-dimensional " }
}{PARA 0 "" 0 "" {TEXT -1 1 "R" }{MPLTEXT 1 0 0 "" }{TEXT -1 66 "ieman
nian manifold (M,g) in terms of the local coordinates x on M." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Author: S
igmundur Gudmundsson, Lund University, Sweden " }}{PARA 0 "" 0 "" 
{TEXT -1 39 "email: Sigmundur.Gudmundsson@math.lu.se" }}{PARA 0 "" 0 "
" {TEXT -1 66 "URL: http://www.maths.lth.se/matematiklu/personal/sigma
/index.html" }}{PARA 0 "" 0 "" {TEXT -1 61 "**************************
***********************************" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 35 " INPUT:   m : the dimension of M,  " }}
{PARA 0 "" 0 "" {TEXT -1 44 "                x : the local coordinates
,  " }}{PARA 0 "" 0 "" {TEXT -1 41 "                g : the Riemannian
 metric" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=3:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x:=array(1..m,[x1,x2,x3]):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "L:=4/(1-(x1^2+x2^2+x3^2))^2:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g:=matrix(m,m,[L,0,0,0,L,0,0,0,L]):
" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 53 "He
re we calculate the inverse of the metric gi[i,j] ;" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 15 "gi:=inverse(g):" }}{PARA 0 "" 0 "" {TEXT -1 72 " H
ere we calculate the areas of the sectional parallelograms area[i,j] ;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to m do" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "   for j from 1 to m do" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "   area[i,j]:=g[i,i]*g[j,j]-g[i,j]*g[j,i] " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "   od; od;" }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 67 "Here we calculate the \+
partial derivatives of the metric dg[i,j,k] ;" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to m do " }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 25 "    for j from 1 to m do " }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "       for k from 1 to m do " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "          dg[i,j,k]:=diff(g[i,j],x[k])" }{TEXT -1 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "          od; od; od;" }}{PARA 
0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 54 "Here we calc
ulate the Christoffel symbols Gam[k,i,j] ;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "for k from 1 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "   for i from 1 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  \+
    for j from 1 to m do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "      \+
Gam[k,i,j]:=add(gi[k,s]*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "       \+
  (dg[s,j,i]+dg[s,i,j]-dg[i,j,s])/2,s=1..m);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "      od; od; od;" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{MPLTEXT 1 0 0 "" }{TEXT -1 83 "Here we calculate the partial derivati
ves dGam[k,i,j,l] of the Christoffelsymbols ;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "for i from 1 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "   for j from 1 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "  \+
    for k from 1 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "       \+
  for l from 1 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "         \+
dGam[k,i,j,l]:=diff(Gam[k,i,j],x[l])" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "         od; od; od; od;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 52 " Here we calculate the curvature tensor R[i,j,k,l] ;" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to
 m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "   for j from 1 to m do" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "      for k from 1 to m do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "         for l from 1 to m do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "R[i,j,k,l]:=add(g[i,a]*(dGam[a,k,j,
l]-dGam[a,l,j,k] +" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "   add(Gam[b,
k,j]*Gam[a,l,b]-Gam[b,l,j]*Gam[a,k,b],b=1..m)),a=1..m)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "         od; od; od; od;" }}{PARA 0 "" 0 "" 
{TEXT -1 50 " Here we calculate the sectional curvatures K[i,j]" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 1 to
 m-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "   for j from i+1 to m d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "   K[i,j]:=simplify(R[i,j,j,i]
/area[i,j]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "   K[j,i]:=K[i,j]" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "   od;od;" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from \+
1 to m do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "   for i from 1 to m d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "      for j from 1 to m do " }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "      print(Gam[k,i,j]);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "      od; od; od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&%#x1G\"\"\",*\"\"\"F(*$)F%\"\"#F&F(*$)%#x2GF+F&F(*$
)%#x3GF+F&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x2G\"\"
\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)F%F,F&F(*$)%#x3GF,F&F(!\"\"!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x3G\"\"\",*\"\"\"F(*$)%#x1G\"\"#
F&F(*$)%#x2GF,F&F(*$)F%F,F&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&%#x2G\"\"\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)F%F,F&F(*$)%#x3GF,F
&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x1G\"\"\",*\"\"
\"F(*$)F%\"\"#F&F(*$)%#x2GF+F&F(*$)%#x3GF+F&F(!\"\"F+" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x3G
\"\"\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)%#x2GF,F&F(*$)F%F,F&F(!\"\"!\"#" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&%#x1G\"\"\",*\"\"\"F(*$)F%\"\"#F&F(*$)%#x2GF+F&F(*$)%#x3GF+F
&F(!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x2G\"\"\",*\"\"\"F
(*$)%#x1G\"\"#F&F(*$)F%F,F&F(*$)%#x3GF,F&F(!\"\"F," }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$*&%#x1G\"\"\",*\"\"\"F(*$)F%\"\"#F&F(*$)%#x2GF+F&F(
*$)%#x3GF+F&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x1G\"\"\",*\"\"\"F(*$)F%\"\"#F&F
(*$)%#x2GF+F&F(*$)%#x3GF+F&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&%#x2G\"\"\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)F%F,F&F(*$)%#x3GF,F
&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x3G\"\"\",*\"\"
\"F(*$)%#x1G\"\"#F&F(*$)%#x2GF,F&F(*$)F%F,F&F(!\"\"!\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x
3G\"\"\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)%#x2GF,F&F(*$)F%F,F&F(!\"\"!\"#
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x2G\"\"\",*\"\"\"F(*$)%#x1G
\"\"#F&F(*$)F%F,F&F(*$)%#x3GF,F&F(!\"\"F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&%#x3G\"\"\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)%#x2GF,F&F
(*$)F%F,F&F(!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&%#x1G\"\"\",*\"\"\"F(*$)F%\"\"#F&F(*$)%#
x2GF+F&F(*$)%#x3GF+F&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x3G\"\"\",*\"\"\"F(*$)%#x
1G\"\"#F&F(*$)%#x2GF,F&F(*$)F%F,F&F(!\"\"F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&%#x2G\"\"\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)F%F,F&F(*$
)%#x3GF,F&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x1G\"\"
\",*\"\"\"F(*$)F%\"\"#F&F(*$)%#x2GF+F&F(*$)%#x3GF+F&F(!\"\"!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#x2G\"\"\",*\"\"\"F(*$)%#x1G\"\"#
F&F(*$)F%F,F&F(*$)%#x3GF,F&F(!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&%#x3G\"\"\",*\"\"\"F(*$)%#x1G\"\"#F&F(*$)%#x2GF,F&F(*$)F%F,F
&F(!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "K[1,2];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 16 0" 24 }{VIEWOPTS 1 1 0 3 2 1804 }