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#### 20.2.2.6 Torsion und Nichtmetrizität

Maxima can now take into account torsion and nonmetricity. When the flag `itorsion_flag` is set to `true`, the contribution of torsion is added to the connection coefficients. Similarly, when the flag `inonmet_flag` is true, nonmetricity components are included.

Variable: inm

  The nonmetricity vector. Conformal nonmetricity is defined through the covariant derivative of the metric tensor. Normally zero, the metric tensor’s covariant derivative will evaluate to the following when `inonmet_flag` is set to `true`:

      g     =- g  inm
       ij;k     ij   k

Variable: inmc1

  Covariant permutation of the nonmetricity vector components. Defined as

                 g   inm  - inm  g   - g   inm
                  ab    c      a  bc    ac    b
      inmc1    = ------------------------------
           abc                 2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: inmc2

  Contravariant permutation of the nonmetricity vector components. Used in the connection coefficients if `inonmet_flag` is `true`. Defined as:

                            c         c         cd
                -inm  kdelta  - kdelta  inm  + g   inm  g
           c        a       b         a    b          d  ab
      inmc2   = -------------------------------------------
           ab                        2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: ikt1

  Covariant permutation of the torsion tensor (also known as contorsion). Defined as:

                        d           d       d
                -g   itr  - g    itr   - itr   g
                  ad    cb    bd    ca      ab  cd
      ikt1    = ----------------------------------
          abc                   2

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: ikt2

  Contravariant permutation of the torsion tensor (also known as contorsion). Defined as:

          c     cd
      ikt2   = g   ikt1
          ab           abd

  (Substitute `ifg` in place of `g` if a frame metric is used.)

Variable: itr

  The torsion tensor. For a metric with torsion, repeated covariant differentiation on a scalar function will not commute, as demonstrated by the following example:

      (%i1) load(itensor);
      (%o1)      /share/tensor/itensor.lisp
      (%i2) imetric:g;
      (%o2)                                  g
      (%i3) covdiff( covdiff( f( [], []), i), j)
                            - covdiff( covdiff( f( [], []), j), i)$
      (%i4) ishow(%)$
                                         %4              %2
      (%t4)                    f    ichr2    - f    ichr2
                                ,%4      j i    ,%2      i j
      (%i5) canform(%);
      (%o5)                                  0
      (%i6) itorsion_flag:true;
      (%o6)                                true
      (%i7) covdiff( covdiff( f( [], []), i), j)
                            - covdiff( covdiff( f( [], []), j), i)$
      (%i8) ishow(%)$
                                 %8             %6
      (%t8)             f    icc2    - f    icc2    - f     + f
                         ,%8     j i    ,%6     i j    ,j i    ,i j
      (%i9) ishow(canform(%))$
                                         %1             %1
      (%t9)                     f    icc2    - f    icc2
                                 ,%1     j i    ,%1     i j
      (%i10) ishow(canform(ev(%,icc2)))$

                                         %1             %1
      (%t10)                    f    ikt2    - f    ikt2
                                 ,%1     i j    ,%1     j i

      (%i11) ishow(canform(ev(%,ikt2)))$
                            %2 %1                    %2 %1
      (%t11)          f    g      ikt1       - f    g      ikt1
                       ,%2            i j %1    ,%2            j i %1
      (%i12) ishow(factor(canform(rename(expand(ev(%,ikt1))))))$
                                 %3 %2            %1       %1
                           f    g      g      (itr    - itr   )
                            ,%3         %2 %1     j i      i j
      (%t12)               ------------------------------------
                                            2
      (%i13) decsym(itr,2,1,[anti(all)],[]);
      (%o13)                               done
      (%i14) defcon(g,g,kdelta);
      (%o14)                               done
      (%i15) subst(g,nounify(g),%th(3))$
      (%i16) ishow(canform(contract(%)))$
                                                 %1
      (%t16)                           - f    itr
                                          ,%1    i j

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