These two functions computes respectively the covariant connection coefficients and the covariant Riemann tensor in a rigid frame which may be null or orthonormal, given the constant metric and its inverse, and the tetrad (and its inverse) that transforms the natural-basis metric to the given constant metric.

These two routines can be used on spaces of arbitrary dimensions and signature.

For a diagonal constant metric, the tensor package function tensor[frame] can be used to obtain the required tetrad (and its inverse).

Simplification:

connexF() has one simplifier, `tensor/connexF/simp`, for simplifying algebraic expressions during the course of its computation.

RiemannF() has also only one simplifier, `tensor/RiemannF/simp`, for simplifying algebraic expressions.

Both simplifiers are initialized to `tensor/simp`, but it is recommended that each be customized to suit the needs of a particular problem.

These two functions are part of the tensor package, and can be used in the form connexF(..), RiemannF(..) only after performing the command with(tensor), or with(tensor, connexF), etc.  The functions can always be accessed in the long form tensor[connexF], and tensor[RiemannF].

$\mathrm{with}\left(\mathrm{tensor}\right)\:$

$\mathrm{coord}≔\left[u\,x\,y\,v\right]\:$

$\mathrm{h\_compts}≔\mathrm{array}\left(1..4\,1..4\,\left[\left(2\,3\right)=0\,\left(2\,4\right)=0\,\left(3\,1\right)=0\,\left(3\,2\right)=0\,\left(3\,3\right)=-1\,\left(1\,2\right)=0\,\left(4\,1\right)=0\,\left(2\,1\right)=1\,\left(4\,2\right)=-1\,\left(4\,4\right)=0\,\left(1\,1\right)=a{x}^2+byx+c{y}^2\,\left(1\,3\right)=0\,\left(1\,4\right)=1\,\left(4\,3\right)=0\,\left(3\,4\right)=0\,\left(2\,2\right)=0\right]\right)\:$

$h≔\mathrm{create}\left(\left[1\,-1\right]\,\mathrm{op}\left(\mathrm{h\_compts}\right)\right)$

$h≔table\left(\left[\mathrm{index\_char}=\left[1\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{cccc}a{x}^2+bxy+c{y}^2& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& \mathrm{-1}& 0& 0\end{array}\right]\right]\right)$

$\mathrm{hinv}≔\mathrm{invert}\left(h\,\mathrm{DETh}\right)$

$\mathrm{hinv}≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,1\right]\,\mathrm{compts}=\left[\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& -a{x}^2-bxy-c{y}^2\\ 0& 0& \mathrm{-1}& 0\\ 0& \mathrm{-1}& 0& 0\end{array}\right]\right]\right)$

$\mathrm{g\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,1..4\,1..4\right)\:$

$$\begin{gathered} \mathbf{for}i\mathbf{to}4\mathbf{do} \\ \mathbf{for}j\mathbf{from}i\mathbf{to}4\mathbf{do} \\ \mathrm{g\_compts}\left[i,j\right]≔0 \\ \mathbf{end}\mathbf{do} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$$\mathrm{g\_compts}\left[1,2\right]≔1\:$$\mathrm{g\_compts}\left[3,4\right]≔-1\:$$g≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{op}\left(\mathrm{g\_compts}\right)\right)$

$g≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& \mathrm{-1}\\ 0& 0& \mathrm{-1}& 0\end{array}\right]\right]\right)$

$\mathrm{ginv}≔\mathrm{invert}\left(g\,\mathrm{DETg}\right)$

$\mathrm{ginv}≔table\left(\left[\mathrm{index\_char}=\left[1\,1\right]\,\mathrm{compts}=\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& \mathrm{-1}\\ 0& 0& \mathrm{-1}& 0\end{array}\right]\right]\right)$

$\mathrm{Gamma}≔\mathrm{connexF}\left(\mathrm{coord}\,g\,h\right)\:$

$\mathrm{act}\left('\mathrm{display}'\,\mathrm{Gamma}\right)$

  NON-ZERO INDEPENDENT COMPONENTS :"[2, 3, 2] ="

"[3, 2, 2] ="

"[2, 4, 2] ="

"[4, 2, 2] ="

  CHARACTER :

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$bx+2cy$

$-bx-2cy$

$2ax+by$

$-2ax-by$

$\left[\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Rm}≔\mathrm{RiemannF}\left(\mathrm{coord}\,\mathrm{ginv}\,\mathrm{hinv}\,\mathrm{Gamma}\right)\:$

$\mathrm{act}\left('\mathrm{display}'\,\mathrm{Rm}\right)$

  NON-ZERO INDEPENDENT COMPONENTS :"[2, 3, 2, 4] ="

"[2, 4, 2, 4] ="

"[2, 3, 2, 3] ="

  CHARACTER :

  INDEXING FUNCTION :

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$-b$

$-2a$

$-2c$

$\left[\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\right]$

$\mathrm{cov\_riemann}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$