There are many situations in differential geometry where computations are tremendously simplified by using a frame or co-frame other than the standard coordinate frame or co-frame. We refer to a general (local) frame or co-frame on a manifold as an anholonomic frame.  Important examples of anholonomic frames are: the orthogonal frames constructed from a metric on a manifold; the null frames used in general relativity; the left (or right) invariant vector fields on a Lie group; the moving frames adapted to a free group action on a manifold.  Cartan's method of equivalence provides an algorithmic approach to constructing adapted co-frames for Pfaffian systems.

All of the commands in the DifferentialGeometry package and the Tensor subpackage work with general anholonomic frames.  At present, the commands in the JetCalculus package work only with the standard coordinate frame.

The structure equations of a frame Fr = [X_1, X_2, ...] (the X_i are vector fields) are the Lie bracket relations

The structure equations for a co-frame Omega = [omega^1,  omega^2, ...] (the omega^k are differential 1-forms) are the exterior derivative formulas

To work in DifferentialGeometry with anholonomic frames on a manifold M, first define an underlying coordinate system on M and define the anholonomic frame or co-frame relative to this coordinate system. Use the command FrameData to generate the structure equations for this frame (along with other data).  Pass the results of the FrameData procedure to the DGsetup procedure.

This command is part of the DifferentialGeometry package, and so can be used in the form FrameData(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-FrameData.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$

$\mathrm{Fr}≔\mathrm{evalDG}\left(\left[x\mathrm{D\_x}+y\mathrm{D\_y}\,y^2\mathrm{D\_x}+x^2\mathrm{D\_y}\right]\right)$

$\mathrm{Fr}≔\left[x\mathrm{D\_x}+y\mathrm{D\_y}\,y^2\mathrm{D\_x}+x^2\mathrm{D\_y}\right]$

$\mathrm{FD}≔\mathrm{FrameData}\left(\mathrm{Fr}\,N\right)$

$\mathrm{FD}≔\left[\left[\mathrm{E1}\,\mathrm{E2}\right]=\mathrm{E2}\right]$

$\mathrm{DGsetup}\left(\mathrm{FD}\,\mathrm{verbose}\right)$

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{E1}\,\mathrm{E2}\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\Θ1}\,\mathrm{\Θ2}\right]$

$\mathrm{frame\; name:\; N}$

$\mathrm{ExteriorDerivative}\left(xy\right)$

$2xy\mathrm{\Θ1}+\left(x^3+y^3\right)\mathrm{\Θ2}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$

$g≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\exp \left(2x\right)\mathrm{dy}\&t\mathrm{dy}\right)$

$g≔\mathrm{dx}\mathrm{dx}+{\ⅇ}^{2x}\mathrm{dy}\mathrm{dy}$

$\mathrm{\Omega }≔\mathrm{evalDG}\left(\left[\mathrm{dx}\,\exp \left(x\right)\mathrm{dy}\right]\right)$

$\mathrm{\Omega }≔\left[\mathrm{dx}\,{\ⅇ}^x\mathrm{dy}\right]$

$\mathrm{coFD}≔\mathrm{FrameData}\left(\mathrm{\Omega }\,N\right)$

$\mathrm{coFD}≔\left[d\mathrm{\Θ1}=0\,d\mathrm{\Θ2}=\mathrm{\Θ1}\bigwedge \mathrm{\Θ2}\right]$

$\mathrm{DGsetup}\left(\mathrm{coFD}\,\left[E\right]\,\left[\mathrm{\omega }\right]\,\mathrm{verbose}\right)$

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{E1}\,\mathrm{E2}\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\ω1}\,\mathrm{\ω2}\right]$

$\mathrm{frame\; name:\; N}$

$g≔\mathrm{evalDG}\left(\mathrm{\ω1}\&t\mathrm{\ω1}+\mathrm{\ω2}\&t\mathrm{\ω2}\right)$

$g≔\mathrm{\ω1}\mathrm{\ω1}+\mathrm{\ω2}\mathrm{\ω2}$

$C≔\mathrm{Christoffel}\left(g\right)$

$C≔-\mathrm{E1}\mathrm{\ω2}\mathrm{\ω2}+\mathrm{E2}\mathrm{\ω1}\mathrm{\ω2}$

$R≔\mathrm{CurvatureTensor}\left(C\right)$

$R≔-\mathrm{E1}\mathrm{\ω2}\mathrm{\ω1}\mathrm{\ω2}+\mathrm{E1}\mathrm{\ω2}\mathrm{\ω2}\mathrm{\ω1}+\mathrm{E2}\mathrm{\ω1}\mathrm{\ω1}\mathrm{\ω2}-\mathrm{E2}\mathrm{\ω1}\mathrm{\ω2}\mathrm{\ω1}$

$\mathrm{R1}≔\mathrm{CovariantDerivative}\left(R\,C\right)$

$\mathrm{R1}≔0\mathrm{E1}\mathrm{\ω1}\mathrm{\ω1}\mathrm{\ω1}\mathrm{\ω1}$

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

$\mathrm{DGsetup}\left(\left[x\,y\,u\,p\,q\,r\,t\right]\,M\right)\:$

$\mathrm{\Omega }≔\mathrm{evalDG}\left(\left[\mathrm{du}-p\mathrm{dx}-q\mathrm{dy}\,\mathrm{dp}-r\mathrm{dx}-\exp \left(u\right)\mathrm{dy}\,\mathrm{dq}-\exp \left(u\right)\mathrm{dx}-t\mathrm{dy}\,\mathrm{dx}\,\mathrm{dr}-p\mathrm{dp}\,\mathrm{dy}\,\mathrm{dt}-q\mathrm{dq}\right]\right)$

$\mathrm{\Omega }≔\left[-p\mathrm{dx}-q\mathrm{dy}+\mathrm{du}\,-r\mathrm{dx}-{\ⅇ}^u\mathrm{dy}+\mathrm{dp}\,-{\ⅇ}^u\mathrm{dx}-t\mathrm{dy}+\mathrm{dq}\,\mathrm{dx}\,-p\mathrm{dp}+\mathrm{dr}\,\mathrm{dy}\,-q\mathrm{dq}+\mathrm{dt}\right]$

$\mathrm{coFD}≔\mathrm{FrameData}\left(\mathrm{\Omega }\,N\right)$

$\mathrm{coFD}≔\left[d\mathrm{\Θ1}=-\mathrm{\Θ2}\bigwedge \mathrm{\Θ4}-\mathrm{\Θ3}\bigwedge \mathrm{\Θ6}\,d\mathrm{\Θ2}=-{\ⅇ}^u\mathrm{\Θ1}\bigwedge \mathrm{\Θ6}-p\mathrm{\Θ2}\bigwedge \mathrm{\Θ4}+\mathrm{\Θ4}\bigwedge \mathrm{\Θ5}\,d\mathrm{\Θ3}=-{\ⅇ}^u\mathrm{\Θ1}\bigwedge \mathrm{\Θ4}-q\mathrm{\Θ3}\bigwedge \mathrm{\Θ6}+\mathrm{\Θ6}\bigwedge \mathrm{\Θ7}\,d\mathrm{\Θ4}=0\,d\mathrm{\Θ5}=0\,d\mathrm{\Θ6}=0\,d\mathrm{\Θ7}=0\right]$

$\mathrm{DGsetup}\left(\mathrm{coFD}\,\left[E\right]\,\left[\mathrm{\θ1}\,\mathrm{\θ2}\,\mathrm{\θ3}\,\mathrm{\π1}\,\mathrm{\π2}\,\mathrm{\π3}\,\mathrm{\π4}\right]\right)$

$\mathrm{frame\; name:\; N}$

$\mathrm{ExteriorDerivative}\left(\mathrm{\θ1}\right)$

$-\mathrm{\θ2}\bigwedge \mathrm{\π1}-\mathrm{\θ3}\bigwedge \mathrm{\π3}$

$\mathrm{ExteriorDerivative}\left(\mathrm{\θ2}\right)$

$-{\ⅇ}^u\mathrm{\θ1}\bigwedge \mathrm{\π3}-p\mathrm{\θ2}\bigwedge \mathrm{\π1}+\mathrm{\π1}\bigwedge \mathrm{\π2}$

$\mathrm{ExteriorDerivative}\left(\mathrm{\θ3}\right)$

$-{\ⅇ}^u\mathrm{\θ1}\bigwedge \mathrm{\π1}-q\mathrm{\θ3}\bigwedge \mathrm{\π3}+\mathrm{\π3}\bigwedge \mathrm{\π4}$