This command will create the standard (flat) metric, symplectic form or complex structure.

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

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

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,\left[\mathrm{u1}\,\mathrm{u2}\,\mathrm{u3}\,\mathrm{u4}\,\mathrm{u5}\,\mathrm{u6}\right]\,M\right)\:$

$\mathrm{g1}≔\mathrm{CanonicalTensors}\left(''Metric''\,''bas''\,3\,1\right)$

$\mathrm{g1}≔\mathrm{dx1}\mathrm{dx1}+\mathrm{dx2}\mathrm{dx2}+\mathrm{dx3}\mathrm{dx3}-\mathrm{dx4}\mathrm{dx4}$

$\mathrm{g1}≔\mathrm{CanonicalTensors}\left(''Metric''\,''vrt''\,6\,0\right)$

$\mathrm{g1}≔\mathrm{du1}\mathrm{du1}+\mathrm{du2}\mathrm{du2}+\mathrm{du3}\mathrm{du3}+\mathrm{du4}\mathrm{du4}+\mathrm{du5}\mathrm{du5}+\mathrm{du6}\mathrm{du6}$

$\mathrm{\ω1}≔\mathrm{CanonicalTensors}\left(''SymplecticForm''\,''bas''\right)$

$\mathrm{\ω1}≔\mathrm{dx1}\bigwedge \mathrm{dx3}+\mathrm{dx2}\bigwedge \mathrm{dx4}$

$\mathrm{\ω2}≔\mathrm{CanonicalTensors}\left(''ComplexStructure''\,''vrt''\right)$

$\mathrm{\ω2}≔-\mathrm{du1}\mathrm{D\_u4}-\mathrm{du2}\mathrm{D\_u5}-\mathrm{du3}\mathrm{D\_u6}+\mathrm{du4}\mathrm{D\_u1}+\mathrm{du5}\mathrm{D\_u2}+\mathrm{du6}\mathrm{D\_u3}$