The Christoffel symbol of the second kind for a metric $g$ is the unique torsion-free connection such that the associated covariant derivative operator $\nabla$ satisfies $\nabla g\=0$. It can be represented as a 3-index set of coefficients:

The Christoffel symbol of the first kind is the non-tensorial quantity obtained from the Christoffel symbol of the second kind by lowering its upper index with the metric:

The default value for the keyword is "SecondKind", that is, the calling sequence Christoffel(g) computes the Christoffel symbol of the second kind.

The inverse of the metric can be computed using InverseMetric.

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

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

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

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

$\mathrm{g1}≔\mathrm{evalDG}\left(\frac{1}{y^2}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)\right)$

$\mathrm{g1}:=\frac{\mathrm{dx}\mathrm{dx}}{y^2}\+\frac{\mathrm{dy}\mathrm{dy}}{y^2}$

$\mathrm{C1}≔\mathrm{Christoffel}\left(\mathrm{g1}\,''FirstKind''\right)$

$\mathrm{C1}:=-\frac{\mathrm{dx}\mathrm{dx}\mathrm{dy}}{y^3}-\frac{\mathrm{dx}\mathrm{dy}\mathrm{dx}}{y^3}\+\frac{\mathrm{dy}\mathrm{dx}\mathrm{dx}}{y^3}-\frac{\mathrm{dy}\mathrm{dy}\mathrm{dy}}{y^3}$

$\mathrm{C2}≔\mathrm{Christoffel}\left(\mathrm{g1}\,''SecondKind''\right)$

$\mathrm{C2}:=-\frac{\mathrm{D\_x}\mathrm{dx}\mathrm{dy}}{y}-\frac{\mathrm{D\_x}\mathrm{dy}\mathrm{dx}}{y}\+\frac{\mathrm{D\_y}\mathrm{dx}\mathrm{dx}}{y}-\frac{\mathrm{D\_y}\mathrm{dy}\mathrm{dy}}{y}$

$\mathrm{CovariantDerivative}\left(\mathrm{g1}\,\mathrm{C2}\right)$

$0\mathrm{dx}\mathrm{dx}\mathrm{dx}$

$\mathrm{TorsionTensor}\left(\mathrm{C2}\right)$

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{\mathrm{dx}}{1+x^2+y^2}\,\frac{\mathrm{dy}}{1+x^2+y^2}\right]\,\mathrm{M1}\right)$

$\mathrm{FR}:=\left[d\mathrm{\Θ1}\=2y\mathrm{\Θ1}\bigwedge \mathrm{\Θ2}\,d\mathrm{\Θ2}\=-2x\mathrm{\Θ1}\bigwedge \mathrm{\Θ2}\right]$

$\mathrm{DGsetup}\left(\mathrm{FR}\,\left[E\right]\,\left[\mathrm{\sigma }\right]\right)$

$\mathrm{frame\; name:\; M1}$

$\mathrm{g2}≔\mathrm{evalDG}\left(\mathrm{\σ1}\&t\mathrm{\σ1}+\mathrm{\σ2}\&t\mathrm{\σ2}\right)$

$\mathrm{g2}:=\mathrm{\σ1}\mathrm{\σ1}\+\mathrm{\σ2}\mathrm{\σ2}$

$C≔\mathrm{Christoffel}\left(\mathrm{g2}\right)$

$C:=-2y\mathrm{E1}\mathrm{\σ2}\mathrm{\σ1}\+2x\mathrm{E1}\mathrm{\σ2}\mathrm{\σ2}\+2y\mathrm{E2}\mathrm{\σ1}\mathrm{\σ1}-2x\mathrm{E2}\mathrm{\σ1}\mathrm{\σ2}$

$\mathrm{CovariantDerivative}\left(\mathrm{g2}\,C\right)$

$0\mathrm{\σ1}\mathrm{\σ1}\mathrm{\σ1}$

$\mathrm{TorsionTensor}\left(C\right)$

$0\mathrm{E1}\mathrm{\σ1}\mathrm{\σ1}$