Let $M$ be a manifold and let $\mathrm{\χ}\left(M\right)$ be the module (over the ring $C^{\infty }\left(M\right)$ of all smooth functions on $M$) of vector fields on$M$. Then a linear connection $\nabla$ on the tangent bundle of $M$ is a mapping $\mathrm{\χ}\left(M\right)\times \mathrm{\χ}\left(M\right)\to \mathrm{\χ}\left(M\right)$which is $C^{\infty }\left(M\right)$ linear in its first argument and a derivation on its second argument. If vector fields $X_1\, X_2\, ..\.\, X_n$ define a local frame on$M$, then the coefficients ${\mathrm{\Γ}}_{ \mathrm{ij}}^k$ of $\nabla$with respect to this frame are defined by

More generally, let $E\to M$ be a vector bundle and let $\mathrm{\Σ}\left(M\right)$ be the module (over $C^{\infty }\left(M\right)$) of sections of $E$. Then a connection $\nabla$on $E$ is a mapping $\mathrm{\χ}\left(M\right)\times \mathrm{\Σ}\left(M\right)\to \mathrm{\Σ}\left(M\right)$which is linear in its first argument and a derivation on it second argument. If vector fields $X_1\, X_2\, ..\.\, X_n$ define a local frame on $M$ and $Z_1\,Z_2\, ..\.\,Z_r$ define a local basis for the sections of $E$, then the coefficients ${\mathrm{\Γ}}_{ \mathrm{bi}}^a$ of $\nabla$ with respect to these frames are defined by

Within the DifferentialGeometry package, connections are displayed using the tensor notation ${\mathrm{\Γ}}_{\mathrm{ji}}^k{\mathrm{\ω}}^j{X}_k{\mathrm{\ω}}^i$ or ${\mathrm{\Γ}}_{\mathrm{bi}}^a{\mathrm{\η}}^b{Z}_a{\mathrm{\ω}}^i$, where the ${\mathrm{\ω}}^j$are the dual coframe to the $X_i$ and the ${\mathrm{\η}}^b$ are the dual coframe to the $Z_a$.

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

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

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

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

$C≔\mathrm{Connection}\left(x\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dy}-y^2\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dy}\right)$

$C:=x\mathrm{D\_x}\mathrm{dx}\mathrm{dy}-y^2\mathrm{D\_x}\mathrm{dy}\mathrm{dy}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(C\,''ObjectType''\right)$

$''connection''$

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

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

$\mathrm{DGsetup}\left(\mathrm{FR}\right)$

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

$C≔\mathrm{Connection}\left(\left(\mathrm{E2}\&t\mathrm{\Θ1}\right)\&t\mathrm{\Θ2}\right)$

$C:=\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\,w\right]\,E\right)$

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

$C≔\mathrm{Connection}\left(x\left(\mathrm{D\_v}\&t\mathrm{du}\right)\&t\mathrm{dy}-y\left(\mathrm{D\_u}\&t\mathrm{dv}\right)\&t\mathrm{dx}\right)$

$C:=-y\mathrm{D\_u}\mathrm{dv}\mathrm{dx}\+x\mathrm{D\_v}\mathrm{du}\mathrm{dy}$