A metric tensor $g_{}$ is a symmetric, covariant, rank 2, non-degenerate tensor. The inverse of $g_{}$ is a symmetric, contravariant, rank 2 non-degenerate tensor.

If $X$ is a vector field (or rank 1 contravariant tensor), then we can use the metric $g$ to define a rank 1 covariant tensor field $\mathrm{\α}\={\mathit{L}}_g\left(X\right)$by $\mathrm{\α}\left(Y\right)\=g\left(X\,Y\right)$, where $Y$ is any vector field. In terms of components, if $X\=X^i{\partial }_i$ and $g\=g_{\mathrm{ij} }{\mathrm{dx}}^i{\mathrm{dx}}^j$, then $\mathrm{\α}\={\mathrm{\α}}_{i }{\mathrm{dx}}^i$ where ${\mathrm{\α}}_i \=g_{\mathrm{ij} }{X}^j$. The mapping ${\mathit{L}}_g$, called the lowering map, is easily extended to higher rank tensors. For example, if $T$ is a contravariant rank 3 tensor, then a mixed type tensor $S$ of covariant rank 2 and contravariant rank 1 can be defined by lowering, say the 1st and 3rd indices of $T$, by setting $S\left(X\, \mathrm{\β}\,Y\right)\=T\left({\mathit{L}}_{g}X\,\mathrm{\β}\,{\mathit{L}}_{g}Y\right)\.$ In terms of components, $S_{i k}^{ j} \= g_{\mathrm{ih} }{g}_{\mathrm{kl} }{T}^{\mathrm{hjl}}\.$

The lowering map ${\mathit{L}}_g$ from rank 1 contravariant tensors to rank 1 covariant tensors is invertible. The inverse map from rank 1 covariant tensors to rank 1 contravariant tensors is called the raising map ${\mathit{R}}_h$ and is defined in terms of the inverse $h$ of the metric $g$. If $X\={\mathit{R}}_h\left(\mathrm{\α}\right)$, then in terms of components $X^i\=h^{\mathrm{ij} }{\mathrm{\α}}_j$, where $h^{\mathrm{ij}}$ is the inverse matrix for $g_{\mathrm{ij} }$. As in the case of the lowering map, the raising map can be extended to higher rank tensors.

With the first calling sequence, RaiseLowerIndices(g, T, Indices) will use the metric g to lower the indices/arguments of T given in the list Indices. Each of these indices must be a contravariant index.

With the second calling sequence, RaiseLowerIndices(h, T, Indices) will use the inverse metric h to raise the indices/arguments of T given in the list Indices. Each of these indices must be a covariant index.

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

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

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

$g≔\mathrm{evalDG}\left(a\mathrm{dx}\&t\mathrm{dx}+b\left(\mathrm{dy}\&t\mathrm{dz}+\mathrm{dz}\&t\mathrm{dy}\right)\right)$

$g:=a\mathrm{dx}\mathrm{dx}\+b\mathrm{dy}\mathrm{dz}\+b\mathrm{dz}\mathrm{dy}$

$h≔\mathrm{InverseMetric}\left(g\right)$

$h:=\frac{\mathrm{D\_x}\mathrm{D\_x}}{a}\+\frac{\mathrm{D\_y}\mathrm{D\_z}}{b}\+\frac{\mathrm{D\_z}\mathrm{D\_y}}{b}$

$X≔\mathrm{evalDG}\left(\mathrm{D\_x}+3\mathrm{D\_y}\right)$

$X:=\mathrm{D\_x}\+3\mathrm{D\_y}$

$\mathrm{RaiseLowerIndices}\left(g\,X\,\left[1\right]\right)$

$a\mathrm{dx}\+3b\mathrm{dz}$

$T≔\mathrm{evalDG}\left(\left(\left(\mathrm{D\_z}\&t\mathrm{dy}\right)\&t\mathrm{D\_x}\right)\&t\mathrm{dz}\right)$

$T:=\mathrm{D\_z}\mathrm{dy}\mathrm{D\_x}\mathrm{dz}$

$\mathrm{RaiseLowerIndices}\left(g\,T\,\left[1\,3\right]\right)$

$ab\mathrm{dy}\mathrm{dy}\mathrm{dx}\mathrm{dz}$

$\mathrm{RaiseLowerIndices}\left(h\,T\,\left[2\,4\right]\right)$

$\frac{\mathrm{D\_z}\mathrm{D\_z}\mathrm{D\_x}\mathrm{D\_y}}{b^2}$

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

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

$g≔\mathrm{evalDG}\left(x\mathrm{du}\&t\mathrm{du}+y\mathrm{dv}\&t\mathrm{dv}\right)$

$g:=x\mathrm{du}\mathrm{du}\+y\mathrm{dv}\mathrm{dv}$

$T≔\mathrm{evalDG}\left(2\mathrm{D\_u}\&t\mathrm{D\_v}+\mathrm{D\_v}\&t\mathrm{D\_v}\right)$

$T:=2\mathrm{D\_u}\mathrm{D\_v}\+\mathrm{D\_v}\mathrm{D\_v}$

$\mathrm{RaiseLowerIndices}\left(g\,T\,\left[1\right]\right)$

$2x\mathrm{du}\mathrm{D\_v}\+y\mathrm{dv}\mathrm{D\_v}$

$\mathrm{RaiseLowerIndices}\left(g\,T\,\left[1\,2\right]\right)$

$2xy\mathrm{du}\mathrm{dv}\+y^2\mathrm{dv}\mathrm{dv}$